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+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nActive Galactic Nuclei have been always known as strongly variable sources in most of their broad band spectra (e.g. IR: \\citealt{edelson1987,kozlowski2016}; optical: \\citealt{ulrich1997,kawaguchi1998,sesar2007}; X-ray: \\citealt{ulrich1997,lawrence1993}). Most of the variability can be attributed to variations of the red noise character, both in the optical and in the X-ray band \\citep{mchardy1987,lehto1993,czerny1999,gaskell2003}. However, some of the observed changes lead to far more dramatic changes than expected from the red noise trend. These changes sometimes are revealed  in the temporary change of the source classification, and these sources started to be known as Changing-Look AGN (CL AGN; \\citealt{matt2003}).\nThere is no well established definition what can - or cannot - be classified as a CL AGN, and we adopted the view that the name can be used for the broad class of objects, not necessarily showing confirmed changes in the optical flux. With progressing understanding of the mechanisms, proper classification will be certainly introduced.\n\nCL AGN phenomenon was once considered as rather rare.  The changes corresponded either to a drastic change in X-ray spectrum, or in the optical\/UV emission lines and continuum, depending on the studied wavelength range\n\\citep{bianchi2005,denney2014,Shappee2014}. On the other hand, historical lightcurves of nearby sources, including well studies AGN  \\citep[e.g.][]{cohen1986,iijima1992,storchibergmann1993,bon2016,oknyanski2016,shapovalova2019} indicated that such episodes do happen. With more and more optical and X-ray surveys, the number of CL AGN is rapidly growing \\citep{ruan2016,ross2018,yang2018,stern2018,trakhtenbrot2019,macleod2019}, and the question about the mechanism of the phenomenon must be addressed. The most extreme case of such phenomenon in the form of Quasi-Periodic Eruptions QPE) has been recently discovered by \\citet{Miniutti2019} and \\citet{2020giustini}.\n\nThere is still an on-going discussion whether the phenomenon is intrinsic to the central engine of the active galaxy, or it is just a result of a temporary obscuration or disappearance of such obscuration. While for some CL AGN phenomenon the obscuration mechanism can work, for most of the sources there are strong arguments in favor of the intrinsic changes:\n\n\n $\\bullet$ complex multi-band recovery, inconsistent with obscuration \\citep[e.g.][]{mathur2018}\n \n  $\\bullet$ strong changes seen in the IR, where the obscuration should not play a role \\citep{sheng2017,stern2018}\n  \n $\\bullet$ low level of polarization in CL AGN which argues against the scattering (and obscuration) scenario \\citep{2019arXiv190403914H}\n \n $\\bullet$ different variability behaviors of the observed emission lines in spectra of CL AGN \\citep[e.g.][]{kynoch2019}\n \n $\\bullet$ regular QPE behavior cannot be due obscuration because of the characteristic spectral evolution during outbursts \\citep{Miniutti2019,2020giustini}. \n \n Thus in most sources the intrinsic change in the bolometric luminosity later affects the X-ray and broad line region (BLR) appearance. These intrinsic changes can be either related to Tidal Disruption Event (TDE), or be a result of the spontaneous unforced behavior of the accretion flow close to a black hole. In some cases perhaps TDE provides the answer but in sources with repeated events the TDE is statistically unlikely. \n\nIn the present paper we concentrate on the discussion of the plausible mechanism which can lead to regular or semi-regular repeating outbursts intrinsic to the nucleus. In such case the source behavior should be related to some instabilities in the accretion flow. However, the radiation pressure instability expected to be operational in the innermost part of an AGN accretion disk does not provide the proper timescales \\citep[e.g.][]{gezari2017}. Convenient formulae for the duration of such outbursts given in \\citep{grzedzielski2017} give timescales of hundreds of years for a black hole mass of $10^7 M_{\\odot}$. \\citet{dexter2019} suggested that strong magnetization can shorten the estimated timescales. On the other hand, we can look for another mechanism related to the complexity of the innermost part of the flow, and \\citet{noda2018} proposed that the CL behavior in the source Mrk 1018 is related to the temporary disappearance of the warm corona. The source NGC 1566 notable for numerous CL outbursts \\citep[e.g.][]{alloin1986, baribaud1992, oknyansky2019} does not show the presence of the warm corona component before the outburst \\citep{parker2019}. The present observations cannot resolve directly any of these issues since they show at best the presence of the gas reservoir at a distance of 60 pc from the black hole (Mkn 590; \\citealt{raimundo2019}). They only show that the phenomenon is complex, for example the reappearance of broad lines in Mkn 590 is not accompanied by the full recovery of the continuum \\citep{raimundo2019}.\n\nIn this paper we propose a new mechanism which is suitable for explaining regular outbursts in sources which are not very close to the Eddington ratio. Using highly simplified toy model we aim at discussion whether the mechanism is likely to reproduce the observed timescales and therefore deserves the effort of more detailed description in the future.\n\n\n\n\\section{Analytical estimates and the model geometry}\n\\label{sec:analitical}\n\nThe character of the accretion flow in AGN strongly depends on the Eddington ratio of the source. In sources with the Eddington ratio above a few per cent, optically thick, geometrically thin disk extends down to the Innermost Stable Circular Orbit (ISCO). Modelling of the optical\/UV emission of quasars support this view \\citep[e.g.][]{capellupo2015}, although warm corona seems to be needed to explain the soft X-ray excess. However, low luminosity AGN, showing low-ionization nuclear emission line region (LINERS) do not show such a component in the optical\/UV spectra, and it is generally accepted that in these sources the innermost part of the accretion flow proceeds in a form of an optically thin advection-dominated accretion flow (ADAF). \n\nFor simplicity, we introduce here a definition of the Eddington accretion rate based on Newtonian physics:\n\\begin{equation}\n\\dot M_{Edd} = {48\\pi GM_{BH} m_p \\over \\sigma_T c},\n\\end{equation}\nwhere $M_{BH}$ is the black hole mass, $m_p$ is the proton mass, and $\\sigma_T$ is the Thomson cross-section. We thus measure the ratio of the accretion rate to the Eddington accretion rate using $\\dot m = \\dot M \/\\dot M_{Edd}$.\n\nIn those units, the transition between an inner ADAF flow and an outer standard accretion disk \\citep{abramowicz1995,czerny2019} takes place at\n\\begin{equation}\nR_{ADAF} = 2 \\alpha_{0.1}^4 \\dot m^{-2} R_{Schw},\n\\end{equation}\nwhere $\\alpha_{0.1}$ is the viscosity parameter introduced by \\citet{ss73}, in units of 0.1, and $R_{Schw} = 2 GM_{BH}\/c^2$ is the Schwarzschild radius of the black hole.\n\nStandard accretion disk is unstable in the innermost part, when the radiation pressure dominates \\citep{le74,pringle1973,ss76}, and the transition from the outer stable to the inner unstable radius takes place at: \n\\begin{equation}\nR_{tr} = 1522 (\\alpha_{0.1}m_7)^{2\/21} \\dot m^{16\/21} R_{Schw},   \n\\end{equation}\n\\citep{ss73}. Here $m$ is the black hole mass expressed in units of $10^7 M_{\\odot}$.  \nThe two lines cross at the specific accretion rate, $\\dot m_{st}$, \n\\begin{equation}\n\\label{eq:limit}\n\\dot m_{st} = 0.0905 \\alpha_{0.1}^{41\/29} m_7^{-1\/29},\n\\end{equation}\nwhere the dependence on the black hole mass is negligible, but the dependence on the viscosity coefficient is stronger than linear. The radius where it happens is given by:\n\\begin{equation}\nR_{st} = 244 \\alpha_{0.1}^{43\/29}  m_7^{2\/29} R_{Schw}.\n\\end{equation}\n\nIf the accretion rate of the flow is smaller than this limiting value, $\\dot m_{st}$, the whole flow is stable since both the gas-dominated cold outer disk and the inner ADAF flow are stable solutions. On the other hand, if the accretion rate is higher that $\\dot m_{st}$, there is a disk range, dominated by the radiation pressure which is unstable and could lead to a limit cycle behavior.\n\nThe timescale for such oscillations is generally set by the viscous timescale of the Shakura-Sunyaev disk, \n\\begin{equation}\n\\tau_{visc,SS} = {1 \\over \\alpha} ({R \\over H})^2 ( {R^3 \\over G M_{BH}})^{1\/2}\n\\end{equation}\n(\\citealt{ss73}; see e.g. a review by \\citealt{czerny2006}) \nwhich is long for the case of AGN, such as hundreds of years. However, if $\\dot m$ is just above the treshold defined by Equation~\\ref{eq:limit}, then the radial extension of the unstable zone, $\\delta R$ is much smaller than the radius $R$ itself. We illustrate schematically such a geometry in Figure~\\ref{fig:schematic}. In that case, the time needed to empty the zone is reduced, and the viscous evolution will happen in a timescale\n\\begin{equation}\n\\tau_{visc} = \\tau_{visc,SS} {\\Delta R \\over R}.\n\\end{equation}\n\nThus, for sources at low Eddington accretion rate the radiation pressure instabillity, operating in a very narrow zone at the border between the outer standard disk and an inner ADAF flow can provide a viable mechanisms explaining repeating outbursts in some CL AGN in timescales of a few years. The schematic view of our new model of CL AGN is shown in Figure~\\ref{fig:schematic}.\n\nThe small radial extension of the instability zone reduces also the amount of variable radiation flux by the same factor. On the other hand, the zone regulates the accretion flow in the inner ADAF, and most of the radiation is actually produced there, in the form of X-rays. The efficiency of the inner ADAF flow is not well known, but significant part of the energy goes directly to electrons, and subsequently, to radiation \\citep[e.g.][]{bisnovatyi1997,yuan2014,marcel2018}. The outburst thus should be clearly seen in X-rays, but in addition X-ray irradiation of the cold disk will lead to enhancement of the disk emission in optical\/UV band.\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.45\\textwidth]{drawing1-eps-converted-to.pdf}\n\\caption{The schematic view of the innermost part of the flow: outer cold stable disk (green), intermediate zone (disk part unstable due to radiation pressure instability), and inner hot ADAF which illuminates the outer disk.}\n \\label{fig:schematic}\n\\end{figure}\n\n\n\n\n\\section{One-zone time-dependent toy model}\n\\label{sect:toy_model}\n\nWe construct a simple toy model in order to check whether the mechanism may indeed give repeated outbursts of the observationally required properties. We basically follow the 1-D model of time evolution of accretion disk under radiation pressure instability \\citet{janiuk2002}, but we simplify it further by concentrating on a single zone approximation, thus reducing the numerical problem to ordinary differential equation in time. Instead of solving for vertically averaged disk structure as functions of both time and disk radius, we follow the time-dependent evolution of a single zone, representing the radiation pressure dominated region. It is a good approximation if the radial extension of the zone is small, i.e. the zone is narrow and comparable to the disk thickness.\n\nThe evolution of the zone in thermal timescale is very similar to the one in multi-radius approach, as the thermal evolution results from the net effect of the heating, radiative cooling, and advection cooling. We assume that the disk is in hydrostatic equilibrium.\nHowever, the viscous evolution is now set simply by the boundary conditions: (i) between the zone and outer stationary disk (ii) between the zone and inner ADAF:\n\\begin{itemize}\n\\item at the border between the zone and the outer disk, we assume a constant inflow rate of the material provided by the stable outer disk, $\\dot M_0$. \n\\item at the border between the zone and the inner ADAF, the material is removed from the disk at a rate $\\dot M$ by evaporation due to the electron conduction. \n\\end{itemize}\n\nSince this is a simple toy model, we do not use any advanced description of this process which would require the knowledge of the ADAF density, ion and electron temperature \\citep[e.g.][]{rozanska00,2007liu,2020qiao}. Instead, we postulate that the efficiency of the process should be proportional to the zone height, since the hot inner ADAF is geometrically thick so the interacting surface is set by the cold disk state. We additionally assume that evaporation is more efficient when there is more mass in the unstable zone. If the zone and ADAF happen to be in equilibrium, then the outflow rate from the zone should be equal to the inflow rate. So introducing the equilibrium zone thickness $H_0$, and equilibrium surface density in the disk, $\\Sigma_0$ our approach allows us to specify the evaporation rate in general as\n\\begin{equation}\n\\dot M = \\dot M_0 {H \\over H_0} {\\Sigma \\over \\Sigma_0},\n\\label{eq:evap}\n\\end{equation}\nwhere the quantities $H$ and $\\Sigma$ describe the height and surface density of the evolving zone. We always start our time-dependent evolution from an equilibrium model, but if the solution corresponds to unstable one, the disk will perform the limit cycle.\n\n\nFrom the two assumptions above, we obtain the time evolution of the surface density in the zone\n\\begin{equation}\n\\label{eq:sigma}\n{d \\Sigma \\over dt} = {\\dot M_0 - \\dot M \\over 2 \\pi R \\Delta R,}\n\\end{equation}\ni.e. it is given by the imbalance between a constant inflow rate into the zone, $\\dot M_0$, and variable outflow rate $\\dot M$ from the zone to inner ADAF flow. \n  \nThe evolution of the zone in the thermal timescale, given by Equation 33 in \\citet{janiuk2002}, under our assumptions, for a narrow zone, reduces to the following equation for the equatorial disk temperature, $T$: \n\\begin{eqnarray}\n\\label{eq:temp}\n{d \\log T \\over dt}& = &{(Q^+ - Q^{-} -Q_{adv})(1 + \\beta) \\over P H [(12 - 10.5 \\beta)(1 + \\beta) + (4 - 3\\beta)^2]} \\nonumber \\\\\n&+&2 {d \\log \\Sigma \\over dt}{4 - 3\\beta \\over (12-10.5 \\beta)(1+\\beta) + (4 - 3\\beta)^2}.\n\\end{eqnarray}\nHere the calculation of the derivatives of the disk thickness $H$ are already included in the expression. The values of the disk thickness, total pressure $P$, gas to the total pressure ratio, $\\beta$, viscous heating $Q^{+}$, radiative cooling $Q^{-}$  are determined from the standard equations of the vertically averaged disk structure in hydrostatic equilibrium as in \\citet{janiuk2002}, but here we do not introduce any additional correction coefficients related to the disk vertical structure (like $C_1$, $C_2$) since the current model is very simple. The advection cooling term $Q_{adv}$ is determined as\n\\begin{equation}\n\\label{eq:adv}\n Q_{adv} = {\\dot M P H \\over 2 \\pi R \\Delta R \\Sigma},   \n\\end{equation}\nso we include only advection term related to the inflow from the zone to inner ADAF, and we neglect the energy carried into the zone from the outer disk, which should be negligible.\n\nThus time-dependent partial differential equations (26) and (33) from \\citet{janiuk2002} reduce to ordinary differential equation for the time evolution of a surface density and temperature in the equatorial plane of a single zone.\n\nThe geometrically narrow instability zone evolves fast, but the amount of energy dissipated in this zone is also correspondingly small. Therefore, the changes in the zone luminosity by itself does not change significantly the system luminosity. However, the zone acts as a regulator of the accretion flow in the innermost ADAF. \n\nADAF flow was frequently considered as inefficient, but most estimates of the ion-electron coupling and of the Ohmic heating imply that actually ADAF flow in energetically quite efficient, at least when the accretion rate is not many orders of magnitude below the Eddington accretion rate. \n\\citep[e.g.][]{bisnovatyi1997, 2011ferreira, 2018Galax...6..122H}. Therefore, the inner part of the flow generates more energy than the outer part of the disk and the transition zone (the exact number would depend on the black hole spin). This energy is emitted in X-rays but part of the produced X-ray radiation will illuminate the disk and enhance the disk emission. \n\nWe thus assume the typical flow efficiency of 10\\%  in ADAF and calculate the result of the disk irradiation. ADAF is an extended medium so in principle this is a complex 2-D issue but in our simple model we represent the ADAF emission by emission localized along the symmetry axes since that allows us to calculate the effect in a simple way (we used the method and the code developed in \\citet{loska2004}. This irradiation is very important, strong illumination is observed in reverberation-studied sources like NGC 5548  \\citet{2015edelson, 2015ApJ...806..128D, 2016ApJ...824...11G,2017edelson, 2016ApJ...821...56F, 2017ApJ...835...65S, 2018mchardy, 2019kriss} where the variable X-ray emission drives the accretion disk continuum variability, although the correlation is not always perfect. \n\nIn our toy model we assume, for simplicity, that the inner region luminosity is equal to the total (time-dependent) bolometric luminosity\n\\begin{equation}\nL_{ADAF} = \\eta \\dot M c^2,\n\\end{equation}\nwith the flow efficiency $\\eta$ equal 0.1 like in radiatively efficient flow. The illumination of the outer disk is calculated semi-analytically as in \\citet{loska2004}, assuming that the emission is localized along the symmetry axis; otherwise 3-D computations would be necessary. The emissivity is adopted as a power law with index $\\beta = 2.0$, the maximum distance is equal to the transition radius $R_{tr}$, and the minimum distance along the axis is set at 1\/3 of this value. We assume complete local thermalization of the incident flux by the cold disk. \n\nIn principle, free parameters of our models are the black hole mass, the accretion rate and the viscosity as free parameters, since the location of the transition and the extension of the unstable zone should results from the computations of the disk structure. However, our toy model does not have all the ingredients (like proper description of the vertical structure, opacity, convection, irradiation etc., see for example \\citealt{rozanska1999}). So we additionally treat the radius and the zone width as independent parameters.\n  \n\n\\section{Results}\n\\label{sect:results}\n\nWe use now our toy model of radiation pressure instability in a narrow zone between the outer cold disk and an inner hot flow to model the repeating outbursts observed in some AGN. The model parameters are: the external accretion rate, $\\dot M_0$, the radius, $R$, the width of the unstable zone, $\\Delta R$, and the viscosity parameter, $\\alpha$, in the zone. The two remaining parameters, $\\Sigma_0$ and $H_0$ are determined self-consistently from the equilibrium (unstable) solution located at the stability curve.\n\n\n\\subsection{Stability curve}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.45\\textwidth]{Sigma_T_6_92_30.pdf}\n\\caption{Accretion rate vs. surface density in the transition zone between the cold SS outer disk and the inner ADAF in a stationary model in two cases $\\Delta R = 0.003 R$ (blue points), and $\\Delta R = 0.03 R$  (magenta points). Other parameters: log M = 6.92, $R = 30 R_{Schw}$, $\\alpha = 0.02$. }\n\\label{fig:zone}\n\\end{figure}\nStability curve is built of solutions to equations \\label{eq:sigma} and \\label{eq:temp}, assuming that all time derivatives are equal 0. They are conveniently plotted as a function of the external accretion rate, $\\dot M$. We express it in dimensionless units. In the case of the stationary solution, the accretion rate inside the zone is coupled to the zone properties as in a standard stationary disk:\n\\begin{equation}\n\\dot M = 4 \\pi \\alpha P H\/\\Omega_K,\n\\end{equation}\nas in the standard disk of \\citet{ss73}. \n\n The result is shown in Figure~\\ref{fig:zone} (blue line). Here we adopt parameters appropriate for NGC 1566. For black hole mass we assume $\\log M = 6.92$ after \\citet{woo2002}, we adopt the viscosity paramater $\\alpha_{0.1} = 0.2$ (i.e. $\\alpha = 0.02$) after \\citet{grzedzielski2017}, and we take 30 $R_{Schw}$ for the radius. The zone width is assumed to be very narrow, 0.003 R, comparable to the disk thickness.\n \n Our stability curves in their high accretion rate parts depend on the adopted width of the zone since the advection term in our model explicitly contains it (see Equation \\ref{eq:adv}).\nWhen the zone is narrow, the advection works particularly efficiently. \n\nThe negative slope of the stability curve implies that the solution is unstable. So, for the assumed radius and the black hole mass, the flow with accretion rate higher than $\\dot m \\sim 0.01$ is unstable. The upper stable branch starts quite early for a narrow zone, so the instability is expected to operate for $\\dot m$ between 0.01 and 0.1 in this case. If the zone width increases, the branch stabilized by advection starts at higher accretion rates, and for $\\Delta R$ of order of $R$ the stabilization would happen above the Eddington accretion rate, as in a standard slim disk \\citep{abramowicz1988}. However, our toy model is not expected to work for geometrically broad zone.\n\n\\subsection{Time evolution of the accretion rate through the zone}\n\nWe compute the time evolution of the zone by assuming the value of the black hole mass, the radius, the radial width of the zone, and the viscosity parameter. We then choose the external accretion rate from the range corresponding to the unstable branch.  The disk irradiation parameters are fixed, as described in Section~\\ref{sect:toy_model}.\n\n\nExemplary time evolution of the accretion rate regulated by the unstable zone is given in Figure~\\ref{fig:zestaw1}. We fixed the black hole mass there at the value corresponding to NGC 1566, but we varied the accretion rate, the viscosity parameters and the radial size of the unstable zone.  We see there that the timescales of outbursts, and the outbursts amplitudes are very sensitive to these parameters. The shape of the outburst vary less, and in our model the duration of the bright phase (outburst) is always longer then the outburst separation. This is because the accumulation phase is longer than the evaporation rate and the transfer through ADAF. Outbursts are regular since our toy model is very simple. More advanced models of disk instabilities, which include the wind, irradiation, magnetic field or tidal interaction with a companion in a binary system frequently lead to much more complex outbursts \\citep[see e.g.][and the references therein]{hameury2019}. These, models however, do not consider the inner ADAF and narrow instability zone, so our toy model gives interesting estimates of the timescales, and further development may easily lead to more complex lightcurve shapes.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.45\\textwidth]{zestaw1.pdf}\n\\caption{The dependence of the time-dependent accretion rate on the external steady accretion rate $\\dot m_0$, viscosity parameter $\\alpha$, and the geometrical thickness of the unstable zone, $\\Delta R$. Parameters are marked in each panel, the default parameters are:  $\\dot m = 0.0122$, $\\alpha = 0.02$, $\\Delta R = 0.003 R$. Fixed parameters: black hole mass $\\log M = 6.92$, inner radius of the disk $R = 30 R_{Schw}$. }\n\\label{fig:zestaw1}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.45\\textwidth]{zestaw2a.pdf}\n\\caption{The dependence of the time-dependent accretion rate on the external steady accretion rate $\\dot m_0$, viscosity parameter $\\alpha$, and the geometrical thickness of the unstable zone, $\\Delta R$. Parameters are marked in each panel, the default parameters are:  $\\dot m = 0.006$, $\\alpha = 0.2$, $\\Delta R = 0.003 R$. Fixed parameters: black hole mass $\\log M = 7.94$, inner radius of the disk $R = 20 R_{Schw}$. }\n\\label{fig:zestaw2}\n\\end{figure}\n\nThe evolution is significantly slower for more massive black holes. Therefore, in Figure~\\ref{fig:zestaw2} we show a set of lightcurves for a black hole mass more appropriate for sources like NGC 5548. In order to model frequent outbursts we have to request values of the higher viscosity parameter.\n\n\\subsection{Irradiation of the cold outer disk}\n\nAs discussed in Section~\\ref{sect:toy_model}, the variable accretion rate in the unstable zone and in the innermost part of the flow strongly affects the outer disk.\nThus, the variable accretion rate as shown in Figure~\\ref{fig:zestaw1} has to be used to receive the time evolution of the monochromatic flux and the line luminosity. \n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.45\\textwidth]{snapshot_nuLnu.pdf}\n\\caption{Two extreme states of the accretion disk in the source: between outbursts (blue line) and during outburst (red line). Here we neglect the contribution from the starlight.}\n\\label{fig:spectra}\n\\end{figure}\n\nIn Figure~\\ref{fig:spectra} we show two extreme examples of the spectra from an illuminated disk: between the outburst and at the peak of the outburst. For the chosen parameters, given in the figure caption, the flux at V band has changed by an order of magnitude, and the spectrum became much bluer in the far UV. Parameters which were used in this case are: $\\dot m = 0.012$ ,  $\\log M = 6.92$ , appropriate for NGC 1566.\n\nWe thus compare the bolometric lightcurve resulting from the instability to the corresponding monochromatic lightcurve. As explained in Section~\\ref{sect:toy_model}, \nwe derive the monochromatic lightcurve  taking into account the disk plus transition zone flux at V band for all the time steps of the evolution, using always the current value of $\\dot M$ to calculate the disk illumination. The shape of the curve is similar, but not identical, with the shape of the accretion rate variability. An example is shown in Figure~\\ref{fig:mdot_V_band}. Such lightcurve can be directly compared to the continuum lightcurve of a given source, but the observed lightcurve should be corrected for the starlight contamination.\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.45\\textwidth]{comparison-modelled-variations22.pdf}\n\\caption{The comparison of the modeled variations of the bolometric luminosity measured as the Eddington ration (blue line) with the modeled variations of the monochromatic disk luminosity in V band (red line), with irradiation included.}\n\\label{fig:mdot_V_band}\n\\end{figure}\n\nIf the line luminosity lightcurve is available, in principle we should compute the structure of the BLR but in our toy model we can assume that the line follows the bolometric luminosity of the source which is well represented by the varying accretion rate. \n\n\\subsection{Comparison with the observational data}\n\nOur toy model is not yet ready for detailed fitting of the observed lightcurves. What is more, such a comparison would be always inherently difficult since the observed variability in AGN is never strictly periodic. Thus our aim is to test if the model can roughly cover the characteristic variability timescales in the few exemplary sources.\nPhysical parameters which we obtained for each object are shown in Table \\ref{table:fits-parameters}.\n\n\\subsubsection{NGC 1566}\n\nThis source, usually classified as Seyfert 1.5 galaxy (z = 0.005017 after NED\\footnote{https:\/\/ned.ipac.caltech.edu\/classic\/}) is a well known CL AGN. Its semi-regular outbursts were already observed by \\citet{alloin1986}. Later, in September 2017 the source started a spectacular brightening in optical band \\citep{stanek2018}, and in X-rays \\citet{parker2019}. The bolometric luminosity of the source thus strongly vary, from $\\log L_{bol}= 41.4$ reported in \\citet{combes2019}, up to $\\log L_{bol}= 44.45$ \\citep{woo2002} . \\citet{parker2019} reported the Eddington ratio of 0.05 during the outburst and 0.002 between the outbursts.\n\nMostly concentrating on the old data showing multiple outbursts (see Figure~\\ref{fig:V_band_curve}) we assume that the characteristic timescale in this source is  5 years. For the black hole mass we assume the value  $\\log M = 6.92$ from \\citet{woo2002} (it is consistent with the value $6.8 \\pm 0.3$ derived from molecular gas dynamics by \\citealt{combes2019}).  We assume the mean accretion rate of $\\dot m =  0.012$ in Eddington units, corresponding the mean value. \n\nWe can find an example of the unstable solution for these input parameters assuming the value of 25 $R_{Schw}$ for the radius. The zone width is assumed to be very narrow, 0.002 R, comparable to the disk thickness. \nThe required value, $\\alpha = 0.04$ is by a factor 2 larger than $\\alpha = 0.02$ used by \\citet{grzedzielski2017}. The solution roughly corresponds to the middle panel of Figure~\\ref{fig:zestaw1} \n\nThe source behavior, however, is not regular, the last outburst appeared earlier than expected and had higher amplitude than the remaining three. \nThe optical V-band lightcurve reported by \\citet{stanek2018} shows a small outburst lasting about one year, at around 2014, thus shorter by a factor of a few than the outbursts observed by \\citet{alloin1986}. \n\nThe duration of the outburst seems too short in comparison to the time separation. This is a characteristic property of the current version of the model, particularly for lower accretion rates, and large outburst amplitudes.\n\n\n\n\\subsubsection{NGC 4151}\nFor this source we assume mass from  \\citet{woo2002} (log($M_{BH}\/M\\odot$) = 7.12) and bolometric luminosity 43.73 from \\citet{2005ApJ...629...61K}, z = 0.003262 after NED. We estimate $\\dot{m}$ as 0.027.\n\\citet{guo2014} suggest three possible periodicities for that source (P$_1$ = 4 $\\pm$ 0.1, P$_2$ = 7.5 $\\pm$ 0.3 and P$_3$ = 15.9 $\\pm$ 0.3 yr). \n\\citet{bon2012} also derive P = 15.9 yr for that source.  The same periodicity is also found in radial velocity curves of H$\\alpha$ broad line.\nSimilar values were suggested by \\citet{2018MNRAS.475.2051K} \n($\\sim$ 5 and $\\sim$ 8 years).\n\\citet{2007oknyanskij} suggest period about 15.6 years obtained using power spectrum. However, as \\citet{czerny_power2003} shows, changes are not strictly periodic and possible period vary between 1\/100 days and 1\/10 years.\n\nPhotometry continuum flux data set includes data from: \n\\citet{2006Bentz}, \\citet{2008A&A...486...99S}, \nAGN Watch provided by \\citet{1996ApJ...470..336K} and \\citet{1997A&A...324..904M}.\nTo reduce the influence of the longest timescale systematic trend (which is probably due to different mechanism), we rebin the data. The result is shown in Figure~\\ref{fig:V_band_curve_4151}. \n\n\nIf we adopt the value of 10 years for a characteristic timescale in this source we require similar values of the remaining parameters as in the case of NGC 1566. For such parameters outbursts amplitudes are large, and outbursts rather short lasting.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.45\\textwidth]{dane-26-06-1566.pdf}\n\n\\caption{The H$\\beta$ line flux evolution in NGC 1566 from \\citet{alloin1986}.}\n\n\n\\label{fig:V_band_curve}\n\\end{figure}\n\n\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.45\\textwidth]{dane-26-06-4151.pdf}\n\\caption{The continuum flux evolution in NGC 4151 points.}\n\n\\label{fig:V_band_curve_4151}\n\\end{figure}\n\n\n\n\n\\subsubsection{NGC 5548}\nNGC 5548 is object with long-term and dense data coverage in various wavelengths \\citep{2003chiang, 2015A&A...575A..22M, Mathur_2017}. Optical reverberation campaigns determined mass of its black hole \\citep{peterson2004, 2015PASP..127...67B}.\nThis complex source is known by the changes of the BLR, which may not be linked with the only one physical origin.\n NGC 5548 showed the obscuration in X-ray and UV range \\citet{2019kriss}, both the intrinsic continuum and the obscurer are variable \\citet{2015A&A...579A..42D}. \n\\citet{2019ApJ...877..119D} and references therein suggest that cloud shadowing should be considered as an appropriate explanation for variability in observation.\n\nWe assume physical parameters for this source as follows:\n$L_{BOL}$ = 44.45 from \\citet{2016A&A...587A.129E}, M = $8.71^{+3.21}_{-2.61} \\times 10^7 M_{\\odot}$ from \\citet{2016lu}.\n\\citet{2018MNRAS.475.2051K} suggest for that source period 13.3 $\\pm$ 2.26 yr,\naccretion rate 0.01 from \\citet{papadakis2019}.\n\\citet{bon2016} suggested slightly longer period of $\\sim$ 5700 days.\nThe continuum flux data set for NGC 5548 includes data from \\citet{bon2016} is shown in Figure \\ref{fig:5548_data}.  \n\nWe decided to model outbursts with period around 13 years. In that case, for the adopted mass and accretion rate as described above we can find the proper representation of the outbursts assuming much higher viscosity and somewhat smaller radius since the timescale is similar than in the two previous sources while the black hole mass is an order of magnitude higher. \n\nIt is interesting to note that the location of the unstable zone in this source, at a distance of 0.23 light days from the center is nicely consistent with the location of the obscurer (below 0.5 light days) discussed by \n\\citet{2019ApJ...882L..30D}.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.45\\textwidth]{dane-26-06-5548.pdf}\n\\caption{The continuum flux evolution in NGC 5548. Gray points represent observational data, black points represent data rebinned to 300 bins in total.}\n\\label{fig:5548_data}\n\\end{figure}\n\n\n\\subsubsection{GSN 069}\n\nThis relatively low mass Seyfert galaxy ($M_{BH} = 4.5 \\times 10^5 M_{\\odot}$, \\citealt{Miniutti2019}) was inactive when measured by ROSAT. In 2010 it showed a spectacular rise in the nuclear luminosity, followed by a slow decay. During the late decay phase, in December 2018, the source showed spectacular rapid large amplitude oscillations with the period of roughly 9 hours \\citep{Miniutti2019}. The behavior was still observed in February 2019. The nature of these Quasi-Periodic Eruptions (QPE) is not clear but the spectral changes strongly suggest the coupling with the corona formation and likely the coronal inflow. The outbursts are shown in Figure~\\ref{fig:gsn}.\n\nWe represented the variations in the disk luminosity using our toy model. \nWe assumed the black hole mass of  $4.5 \\times 10^5 M_{\\odot}$, after \\citet{Miniutti2019}, and we adjusted the remaining parameters to reproduce the timescale. QPE time separation can be indeed reproduced, although it requires small radius and large value of the viscosity coefficient. The external accretion rate favored by our model (0.013 in Eddington units) is much lower than the bolometric luminosity 0.46 estimated by \\citet{Miniutti2019}. The low accretion rate was implied by the instability zone present very close to the black hole. Clearly the current toy model does not describe yet the source behavior, and most likely the source performed just disk\/corona pulsations, as suggested by \\citet{Miniutti2019}. If so, more complex model with two-phase disk\/corona medium is needed to represent well this source.\n\n\\begin{table*}[]\n\\centering\n\\caption{Information about sources: Name of the source, redshift, luminosity distance [Mpc], assuming the cosmology: $H_O = 67$km s$^{-1}$, $\\Omega_m = 0.32$,$\\Omega_L = 0.68 $ \\citep{2014planck}, black hole mass, bolometric luminosity, time span for the optical data coverage, the amplitude expected from the stochastic behavior of AGN, the observed amplitude from line or continuum.}\n\\begin{tabular}{lllllllll}\nName & redshift &d$_{L}$(z)   & log(M$_{BH}$) & L$_{BOL}$   & data coverage       & $\\sigma$ exp & \\begin{tabular}[c]{@{}l@{}}$\\sigma$ obs \\\\ H$\\beta$ \\end{tabular} & \\begin{tabular}[c]{@{}l@{}}$\\sigma$ obs \\\\ cont\\end{tabular} \\\\ \\hline\n\\hline\nNGC 1566 &0.005017 & 22.5 & 6.92    & $2.5 \\times 10^{42}$ & 1972-1987 & 0.18  & 0.73                                                          & -                                                         \\\\\nNGC 4151 &0.003319 & 14.9 & 7.12     & $7\\times 10^{43}$   & 1988-2012 &  0.27     & -                                                             & 0.36                                                      \\\\\nNGC 5548 &0.017175 & 77.8 & 7.94    & $2.8 \\times 10^{44}$  & 1972-2015 &   0.20    & 0.40                                                          & 0.33 \\\\\nGSN 069 & 0.018 & 81.6  &  5.65   & $2.7 \\times 10^{43}$  & January 16\/17, 2019   &   0.21     &                        -                &  1.85  \\\\ \n          & &  &    & & (130 [ks]) &      &  &    \\\\                   \\hline                        \n\\end{tabular}\n\\label{table:observations}\n\\end{table*}\n\n\n\\begin{table*}[]\n\\centering\n\\caption{Summary of the results for each object. Name of the source, inner radius in $R_{Schw}$, the thickness of the unstable zone, viscosity parameter, period in years (except of the last object) and amplitude. Black hole mass used in the model was taken from Table~\\ref{table:observations}}.\n\\begin{tabular}{lllllll}\nName     &  R$_{in}$ & $\\Delta_R$ & $\\alpha$ & $\\dot{m}$ & period & amplitude \\\\ \\hline\n\\hline\nNGC 1566        & 25     & 0.002    & 0.04   & 0.015   & 5 yr  & 62.05     \\\\\nNGC 4151        & 26     & 0.006    & 0.05   & 0.027   & 10 yr  & 341.3     \\\\\nNGC 5548       & 20     & 0.003    & 0.20   & 0.0055  & 13 yr & 4892.36   \\\\\nGSN 069         & 5      & 0.03     & 0.25   & 0.013   & 0.387 day & 3.95 \\\\ \\hline\n\\end{tabular}\n\\label{table:fits-parameters}\n\\end{table*}\n\n\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.45\\textwidth]{dane-30-06-gsn.pdf}\n\\caption{GSN 069 disk contribution in 0.2-2 keV from \\citet{Miniutti2019}}\n\n\n\n\n\\label{fig:gsn}\n\\end{figure}\n\\section{Discussion} \\label{sec:discussion}\n\nThe Changing-Look behavior - rapid large amplitude changes in active galaxies - is reported now with increasing frequency but the mechanism is still unknown. Part of the phenomena can be actually related to tidal disruptions, particularly in the case of a single, long lasting large amplitude event. On the other hand, if multiple event takes lace in a single source, particuarly in a semi-regular form, TDE mechanism is ruled out. There is al least one source where we observe a combination of the two phenomena: a galaxy GSN 069. Classified as Seyfert 2 galaxy, GSN 069 was undetected in ROSAT All Sky Survey, but in 2010 the source showed a spectacular brightening, and when the dimming continued, the source showed very regular outbursts \\citep{Miniutti2019}.\n\nIn this paper we concentrate of modelling the repeating semi-regular outbursts observed in the sources radiating at a few per cent of the Eddington ratio.\nWe propose that the radiation pressure instability operating in the narrow zone between the outer gas-dominated stable accretion disk and an inner hot ADAF flow may be responsible for repeating outbursts in some CL AGN, as NGC 1566. We show that the proposed mechanism can lead to outbursts on a timescale much shorter than the usual viscous timescale in a cold disk. \n\nFor this mechanism to operate we require that the mean accretion rate in the source is relatively low so the inner ADAF flow extends to a radius which is not much smaller than the radius where the radiation pressure in a standard Keplerian disk dominates. The amplitudes of the outbursts in the optical band are large due to the irradiation of the outer disk by the enhanced inner hot flow.\n\nThe generic prediction of the model is that the spectrum of the nucleus should become much bluer during outburst, and the outbursts in X-ray band should have comparable or larger amplitude. On the other hand, the current model does not  give the outburst shape consistent with the observational data. Our toy model relies  only on the viscous timescale in the unstable ring close to ADAF, without addressing the full complexity of the standard disk\/ADAF transition. \n\nThe model predicts strong semi-periodic variations in the emitted flux for sources at a few per cent of the Eddington ratio but these changes may, or may not be revealed in the properties of the BLR. If the variability timescale is long in comparison to the time required for the adjustment of the BLR structure to the change of the nuclear emission then the BLR will follow the changes in the nucleus in a quasi-stationary way, and we should see the classical Changing-Look AGN phenomenon. On the other hand, if the nuclear changes are lasting too shortly, then the BLR may not fully adjust. For example, in sources like GSN 069 the eruption lasts about an hour while the distance to the BLR is likely a few hours, like in another low mass Seyfert galaxy NGC 4395 \\citep{peterson2005}. What is more, the light travel time describes just a change in irradiation while BLR structure adjusts more slowly \\citep[see e.g.][]{hryniewicz2010}. \n\n\\citet{ross2018} considered a possibility that the behavior of the quasar J1100-0053 is related to instability in a cold disk\/ADAF  transition zone but argues against it since in other objects (e.g. NGC 1097) the transition zone is stable. Indeed, the position of the transition radius determined by balancing the cold disk evaporation rate and the inner hot flow depends on the global accretion rate \\citep[e.g.][]{rozanska00,spruit2002,taam2012} seems rather stable. Our solution to the problem comes from introducing the radiation pressure instability. It also implies that for lower Eddington ratio objects the instability would not operate while for higher Eddington objects this mechanism would lead to outbursts of much larger part of the disk and it will operate on a timescale of thousands of years, as typically predicted for the radiation pressure instability \\citep{janiuk2002,czerny2009,wu2016,grzedzielski2017}. However, if the evolution includes the time-dependent coronal flow \\citep[e.g.][]{2007janiuk} or time-dependent vertical stratification of the disk into cold standard disk and the warm corona \\citep[e.g.][]{2003corona,2020gronkiewicz,2020pop}, the timescales will be strongly affected due to quadratic dependence of the viscous timescale on ratio of the local medium geometrical thickness to the local radius.\n\nIn most cases the outbursts we model are not clearly quasi-periodic (the behavior of the source GSN 069 discovered by \\citealt{Miniutti2019} is a nice exception) so there is a danger that we try to model the source behavior using a dedicated mechanism while in reality all AGN show a stochastic variability, and this stochastic variability may lead sometimes, with certain statistical probability, to a behavior which looks like quasi-periodic \\citep{vaughan2016}. However, stochastic variability has a well defined power spectrum shape and normalization, both in X-rays \\citep{mchardy2004} and in the optical band \\citep[e.g.][]{czerny1999,czerny_power2003}.  Thus the amplitude for a given time span is limited. For the studied sources we thus report the observed variability amplitude for a given time period and we compare it from the amplitude expected from the stochastic behavior of AGN. For NGC 1566, NGC 4151 and NGC 5548 this stochastic amplitude was predicted assuming the power spectrum from the recent work of \\citet{2010MNRAS.403..605B}, including the scaling by a factor 100 between the optical and the X-ray power spectrum, and the break in the X-ray spectrum for each source was estimated following \\citet{2006Natur.444..730M}. Knowing the optical power spectrum we could predict the source variance expected from the standard stochastic variability. For GSN 069 we estimated the expected X-ray variability from the typical X-ray variability level of AGN \\citet{2002MNRAS.332..231U} by averaging the provided values of $\\sigma$ for four sources for timescales of order of $10^6$ s. The dispersion in those values was small, and we took the timescales longer than the length of the used GSN 069 lightcurve since the level of X-ray variability at a given timescale scales with the black hole mass \\citep[e.g.][]{nikolajuk2004}. All values are reported in Table~\\ref{table:observations}. We see that the observed dispersion is much larger than expected from the stochastic variations. Therefore, invoking a separate mechanism to explain this phenomenon is justified.\n\nThe presented toy model is still too simplistic to account quantitatively for the observed outbursts. The comparison shows that the duration of the outburst in some models is far too short in comparison with the rising phase while is some cases (GSN 069) they are too long. This is partially because the model does not account properly for the evaporation mechanism of the zone. In a realistic model Equation~\\ref{eq:evap} should be replaced with the physically motivated equation containing the additional timescale for the process. However, this is not simple. The spectral changes observed in GSN 069 during outbursts \\citep{Miniutti2019} suggest that a comptonizing corona forms above the disk, and it may be that the real mechanism is actually a two-step mechanism, with corona formation as a stage one, and then the corona inflow as a stage two, finishing outbursts. Thus the future model should have both radial and vertical stratification, perhaps actually a full 2-D since the height of the zone is comparable to its radial extension, and it should include full time-dependence of the outer disk since the irradiation would couple to the stability properties. However, such a model is far beyond the aim of the current project.\n\nThe second equally important aspect is the time delay in the signal propagation. The current model assumes that the change in the irradiation patters happens without any time delay while actually the outer parts of the disk react with significant time delay of days, as is well known from reverberation studies of AGN continua (e.g. \\citealt{collier1998,sergeev2005,cackett2007,edelson2015,cackett2020}, and the references therein). The response of the emission lines is delayed even more strongly as showed by numerous campaigns (e.g. \\citealt{liutyi1977,collier1998,kaspi2000,peterson2004,grier2017,du2018}), thus the final model has to include these effects, particularly in the case of relatively fast variations in the source.\n\n\\section*{Acknowledgements}\nWe thank Alex Markowitz for helpful discussion and Giovanni Miniutti for providing the data for the source GSN 069 and very helpful comments to the manuscript.\nThe project was partially supported by National Science Centre, Poland, grant No. 2017\/26\/A\/ST9\/00756 (Maestro 9),  and by the MNiSW grant DIR\/WK\/2018\/12. Part of the work was done when BC was supported by a Durham Senior Research Fellowship COFUNDed between Durham\nUniversity and the European Union under grant agreement number 609412.\nE.B. and N.B. acknowledge the support of Serbian Ministry of Education,\nScience and Technological Development, contract number 451-03-68\/2020\/14\/20002.\n\n\n\\section*{Software}\n\\citet{r}\n\n\n\\bibliographystyle{aa}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nAssume throughout that $\\FF$ is a field of arbitrary characteristic, not necessarily algebraically closed, with group of units $\\FF^*$. Fix $q\\in\\FF^*$ with $q\\neq 1$. The \\emph{quantum plane}  is the unital associative algebra \n\\begin{equation}\n\\qp=\\FF\\{ x, y\\}\/(yx-qxy)\n\\end{equation}\nwith generators $x$ and $y$ subject to the relation $yx=qxy$.\n\nConsider the operators $\\tau_q$ and $\\partial_q$ defined on the polynomial algebra $\\FF[t]$ by\n\\begin{equation}\n \\tau_q (p)(t)=p(qt), \\quad \\mbox{and} \\quad \\partial_q (p)(t)=\\frac{p(qt)-p(t)}{qt-t}, \\quad \\mbox{for $p\\in\\FF[t]$}.\n\\end{equation}\nThen the assignment $x\\mapsto \\tau_q$, $y\\mapsto \\partial_q$ yields a (reducible) representation $\\qp\\rightarrow \\mathrm{End}_\\FF (\\FF[t])$ of $\\qp$, which is faithful if and only if $q$ is not a root of unity. The operators $\\tau_q$ and $\\partial_q$ are central in the theory of linear $q$-difference equations and $\\partial_q$ is also known as the \\emph{Jackson derivative}, as it appears in \\cite{fJ10}. See e.g.\\ \\cite{yM88}, \\cite[Chap.\\ IV]{cK95} and references therein for further details.\n\nThe irreducible representations of the quantum plane $\\qp$ have been classified in~\\cite{vB97} using results from~\\cite{BvO97}. Following \\cite{vB97} we say that a representation of $\\qp$ is a \\emph{weight representation} if it is semisimple as a representation of the polynomial subalgebra $\\FF[H]$ generated by the element $H=xy$. When $q$ is a root of unity all irreducible representations of $\\qp$ are finite-dimensional weight representations, and these are well understood. For example, if $\\FF$ is algebraically closed and $q$ is a primitive $n$-th root of unity then the irreducible representations of $\\qp$ are either $1$ or $n$ dimensional. When $q$ is not a root of unity there are irreducible representations of $\\qp$ that are not weight representations, and in particular are not finite dimensional. These turn out to be the \\emph{$\\FF[H]$-torsionfree} irreducible representations of $\\qp$, as they remain irreducible (i.e. nonzero) upon localizing at the nonzero elements of $\\FF[H]$. In~\\cite[Cor.\\ 3.3]{vB97} the torsionfree representations of $\\qp$ are classified in terms of elements satisfying certain conditions, but no explicit construction of these representations is given.\n\nWe assume $q$ is not a root of unity, and we give an explicit construction of a 3-parameter family $\\V{m, n}{f}$ of infinite-dimensional representations of $\\qp$ having the following properties (compare Propositions \\ref{P:isoclass}, \\ref{P:dec} and \\ref{P:weight}):\n\\begin{itemize}\n\\item $m$ and $n$ are positive integers, and $f:\\mathbb{Z} \\rightarrow \\FF^{*}$ satisfies condition \\eqref{prop} below, which essentially encodes $n$ independent parameters from $\\FF^*$;\n\\item $\\V{m, n}{f}$ is irreducible if and only if $\\gcd(m, n)=1$;\n\\item if $(m, n)\\neq (m', n')$ then $\\V{m, n}{f}$ and $\\V{m', n'}{f'}$ are not isomorphic;\n\\item $\\V{m, n}{f}$ is a weight representation if and only if $m=n$;\n\\item if $\\FF$ is algebraically closed and $V$ is an irreducible weight representation of $\\qp$ that is infinite dimensional, then $V\\simeq\\V{1, 1}{f}$ for some $f:\\mathbb{Z} \\rightarrow \\FF^{*}$.\n\\end{itemize}\nThus, in some sense weight and non-weight representations of $\\qp$ are rejoined in the family $\\V{m, n}{f}$.\n\nThe localization of $\\qp$ at the multiplicative set generated by $x$ contains a copy of the \\emph{$q$-Weyl algebra}, which is the algebra \n\\begin{equation}\\label{E:qwa}\n\\mathbb{A}_1(q)=\\FF\\{ X, Y\\}\/(YX-qXY-1)\n\\end{equation}\nwith generators $X$ and $Y$ subject to the relation $YX-qXY=1$ (see~\\eqref{E:qwainqp} for details about this embedding). This is used in Subsection~\\ref{SS:qwa} to regard the representations $\\V{m, n}{f}$ as infinite-dimensional irreducible representations of $\\mathbb{A}_1(q)$. In contrast with the action of $\\qp$ on $\\V{m, n}{f}$ when $m=n$, it turns out that $\\V{m, n}{f}$ is never a weight representation of $\\mathbb{A}_1(q)$ in the sense of~\\cite{vB97}. In Subsection~\\ref{SS:restriction} we pursue a dual approach by constructing representations $\\W{n}{}$ of $\\mathbb{A}_1(q)$ and then restricting the action from the $q$-Weyl algebra to two distinct subalgebras of $\\mathbb{A}_1(q)$ isomorphic to $\\qp$.\n\n\n\n\\section{A family $\\V{m, n}{f}$ of infinite-dimensional irreducible representations of $\\qp$ for $q$ not a root of unity}\\label{S:qnru}\n\nAssume $q\\in\\FF^*$ is not a root of unity. We introduce a family $\\V{m, n}{f}$ of infinite-dimensional representations of $\\qp$ which are not in general weight representations in the sense of~\\cite{vB97}, but which includes all irreducible infinite-dimensional weight representations of $\\qp$ if we further assume $\\FF$ to be algebraically closed.\n\n\\subsection{Structure of the representations $\\V{m, n}{f}$}\n \nFix positive integers $m, n\\in\\ZZ_{>0}$ and a function $f:\\mathbb{Z} \\rightarrow \\FF^{*}$ satisfying \n\\begin{equation}\\label{prop}\nf(i+n)=qf(i), \\quad \\quad \\text{for all $i\\in\\ZZ$.}\n\\end{equation}\nSuch functions are in one-to-one correspondence with elements of $\\left(\\FF^*\\right)^n$. Let $\\V{m, n}{f}$ denote the representation of $\\qp$ on the space $\\FF[t^{\\pm 1}]$ of Laurent polynomials in $t$ given by \n\\begin{equation}\\label{action}\nx.t^{i}=t^{i+n}, \\quad \\quad y.t^{i}=f(i)t^{i-m},\\quad \\quad \\text{for all $i\\in\\ZZ$.}\n\\end{equation}\nCondition~\\eqref{prop} ensures that the expressions~\\eqref{action} do define an action of $\\qp$ on $\\FF[t^{\\pm 1}]$ as, for all $i\\in\\ZZ$,\n\\begin{equation*}\n(yx-qxy).t^{i}=(f(i+n)-qf(i))t^{i+n-m}=0.\n\\end{equation*}\n\n\\begin{exam}\\label{Ex:floor}\nFix $\\mu\\in\\FF^{*}$ and $m, n\\in\\ZZ_{>0}$. For $i\\in\\ZZ$ let $f(i)=\\mu q^{\\left\\lfloor \\frac{i}{n}\\right\\rfloor}$, where $\\left\\lfloor \\frac{i}{n}\\right\\rfloor$ denotes the largest integer not exceeding $\\frac{i}{n}$. Then $f:\\mathbb{Z} \\rightarrow \\FF^{*}$ satisfies condition~\\eqref{prop} and thus there is a representation $\\V{m, n}{f}$ of $\\qp$ on $\\FF[t^{\\pm 1}]$ with action \n\\begin{equation*}\nx.t^{i}=t^{i+n}, \\quad \\quad y.t^{i}=\\mu q^{\\left\\lfloor \\frac{i}{n}\\right\\rfloor}t^{i-m},\\quad \\quad \\text{for all $i\\in\\ZZ$.}\n\\end{equation*}\n\\end{exam}\n\n\nWe begin the study of the representations $\\V{m, n}{f}$ by first considering the case that the parameters $m$ and $n$ are coprime. The following consequence of~\\eqref{prop} will be helpful.\n\n\n\n\n\n\\begin{lemma}\\label{L:NT}\nAssume  $\\gcd(m, n)=1$ and $f:\\mathbb{Z} \\rightarrow \\FF^{*}$ satisfies~\\eqref{prop}. For $k\\in\\ZZ$ define \n\\begin{equation}\n\\s{f}(k)=\\prod_{i=0}^{n-1} f(k-im).\n\\end{equation}\nThen $\\s{f}(k)=\\s{f}(0)q^{k}$.\n\\end{lemma}\n\n\n\\begin{proof}\nFor $j\\in\\ZZ$ let $0\\leq \\overline{\\jmath}<n$ be the unique integer such that  $\\overline{\\jmath}\\equiv j \\, \\mathsf{mod} \\, n$. Then the formula $f(j)=f(\\overline{\\jmath}) q^{\\frac{j-\\overline{\\jmath}}{n}}$ can be verified by induction on $\\left| \\frac{j-\\overline{\\jmath}}{n} \\right|$. Thus,\n\\begin{equation*}\n \\s{f}(k)=\\prod_{i=0}^{n-1} f(k-im)=\\prod_{i=0}^{n-1} f\\left(\\overline{k-\\imath m}\\right) \\prod_{i=0}^{n-1}q^{\\frac{k-im-\\overline{k-\\imath m}}{n}}. \n\\end{equation*}\nSince $m$ and $n$ are coprime,  the set $\\left\\{ \\overline{k-\\imath m} \\mid 0\\leq i<n \\right\\}$  consists of all the integers from $0$ to $n-1$, and is thus independent of $k$. Moreover,\n\\begin{equation*}\n\\sum_{i=0}^{n-1}{\\frac{k-im-\\overline{k-\\imath m}}{n}} = k+ \\sum_{i=0}^{n-1}{\\frac{-im-\\overline{k-\\imath m}}{n}} = k+ \\sum_{i=0}^{n-1}{\\frac{-im-\\overline{(-\\imath m)}}{n}}.\n\\end{equation*}\nHence,\n\\begin{equation*}\n \\s{f}(k) =q^{k}\\prod_{i=0}^{n-1} f\\left(\\overline{-\\imath m}\\right) \\prod_{i=0}^{n-1}q^{\\frac{-im-\\overline{(-\\imath m)}}{n}}=q^{k}\\s{f}(0).\n\\end{equation*}\n\\end{proof}\n\n\\begin{prop}\\label{P:irred}\n Assume  $\\gcd(m, n)=1$ and $f:\\mathbb{Z} \\rightarrow \\FF^{*}$ satisfies~\\eqref{prop}. Then the representation $\\V{m, n}{f}$ defined by~\\eqref{action} is an irreducible representation of $\\qp$.\n\\end{prop}\n\n\\begin{proof}\nWe begin with a computation: for $k\\in\\ZZ$ we have, by Lemma~\\ref{L:NT},\n\\begin{equation}\\label{E:eigen}\nx^m y^n . t^k=x^m\\left(\\prod_{i=0}^{n-1}f(k-im)\\right) t^{k-nm}=\\s{f}(k) t^{k}=\\s{f}(0)q^{k} t^{k}.\n\\end{equation}\nHence,  $x^m y^n . p(t)=\\s{f}(0)p(qt)$ for all $p\\in\\FF[t^{\\pm 1}]$.\n\nLet $\\mathsf{W}\\subseteq \\V{m, n}{f}$ be a nonzero subrepresentation. If $p(t)\\in\\mathsf{W}$ then also $p(qt)\\in\\mathsf{W}$, by~\\eqref{E:eigen}. As $q$ is not a root of unity, the latter implies that $t^{\\ell}\\in\\mathsf{W}$ for some $\\ell\\in\\ZZ$. The coprimeness of $m$ and $n$ shows the existence of integers $a$ and $b$ so that $an-bm=1$. By replacing $a$ and $b$ with $a+jm$ and $b+jn$ for a sufficiently large integer $j$, we can assume $a, b\\in\\ZZ_{>0}$. Then $x^{a} y^{b}.t^{k}=\\lambda_{k}t^{k+1}$ for some $\\lambda_{k}\\in\\FF^{*}$, showing that $t^{k}\\in\\mathsf{W}$ for all $k\\geq \\ell$. A similar argument shows that $t^{k}\\in\\mathsf{W}$ for all $k\\leq \\ell$. Hence $\\mathsf{W}=\\V{m, n}{f}$, establishing the irreducibility of $\\V{m, n}{f}$.\n\\end{proof}\n\nNext we describe $\\V{m, n}{f}$ in terms of a maximal left ideal of $\\qp$. Recall that for a representation $\\mathsf{V}$ of $\\qp$ and an element $v\\in\\mathsf{V}$, the annihilator of $v$ in $\\qp$ is $\\mathsf{ann}_{\\qp}(v)=\\{ r\\in\\qp \\mid r.v=0 \\}$, a left ideal of $\\qp$.\n\n\\begin{prop}\\label{P:isoclass}\nAssume  $\\gcd(m, n)=1$ and $f:\\mathbb{Z} \\rightarrow \\FF^{*}$ satisfies~\\eqref{prop}. \n\\begin{enumerate}\n\\item[\\textup{(a)}] For $1\\in\\V{m, n}{f}$, $\\mathsf{ann}_{\\qp}(1)=\\qp\\left(x^{m}y^{n}-\\s{f}(0)\\right)$ and $$\\V{m, n}{f}\\simeq \\qp\/\\qp\\left(x^{m}y^{n}-\\s{f}(0)\\right).$$\n\\item[\\textup{(b)}] For positive integers $m', n'$, and $f':\\mathbb{Z} \\rightarrow \\FF^{*}$ satisfying \\eqref{prop} (with $n$ replaced by $n'$), \nwe have $\\V{m, n}{f}\\simeq \\V{m', n'}{f'}$ if and only if $m=m'$, $n=n'$ and $\\s{f'}(0)=q^{k}\\s{f}(0)$ for some $k\\in\\ZZ$.\n\\end{enumerate}\n\\end{prop}\n\n\\begin{proof}\n(a)\\ Let $\\theta=x^m y^n$. First we show that \n\\begin{equation}\\label{E:claim_ann}\n\\mathsf{ann}_{\\qp}(1)=\\qp\\left(\\FF[\\theta]\\cap\\mathsf{ann}_{\\qp}(1) \\right).\n\\end{equation}\nThe inclusion $\\supseteq$ is clear, so suppose $u\\in\\mathsf{ann}_{\\qp}(1)$. Write $u=\\sum_{i\\geq 0}\\mu_i x^{a_i}y^{b_i}=\\sum_{k\\in\\ZZ}u_k$, where $\\displaystyle u_k=\\sum_{na_i-mb_i=k}\\mu_i x^{a_i}y^{b_i}$. Since $u_k.1$ is in $\\FF t^k$, it follows that $u_k\\in\\mathsf{ann}_{\\qp}(1)$ for all $k\\in\\ZZ$, and it suffices to prove $u_k\\in\\qp\\left(\\FF[\\theta]\\cap\\mathsf{ann}_{\\qp}(1) \\right)$.\n\nIf $na_i-mb_i=na_j-mb_j$ then, as $\\gcd(m, n)=1$, we deduce that $(a_i, b_i)=(a_j, b_j)+\\xi (m, n)$ for some $\\xi\\in\\ZZ$. Thus, by the normality of $x$ and $y$, there are  $a, b\\geq 0$ with $na-mb=k$ such that $u_k=x^a y^b w_0$, where $w_0=\\sum_{j\\geq 0}\\nu_j x^{\\xi_j m}y^{\\xi_j n}\\in\\FF[\\theta]$. Notice that for any $\\ell\\in\\ZZ$, $x^a y^b.t^\\ell$ is a nonzero scalar multiple of $t^{\\ell+k}$, so $x^a y^b w_0=u_k\\in\\mathsf{ann}_{\\qp}(1)$ implies that $w_0\\in\\mathsf{ann}_{\\qp}(1)$. Hence, $u_k\\in\\qp\\left(\\FF[\\theta]\\cap\\mathsf{ann}_{\\qp}(1) \\right)$ and \\eqref{E:claim_ann} is established.\n\nNow \\eqref{E:eigen} implies that $\\theta-\\s{f}(0)\\in\\FF[\\theta]\\cap\\mathsf{ann}_{\\qp}(1)$. Since $\\FF[\\theta]\\left( \\theta-\\s{f}(0) \\right)$ is a maximal ideal of $\\FF[\\theta]$ it follows that $\\FF[\\theta]\\cap\\mathsf{ann}_{\\qp}(1)=\\FF[\\theta]\\left( \\theta-\\s{f}(0) \\right)$ and $\\mathsf{ann}_{\\qp}(1)=\\qp\\left(\\theta-\\s{f}(0)\\right)$. This proves (a) as $1\\in\\V{m, n}{f}$ generates $\\V{m, n}{f}$.\n\n(b)\\ We observe that the arguments above also show that for $t^k\\in\\V{m, n}{f}$, $\\mathsf{ann}_{\\qp}(t^k)=\\qp\\left(\\theta-q^k \\s{f}(0)\\right)$ and \n\\begin{equation*}\n\\V{m, n}{f}\\simeq \\qp\/\\qp\\left(x^{m}y^{n}-q^k\\s{f}(0)\\right),\n\\end{equation*}\nfor any $k\\in\\ZZ$. This establishes the \\textit{if} part of (b). For the direct implication, suppose $\\V{m, n}{f}\\simeq \\V{m', n'}{f'}$. We have, for $a, b\\geq 0$ and $t^k\\in\\V{m, n}{f}$, \n$$\nx^a y^b.t^k=\\left( \\prod_{i=0}^{b-1}f(k-im)\\right)t^{k+na-mb}\n$$\nand $\\prod_{i=0}^{b-1}f(k-im)\\neq 0$. This implies that $x^a y^b$ is diagonalizable on $\\V{m, n}{f}$ if and only if $na=mb$. As $\\gcd(m, n)=1$ this amounts to having $(a, b)=\\xi(m, n)$ for some $\\xi\\geq 0$. \n\nSince $\\V{m, n}{f}\\simeq \\V{m', n'}{f'}$, then $x^{m'}y^{n'}$ is diagonalizable on $\\V{m, n}{f}$ and similarly $x^{m}y^{n}$ is diagonalizable on $\\V{m', n'}{f'}$. By the relation above we conclude that $(m, n)=(m', n')$. Moreover, the eigenvalues of $x^{m}y^{n}$ on $\\V{m, n}{f}$ are of the form $q^{k}\\s{f}(0)$, whereas $\\s{f'}(0)$ is an eigenvalue of $x^{m'}y^{n'}=x^{m}y^{n}$ on $\\V{m', n'}{f'}$. Hence $\\s{f'}(0)=q^{k}\\s{f}(0)$ for some $k\\in\\ZZ$, which concludes the proof.\n\\end{proof}\n\n\\begin{remark}\\label{R:f}\nBy  Proposition~\\ref{P:isoclass} above, for $\\gcd(m, n)=1$ and $f:\\mathbb{Z} \\rightarrow \\FF^{*}$ satisfying~\\eqref{prop}, the isomorphism class of $\\V{m, n}{f}$ depends only on $m$, $n$ and $\\s{f}(0)\\in\\FF^*$. \n\nFix $\\lambda\\in\\FF^*$. Since $\\gcd(m, n)=1$ there is a unique $f_\\lambda:\\mathbb{Z} \\rightarrow \\FF^{*}$ such that \\eqref{prop} holds and $f_\\lambda(km)=\\lambda$ if $k=0$ and $f_\\lambda(km)=1$ if $-(n-1)\\leq k\\leq -1$. Then $\\s{f_\\lambda}(0)=\\lambda$, $\\V{m, n}{f_\\lambda}\\simeq \\qp\/\\qp\\left(x^{m}y^{n}-\\lambda\\right)$ and, for $\\lambda'\\in\\FF^*$, $\\V{m, n}{f_\\lambda}\\simeq \\V{m, n}{f_{\\lambda'}}$ if and only if $\\lambda\/\\lambda'\\in \\langle q\\rangle$, where $\\langle q\\rangle$ is the  subgroup of $\\FF^*$ generated by $q$.\n\nIf $\\FF$ contains an $n$-th root of $\\lambda$, say $\\mu$, there is a more natural construction for the irreducible representation $\\qp\/\\qp\\left(x^{m}y^{n}-\\lambda\\right)$. Define $f^{\\mu}(i)=\\mu q^{\\left\\lfloor \\frac{i}{n}\\right\\rfloor}$, as in Example~\\ref{Ex:floor}. Then $\\s{f^\\mu}(0)=q^k \\mu^n=q^k \\lambda$, for some $k\\in\\ZZ$. It follows from Proposition~\\ref{P:isoclass} that $\\V{m, n}{f^\\mu}\\simeq \\qp\/\\qp\\left(x^{m}y^{n}-\\lambda\\right)$ and $\\V{m, n}{f^\\mu}$ depends only on $m$, $n$ and $\\lambda$, and not on the particular $n$-th root of $\\lambda$ that was chosen.\n\\end{remark}\n\nFinally we consider the general case of arbitrary $m, n\\in\\ZZ_{>0}$.\n\n\n\\begin{prop}\\label{P:dec}\nLet  $m, n\\in\\ZZ_{>0}$ be arbitrary, with $d=\\gcd(m, n)$, and assume $f:\\mathbb{Z} \\rightarrow \\FF^{*}$ satisfies~\\eqref{prop}. Then there is a direct sum decomposition\n\\begin{equation}\\label{dec}\n\\V{m, n}{f}\\simeq \\bigoplus_{k=0}^{d-1}\\V{m\/d, n\/d}{f_k}\n\\end{equation}\ninto irreducible representations, where $f_k(i)=f(k+id)$, for $0\\leq k<d$ and $i\\in\\ZZ$.\n\nMoreover, suppose $m', n'\\in\\ZZ_{>0}$, and $f':\\mathbb{Z} \\rightarrow \\FF^{*}$ satisfies~\\eqref{prop} (with $n$ replaced by $n'$). If  $\\V{m, n}{f}\\simeq \\V{m', n'}{f'}$ then $m=m'$ and $n=n'$.\n\\end{prop}\n\n\\begin{proof}\nFor $0\\leq k<d$,  the subspace $t^k \\FF[t^{\\pm d}]$ of $\\V{m, n}{f}$ is readily seen to be invariant under the actions of $x$ and $y$, and we have $\\V{m, n}{f} = \\bigoplus_{k=0}^{d-1}t^k \\FF[t^{\\pm d}]$. Thus, next we argue that the subrepresentation $t^k \\FF[t^{\\pm d}]$ is isomorphic to $\\V{m\/d, n\/d}{f_k}$, where $f_k(i)=f(k+id)$ for all $i\\in\\ZZ$. First notice that $f_k(i+n\/d)=f(k+id+n)=qf(k+id)=qf_k(i)$, so $\\V{m\/d, n\/d}{f_k}$ is defined. Consider the map $\\phi: \\V{m\/d, n\/d}{f_k}\\rightarrow t^k \\FF[t^{\\pm d}]$ given by $\\phi(p)(t)=t^kp(t^d)$, for all $p\\in\\FF[t^{\\pm 1}]$. In particular, $\\phi(t^i)=t^{k+id}$ for $i\\in\\ZZ$. Still viewing $t^k \\FF[t^{\\pm d}]$ as a subrepresentation of $\\V{m, n}{f}$, we have:\n\\begin{align*}\n \\phi(x.t^i) &=\\phi(t^{i+n\/d})=t^{k+id+n}=x.t^{k+id}=x.\\phi(t^i),\\\\\n \\phi(y.t^i) &=\\phi(f_k(i)t^{i-m\/d})=f(k+id)t^{k+id-m}=y.t^{k+id}=y.\\phi(t^i).\n\\end{align*}\nSince $\\phi$ is clearly bijective, the calculations above show that $\\phi$ is an isomorphism of representations, and $\\V{m, n}{f}\\simeq \\bigoplus_{k=0}^{d-1}\\V{m\/d, n\/d}{f_k}$. The fact that each summand $\\V{m\/d, n\/d}{f_k}$ is irreducible follows from $\\gcd(m\/d, n\/d)=1$ and Proposition~\\ref{P:irred}, which will be established independently. \n\nFinally, assume $\\V{m, n}{f}\\simeq \\V{m', n'}{f'}$ for positive integers $m'$ and $n'$, and $f':\\mathbb{Z} \\rightarrow \\FF^{*}$ satisfying $f'(i+n')=qf'(i)$, for all $i\\in\\ZZ$. Then, up to isomorphism, $\\V{m, n}{f}$ and $\\V{m', n'}{f'}$ have the same composition factors, and in particular the same composition length. This proves that $d=\\gcd(m, n)=\\gcd(m', n')$ and that $\\V{m\/d, n\/d}{f_0}\\simeq \\V{m'\/d, n'\/d}{f'_k}$ for some $k$. By Proposition~\\ref{P:isoclass}, which will also be established independently, we have $m\/d=m'\/d$ and $n\/d=n'\/d$, so $m=m'$ and $n=n'$.\n\n\\end{proof}\n\n\n\\subsection{Weight representations of the form $\\V{m, n}{f}$}\\label{SS:weight}\n\nLet us now determine when $\\V{m, n}{f}$ is a weight representation in the sense of~\\cite{vB97}. Recall that this occurs when $\\V{m, n}{f}$ is semisimple as a representation over the polynomial subalgebra $\\FF[H]$, where $H=xy$. Assume first that $m=n=1$ and fix $\\lambda\\in\\FF^*$. The map $f_\\lambda$ defined in Remark~\\ref{R:f} is given by $f_\\lambda (i)=\\lambda q^{i}$ for all $i\\in\\ZZ$, and the corresponding representation $\\V{1, 1}{f_\\lambda}\\simeq \\qp\/\\qp\\left(H-\\lambda\\right)$ is irreducible. Since $H.t^i=xy.t^i=\\lambda q^i t^i$ for all $i$, the decomposition $\\V{1, 1}{f_\\lambda}=\\bigoplus_{i\\in\\ZZ}\\FF\\, t^i$ shows that $\\V{1, 1}{f_\\lambda}$ is semisimple over $\\FF[H]$. Moreover, for $\\nu\\in\\FF^*$, $\\V{1, 1}{f_\\lambda}\\simeq \\V{1, 1}{f_\\nu}$ if and only if $\\lambda\/\\nu\\in\\langle q\\rangle$, the multiplicative subgroup of $\\FF^*$ generated by $q$, by Proposition~\\ref{P:isoclass}. In case $\\FF$ is algebraically closed, these are all the infinite-dimensional irreducible weight representations of $\\qp$, by \\cite[Cor. 3.2]{vB97}. Combined with Proposition~\\ref{P:isoclass}(b) the above yields the classification of irreducible weight representations in the family $\\V{m, n}{f}$.\n\n\n\\begin{prop}\\label{P:weight}\nAssume  $\\gcd(m, n)=1$ and $f:\\mathbb{Z} \\rightarrow \\FF^{*}$ satisfies~\\eqref{prop}. Then $\\V{m, n}{f}$ is a weight representation if and only if $m=n=1$.\n\\end{prop}\n\nFor completeness, we include a brief and direct proof of Proposition~\\ref{P:weight} not assuming that $\\FF$ is algebraically closed, a condition that was used implicitly at the end of the previous paragraph.\n\n\\begin{proof}\nAssume first that $m=n=1$. Then since $f$ satisfies~\\eqref{prop} we have $f=f_\\lambda$ for $\\lambda=f(0)$ and the discussion above shows that $\\V{m, n}{f}$ is a weight representation of $\\qp$. Conversely, suppose $\\V{m, n}{f}$ is a weight representation of $\\qp$. Then clearly  $\\dim_\\FF \\FF[H].v<+\\infty$ for any $v\\in\\V{m, n}{f}$. Notice that, for all $i\\in\\ZZ$, $H.t^i=xy.t^i=f(i)t^{i+n-m}$. Thus, for $\\ell\\in\\ZZ$, $H^\\ell.t^i=\\zeta t^{i+\\ell(n-m)}$ for some $\\zeta\\in\\FF^*$. But then the condition $\\dim_\\FF \\FF[H].1<+\\infty$ immediately implies $m=n$, and hence $m=n=1$, as $\\gcd(m, n)=1$.\n\\end{proof}\n\n\\begin{remark}\nGiven arbitrary positive integers $m$ and $n$, and  $f$ satisfying~\\eqref{prop}, the representation $\\V{m, n}{f}$ is a weight representation if and only if $m=n$. The direct implication follows from the proof of Proposition~\\ref{P:weight}. For the converse implication, recall that $\\V{m, m}{f}$ is the direct sum of $m$ representations of the form $\\V{1, 1}{f_k}$, for $0\\leq k<m$, by Proposition~\\ref{P:dec}, so the claim follows as each of these is a weight representation.\n\\end{remark}\n\n\n\\section{Connections with the representation theory of the $q$-Weyl algebra $\\mathbb{A}_1(q)$}\\label{S:conn}\n\n\nWe continue to assume $q\\in\\FF^*$ is not a root of unity. Let $\\mathbb{A}_1(q)$ be the $q$-Weyl algebra given by generators $X$ and $Y$ and defining relation $YX-qXY=1$, as in~\\eqref{E:qwa}. It is straightforward to show that $\\{x^k \\mid k\\geq 0\\}$ is a right and left Ore set consisting of regular elements of $\\qp$, and we denote the corresponding localization by $\\FF_q[x^{\\pm 1}, y]$. The calculation\n\\begin{equation*}\n\\big(x^{-1}(y-1)\\big)x-qx\\big(x^{-1}(y-1)\\big)=x^{-1}yx -q(y-1)-1= qy -q(y-1)-1=q-1\n\\end{equation*}\nshows that there is an algebra map \n\\begin{equation}\\label{E:qwainqp}\n\\mathbb{A}_1(q)\\rightarrow \\FF_q[x^{\\pm 1}, y], \\quad \\mbox{with \\quad $X\\mapsto x,\\quad Y\\mapsto \\frac{1}{q-1}x^{-1}(y-1)$.}\n\\end{equation}\nTo see that the map in \\eqref{E:qwainqp} is injective we can argue as follows. The multiplicative subset $\\{X^k \\mid k\\geq 0\\}$ of $\\mathbb{A}_1(q)$ is a right and left Ore set of regular elements and we denote the corresponding localization by $\\widehat{\\mathbb{A}}_1(q)$. Then the map in \\eqref{E:qwainqp} extends to a map $\\widehat{\\mathbb{A}}_1(q)\\rightarrow \\FF_q[x^{\\pm 1}, y]$, which has an inverse $\\FF_q[x^{\\pm 1}, y] \\rightarrow \\widehat{\\mathbb{A}}_1(q)$ with $x^{\\pm 1}\\mapsto X^{\\pm 1}$ and $y\\mapsto (q-1)XY+1$. It follows that \\eqref{E:qwainqp} induces an isomorphism $\\widehat{\\mathbb{A}}_1(q)\\simeq \\FF_q[x^{\\pm 1}, y]$, and in particular \\eqref{E:qwainqp} is injective. In view of the above we will identify $X$ with $x$, $Y$ with $\\frac{1}{q-1}x^{-1}(y-1)$ and $\\mathbb{A}_1(q)$ with the corresponding subalgebra of $\\FF_q[x^{\\pm 1}, y]$. Since $y=(q-1)XY+1=YX-XY$, we have the embeddings\n\\begin{equation}\\label{E:embed}\n\\qp \\subseteq \\mathbb{A}_1(q)\\subseteq \\FF_q[x^{\\pm 1}, y]=\\widehat{\\mathbb{A}}_1(q).\n\\end{equation}\n\n\\subsection{Extension of the representations $\\V{m, n}{f}$ to $\\mathbb{A}_1(q)$}\\label{SS:qwa}\n\nOur aim in this subsection is to extend the action of $\\qp$ on $\\V{m, n}{f}$ to an action of the $q$-Weyl algebra $\\mathbb{A}_1(q)$. Assume thus that $m, n$ are positive integers and $f:\\mathbb{Z} \\rightarrow \\FF^{*}$ satisfies~\\eqref{prop}. If $\\rho^{m, n}_{f}:\\qp\\rightarrow \\mathrm{End}_\\FF (\\V{m, n}{f})$ is the representation of $\\qp$ on $\\V{m, n}{f}$, we first observe that $\\rho^{m, n}_{f}(x)$ is an invertible linear map on $\\V{m, n}{f}$, a fact which is clear from \\eqref{action}. Therefore $\\rho^{m, n}_{f}$ extends to the localization $\\FF_q[x^{\\pm 1}, y]$, and  $\\V{m, n}{f}$ can be seen as a representation of $\\FF_q[x^{\\pm 1}, y]$ with $x^{-1}.t^{i}=t^{i-n}$ for all $i\\in\\ZZ$. Now we get an action of $\\mathbb{A}_1(q)$ on $\\V{m, n}{f}=\\FF[t^{\\pm 1}]$ by restricting $\\rho^{m, n}_{f}$:\n\\begin{align}\\label{E:qwaaction}\\nonumber\n&X.t^{i} =x.t^{i}=t^{i+n},\\\\  \n&Y.t^{i} =\\frac{1}{q-1}x^{-1}(y-1).t^{i}=\\frac{1}{q-1}(f(i)t^{i-m-n}-t^{i-n}),\\quad \\quad \\text{for all $i\\in\\ZZ$.}\n\\end{align}\n\n\nIn our next result we view $\\V{m, n}{f}$ as a representation of $\\mathbb{A}_1(q)$, as above.\n\n\\begin{prop}\n Assume  $\\gcd(m, n)=1$ and $f:\\mathbb{Z} \\rightarrow \\FF^{*}$ satisfies~\\eqref{prop}. Then:\n \\begin{enumerate}\n \\item[\\textup{(a)}] $\\V{m, n}{f}$ defined by~\\eqref{E:qwaaction} is an irreducible representation of $\\mathbb{A}_1(q)$.\n\\item[\\textup{(b)}]  For positive integers $m', n'$, and $f':\\mathbb{Z} \\rightarrow \\FF^{*}$ satisfying \\eqref{prop} (with $n$ replaced by $n'$), \nwe have $\\V{m, n}{f}\\simeq \\V{m', n'}{f'}$ as representations of $\\mathbb{A}_1(q)$ if and only if $m=m'$, $n=n'$ and $\\s{f'}(0)=q^{k}\\s{f}(0)$ for some $k\\in\\ZZ$.\n\\item[\\textup{(c)}] $\\V{m, n}{f}$ is not semisimple as a representation over the polynomial subalgebra of $\\mathbb{A}_1(q)$ generated by $XY$; hence, $\\V{m, n}{f}$ is not a weight representation of $\\mathbb{A}_1(q)$ in the sense of \\cite{vB97}.\n\\end{enumerate}\n\\end{prop}\n\\begin{proof}\nPart (a) and the direct implication in (b) follow from the embedding \\eqref{E:embed}, and from Propositions \\ref{P:irred} and \\ref{P:isoclass}.\n \nSuppose now $f':\\mathbb{Z} \\rightarrow \\FF^{*}$ satisfies \\eqref{prop}, and there is $k\\in\\ZZ$ so that $\\s{f'}(0)=q^{k}\\s{f}(0)$. Then by  Proposition~\\ref{P:isoclass} there is an isomorphism $\\phi : \\V{m, n}{f}\\rightarrow \\V{m, n}{f'}$ as representations of $\\qp$. For $v\\in\\V{m, n}{f}$ we have $\\phi(v)=\\phi(xx^{-1}.v)=x.\\phi(x^{-1}.v)$, thus $\\phi(x^{-1}.v)=x^{-1}.\\phi(v)$. Whence $\\phi$ is an isomorphism of representations of $\\FF_q[x^{\\pm 1}, y]$. The other implication in (b) now follows from \\eqref{E:embed}.\n\nObserve that $XY=\\frac{1}{q-1}(y-1)$, so the polynomial subalgebra of $\\mathbb{A}_1(q)$ generated by $XY$ is just $\\FF[y]$. Given $0\\neq v\\in\\V{m, n}{f}$, the formula $y.t^{i}=f(i)t^{i-m}$ for $i\\in\\ZZ$ implies $\\dim_\\FF \\FF[y].v=+\\infty$. Hence, $\\V{m, n}{f}$ is not semisimple over $\\FF[y]=\\FF[XY]$, and therefore it is not a weight representation of $\\mathbb{A}_1(q)$ in the sense of \\cite{vB97}.\n\\end{proof}\n\n\n\\begin{remark}\nIn~\\cite{BO09} the authors introduce Whittaker representations for generalized Weyl algebras. For the cases covered in this note, a representation $\\mathsf{V}$ is a \\emph{Whittaker representation} for $\\qp$ (respectively, for $\\mathbb{A}_1(q)$) if $\\mathsf{V}$ is generated by an element $v\\in \\mathsf{V}$ which is an eigenvector for the action of $x\\in\\qp$ (respectively, for the action of $X\\in \\mathbb{A}_1(q)$). Since $m, n\\geq 1$, it is immediate that the operators $x, y\\in\\qp$ (respectively, $X, Y\\in\\mathbb{A}_1(q)$) have no eigenvectors in $\\V{m, n}{f}$, so $\\V{m, n}{f}$ is not a Whittaker representation for the quantum plane (respectively, for the $q$-Weyl algebra).\n\\end{remark}\n\n\\subsection{The representations $\\W{n}{}$ of $\\mathbb{A}_1(q)$ and their restriction to $\\qp$}\\label{SS:restriction}\n\nWe will now use a similar idea to construct representations of the $q$-Weyl algebra on the Laurent polynomial algebra $\\FF[t^{\\pm 1}]$. Fix positive integers $m, n\\in\\ZZ_{>0}$ and a function $g:\\mathbb{Z} \\rightarrow \\FF$. Then the formulas \n\\begin{equation}\\label{qwa:action}\nX.t^{i}=t^{i+n}, \\quad \\quad Y.t^{i}=g(i)t^{i-m},\\quad \\quad \\text{for all $i\\in\\ZZ$}\n\\end{equation}\nyield a representation of $\\mathbb{A}_1(q)$ on $\\FF[t^{\\pm 1}]$ if and only if $m=n$ and $g$ satisfies\n\\begin{equation}\\label{qwa:prop}\ng(i+n)=qg(i)+1, \\quad \\quad \\text{for all $i\\in\\ZZ$.}\n\\end{equation}\nWe denote the corresponding representation of $\\mathbb{A}_1(q)$ by $\\W{n}{}$. Notice that for all $i\\in\\ZZ$\n\\begin{equation}\\label{qwa:xy:com}\nXY.t^i=g(i)t^i, \\quad \\quad (YX-XY).t^i=\\left(g(i+n)-g(i)\\right)t^i=\\left((q-1)g(i)+1\\right)t^i,\n\\end{equation}\nso $\\W{n}{}$ is a weight representation of $\\mathbb{A}_1(q)$ in the sense of \\cite{vB97}.\n\n\\begin{remark}\nIt follows from the computations at the beginning of Section~\\ref{S:conn} that the element $YX-XY$ is normal in $\\mathbb{A}_1(q)$ and it is sometimes referred to as a Casimir element, in spite of not being central. The equality $YX-XY=(q-1)XY+1$ shows that $YX-XY$ and $(q-1)XY+1$ generate the same subalgebra of $\\mathbb{A}_1(q)$ and thus a weight representation of $\\mathbb{A}_1(q)$ could be defined in an equivalent manner as a representation which is semisimple over the subalgebra generated by the Casimir element $YX-XY$. \n\\end{remark}\n\nOur first observation is the analogue of Proposition~\\ref{P:dec}.\n\n\\begin{lemma}\\label{L:qwa:dec}\nLet  $n\\in\\ZZ_{>0}$ and assume $g:\\mathbb{Z} \\rightarrow \\FF$ satisfies~\\eqref{qwa:prop}. There is a direct sum decomposition\n\\begin{equation}\\label{qwa:dec}\n\\W{n}{}\\simeq \\bigoplus_{k=0}^{n-1}\\W{1}{k},\n\\end{equation}\nwhere $g_k(i)=g(k+in)$, for $0\\leq k<n$ and $i\\in\\ZZ$. \n\\end{lemma}\n\\begin{proof}\nFor  $0\\leq k<n$, the subspace $t^k \\FF[t^{\\pm n}]$ is invariant under the actions of $X$ and $Y$ and $\\W{n}{}=\\bigoplus_{k=0}^{n-1}t^k \\FF[t^{\\pm n}]$. Moreover, the map $\\phi: \\W{1}{k}\\rightarrow t^k \\FF[t^{\\pm n}]$ given by $\\phi(p)(t)=t^kp(t^n)$, for all $p\\in\\FF[t^{\\pm 1}]$ is easily checked to be an isomorphism.\n\\end{proof}\n\nIn view of the above, it is enough to study the structure of the representations $\\W{1}{}$, where $g:\\mathbb{Z} \\rightarrow \\FF$ satisfies $g(i+1)=qg(i)+1$ for all $i\\in\\ZZ$. Equivalently, $g(i)=g(0)q^i + [i]_q$, where $[i]_q=\\frac{q^i-1}{q-1}$ for all $i\\in\\ZZ$. \n\n\\begin{prop}\\label{P:qwa:w:irriso}\nLet $g, g':\\mathbb{Z} \\rightarrow \\FF$ satisfy~\\eqref{qwa:prop} with $n=1$. Then:\n\\begin{enumerate}\n\\item[\\textup{(a)}] $\\W{1}{}\\simeq\\mathsf{W}^{1}_{g'}$ if and only if $g(0)=g'(i)$ for some $i\\in\\ZZ$;\n\\item[\\textup{(b)}] $\\W{1}{}$ is irreducible if and only if $g(0)\\notin \\{ [i]_q \\mid i\\in\\ZZ\\}\\cup\\left\\{ -\\frac{1}{q-1}\\right\\}$.\n\\end{enumerate}\n\\end{prop}\n\n\\begin{proof}\nFor (a), suppose $\\W{1}{}\\simeq\\mathsf{W}^{1}_{g'}$. By~\\eqref{qwa:xy:com} the eigenvalues of $XY$ on $\\W{1}{}$ are $g(i)$, for $i\\in\\ZZ$ and similarly the eigenvalues of $XY$ on $\\mathsf{W}^{1}_{g'}$ are $g'(i)$, for $i\\in\\ZZ$. Thus, $g$ and $g'$ must have the same image and in particular $g(0)=g'(i)$ for some $i\\in\\ZZ$. Conversely, if the latter holds then the map $\\phi:\\W{1}{}\\rightarrow\\mathsf{W}^{1}_{g'}$ given by $\\phi(p)(t)=t^ip(t)$ for all $p\\in\\FF[t^{\\pm 1}]$ is an isomorphism.\n\nFor (b), first observe that for $i\\in\\ZZ$ we have $g(0)=[i]_q\\iff g(-i)=0$. Thus, if $g(0)=[i]_q$ for some $i\\in\\ZZ$, then $t^{-i}\\FF[t]$ is invariant under the actions of $X$ and $Y$, so $\\W{1}{}$ is not irreducible in this case. Next observe that $g(0)=-\\frac{1}{q-1}\\iff g$ is not injective $\\iff g$ is constant. It follows that if $g(0)=-\\frac{1}{q-1}$, then $(t-1)\\FF[t^{\\pm 1}]$ is a proper subrepresentation and hence $\\W{1}{}$ is not irreducible. This proves the direct implication in (b). For the converse, by the observations above, we can assume that $g(i)\\neq 0$ for all $i\\in\\ZZ$ and that $g$ is injective. Let $\\mathsf{S}$ be a nonzero subrepresentation of $\\W{1}{}$. By repeatedly applying the operator $X$ to a chosen nonzero element of $\\mathsf{S}$, we will obtain a nonzero element of $\\mathsf{S}\\cap\\FF[t]$. Let $p$ be one such element, chosen so that it has minimum degree, say $p=\\sum_{k=0}^d a_k t^k$, with $a_d\\neq 0$. Since $g(i)\\neq 0$ for all $i\\in\\ZZ$, the minimality of $p$ implies that $a_0\\neq 0$. Then \n\\begin{equation*}\n\\mathsf{S}\\cap\\FF[t]\\ni (XY-g(d)).p=\\sum_{k=0}^{d-1} (g(k)-g(d))a_k t^k.\n\\end{equation*}\nBy the minimality of $p$ we must have $(XY-g(d)).p=0$. Hence, $g(0)=g(d)$ and the injectivity of $g$ gives $d=0$. It follows that $t^0\\in\\mathsf{S}$ and thus $\\mathsf{S}=\\W{1}{}$.\n\\end{proof}\n\nNow that we understand the representations $\\W{n}{}$, we will consider their restriction to $\\qp$ via each of the two embeddings\n\\begin{align}\\label{E:emb:sigma}\n\\sigma &: \\qp\\rightarrow \\mathbb{A}_1(q),  \\quad\\quad x\\mapsto X, \\quad y\\mapsto YX-XY=(q-1)XY+1;\\\\ \\label{E:emb:tau}\n\\tau &: \\qp\\rightarrow \\mathbb{A}_1(q),  \\quad\\quad x\\mapsto YX-XY=(q-1)XY+1, \\quad y\\mapsto Y.\n\\end{align}\n\nWe consider first the restriction relative to $\\sigma$. In this case, the action of $\\qp$ on $\\W{n}{}$ is given by\n\\begin{equation}\\label{rest:w:qp}\nx.t^{i}=t^{i+n}, \\quad \\quad y.t^{i}=\\left((q-1)g(i)+1\\right)t^i,\\quad \\quad \\text{for all $i\\in\\ZZ$.}\n\\end{equation}\n\n\\begin{lemma}\\label{L:qwa:rest:sigma}\nConsider the restriction map $\\sigma$ given in~\\eqref{E:emb:sigma} to view the representations $\\W{n}{}$ as representations of $\\qp$.\n\\begin{enumerate}\n\\item[\\textup{(a)}] Let  $n\\in\\ZZ_{>0}$ and assume $g:\\mathbb{Z} \\rightarrow \\FF$ satisfies~\\eqref{qwa:prop}. Then $\\W{n}{}\\simeq \\bigoplus_{k=0}^{n-1}\\W{1}{k}$ as representations of $\\qp$, where $g_k(i)=g(k+in)$, for $0\\leq k<n$ and $i\\in\\ZZ$.\n\n\\item[\\textup{(b)}] Let $g, g':\\mathbb{Z} \\rightarrow \\FF$ satisfy~\\eqref{qwa:prop} with $n=1$. Then $\\W{1}{}\\simeq\\mathsf{W}^{1}_{g'}$ as representations of $\\qp$ if and only if $g(0)=g'(i)$ for some $i\\in\\ZZ$.\n\n\\item[\\textup{(c)}] Assume $g:\\mathbb{Z} \\rightarrow \\FF$ satisfies~\\eqref{qwa:prop} with $n=1$. Then $\\W{1}{}$ has trivial socle as a representation of $\\qp$, i.e., it has no irreducible $\\qp$-subrepresentations.\n\\end{enumerate}\n\\end{lemma}\n\\begin{proof}\nPart (a) follows directly from Lemma~\\ref{L:qwa:dec} and part (b) follows from the proof of Proposition~\\ref{P:qwa:w:irriso}(a), as the argument for the direct implication in Proposition~\\ref{P:qwa:w:irriso}(a) used only the restriction of the action to the subalgebra generated by $XY$, which coincides with the subalgebra generated by $YX-XY$.\n\nFor part (c), suppose by way of contradiction that $\\mathsf{S}$ is an irreducible $\\qp$-subrepresentation of $\\W{1}{}$. Let $0\\neq s\\in\\mathsf{S}$. Then $x.s\\neq 0$ and thus $\\qp x.s=\\mathsf{S}$, which is a contradiction as $s\\notin \\qp x.s$. \n\\end{proof}\n\n\\begin{remark}\nIn the conditions of Lemma~\\ref{L:qwa:rest:sigma}, it can be checked that $\\W{1}{}$ has maximal $\\qp$-subrepresentations if and only if $g$ is constant.\n\\end{remark}\n\nNow we consider the restriction of $\\W{n}{}$ to $\\qp$ relative to the map $\\tau$ defined in~\\eqref{E:emb:tau}. In this case, the action of $\\qp$ on $\\W{n}{}$ is given by\n\\begin{equation}\\label{rest:w:qp:tau}\nx.t^{i}=\\left((q-1)g(i)+1\\right)t^i, \\quad \\quad y.t^{i}=g(i)t^{i-n},\\quad \\quad \\text{for all $i\\in\\ZZ$.}\n\\end{equation}\n\n\\begin{lemma}\\label{L:qwa:rest:tau}\nConsider the restriction map $\\tau$ given in~\\eqref{E:emb:tau} to view the representations $\\W{n}{}$ as representations of $\\qp$.\n\\begin{enumerate}\n\\item[\\textup{(a)}] Let  $n\\in\\ZZ_{>0}$ and assume $g:\\mathbb{Z} \\rightarrow \\FF$ satisfies~\\eqref{qwa:prop}. Then $\\W{n}{}\\simeq \\bigoplus_{k=0}^{n-1}\\W{1}{k}$ as representations of $\\qp$, where $g_k(i)=g(k+in)$, for $0\\leq k<n$ and $i\\in\\ZZ$.\n\n\\item[\\textup{(b)}] Let $g, g':\\mathbb{Z} \\rightarrow \\FF$ satisfy~\\eqref{qwa:prop} with $n=1$. Then $\\W{1}{}\\simeq\\mathsf{W}^{1}_{g'}$ as representations of $\\qp$ if and only if $g(0)=g'(i)$ for some $i\\in\\ZZ$.\n\n\\item[\\textup{(c)}] Assume $g:\\mathbb{Z} \\rightarrow \\FF$ satisfies~\\eqref{qwa:prop} with $n=1$. If $g(0)\\notin \\{ [i]_q \\mid i\\in\\ZZ\\}$ then $\\W{1}{}$ has trivial socle as a representation of $\\qp$, i.e., it has no irreducible $\\qp$-subrepresentations. If $g(0)=[i]_q$ for some $i\\in\\ZZ$ then $\\FF t^{-i}$ is the unique irreducible $\\qp$-subrepresentation of $\\W{1}{}$.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nThe proof is the same as the proof of Lemma~\\ref{L:qwa:rest:sigma}, except for part (c). For this part, suppose that $\\mathsf{S}$ is an irreducible $\\qp$-subrepresentation of $\\W{1}{}$. If there is $0\\neq s\\in\\mathsf{S}$ such that $y.s\\neq 0$, then we obtain a contradiction as in the proof of Lemma~\\ref{L:qwa:rest:sigma}(c), showing that no such irreducible $\\qp$-subrepresentation of $\\W{1}{}$ exists. If $g(0)\\notin \\{ [i]_q \\mid i\\in\\ZZ\\}$ then $g(i)\\neq 0$ for all $i\\in\\ZZ$, so $y.s\\neq 0$ for all $s\\neq 0$ and the first claim follows. Now suppose $g(0)=[i]_q$ for some $i\\in\\ZZ$. Then $g(k)=0\\iff k=-i$. In particular, $x.t^{-i}=t^{-i}$ and $y.t^{-i}=0$, so that $\\FF t^{-i}$ is an irreducible $\\qp$-subrepresentation of $\\W{1}{}$. If $\\mathsf{S}$ is any irreducible $\\qp$-subrepresentation of $\\W{1}{}$, then the argument above implies that $y.s=0$ for all $s\\in\\mathsf{S}$, and this in turn implies that $\\mathsf{S}\\subseteq\\FF t^{-i}$, which establishes the second claim in (c).\n\\end{proof}\n\n\\medskip\n\n\\begin{flushleft}\n\\textbf{Acknowledgments.} The authors wish to thank G.~Benkart and M.~Ondrus for helpful comments and suggestions on a preliminary version of this manuscript. They would also like to thank the anonymous referees for their valuable comments and for suggesting the approach in Subsection~\\ref{SS:restriction}.\n\\end{flushleft}\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nSuppose you have encrypted some data, you intend to compute with the data, but you do not want to decrypt them.\nIs it possible to compute with the encrypted data without decryption?\n\nThis problem is related with the study ``information processing with encrypted data'' or ``privacy-preserving quantum computing''. It has been investigated a lot in modern cryptography, and includes the studies of homomorphic encryption \\cite{rivest1978} and blind computing \\cite{feigenbaum1986,abadi1989}. It was first considered by Rivest et al. who suggest some homomorphic encryption schemes \\cite{rivest1978}. However, these schemes are insecure \\cite{brickell1987}. Later, more results are proposed about homomorphic encryption and fully homomorphic encryption \\cite{gentry2009,gentry2010}. These are constructed based on some hard computational problems in mathematics, and their security relies on the computational difficulty of these mathematical problems. Thus they have just computational security. In addition, symmetric homomorphic encryption \\cite{castelluccia2009,hessler2012,armknecht2011}\nis also an useful and interesting research, and it has higher efficiency. Though its encryption key is the same as the decryption key, it is sufficient for many application.\n\nBased on the theory of quantum mechanics, lots of quantum cryptographic protocols \\cite{bennett1984,ekert1991,tamaki2003} have been proposed, and have unconditional security. This stimulates us to consider the above problem in the context of quantum information, and intend to get a more secure solution to the privacy-preserving quantum computing. The closely related with the problem includes: 1) blind quantum computing \\cite{childs2005,arrighi2006,aharonov2008,broadbent2009,sueki2012,barz2012,vedral2012,morimae2012a,morimae2012b,morimae2010,fitzsimons2012}, where one party delegates quantum computation to another party without revealing his data and result; 2) quantum homomorphic encryption (QHE) \\cite{rohde2012}, which allows quantum data to be manipulated without decrypting; 3) quantum private query \\cite{giovannetti2008}, which is used to query a database while keeping the query secret; and 4) quantum private comparison \\cite{tseng2012,lin2013,guo2013,li2013,liu2013}, which allows two parties to compare their secret data without revealing the secret.\n\nThis paper considers the problem of homomorphic encryption in the context of quantum information processing: suppose arbitrary quantum plaintext $\\sigma$ has been encrypted (the ciphertext is $\\mathcal{E}(\\sigma)$), can we perform any quantum operator directly on the ciphertext (without decrypting the ciphertext) and obtain the desired state $\\mathcal{E}(T(\\sigma))$? The state $\\mathcal{E}(T(\\sigma))$ is the ciphertext of the result of performing quantum operator $T$ on the original plaintext $\\sigma$.\n\nRohde et al. \\cite{rohde2012} studied quantum walk with encrypted data, and proposed a limited QHE scheme using the Boson sampling and multi-walker quantum walk models. This QHE scheme can be used in blind computing of quantum walk. However, QHE has still not been defined and quantum fully homomorphic encryption (QFHE) scheme has not been constructed.\n\nWe study the quantum information processing with encrypted data from the aspect of QHE. The definitions of QHE and QFHE have been presented, and some concrete schemes including QFHE scheme have been constructed. All these schemes are constructed based on quantum one-time pad (QOTP)  \\cite{boykin2000,ambainis2000}, and have perfect security.\n\n\\section{Concepts}\nIn Ref. \\cite{rohde2012}, a limited QHE scheme was presented using the Boson sampling and multi-walker quantum walk models. However, no definition of QHE or QFHE has been given, and no QFHE scheme has been constructed.\n\nQHE can be either symmetric or asymmetric. Both of the two kinds will be defined, but we will focus on the symmetric QHE. All the schemes constructed in the next sections are symmetric QHE schemes.\n\nQuantum states are usually written in the form of density matrices, denoted as $\\rho,\\sigma$. The output states of quantum operators $T,\\mathcal{E}$ are also represented as density matrices.\n\n{\\bf Definition 1:} A symmetric quantum homomorphic encryption scheme $\\Delta$ has the following four algorithms:\n\\begin{enumerate}\n  \\item Key Generating algorithm $KeyGen_\\Delta$ is used to generate a key $key$;\n  \\item $Encrypt_\\Delta$ is the encryption algorithm: $\\rho=\\mathcal{E}(key,\\sigma)$, where $\\sigma$ is the quantum plaintext;\n  \\item $Decrypt_\\Delta$ is the decryption algorithm: $\\sigma=\\mathcal{D}(key,\\rho)$, where $\\rho$ is the quantum ciphertext;\n  \\item The algorithm $Evaluate_\\Delta$ is used to process the quantum ciphertext without decryption. $Evaluate_\\Delta$ is associated to a set $\\mathcal{F}_\\Delta$ of permitted quantum operators. For any quantum operator $T\\in\\mathcal{F}_\\Delta$, according to the key $key$ and quantum ciphertext $\\rho$, it performs the algorithm $Evaluate_\\Delta(key, T, \\rho)$, and can output a quantum state which is just the ciphertext $\\mathcal{E}(key,T(\\sigma))$.\n\\end{enumerate}\n\nIt should be noticed that there exists a case that the algorithm $Evaluate_\\Delta$ is independent of the key. In this case, it should be written as\n$Evaluate_\\Delta(T,\\rho)$ in the above definition. So, the symmetric QHE can be divided into two classes: 1) symmetric QHE with encryption key, where the algorithm $Evaluate_\\Delta$ is executed using the encryption key; and 2) symmetric QHE without encryption key, where the algorithm $Evaluate_\\Delta$ is executed without using the encryption key.\n\n{\\bf Definition 2:} An asymmetric quantum homomorphic encryption scheme $\\Delta$ has the following four algorithms:\n\\begin{enumerate}\n  \\item Key Generating algorithm $KeyGen_\\Delta$ is used to generate two keys -- a public key $pk$ and a secret key $sk$;\n  \\item $Encrypt_\\Delta$ is the encryption algorithm: $\\rho=\\mathcal{E}(pk,\\sigma)$, where $\\sigma$ is the quantum plaintext;\n  \\item $Decrypt_\\Delta$ is the decryption algorithm: $\\sigma=\\mathcal{D}(sk,\\rho)$, where $\\rho$ is the quantum ciphertext;\n  \\item The algorithm $Evaluate_\\Delta$ is used to process the quantum ciphertext without decryption. $Evaluate_\\Delta$ is associated to a set $\\mathcal{F}_\\Delta$ of permitted quantum operators. For any quantum operator $T\\in\\mathcal{F}_\\Delta$, according to the public key $pk$ and quantum ciphertext $\\rho$, it performs the algorithm $Evaluate_\\Delta(pk, T, \\rho)$, and can output a quantum ciphertext which can be decrypted as $T(\\sigma)$ with the secret key $sk$.\n\\end{enumerate}\n\nCompared with the usual encryption scheme, the QHE scheme has a fourth algorithm $Evaluate_\\Delta$, which is used to process the encrypted data. For example, the algorithm $Evaluate_\\Delta$ may be described as this: from the key and the quantum operator $T\\in\\mathcal{F}_\\Delta$, it generates another quantum operator $T'$ and performs it on the quantum ciphertext $\\mathcal{E}(key,\\sigma)$. In this case, the operator $T'$ is related with the operator $T$ and the key, such that\n\\begin{equation}\\label{equ8}\nT'(\\mathcal{E}(key,\\sigma))=\\mathcal{E}(key,T(\\sigma)).\n\\end{equation}\nThe operator $T$ can be regarded as an desired operation on the quantum plaintext. The operation $T'$ corresponding to $T$ is performed on the quantum ciphertext, and can implement the desired operation on the plaintext.\n\nFrom the definitions of symmetrc\/asymmetric QHE scheme, we say the scheme $\\Delta$ can handle all the quantum operators in $\\mathcal{F}_\\Delta$.\n\nThe scheme $\\Delta$ is (symmetric or asymmetric) quantum fully homomorphic encryption scheme, if it can handle all quantum operators and $Evaluate_\\Delta$ is efficient in a similar way as in Ref.\\cite{gentry2010}. The quantum operator $T\\in\\mathcal{F}_\\Delta$ may be uncomputable in polynomial time, and suppose its running time is $S_T$. We say $Evaluate_\\Delta$ is efficient if there exists a polynomial $g$ such that, for any quantum operator $T\\in\\mathcal{F}_\\Delta$ that can be implemented in time $S_T$, $Evaluate_\\Delta$ can be implemented in time at most $S_T\\cdot g(\\lambda)$, where $\\lambda$ is the security parameter.\n\nThe security of QHE scheme depends on the security of the encryption algorithm $Encrypt_\\Delta$. In the asymmetric QHE scheme, the security also depends on the privacy of the secret key $sk$ while the key $pk$ is public.\n\n\\section{Quantum homomorphic encryption}\nSome operators are introduced firstly. Four single-qubit operators $X,Y,Z,H$ are shown as follows:\n$X=\\left(\\begin{array}{cc}\n  0 & 1\\\\\n  1 & 0\n\\end{array}\\right)$,\n$Y=\\left(\\begin{array}{cc}\n  0 & -i\\\\\n  i & 0\n\\end{array}\\right)$,\n$Z=\\left(\\begin{array}{cc}\n  1 & 0\\\\\n  0 & -1\n\\end{array}\\right)$,\n$H=\\frac{1}{\\sqrt{2}}\\left(\\begin{array}{cc}\n  1 & 1\\\\\n  1 & -1\n\\end{array}\\right)$.\nTwo-qubit operator controlled-NOT is denoted as $CNOT$.\nThe rotation operators about the $\\hat{z}$ and $\\hat{y}$ axes are defined by $R_z(\\theta)=e^{-i\\theta Z\/2}$ and $R_y(\\theta)=e^{-i\\theta Y\/2}$.\nSee the Appendix for some of their commutation rules. All the equations used in this paper can be deduced from those commutation rules.\n\nIn this section, we will firstly present three QHE schemes on single qubit, whose permitted quantum operators are in the set $\\{R_z(\\theta)|\\theta\\in[0,2\\pi)\\}$, $\\{R_y(\\theta)|\\theta\\in[0,2\\pi)\\}$ or the union of them. Then a QHE scheme for $CNOT$ is also constructed.\n\nIn the first QHE scheme, the set of permitted quantum operators is $\\{R_z(\\theta)|\\theta\\in[0,2\\pi)\\}$. The scheme is shown as follows.\n\\begin{description}\n  \\item[{\\bf QHE scheme 1}]\n  \\item[$KeyGen_\\Delta$:] Randomly select two bits $k\\in\\{0,1\\},j\\in\\{0,1\\}$;\n  \\item[$Encrypt_\\Delta$:] Compute $\\rho_c=X^jZ^k\\sigma_m Z^kX^j$;\n  \\item[$Decrypt_\\Delta$:] Compute $\\sigma_m=Z^kX^j\\rho_c X^jZ^k$;\n  \\item[$Evaluate_\\Delta$:] According to $k,j,R_z(\\theta)$, it performs quantum operator $R_z(\\theta')$ on the given quantum ciphertext $\\rho_c$, where $\\theta'=(-1)^j\\theta$.\n\\end{description}\n\nIt can be verified that\n\\begin{equation}\\label{equ2}\nR_z((-1)^j\\theta)X^jZ^k=X^jZ^kR_z(\\theta),\n\\end{equation}\nwhere $k\\in\\{0,1\\},j\\in\\{0,1\\}$. According to Eq.(\\ref{equ2}), the output state of the algorithm $Evaluate_\\Delta$ is\n$$R_z((-1)^j\\theta)\\rho_c R_z((-1)^j\\theta)^\\dagger=X^jZ^k(R_z(\\theta)\\sigma_m R_z(\\theta)^\\dagger)Z^kX^j,$$\nwhich is just the ciphertext of the state $R_z(\\theta)\\sigma_m R_z(\\theta)^\\dagger$. Moreover, no decryption is performed during the computing of $Evaluate_\\Delta$.\nThus the scheme satisfies the definition of symmetric QHE, and all the unitary transformations in $\\{R_z(\\theta)|\\theta\\in[0,2\\pi)\\}$ are permitted quantum operators of the QHE scheme.\n\nIn the second QHE scheme, the set of permitted quantum operators is $\\{R_y(\\theta)|\\theta\\in[0,2\\pi)\\}$. The scheme is shown as follows.\n\\begin{description}\n  \\item[{\\bf QHE scheme 2}]\n  \\item[$KeyGen_\\Delta$:] Randomly select two bits $k\\in\\{0,1\\},j\\in\\{0,1\\}$;\n  \\item[$Encrypt_\\Delta$:] Compute $\\rho_c=X^jZ^k\\sigma_m Z^kX^j$;\n  \\item[$Decrypt_\\Delta$:] Compute $\\sigma_m=Z^kX^j\\rho_c X^jZ^k$;\n  \\item[$Evaluate_\\Delta$:] According to $k,j,R_y(\\theta)$, it performs quantum operator $R_y(\\theta')$ on the given quantum ciphertext $\\rho_c$, where $\\theta'=(-1)^{k+j}\\theta$.\n\\end{description}\n\nIt can be verified that\n\\begin{equation}\\label{equ3}\nR_y((-1)^{k+j}\\theta)X^jZ^k=X^jZ^k R_y(\\theta),\n\\end{equation}\nwhere $k\\in\\{0,1\\},j\\in\\{0,1\\}$. According to Eq.(\\ref{equ3}), the output state of the algorithm $Evaluate_\\Delta$ is\n\\begin{equation}\nR_y((-1)^{k+j}\\theta)\\rho_c R_y((-1)^{k+j}\\theta)^\\dagger = X^jZ^k(R_y(\\theta)\\sigma_m R_y(\\theta)^\\dagger)Z^kX^j,\n\\end{equation}\nwhich is just the ciphertext of the state $R_y(\\theta)\\sigma_m R_y(\\theta)^\\dagger$. Moreover, no decryption is performed during the computing of $Evaluate_\\Delta$.\nThus the scheme also satisfies the definition of symmetric QHE, and all the unitary transformations in $\\{R_y(\\theta)|\\theta\\in[0,2\\pi)\\}$ are permitted quantum operators of the QHE scheme.\n\n\n{\\bf Remark 1:} The relation $R_y((-1)^j\\theta)H^jY^k=H^jY^kR_y(\\theta)$ can be easily verified. So, in the QHE scheme 2, the encryption\/decryption algorithm can also be modified by replacing $X^jZ^k$ with $H^jY^k$, and the algorithm $Evaluate_\\Delta$ performs quantum operator $R_y((-1)^j\\theta)$ on the given quantum ciphertext.\n\n\nThe third QHE scheme can be constructed by combining the scheme 1 and scheme 2, and its permitted quantum operators are in $\\{R_z(\\theta),R_y(\\theta)|\\theta\\in[0,2\\pi)\\}$. The only modification is about the algorithm $Evaluate_\\Delta$, and the modified $Evaluate_\\Delta$ is described as follow: according to the key $k,j$ and $R_z(\\theta)$ (or $R_y(\\theta)$), perform quantum operator $R_z((-1)^j\\theta)$ (or $R_y((-1)^{k+j}\\theta)$) on the given quantum ciphertext $\\rho_c$.\n\nUntil now, we have presented three symmetric QHE schemes, such that the permitted quantum operators are in $\\{R_z(\\theta)|\\theta\\in[0,2\\pi)\\}$, $\\{R_y(\\theta)|\\theta\\in[0,2\\pi)\\}$, or the union of them. All these schemes are constructed based on QOTP. In these constructions, each permitted quantum operator $T\\in\\{R_z(\\theta),R_y(\\theta)|\\theta\\in[0,2\\pi)\\}$ is corresponding to another quantum operator $T'$ which is directly performed on the quantum ciphertext. Finding the operator $T'$ is the key point to construct a QHE scheme.\n\nNext, we construct a symmetric QHE scheme that permits $CNOT$ operator. It can be verified that\n\\begin{equation}\\label{equ1}\n(X^jZ^k\\otimes X^lZ^m)CNOT = CNOT((-1)^{j\\cdot m}Z^m\\otimes X^j)(X^jZ^k\\otimes X^lZ^m),\n\\end{equation}\nwhere $j,k,l,m$ are arbitrary four bits.\nDenote $CNOT'=CNOT((-1)^{j\\cdot m}Z^m\\otimes X^j)$, then the Eq.(\\ref{equ1}) can be written as $CNOT'(X^jZ^k\\otimes X^lZ^m)=(X^jZ^k\\otimes X^lZ^m)CNOT$, so the operator $CNOT'$ is corresponding to $CNOT$. From this relation, the QHE scheme can be constructed in the same way as follows. In the encryption\/decryption algorithm, the QOTP is used to encrypt\/decrypt two qubits with $4$-bit key $j,k,l,m$. According to the key $j,k,l,m$, the algorithm $Evaluate_\\Delta$ performs $CNOT'$ on the given quantum ciphertext.\n\n\n\\section{Quantum fully homomorphic encryption}\nBased on the QHE schemes in the previous section, the QFHE scheme can also be constructed for any $n$-qubit quantum computation (without quantum measurements).\n\nFrom the Ref. \\cite{nielsen2000}, any single-qubit unitary transformation can be written as the form\n\\begin{equation}\\label{equ4}\nU(\\alpha,\\beta,\\gamma,\\delta)=e^{i\\alpha}R_z(\\beta)R_y(\\gamma)R_z(\\delta),\n\\end{equation}\nwhere $\\alpha,\\beta,\\gamma,\\delta$ are real numbers in the $[0,2\\pi)$. It means all the single-qubit unitary transformations can be expressed as the set $\\{U(\\alpha,\\beta,\\gamma,\\delta)|\\alpha,\\beta,\\gamma,\\delta\\in[0,2\\pi)\\}$, where $U(\\alpha,\\beta,\\gamma,\\delta)=e^{i\\alpha}R_z(\\beta)R_y(\\gamma)R_z(\\delta)$.\n\nAccording to Eq.(\\ref{equ2}) and Eq.(\\ref{equ3}), it can be concluded that\n\\begin{eqnarray}\\label{equ5}\nX^jZ^k U(\\alpha,\\beta,\\gamma,\\delta) &=& e^{i\\alpha}X^jZ^k R_z(\\beta)R_y(\\gamma)R_z(\\delta),\\nonumber\\\\\n&=& e^{i\\alpha}R_z((-1)^j\\beta)X^jZ^k R_y(\\gamma)R_z(\\delta),\\nonumber\\\\\n&=& e^{i\\alpha}R_z((-1)^j\\beta)R_y((-1)^{k+j}\\gamma)X^jZ^k R_z(\\delta),\\nonumber\\\\\n&=& e^{i\\alpha}R_z((-1)^j\\beta)R_y((-1)^{k+j}\\gamma)R_z((-1)^j\\delta)X^jZ^k ,\\nonumber\\\\\n&=& U(\\alpha,(-1)^j\\beta,(-1)^{k+j}\\gamma,(-1)^j\\delta) X^jZ^k,\n\\end{eqnarray}\nwhere $k\\in\\{0,1\\},j\\in\\{0,1\\}$.\n\nFrom Eq.(\\ref{equ5}), any single-qubit unitary operator $U(\\alpha,\\beta,\\gamma,\\delta)$ is corresponding to $U(\\alpha,(-1)^j\\beta,(-1)^{k+j}\\gamma,(-1)^j\\delta)$. So the QFHE scheme for single-qubit computation is constructed in the same way as the construction of QHE schemes in the previous section.\n\\begin{description}\n  \\item[{\\bf QFHE on single qubit}]\n  \\item[$KeyGen_\\Delta$:] Randomly select two bits $k\\in\\{0,1\\},j\\in\\{0,1\\}$;\n  \\item[$Encrypt_\\Delta$:] Compute $\\rho_c=X^jZ^k\\sigma_m Z^kX^j$;\n  \\item[$Decrypt_\\Delta$:] Compute $\\sigma_m=Z^kX^j\\rho_c X^jZ^k$;\n  \\item[$Evaluate_\\Delta$:] According to $k,j$ and $U(\\alpha,\\beta,\\gamma,\\delta)$, it performs the unitary transformation $U(\\alpha,(-1)^j\\beta,(-1)^{k+j}\\gamma,(-1)^j\\delta)$ on the given quantum ciphertext $\\rho_c$.\n\\end{description}\n\nAccording to Eq.(\\ref{equ5}), the output state of the algorithm $Evaluate_\\Delta$ is\n$$X^jZ^k(U(\\alpha,\\beta,\\gamma,\\delta)\\sigma_m U(\\alpha,\\beta,\\gamma,\\delta)^\\dagger)Z^kX^j.$$\nThus the scheme also satisfies the definition of symmetric QHE. Moreover, $Evaluate_\\Delta$ has the same computational complexity with the quantum operator $U(\\alpha,\\beta,\\gamma,\\delta)\\in\\mathcal{F}_\\Delta$, and the set $\\{U(\\alpha,\\beta,\\gamma,\\delta)|\\alpha,\\beta,\\gamma,\\delta\\in[0,2\\pi)\\}$ contains all of the single-qubit unitary transformations, so the above QHE scheme is fully homomorphic for single-qubit computation.\n\nNotice that while intending to perform any given unitary operator on a quantum ciphertext, the unitary operator should be firstly decomposed into the form of Eq.(\\ref{equ4}) (or computing the four parameters $\\alpha,\\beta,\\gamma,\\delta$).\n\nNext, we present another scheme, which will be shown to be a QFHE scheme, and permits any $n$-qubit unitary transformation.\n\nAny $n$-qubit unitary transformation can be decomposed as the combination of some $CNOT$ and single-qubit unitary transformations. Then any $n$-qubit\ncomputation (without quantum measurements) can be described as a quantum circuit $C$, which consists of some $CNOT$ and single-qubit unitary gates \\cite{liang2011}.\nFrom the above constructions of QHE schemes, the $CNOT$ gate or each single-qubit unitary gates $U(\\alpha,\\beta,\\gamma,\\delta)$ can correspond to another quantum gate $CNOT'$ or $U(\\alpha,(-1)^j\\beta,(-1)^{k+j}\\gamma,(-1)^j\\delta)$. By replacing each gate in quantum circuit $C$ with the corresponding quantum gate, we can obtain a new quantum circuit, denoted as $C'$. The two quantum circuits $C$ and $C'$ satisfy the relation\n\\begin{equation}\\label{equ6}\nC'(X^{a_1}Z^{b_1}\\otimes\\cdots\\otimes X^{a_n}Z^{b_n})=(X^{a_1}Z^{b_1}\\otimes\\cdots\\otimes X^{a_n}Z^{b_n})C,\n\\end{equation}\nwhere $a=a_1\\cdots a_n$ and $b=b_1\\cdots b_n$ are arbitrary $n$-bit strings selected from $\\{0,1\\}^n$. The circuit $C'$ is dependent on the circuit $C$ and the two numbers $a,b$.\n\nFrom Eq.(\\ref{equ6}), the QFHE on $n$ qubits can be constructed as follows. QOTP is used as the encryption\/decryption algorithms again, and it needs a $2n$-bit key to pefectly encrypt $n$-qubit plaintext, and obtains an $n$-qubit ciphertext. According to the key, the algorithm $Evaluate_\\Delta$ replaces each gate in quantum circuit $C$ with its corresponding gate, and gets a new quantum circuit $C'$, and then puts the $n$-qubit ciphertext as the inputs of $C'$. The output state of $C'$ can just be decrypted according to QOTP, and get the result of performing quantum circuit $C$ on the original plaintext. Thus, this QFHE scheme permits any $n$-qubit unitary circuit $C$ to be performed on $n$ qubits that have been encrypted, and no decryption is needed.\n\nNext, the size of quantum circuit $C'$ is analyzed. From the relation $CNOT'=CNOT((-1)^{j\\cdot m}Z^m\\otimes X^j)$, the gate $CNOT'$ can be implemented by at most $3$ quantum gates (e.g. $CNOT, Z, X$). So, while constructing the new quantum circuit $C'$, each $CNOT$ gate in the circuit $C$ is replaced with at most $3$ quantum gates. On the other hand, each single-qubit gate in $C'$ is only corresponding to $1$ single-qubit gate. Then, we can conclude that, the size of the new circuit $C'$ is not more than three times of the size of $C$. Thus, the complexity of $Evaluate_\\Delta$ is at most three times than that of the circuit $C$, and then the algorithm $Evaluate_\\Delta$ is efficient.\n\nIt follows from the above analysis that, this scheme is indeed a QFHE scheme, which permits any $n$-qubit unitary transformation.\n\n\\section{Analysis}\nThe security of these QHE and QFHE schemes is analyzed.\n\n{\\bf Theorem 1:}\nAll the QHE and QFHE schemes proposed in the previous sections are perfectly secure.\n\n{\\bf Proof:}\nFirstly, it can be verified that \\cite{boykin2000}\n\\begin{equation}\\label{equ7}\\frac{1}{2^{2n}}\\sum_{a,b\\in\\{0,1\\}^n}X^aZ^b\\tau Z^b X^a=\\frac{1}{2^n}I_{2^n},\\end{equation}\nwhere $\\tau$ is arbitrary $n$-qubit state.\n\nIn all these schemes, QOTP is used as the encryption\/decryption algorithm. Suppose the single-qubit plaintext $\\sigma_m$ is encrypted by QOTP using secret key $j,k\\in\\{0,1\\}$, which is unknown to the attacker. With regard to the attacker, the output of the algorithm $Encrypt_\\Delta$ is $\\frac{1}{2^2}\\sum_{j,k\\in\\{0,1\\}}X^jZ^k\\sigma_m Z^kX^j$, which is just the totally mixed state according to Eq.(\\ref{equ7}). Moreover, the output of $Evaluate_\\Delta$ with regard to the attacker is\n\\begin{eqnarray}\n&& \\frac{1}{2^2}\\sum_{j,k\\in\\{0,1\\}}U(\\alpha,(-1)^j\\beta,(-1)^{k+j}\\gamma,(-1)^j\\delta)\\rho_c U(\\alpha,(-1)^j\\beta,(-1)^{k+j}\\gamma,(-1)^j\\delta)^\\dagger \\nonumber\\\\\n&&= \\frac{1}{2^2}\\sum_{j,k\\in\\{0,1\\}}X^jZ^kU(\\alpha,\\beta,\\gamma,\\delta)\\sigma_m U(\\alpha,\\beta,\\gamma,\\delta)^\\dagger Z^kX^j =\\frac{1}{2}I_{2}, \\nonumber\n\\end{eqnarray}\nwhich is also the totally mixed state. Similarly, the same result can be obtained for the $n$-qubit QFHE scheme. Thus, with regard to the attacker, the outputs of the algorithms $Encrypt_\\Delta$ and $Evaluate_\\Delta$ in each QHE\/QFHE scheme are totally mixed states.\nIt means these QHE and QFHE schemes are also perfectly secure, and the attacker can obtain nothing about the plaintext and the result of evaluation.\n$\\hfill{~}\\Box$\n\nAll the QHE schemes presented here are constructed based on QOTP. It should be noticed that, the algorithm $Evaluate_\\Delta$ is dependent on the secret key.\nThis may restrict the application of these schemes, because only the owner of the secret key can process the encrypted qubits correctly. Does there exist a QOTP-based symmetric QHE scheme, in which the algorithm $Evaluate_\\Delta$ is independent of the secret key? Next, we can present a positive answer to this question.\n\nFor convenient, suppose $C$ is a permitted unitary operator in the QHE scheme, and $C'$ is the unitary operator which corresponds to $C$. The algorithm $Evaluate_\\Delta$ performs $C'$ on the ciphertext $X^jZ^k|\\varphi\\rangle$, and can get the result $X^jZ^kC|\\varphi\\rangle$.\n\n{\\bf Lemma 1:}\nIn the QOTP-based QHE scheme, if the operator $C'$ is independent of the secret key, and satisfies the condition $(X^jZ^k)C=C'(X^jZ^k),\\forall j,k$, then $C\\in\\{e^{i\\theta}I|\\theta\\in[0,2\\pi)\\}$.\n\n{\\bf Proof:}\nFrom the above condition, we know $(X^jZ^k)C(Z^kX^j)$ is independent of the secret key $j,k$. Because $\\{X^jZ^k|k,j\\in\\{0,1\\}^n\\}$ is a complete orthogonal basis in the $n$-qubit Hilbert space, any $n$-qubit unitary operator $C$ can be represented as\n$C=\\sum_{a,b\\in\\{0,1\\}^n}\\alpha_{a,b}X^aZ^b$. Then $$X^jZ^kCZ^kX^j=\\sum_{a,b\\in\\{0,1\\}^n}\\alpha_{a,b}(-1)^{a\\cdot k\\oplus b\\cdot j}X^aZ^b,$$ where ``$\\cdot$'' denotes inner product modular 2. Because $X^jZ^kCZ^kX^j$ is independent of $k,j$, it can be deduced that, $\\forall (k_1,j_1)\\neq(k_2,j_2)$, $$\\sum_{a,b}\\alpha_{a,b}\\left((-1)^{a\\cdot k_1\\oplus b\\cdot j_1}-(-1)^{a\\cdot k_2\\oplus b\\cdot j_2}\\right)X^aZ^b=0.$$\nSo, $$\\alpha_{a,b}\\left((-1)^{a\\cdot k_1\\oplus b\\cdot j_1}-(-1)^{a\\cdot k_2\\oplus b\\cdot j_2}\\right)=0,\\forall a,b,\\forall (k_1,j_1)\\neq(k_2,j_2).$$ Because it is impossible that $\\alpha_{a,b}=0,\\forall a,b$ (otherwise $C=0$), we have $$a\\cdot k_1\\oplus b\\cdot j_1=a\\cdot k_2\\oplus b\\cdot j_2, \\exists a,b,\\forall (k_1,j_1)\\neq(k_2,j_2).$$\nThen $$(a,b)\\cdot (k_1-k_2,j_1-j_2)=0, \\exists a,b,\\forall (k_1-k_2,j_1-j_2)\\neq (0,0).$$\nThus it concludes that $(a,b)\\equiv (0,0)$, and $C=\\alpha_{0,0}X^0Z^0=e^{i\\theta}I_{2^n}$, $\\forall\\theta\\in[0,2\\pi)$. Thus complete the proof.\n$\\hfill{~}\\Box$\n\nBecause the global phase in a quantum state does not influence the result of measurement (the global phase has no influence on the density matrix), the condition in Lemma 1 can be relaxed as this: $e^{i\\beta}(X^jZ^k)C=C'(X^jZ^k),\\forall j,k$, where $\\beta$ depends on $j,k$.\n\n{\\bf Lemma 2:}\nIn the QOTP-based QHE scheme, if the operator $C'$ is independent of the secret key, and satisfies the condition $e^{i\\beta}(X^jZ^k)C=C'(X^jZ^k),\\forall j,k$, then all the operators in $\\{e^{i\\theta}X^aZ^b|a,b\\in\\{0,1\\}^n,\\theta\\in[0,2\\pi)\\}$ are permitted unitary operators.\n\n{\\bf Proof:}\nIt can be verified that\n$$(-1)^{a\\cdot k\\oplus b\\cdot j}(X^jZ^k)(e^{i\\theta}X^aZ^b)=(e^{i\\theta}X^aZ^b)(X^jZ^k),\\forall j,k.$$\t\nLet $C=C'=e^{i\\theta}X^aZ^b$, which is independent of $j,k$. Thus complete the proof.\n$\\hfill{~}\\Box$\n\nFrom the two lemmas, we can conclude the following result.\n\n{\\bf Theorem 2:}\nIf the algorithm $Evaluate_\\Delta$ is independent of the secret key, then the QOTP-based symmetric QHE scheme permits any of the unitary operators in the set $\\{e^{i\\theta}X^aZ^b|a,b\\in\\{0,1\\}^n,\\theta\\in[0,2\\pi)\\}$.$\\hfill{~}\\Box$\n\nNow we have shown there exists a symmetric QHE scheme, which satisfies the condition: $Evaluate_\\Delta$ is independent of the secret key. However, it is obvious that it cannot be fully homomorphic.\n\nThere is another point that is worth to notice. It can be concluded from Eq.(\\ref{equ8}) that, given any $T$, if the encryption algorithm $\\mathcal{E}$ is replaced by another encryption algorithm $\\mathcal{E}'$, then the operation $T'$ will changes accordingly. The relationship between $T'$ and $T$ depends on the encryption algorithm $\\mathcal{E}$. In all our schemes,\nQOTP is chosen as the encryption algorithm, and this makes for our simple constructions.\n\n\\section{Discussions}\nThe symmetric QFHE scheme has been constructed. Suppose you have encrypted a qubit with this scheme, you can perform any single-qubit unitary operator on the qubit without decryption. If you intend to delegate the unitary operator to another party, you have to preshare the key with that party, thus that party may decrypt it and obtain the original qubit. So the symmetric QFHE scheme proposed here cannot be used in blind computing. However, you can still delegate the computation to the trusted party by using the QFHE scheme, which can prevent the malicious parties from obtaining the data and the result of computation.\n\nThe symmetric QFHE scheme may be used in secure multiparty quantum computation \\cite{dupuis2010,dupuis2012}, where $n$ parties participate in a computation (e.g. an evaluation of a unitary circuit $C$). Suppose the circuit $C$ is given $n$ quantum inputs and outputs $n$ quantum states, each of which is desired by one party. A naive multiparty computation protocol is given as follows. Each participant preshares a secret key with trusted third party (TTP) through quantum key distribution (QKD), and encrypts his inputs with his secret key according to the QFHE scheme, and then sends his ciphertext to TTP. TTP performs quantum circuit $C'$ (which is relative to $C$ and all the keys shared between TTP and every party) on all the ciphertexts, and sends back the results to each party. Then each party can decrypt the received state with his secret key, and obtain the desired result.\n\nBesides the homomorphic encryption, blind computing is another kind of study about privacy-preserving computation. Actually, homomorphic encryption and blind computing are two closely related research directions. Homomorphic encryption scheme can be used in blind computing in the following two cases: (1) Homomorphic encryption scheme is symmetric, and the algorithm $Evaluate_\\Delta$ is independent of the key $key$; (2) Homomorphic encryption scheme is asymmetric. In blind computing, the decryption $f(c)\\rightarrow f(m)$ is unnecessarily  the inverse of the encryption $m\\rightarrow c$ (See Refs.\\cite{feigenbaum1986,childs2005}). However, in the homomorphic encryption scheme, the decryption algorithm is just the inverse of the encryption algorithm.\n\nIn the QFHE scheme, $2n$ bits of key are needed to encrypt $n$ qubits with perfect security. To reduce the length of key, the QOTP can be replaced with approximate QOTP \\cite{ambainis2004}. However, the security would be slightly weaker.\n\nThis paper considers the symmetric QHE, and the asymmetric QHE is only defined but not constructed. So the open problem is: how to construct an asymmetric QHE scheme, where the algorithm $Evaluate_\\Delta$ depends only on the public key $pk$ but not the secret key $sk$? Or you can consider how to modify the quantum public-key encryption scheme in Ref.\\cite{liang2012}, such that it becomes an asymmetric QHE scheme.\nIf this goal were achieved, the computing on the quantum ciphertext could be securely outsourced, and then the blind quantum computing would be implemented in this way.\n\n\\section{Conclusions}\nThis paper defines symmetric and asymmetric QHE, and proposes four symmetric QHE schemes that permit all the quantum operators in the set $\\{R_z(\\theta)|\\theta\\in[0,2\\pi)\\}$, $\\{R_y(\\theta)|\\theta\\in[0,2\\pi)\\}$ or $\\{CNOT\\}$. Moreover, we construct a symmetric QFHE scheme, which permits any unitary operator on single qubit. Then the QFHE scheme is extended to permit any $n$-qubit unitary operator. All these schemes are constructed based on QOTP, and have perfect security. In the QFHE scheme, the algorithm $Evaluate_\\Delta$ depends on the secret key. Finally, we proposed a symmetric QHE scheme, in which $Evaluate_\\Delta$ is independent of the secret key.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Part~I. A review of kinematics via Cayley--Klein geometries}\n\n\n\\section{Possible kinematics}\n\nAs noted by Inonu and Wigner in their work \\cite{IW53}\non contractions of groups and their representations,\nclassical mechanics is a limiting case of relativistic mechanics,\nfor both the Galilei group as well as its Lie algebra are limits of the\nPoincar\\'{e} group and its Lie algebra.  Bacry and L\\'{e}vy-Leblond\n\\cite{BL67} classif\\\/ied and investigated the nature of\nall possible Lie algebras for kinema\\-tical groups (these\ngroups are assumed to be Lie groups as 4-dimensional spacetime\nis assumed to be continuous) given the three basic principles that\n\\begin{itemize}\\itemsep=0pt\n\\item[(i)] space is isotropic and spacetime is homogeneous,\n\\item[(ii)] parity and time-reversal are automorphisms of the kinematical group, and\n\\item[(iii)] the one-dimensional subgroups generated by the boosts are non-compact.\n\\end{itemize}\n\n\\begin{table}[th]\n \\centering\n \\caption{The 11 possible kinematical groups.}\n \\vspace{1mm}\n  \\begin{tabular}{ c | c  }\n  \\hline\nSymbol &  Name  \\tsep{1ex} \\bsep{1ex}\\\\\n\\hline\n\\tsep{1ex} $dS_1$             & de Sitter group $SO(4,1)$ \\\\\n$dS_2$             & de Sitter group $SO(3,2)$ \\\\\n$P$                   & Poincar\\'{e} group \\\\\n$P^{\\prime}_1$ & Euclidean group $SO(4)$ \\\\\n$P^{\\prime}_2$ & Para-Poincar\\'{e} group \\\\\n$C$                    &Carroll group \\\\\n$N_+$               & Expanding Newtonian Universe group \\\\\n$N_-$                & Oscillating Newtonian Universe group \\\\\n$G$                   & Galilei group \\\\\n$G^{\\prime}$     & Para-Galilei group \\\\\n$St$                   & Static Universe group \\bsep{0.5ex}\\\\\n\\hline\n \\end{tabular}\n \\end{table}\n\n\\begin{figure}[th]\n\\begin{center}\n\\includegraphics[width=12.5cm]{McRae-fig1e}\n\\end{center}\n\\caption{The contractions of the kinematical groups.}\n\\end{figure}\n\nThe resulting possible Lie algebras give 11 possible kinematics,\nwhere each of the kinematical groups (see Table~1) is generated by its inertial\ntransformations as well as its spacetime translations and spatial rotations.\nThese groups consist of the de Sitter groups and their\nrotation-invariant contractions:  the physical nature of\na contracted group is determined by the nature of the contraction\nitself, along with the nature of the parent de Sitter group.\nBelow we will illustrate the nature of these contractions when\nwe look more closely at the simpler case of a~2-dimensional spacetime.\nFor Fig.~1, note that a ``upper\" face of the cube is transformed under\none type of contraction into the opposite face.\n\nSanjuan \\cite{F84} noted that the methods employed by Bacry\nand L\\'{e}vy-Leblond could be easily applied to 2-dimensional\nspacetimes: as it is the purpose of this paper to investigate\nthese kinematical Lie algebras and groups through Clif\\\/ford algebras,\nwe will begin by explicitly classifying all such possible Lie algebras.\nThis section then is a detailed and expository account of certain parts of Bacry,\nL\\'{e}vy-Leblond, and Sanjuan's work.\n\nLet $K$ denote the generator of the inertial transformations, $H$ the generator\nof time translations, and $P$ the generator of space translations.\nAs space is one-dimensional, space is isotropic.   In the following\nsection we will see how to construct, for each possible kinematical\nstructure, a~spacetime that is a homogeneous space for its kinematical\ngroup, so that basic principle (i) is satisf\\\/ied.\n\nNow let $\\Pi$ and $\\Theta$ denote the respective operations\nof parity and time-reversal:  $K$ must be odd under both $\\Pi$ and $\\Theta$.\nOur basic principle (ii) requires that the Lie algebra is acted upon by\nthe $\\mathbb{Z}_2 \\otimes \\mathbb{Z}_2$ group of involutions generated by\n\\[ \\Pi \\, : \\, \\left( K, H, P \\right) \\rightarrow \\left( -K, H, -P \\right) \\qquad\n\\mbox{and} \\qquad  \\Theta \\, : \\, \\left( K, H, P \\right) \\rightarrow \\left( -K, -H, P \\right) .\\]\n\nFinally, basic principle (iii) requires that the subgroup\ngenerated by $K$ is noncompact, even though we will allow for\nthe universe to be closed, or even for closed time-like worldlines to exist.\nWe do not wish for $e^{0K} = e^{\\theta K}$ for some non-zero $\\theta$,\nfor then we would f\\\/ind it possible for a boost to be no boost at all!\n\nAs each Lie bracket $[K, H]$, $[K, P]$, and $[H, P]$ is invariant\nunder the involutions $\\Pi$ and $\\Theta$ as well as the involution\n\\[ \\Gamma = \\Pi\\Theta \\, : \\, \\left( K, H, P \\right)\n\\rightarrow \\left( K, -H, -P \\right), \\]\nwe must have that $[K, H] = pP$, $[K, P] = hH$,\nand $[H, P] = kK$ for some constants $k$, $h$, and $p$.\nNote that these Lie brackets are also invariant under the symmetries def\\\/ined by\n\\begin{gather*}\n S_P : \\{ K \\leftrightarrow H, p \\leftrightarrow -p, k \\leftrightarrow h \\}, \\qquad\n S_H : \\{ K \\leftrightarrow P, h \\leftrightarrow -h, k \\leftrightarrow -p \\}, \\qquad\n\\mbox{and}\\\\\n S_K : \\{ H \\leftrightarrow P, k \\leftrightarrow -k, h \\leftrightarrow p \\} ,\n \\end{gather*}\nand that the Jacobi identity is automatically satisf\\\/ied\nfor any triple of elements of the Lie algebra.\n\n\\begin{table}[t]\n \\centering\n  \\caption{The 21 kinematical Lie algebras, grouped into 11 essentially distinct types of kinematics.}\n \\vspace{1mm}\n\n \\begin{tabular}{| r | r | r | r | r | r | r | r | r | r | r | r | r | r | r | r |}\n \\cline{1-2} \\cline{4-5} \\cline{7-8} \\cline{10-11} \\cline{13-14} \\cline{16-16}\n $P$ & $-P$ & & $P$ & $-P$ & & $P$ & $-P$ & & $0$  & $0$    & & $0$ & $0$  & & $0$ \\\\\n $H$ & $-H$ & & $H$ & $-H$ & & $0$ & $0$   & & $H$ & $-H$  & & $0$ & $0$   & & $0$\\\\\n $K$ & $-K$ & & $-K$ & $K$ & & $0$ & $0$   & & $0$  & $0$     & & $K$ & $-K$ & & $0$ \\\\\n \\cline{1-2} \\cline{4-5} \\cline{7-8} \\cline{10-11} \\cline{13-14} \\cline{16-16}\n  \\multicolumn{16}{c}{}       \\\\[-2mm]\n  \\cline{1-2} \\cline{4-5} \\cline{7-8} \\cline{10-11} \\cline{13-14}\n $0$ & $0$   & & $0$  & $0$  & & $P$ & $-P$ & & $P$ & $-P$    & & $P$ & $-P$ & \\multicolumn{2}{c}{} \\\\\n $H$ & $-H$ & & $H$ & $-H$ & & $0$ & $0$ & & $0$   & $0$     & & $H$ & $-H$ & \\multicolumn{2}{c}{} \\\\\n $K$ & $-K$ & & $-K$ & $K$ & & $K$ & $-K$ & & $-K$ & $K$    & & $0$ & $0$   & \\multicolumn{2}{c}{} \\\\\n \\cline{1-2} \\cline{4-5} \\cline{7-8} \\cline{10-11} \\cline{13-14}\n \\end{tabular}\n \\end{table}\n\n   \\begin{table}[t]\n \\centering\n \\caption{6 non-kinematical Lie algebras.}\n \\vspace{1mm}\n   \\begin{tabular}{| r | r | r | r | r | r | r | r | }  \\cline{1-2} \\cline{4-5} \\cline{7-8}\n $P$ & $-P$ & & $P$ & $-P$ & & $-P$ & $P$ \\\\\n $-H$ & $H$ & & $-H$ & $H$ & & $H$ & $-H$ \\\\\n $K$ & $-K$ & & $-K$ & $K$  & & $0$ & $0$ \\\\ \\cline{1-2} \\cline{4-5} \\cline{7-8}\n \\end{tabular}\\vspace{-2mm}\n \\end{table}\n\n\nWe can normalize the constants $k$, $h$, and $p$ by\na scale change so that $k, h, p \\in \\{-1, 0, 1 \\}$,\ntaking advantage of the simple form of the Lie brackets\nfor the basis elements $K$, $H$, and $P$.   There\nare then $3^3$ possible Lie algebras, which we\ntabulate in Tables~2 and~3 with columns that have the following form:\n\\begin{table}[htbp]\n \\centering\n \\begin{tabular}{| r | }  \\hline\n $[K, H]$ \\\\\n $[K, P]$ \\\\\n $[H, P]$ \\\\\\hline\n \\end{tabular}\n \\end{table}\n\n\n\n  \\begin{table}[t]\n \\centering\n \\caption{Some kinematical groups along with their notation and structure constants.}\n\\vspace{1mm}\n  \\begin{tabular}{ c c c c c } \\hline\n             &  Anti-de Sitter & Oscillating Newtonian Universe & Para-Minkowski      & Minkowski   \\\\  \\cline{2-5}\n             &\\tsep{0.2ex} $adS$            & $N_-$                          & $M^{\\prime}$           & $M$ \\\\ \\hline \\hline\n\\tsep{0.2ex}$[K,H]$ & $P$                & $P$                              &  $0$                        &$P$ \\\\\n$[K,P]$ & $H$                & $0$                              & $H$                         & $H$ \\\\\n$[H,P]$ & $K$                & $K$                             & $K$                          & $0$ \\\\\n\\hline\n \\end{tabular}\n \\end{table}\n\n\\begin{table}[t]\n \\centering\n \\caption{Some kinematical groups along with their notation and structure constants.}\n\\vspace{1mm}\n  \\begin{tabular}{ c c c c  } \\hline\n             &  de Sitter       & Expanding Newtonian Universe & Expanding Minkowski Universe     \\\\   \\cline{2-4}\n             & $dS$            & $N_+$      & $M_+$   \\\\  \\hline \\hline\n\\tsep{0.2ex}$[K,H]$ & $P$                & $P$         &  $0$           \\\\\n$[K,P]$ & $H$                & $0$         & $H$               \\\\\n$[H,P]$ & $-K$                & $-K$        & $-K$           \\\\ \\hline\n \\end{tabular}\n  \\end{table}\n\n \\begin{table}[t]\n \\centering\n\\caption{Some kinematical groups along with their notation and structure constants.}\n \\vspace{1mm}\n \\begin{tabular}{ c c c c c } \\hline\n             & Galilei            & Carroll    & Static de Sitter Universe   & Static Universe  \\\\  \\cline{2-5}\n             & $G$               & $C$         & $SdS$         & $St$ \\\\ \\hline \\hline\n$[K,H]$ & $P$                & $0$         &  $0$           &$0$ \\\\\n$[K,P]$ & $0$                & $H$         & $0$                & $0$ \\\\\n$[H,P]$ & $0$                & $0$        & $K$          & $0$ \\\\ \\hline\n \\end{tabular}\n \\end{table}\n\n  \\noindent We also pair each Lie\n algebra with its image under the isomorphism given by\n $P \\leftrightarrow -P$, $H \\leftrightarrow -H$,\n $K \\leftrightarrow -K$, and $[ \\star, \\star \\star] \\leftrightarrow [ \\star \\star, \\star]$,\n for both Lie algebras then give the same kinematics.  There are then 11 essentially\n distinct kinematics, as illustrated in Table~2.\n Also (as we shall see in the next section) each of the other 6 Lie algebras\n (that are given in Table~3) violate the third basic principle,\n generating a compact group of inertial transformations.\n\nThese non-kinematical Lie algebras are the lie algebras for\nthe motion groups for the elliptic, hyperbolic, and Euclidean planes:\nlet us denote these respective groups as $El$, $H$, and $Eu$.\n\n\n\n\nWe name the kinematical groups (that are generated by\nthe boosts and translations) in concert with the 4-dimensional case (see Tables~4,~5, and~6).\nEach of these kinematical groups is either the de Sitter or the anti-de Sitter group,\nor one of their contractions.  We can contract with respect to any subgroup,\ngiving us three fundamental types of contraction: {\\it speed-space, speed-time},\nand {\\it space-time contractions}, corresponding respectively to contracting\nto the subgroups generated by $H$, $P$, and $K$.\n\n\n{\\bfseries \\itshape Speed-space contractions.}\nWe make the substitutions $K \\rightarrow \\epsilon K$ and\n$P \\rightarrow \\epsilon P$ into the Lie algebra and then\ncalculate the singular limit of the Lie brackets as $\\epsilon \\rightarrow 0$.\nPhysically the velocities are small when compared to the speed of light,\nand the spacelike intervals are small when compared to the timelike intervals.\nGeometrically we are describing spacetime near a timelike geodesic,\nas we are contracting to the subgroup that leaves this worldline invariant,\nand so are passing from relativistic to absolute time.  So $adS$\nis contracted to $N_-$ while $dS$ is contracted to $N_+$, for example.\n\n{\\bfseries \\itshape Speed-time contractions.}\nWe make the substitutions $K \\rightarrow \\epsilon K$ and $H \\rightarrow \\epsilon H$\ninto the Lie algebra and then calculate the singular limit of the Lie brackets\nas $\\epsilon \\rightarrow 0$.  Physically the velocities are small when compared\nto the speed of light, and the timelike intervals are small\nwhen compared to the spacelike intervals.\nGeometrically we are describing spacetime near\na spacelike geodesic, as we are contracting to the\nsubgroup that leaves invariant this set of simultaneous events,\nand so are passing from relativistic to absolute space.\nSuch a spacetime may be of limited physical interest,\nas we are only considering intervals connecting events that are not causally related.\n\n{\\bfseries \\itshape Space-time contractions.}\nWe make the substitutions $P \\rightarrow \\epsilon P$\nand $H \\rightarrow \\epsilon H$ into the Lie algebra\nand then calculate the singular limit of the Lie brackets\nas $\\epsilon \\rightarrow 0$.  Physically the spacelike\nand timelike intervals are small, but the boosts are not restricted.\nGeometrically we are describing spacetime near an event,\nas we are contracting to the subgroup that leaves invariant\nonly this one event, and so we call the corresponding kinematical\ngroup a {\\it local group} as opposed to a {\\it cosmological group}.\n\nFig.~2 illustrates several interesting\nrelationships among the kinematical groups.\nFor example, Table~7 gives important classes of kinematical groups,\neach class corresponding to a face of the f\\\/igure, that  transform to\nanother class in the table under one of the symmetries $S_H$, $S_P$,\nor $S_K$, provided that certain exclusions are made as outlined in Table~8.\n The exclusions are necessary under the given symmetries as\n some kinematical algebras are taken to algebras that are not kinematical.\n\n \\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=12.5cm]{McRae-fig2e}\n\\end{center}\n\\caption{The contractions of the kinematical groups for 2-dimensional spacetimes.}\n\\end{figure}\n\n\\begin{table}[t]\n \\centering\n\\caption{Important classes of kinematical groups and their geometrical conf\\\/igurations in Fig.~2.}\n\\vspace{1mm}\n\n \\begin{tabular}{ l | l } \\hline\n Class of groups & Face \\\\ \\hline \\hline\n Relative-time & $1247$ \\\\\n Absolute-time & $3568$ \\\\\n Relative-space & $1346$ \\\\\n Absolute-space & $2578$ \\\\\n Cosmological & $1235$ \\\\\n Local & $4678$ \\\\ \\hline\n \\end{tabular}\n \\end{table}\n\n\\begin{table}[t]\n \\centering\n\\caption{The 3 basic symmetries are represented by ref\\\/lections of Fig.~2, with some exclusions.}\n\n\\vspace{1mm}\n\n \\begin{tabular}{ c | c } \\hline\n Symmetry & Ref\\\/lection across face \\\\ \\hline \\hline\n $S_H$ & $1378$ (excluding $M_+$) \\\\\n $S_P$ & $1268$ (excluding $adS$ and $N_-$)\\\\\n $S_K$ & $1458$ \\\\ \\hline\n \\end{tabular}\n \\end{table}\n\n\n\n\\section[Cayley-Klein geometries]{Cayley--Klein geometries}\n\nIn this section we wish to review work done by Ballesteros, Herranz, Ortega\nand Santander on homogeneous spaces that are spacetimes\nfor kinematical groups, and we begin with a bit of history concerning\nthe discovery of non-Euclidean geometries.\nFranz Taurinus was the f\\\/irst to explicitly give mathematical\ndetails on how a hypothetical sphere of imaginary radius would have a\nnon-Euclidean geometry, what he called {\\it log-spherical geometry},\n and this was done via hyperbolic trigonometry (see \\cite{G89} or \\cite{K98}).\n Felix Klein\\footnote{Roger Penrose \\cite{P05}\n notes that it was Eugenio Beltrami who f\\\/irst discovered\n both the projective and conformal models of the hyperbolic plane.}\n is usually given credit for being the f\\\/irst to give a complete model of\n a non-Euclidean geometry\\footnote{ Spherical geometry was not historically considered\n to be non-Euclidean in nature, as it can be embedded in a 3-dimensional Euclidean space,\n unlike Taurinus' sphere.}: he built his model by suitably adapting\n Arthur Cayley's metric for the projective plane.  Klein \\cite{K21} (originally published in 1871)\n went on, in a systematic way, to describe nine types of two-dimensional geometries\n (what Yaglom~\\cite{Y79} calls {\\it Cayley--Klein geometries})\n that were then further investigated by Sommerville~\\cite{S10}.\n Yaglom gave conformal models for these geometries, extending what\n had been done for both the projective and hyperbolic planes.\n Each type of geometry is homogeneous and can be determined by\n two real constants ${\\kappa_1}$ and ${\\kappa_2}$ (see Table~9).\nThe names of the geometries when ${\\kappa_2} \\leq 0$ are those as given by Yaglom,\nand it is these six geometries that can be interpreted as spacetime geometries.\n\n\\begin{table}[t]\n \\centering\n \\caption{The 9 types of Cayley--Klein geometries.}\n \\vspace{1mm}\n\n \\begin{tabular}{ l l l l } \\hline\n&  \\multicolumn{3}{c }{Metric Structure}   \\\\  \\cline{2-4}\n Conformal  & Elliptic & Parabolic & Hyperbolic \\\\\nStructure & ${\\kappa_1} > 0$ & ${\\kappa_1} = 0$ & ${\\kappa_1} < 0$ \\\\ \\hline \\hline\n\\tsep{1ex} Elliptic & elliptic & Euclidean & hyperbolic \\\\\n${\\kappa_2} > 0$ & geometries & geometries & geometries \\\\[1mm]\nParabolic & co-Euclidean & Galilean & co-Minkowski  \\\\\n${\\kappa_2} = 0$ & geometries  & geometry & geometries  \\\\[1mm]\nHyperbolic & co-hyperbolic & Minkowski & doubly \\\\\n${\\kappa_2} < 0$ & geometries & geometries & hyperbolic \\\\\n                &                    &                     & geometries \\\\ \\hline\n \\end{tabular}\n \\end{table}\n\nFollowing Taurinus, it is easiest to describe a bit of the geometrical\nnature of these geometries by applying the appropriate kind of trigonometry:\nwe will see shortly how to actually construct a model for each geometry.\nLet $\\kappa$ be a real constant.  The unit circle $a^2 + \\kappa b^2 = 1$\nin the plane ${\\mathbb{R}}^2 = \\{ (a,b) \\}$ with metric $ds^2 = da^2 + \\kappa db^2$ can be used to def\\\/ined the cosine\n\\begin{equation*}\n{C_{\\kappa}}(\\phi) =\n\\begin{cases}\n\\cos{\\left(\\sqrt{\\kappa} \\, \\phi \\right) },                   &\\text{if $\\kappa  > 0$},   \\\\\n1,  &\\text{if $\\kappa = 0$},      \\\\\n\\cosh{\\left( \\sqrt{-\\kappa} \\, \\phi \\right) },   &\\text{if $\\kappa < 0$},           \\\\\n\\end{cases}\n\\end{equation*}\nand sine\n\\begin{equation*}\n{S_{\\kappa}}(\\phi) =\n\\begin{cases}\n\\frac{1}{\\sqrt{\\kappa}}\\sin{\\left( \\sqrt{\\kappa} \\, \\phi \\right) },                   &\\text{if $\\kappa > 0$},   \\\\\n\\phi,  &\\text{if $\\kappa = 0$},      \\\\\n\\frac{1}{\\sqrt{-\\kappa}} \\sinh{\\left( \\sqrt{-\\kappa} \\, \\phi \\right) },   &\\text{if $\\kappa < 0$}           \\\\\n\\end{cases}\n\\end{equation*}\nfunctions:  here $(a,b) = ({C_{\\kappa}}(\\phi), {S_{\\kappa}}(\\phi))$ is a point on\nthe connected component of the unit circle containing the\npoint $(1,0)$, and $\\phi$ is the signed distance from $(1,0)$\nto $(a,b)$ along the circular arc, def\\\/ined modulo the\nlength $\\dfrac{2\\pi}{\\sqrt{\\kappa}}$ of the unit circle when $\\kappa > 0$.\nWe can also write down the power series for these analytic trigonometric functions:\n\\begin{gather*}\n{C_{\\kappa}}(\\phi) = 1 - \\frac{1}{2!}\\kappa \\phi^2 + \\frac{1}{4!} \\kappa^2 \\phi^4 + \\cdots,\n\\\\\n{S_{\\kappa}}(\\phi) = \\phi - \\frac{1}{3!}\\kappa \\phi^3 + \\frac{1}{5!} \\kappa^2 \\phi^5 + \\cdots.\n\\end{gather*}\nNote that ${C_{\\kappa}}^2(\\phi) + \\kappa {S_{\\kappa}}^2(\\phi) = 1$.   So if $\\kappa > 0$ then\nthe unit circle is an ellipse (giving us elliptical trigonometry),\nwhile if $\\kappa < 0$ it is a hyperbola (giving us hyperbolic trigonometry).\nWhen $\\kappa = 0$ the unit circle consists of two parallel straight lines,\nand we will say that our trigonometry is parabolic.\nWe can use such a trigonometry to def\\\/ine the angle $\\phi$\nbetween two lines, and another independently chosen\ntrigonometry to def\\\/ine the distance between two points\n(as the angle between two lines, where each line passes\nthrough one of the points as well as a distinguished point).\n\nAt this juncture it is not clear that such geometries,\nas they have just been described, are of either\nmathematical or physical interest.  That mathematicians\nand physicists at the beginning of the 20th century\nwere having similar thoughts is perhaps not surprising,\nand Walker \\cite{W99} gives an interesting account\nof the mathematical and physical research into non-Euclidean\ngeometries during this period in history.\nKlein found that there was a fundamental unity\nto these geometries, and so that alone made them worth studying.\nBefore we return to physics, let us look at these geometries\nfrom a perspective that Klein would have appreciated, describing their motion groups in a unif\\\/ied manner.\n\nBallesteros, Herranz, Ortega and Santander have constructed\nthe Cayley--Klein geometries as homogeneous\nspaces\\footnote{See \\cite{BH06,HOS96,HS02}, and also \\cite{HOS00},\nwhere a special case of the group law is investigated,\nleading to a plethora of trigonometric identities,\nsome of which will be put to good use in this paper: see Appendix~A.}\nby looking at real representations of their motion groups.\nThese motion groups are denoted by ${SO_{\\ka,\\kb}(3)}$ (that we will refer to as\nthe {\\it generalized} $SO(3)$ or  simply by $SO(3)$)\nwith their respective Lie algebras being denoted by ${so_{\\ka,\\kb}(3)}$\n(that we will refer to as the {\\it generalized} $so(3)$ or simply by $so(3)$),\nand most if not all of these groups are probably familiar to the reader\n(for example, if both ${\\kappa_1}$ and ${\\kappa_2}$ vanish, then $SO(3)$ is the Heisenberg group).\nLater on in this paper we will use Clif\\\/ford algebras to show how we can\nexplicitly think of $SO(3)$ as a rotation group, where each element of $SO(3)$\nhas a well-def\\\/ined axis of rotation and rotation angle.\n\nNow a matrix representation of $so(3)$ is given by the matrices\n\\[\n   H =\n   \\left(\n   \\begin{matrix}\n   0 & -{\\kappa_1} & 0 \\\\\n   1 &     0 & 0 \\\\\n   0 &    0 & 0\n   \\end{matrix}\n   \\right), \\qquad\n      P =\n      \\left(\n      \\begin{matrix}\n      0 & 0 & -{\\kappa_1} {\\kappa_2} \\\\\n      0 & 0 & 0 \\\\\n      1 & 0 & 0\n      \\end{matrix}\n      \\right),  \\qquad \\mbox{and} \\qquad\n         K =\n         \\left(\n         \\begin{matrix}\n         0 & 0 & 0 \\\\\n         0 & 0 & -{\\kappa_2} \\\\\n         0 & 1 & 0\n         \\end{matrix}\n         \\right),\n \\]\nwhere the structure constants are given by the commutators\n\\[ \\left[ K, H \\right] = P,  \\qquad \\left[ K, P \\right]\n= -{\\kappa_2} H, \\qquad \\mbox{and}  \\qquad \\left[ H, P \\right] = {\\kappa_1} K.  \\]\nBy normalizing the constants we obtain matrix representations of the $adS$, $dS$,\n$N_-$, $N_+$, $M$, and $G$ Lie algebras, as well as the Lie algebras for the elliptic, Euclidean,\nand hyperbolic motion groups, denoted $El$, $Eu$, and $H$ respectively.\nWe will see at the end of this section how the Cayley--Klein\nspaces can also be used to give homogeneous spaces for $M^{\\prime}$, $M_+$, $C$,\nand $SdS$ (but not for $St$).  One benef\\\/it of not normalizing\nthe parameters ${\\kappa_1}$ and ${\\kappa_2}$ is that we can easily obtain contractions\nby letting ${\\kappa_1} \\rightarrow 0$ or ${\\kappa_2} \\rightarrow 0$.\n\n\nElements of $SO(3)$ are real-linear, orientation-preserving\nisometries of ${\\mathbb{R}}^3 = \\{ (z, t, x)) \\}$ imbued with the\n(possibly indef\\\/inite or degenerate) metric $ds^2 = dz^2 + {\\kappa_1} dt^2 +  {\\kappa_1} {\\kappa_2} dx^2$.\nThe one-parameter subgroups $\\mathcal{H}$, $\\mathcal{P}$,\nand $\\mathcal{K}$ generated respectively by $H$, $P$, and $K$ consist of matrices of the form\n\\[\n   e^{\\alpha H} =\n   \\left(\n   \\begin{matrix}\n   C_{{\\kappa_1}}(\\alpha) & -{\\kappa_1} S_{{\\kappa_1}}(\\alpha) & 0 \\\\\n   S_{{\\kappa_1}}(\\alpha) & C_{{\\kappa_1}}(\\alpha)        & 0 \\\\\n   0                       &  0                             & 1\n   \\end{matrix}\n   \\right),\n\\qquad\n   e^{\\beta P} =\n   \\left(\n   \\begin{matrix}\n   C_{{\\kappa_1} {\\kappa_2}}(\\beta) & 0 & -{\\kappa_1} {\\kappa_2} S_{{\\kappa_1} {\\kappa_2}}(\\beta) \\\\\n   0                           & 1 & 0        \\\\\n   S_{{\\kappa_1} {\\kappa_2}}(\\beta) & 0 & C_{{\\kappa_1} {\\kappa_2}}(\\beta)\n   \\end{matrix}\n   \\right),\n\\]\n\\bigskip\nand\n\\[\n   e^{\\theta K} =\n   \\left(\n   \\begin{matrix}\n   1 & 0 & 0 \\\\\n   0 & C_{{\\kappa_2}}(\\theta)  & -{\\kappa_2} S_{{\\kappa_2}}(\\theta)  \\\\\n   0 & S_{{\\kappa_2}}(\\theta)  & C_{{\\kappa_2}}(\\theta)\n   \\end{matrix}\n   \\right)\n\\]\n(note that the orientations induced on the coordinate planes\nmay be dif\\\/ferent than expected).   We can now see that in order for $\\mathcal{K}$\nto be non-compact, we must have that ${\\kappa_2} \\leq 0$, which explains the content of Table~3.\n\nThe spaces $SO(3) \/ \\mathcal{K}$, $SO(3) \/ \\mathcal{H}$,\nand $SO(3) \/ \\mathcal{P}$ are homogeneous spaces for $SO(3)$.\nWhen $SO(3)$ is a kinematical group, then $S \\equiv SO(3) \/ \\mathcal{K}$\ncan be identif\\\/ied with the manifold of space-time translations.\nRegardless of the values of ${\\kappa_1}$ and ${\\kappa_2}$ however, $S$ is the Cayley--Klein\ngeometry with parameters ${\\kappa_1}$ and ${\\kappa_2}$, and $S$ can be shown to have\nconstant curvature ${\\kappa_1}$ (also, see \\cite{M06}).   So the angle between\ntwo lines passing through the origin (the point that is invariant under\nthe subgroup $\\mathcal{K}$) is given by the parameter $\\theta$ of the\nelement of $\\mathcal{K}$ that rotates one line to the other (and\nso the measure of angles is related to the parameter ${\\kappa_2}$).\nSimilarly if one point can be taken to another by an element\nof $\\mathcal{H}$ or $\\mathcal{P}$ respectively, then the distance\nbetween the two points is given by the parameter $\\alpha$ or $\\beta$,\n(and so the measure of distance is related to the parameter ${\\kappa_1}$ or to ${\\kappa_1} {\\kappa_2}$).\nNote that the spaces $SO(3) \/ \\mathcal{H}$ and $SO(3) \/ \\mathcal{P}$\nare respectively the spaces of timelike and spacelike geodesics for kinematical groups.\n\nFor our purposes we will also need to model $S$ as a projective geometry.\nFirst, we def\\\/ine the projective quadric $\\bar{\\Sigma}$ as the\nset of points on the unit sphere\n$\\Sigma \\equiv \\{ (z, t, x) \\in {\\mathbb{R}}^3 \\; | \\; z^2 + {\\kappa_1} t^2 + {\\kappa_1} {\\kappa_2} x^2  = 1 \\}$\nthat have been identif\\\/ied by the equivalence relation $(z, t, x) \\thicksim (-z, -t, -x)$.\nThe group $SO(3)$ acts on $\\bar{\\Sigma}$, and the\nsubgroup $\\mathcal{K}$ is then the isotropy subgroup\nof the equivalence class $\\mathcal{O} = [(1, 0, 0)]$.\nThe metric $g$ on ${\\mathbb{R}}^3$ induces a metric on $\\bar{\\Sigma}$ that has ${\\kappa_1}$ as a factor.\nIf we then def\\\/ine the main metric $g_1$ on $\\bar{\\Sigma}$ by setting\n\\[ \\left( ds^2 \\right)_1 = \\frac{1}{{\\kappa_1}} ds^2, \\]\nthen the surface $\\bar{\\Sigma}$, along\nwith its main metric (and subsidiary metric, see below),\nis a projective model for the Cayley--Klein geometry $S$.\nNote that in general $g_1$ can be indef\\\/inite as well as nondegenerate.\n\n\nThe motion $\\exp(\\theta K$) gives a rotation (or boost for a spacetime)\nof $S$, whereas the motions $\\exp(\\alpha H$) and $\\exp(\\beta P$)\ngive translations of $S$ (time and space translations respectively for a~spacetime).\nThe parameters ${\\kappa_1}$ and ${\\kappa_2}$ are, for the spacetimes, identif\\\/ied\nwith the universe time radius $\\tau$ and speed of light $c$ by the formulae\n\\[  {\\kappa_1} = \\pm \\frac{1}{\\tau^2} \\qquad \\mbox{and} \\qquad {\\kappa_2} = - \\frac{1}{c^2}.  \\]\n\n\nFor the absolute-time spacetimes with kinematical groups $N_-$, $G$,\nand $N_+$, where ${\\kappa_2} = 0$ and $c = \\infty$, we foliate $S$\nso that each leaf consists of all points that are simultaneous\nwith one another, and then $SO(3)$ acts transitively on each leaf.\nWe then def\\\/ine the subsidiary metric $g_2$ along each leaf of the foliation by setting\n\\[\n\\left( ds^2 \\right)_2 = \\frac{1}{{\\kappa_2}} \\left( ds^2 \\right)_1.\n\\]\nOf course when ${\\kappa_2} \\neq 0$, the subsidiary metric\ncan be def\\\/ined on all of $\\bar{\\Sigma}$.  The group $SO(3)$ acts\non $S$ by isometries of $g_1$, by isometries of $g_2$ when ${\\kappa_2} \\neq 0$\nand, when ${\\kappa_2} = 0$,  on the leaves of the foliation by isometries of $g_2$.\n\nIt remains to be seen then how homogeneous spacetimes for the\nkinematical groups $M_+,$~$M^{\\prime},$~$C,\\!$ and $SdS$ may be obtained from the Cayley--Klein geometries.\nIn Fig.~3 the face $1346$ contains the motion groups for all nine types of Cayley--Klein geometries,\nand the symmetries $S_H$, $S_P$, and $S_K$ can be represented as\nsymmetries of the cube, as indicated in\nTable~10\\footnote{Santander \\cite{mS01} discusses\nsome geometrical consequences of such symmetries when\napplied to $dS$, $adS$, and $H$: note that $S_H$, $S_P$, and $S_K$ all f\\\/ix vertex $1$.}.\nAs vertices~$1$ and~$8$ are in each of the three planes of ref\\\/lection,\nit is impossible to get $St$ from any one of the Cayley--Klein\ngroups through the symmetries $S_H$, $S_P$, and $S_K$.\nUnder the symmetry $S_K$, respective spacetimes for $M_+$, $M^{\\prime}$,\nand $C$ are given by the spacetimes $SO(3)\/\\mathcal{K}$ for $N_+$, $N_-$, and $G$,\nwhere space and time translations are interchanged.\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=12.5cm]{McRae-fig3e}\n\\end{center}\n\\caption{The 9 kinematical and 3 non-kinematical groups.}\n\\end{figure}\n\n\\begin{table}[t]\n \\centering\n\\caption{The 3 basic symmetries are given as ref\\\/lections of Fig.~3.}\n\\vspace{1mm}\n \\begin{tabular}{ c | c } \\hline\n Symmetry & Ref\\\/lection across face \\\\ \\hline \\hline\n $S_H$ & $1378$ \\\\\n $S_P$ & $1268$ \\\\\n $S_K$ & $1458$ \\\\ \\hline\n \\end{tabular}\n \\end{table}\n\nUnder the symmetry $S_H$, the spacetime\nfor $SdS$ is given by the homogeneous space $SO(3)\/\\mathcal{P}$ for $G$,\nas boosts and space translations are interchanged by $S_H$.\nNote however that there actually are no spacelike geodesics for $G$,\nas the Cayley--Klein geometry $S = SO(3)\/\\mathcal{K}$ for ${\\kappa_1} = {\\kappa_2} = 0$\ncan be given simply by the plane ${\\mathbb{R}}^2 = \\{ (t,x) \\}$ with $ds^2 = dt^2$\nas its line element\\footnote{Yaglom writes in \\cite{Y79} about this geometry,\n{\\it ``\\dots which, in spite of its relative simplicity,\nconfronts the uninitiated reader with many surprising results.''}}.\nAlthough $SO(3)\/\\mathcal{P}$ is a homogeneous space for $SO(3)$, $SO(3)$\ndoes not act ef\\\/fectively on $SO(3)\/\\mathcal{P}$:  since both $[K,P] = 0$ and $[H,P] = 0$,\nspace translations do not act on $SO(3)\/\\mathcal{P}$.   Similarly, inertial transformations\ndo not act on spacetime for $SdS$, or on $St$ for that matter.\nNote that $SdS$ can be obtained from $dS$ by $P \\rightarrow \\epsilon P$,\n$H \\rightarrow \\epsilon H$, and $K \\rightarrow \\epsilon^2 K$,\nwhere $\\epsilon \\rightarrow 0$.  So velocities are negligible even when\ncompared to the reduced space and time translations.\n\nIn conclusion to Part I then, a study of all nine types of Cayley--Klein\ngeometries af\\\/fords us a beautiful and unif\\\/ied study of all 11\npossible kinematics save one, the static kinematical structure.\nIt was this study that motivated the author to investigate another\nunif\\\/ied approach to possible kinematics, save for that of the Static Universe.\n\n\n\\pdfbookmark[1]{Part II.  Another unified approach to possible kinematics}{part2}\n\\section*{Part II.  Another unif\\\/ied approach to possible kinematics}\n\n\n\\section[The generalized Lie algebra $so(3)$]{The generalized Lie algebra $\\boldsymbol{so(3)}$}\n\nPreceding the work of Ballesteros, Herranz, Ortega,\nand Santander was the work of Sanjuan~\\cite{F84}\non possible kinematics and the nine\\footnote{Sanjuan and Yaglom\nboth tacitly assume that both parameters ${\\kappa_1}$ and ${\\kappa_2}$ are normalized.}\nCayley--Klein geometries.   Sanjuan represents each kinematical Lie algebra as\na real matrix subalgebra of $M(2,{\\mathbb{C}})$, where ${\\mathbb{C}}$ denotes the generalized\ncomplex numbers (a description of the generalized complex numbers is given below).\nThis is accomplished using Yaglom's analytic representation of each Caley--Klein\ngeometry as a region of ${\\mathbb{C}}$: for the hyperbolic plane this gives the\nwell-known Poincar\\'{e} disk model.  Sanjuan\nconstructs the Lie algebra for the hyperbolic plane using the standard method,\nstating that this method can be used to obtain the other Lie algebras as well.\nAlso, extensive work has been done by Gromov \\cite{nG90a,nG90b,nG92,nG95,nG96}\non the generalized orthogonal groups $SO(3)$ (which we refer to simply as $SO(3)$),\nderiving representations of the generalized $so(3)$\n(which we refer to simply as $so(3)$) by utilizing the dual\nnumbers as well as the standard complex numbers, where again\nit is tacitly assumed that the parameters ${\\kappa_1}$ and ${\\kappa_2}$ have been normalized.\nAlso, Pimenov has given an axiomatic description of all Cayley--Klein\nspaces in arbitrary dimensions in his paper \\cite{P65} via the dual numbers $ i_k$, $k=1,2,\\dots $,\nwhere $ i_k i_m = i_m i_k \\neq 0$ and $i_k^2=0$.\n\nUnless stated otherwise, we will not assume that the parameters\n${\\kappa_1}$ and ${\\kappa_2}$ have been normalized, as we wish to obtain\ncontractions by simply letting ${\\kappa_1} \\rightarrow 0$ or ${\\kappa_2} \\rightarrow 0$.\n Our goal in this section is to derive representations of $so(3)$\n as real subalgebras of $M(2,{\\mathbb{C}})$, and in the process give a\n conformal model of $S$ as a region of the generalized complex plane ${\\mathbb{C}}$\n along with a hermitian metric, extending what has been done for the\n projective and hyperbolic planes\\footnote{Fjelstad and Gal \\cite{FG01}\n have investigated two-dimensional geometries and physics generated by\n complex numbers from a topological perspective.  Also, see \\cite{CCCZ}.}.\n We feel that it is worthwhile to write\ndown precisely how these representations are obtained in order\nthat our later construction of a Clif\\\/ford algebra is more meaningful.\n\n\nThe f\\\/irst step is to represent the generators\nof $SO(3)$ by M\\\"{o}bius transformations (that is, linear\nfractional transformations) of an appropriately def\\\/ined\nregion in the complex number plane ${\\mathbb{C}}$, where the points\nof $S$ are to be identif\\\/ied with this region.\n\n\\begin{definition}\nBy the complex number plane ${\\mathbb{C}}_{\\kappa}$ we will mean\n$\\{ w = u + i v \\, | \\, (u,v) \\in {\\mathbb{R}}^2 \\ \\mbox{and} \\ i^2 = -\\kappa \\}$ where $\\kappa$ is a real-valued parameter.\n\\end{definition}\n\nThus ${\\mathbb{C}}_{\\kappa}$ refers to the complex numbers,\ndual numbers, or double numbers when $\\kappa$ is\nnormalized to $1$, $0$, or $-1$ respectively (see \\cite{Y79} and \\cite{HH04}).\nOne may check that ${\\mathbb{C}}_{\\kappa}$ is an associative algebra with\na multiplicative unit, but that there are zero divisors when $\\kappa \\leq 0$.\nFor example, if $\\kappa = 0$, then $i$ is a zero-divisor.\nThe reader will note below that $\\frac{1}{i}$ appears in certain\nequations, but that these equations can always be rewritten without\nthe appearance of any zero-divisors in a denominator.\nOne can extend ${\\mathbb{C}}_{\\kappa}$ so that terms like $\\frac{1}{i}$\nare well-def\\\/ined (see \\cite{Y79}).  It is these zero divisors\nthat play a crucial rule in determining the null-cone\nstructure for those Cayley--Klein geometries that are spacetimes.\n\n\\begin{definition}\nHenceforward ${\\mathbb{C}}$ will denote ${\\mathbb{C}}_{{\\kappa_2}}$, as it is\nthe parameter ${\\kappa_2}$ which determines the conformal structure\nof the Cayley--Klein geometry $S$ with parameters ${\\kappa_1}$ and ${\\kappa_2}$.\n\\end{definition}\n\n\n\\begin{theorem}\nThe matrices $\\frac{i}{2} {\\sigma_1}$, $\\frac{i}{2} {\\sigma_2}$, and $\\frac{1}{2i}{\\sigma_3}$\nare generators for the generalized Lie algebra $so(3)$,\nwhere $so(3)$ is represented as a subalgebra of the real matrix algebra $M(2,{\\mathbb{C}})$, where\n\\[\n   {\\sigma_1} =\n    \\left(\n   \\begin{matrix}\n   1 & 0 \\\\\n   0 &  -1\n   \\end{matrix}\n   \\right) , \\qquad\n   {\\sigma_2} =\n    \\left(\n   \\begin{matrix}\n   0 & 1 \\\\\n   {\\kappa_1} &  0\n   \\end{matrix}\n   \\right) \\qquad \\mbox{and} \\qquad\n   {\\sigma_3} =\n    \\left(\n   \\begin{matrix}\n   0 & i \\\\\n   -{\\kappa_1} i &  0\n   \\end{matrix}\n   \\right) .\n\\]\nIn fact, we will show that $\\mathcal{K}$, $\\mathcal{H}$,\nand $\\mathcal{P}$ (the subgroups generated respectively\nby boosts, time and space translations) can be\nrespectively represented by elements of $SL(2,{\\mathbb{C}})$\nof the form $e^{i\\frac{\\theta}{2}{\\sigma_1}}$, $e^{i\\frac{\\alpha}{2}{\\sigma_2}}$, and $e^{\\frac{\\beta}{2i}{\\sigma_3}}$.\n\\end{theorem}\n\n Note that when ${\\kappa_1} =1$ and ${\\kappa_2} = 1$, we recover the\nPauli spin matrices, though my indexing is dif\\\/ferent,\nand there is a sign change as well:  recall that the Pauli spin matrices are typically given as\n\\[\n   {\\sigma_1} =\n    \\left(\n   \\begin{matrix}\n   0 & 1 \\\\\n   1 & 0\n   \\end{matrix}\n   \\right) , \\qquad\n   {\\sigma_2} =\n    \\left(\n   \\begin{matrix}\n   0 & -i \\\\\n   i &  0\n   \\end{matrix}\n   \\right) \\qquad \\mbox{and} \\qquad\n   {\\sigma_3} =\n    \\left(\n   \\begin{matrix}\n   1 & 0 \\\\\n   0 & -1\n   \\end{matrix}\n   \\right) .\n\\]\nWe will refer to ${\\sigma_1}$, ${\\sigma_2}$, and ${\\sigma_3}$\nas given in the statement of Theorem~1 as the generalized Pauli spin matrices.\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=8cm]{McRae-fig4e}\n\\end{center}\n\\caption{The unit sphere $\\Sigma$ and the three complex planes ${\\mathbb{C}}_{{\\kappa_2}}$, ${\\mathbb{C}}_{{\\kappa_1}}$, and ${\\mathbb{C}}_{{\\kappa_1} {\\kappa_2}}$.}\n\\end{figure}\n\n\nThe remainder of this section is devoted to proving the above theorem.\nThe reader may f\\\/ind Fig.~4 helpful.\nThe respective subgroups $\\mathcal{K}$, $\\mathcal{H}$,\nand $\\mathcal{P}$ preserve the $z$, $x$, and $t$ axes\nas well as the ${\\mathbb{C}}_{{\\kappa_2}}$, ${\\mathbb{C}}_{{\\kappa_1}}$, and ${\\mathbb{C}}_{{\\kappa_1} {\\kappa_2}}$\nnumber planes, acting on these planes as rotations.\nAlso, as these groups preserve the unit sphere\n$\\Sigma = \\{ (z, t, x) \\; | \\; z^2 + {\\kappa_1} t^2 + {\\kappa_1} {\\kappa_2} x^2 = 1 \\}$,\nthey preserve the respective intersections of $\\Sigma$ with\nthe ${\\mathbb{C}}_{{\\kappa_2}}$, ${\\mathbb{C}}_{{\\kappa_1}}$, and ${\\mathbb{C}}_{{\\kappa_1} {\\kappa_2}}$ number planes.\nThese intersections are, respectively, circles\nof the form ${\\kappa_1} w\\bar{w} = 1$ (there is no intersection\nwhen ${\\kappa_1} = 0$ or when ${\\kappa_1} < 0$ and ${\\kappa_2} > 0$),\n${\\mathsf{w}}\\bar{{\\mathsf{w}}} = 1$, and ${\\mathfrak{w}}\\bar{{\\mathfrak{w}}} = 1$, where $w$, ${\\mathsf{w}}$,\nand ${\\mathfrak{w}}$ denote elements of ${\\mathbb{C}}_{{\\kappa_2}}$, ${\\mathbb{C}}_{{\\kappa_1}}$,\nand ${\\mathbb{C}}_{{\\kappa_1} {\\kappa_2}}$ respectively.  We will see in the\nnext section how a general element of $SO(3)$ behaves\nin a~manner similar to the generators of $\\mathcal{K}$,\n$\\mathcal{H}$, and $\\mathcal{P}$, utilizing the power of a Clif\\\/ford algebra.\n\nSo we will let the plane $z = 0$ in ${\\mathbb{R}}^3$ represent ${\\mathbb{C}}$\n(recall that ${\\mathbb{C}}$ denotes ${\\mathbb{C}}_{{\\kappa_2}}$).  We may then\nidentify the points of $S$ with a region $\\varsigma$\nof ${\\mathbb{C}}$ by centrally projecting $\\Sigma$ from the point\n$(-1, 0, 0)$ onto the plane $z = 0$, projecting only\nthose points $(z,t,x) \\in \\Sigma$ with non-negative $z$-values.\nThe region $\\varsigma$ may be open or closed or neither, bounded\nor unbounded, depending on the geometry of $S$.\nSuch a construction is well known for both the\nprojective and hyperbolic planes ${\\bf RP}^2$\nand~${\\bf H}^2$ and gives rise to the conformal models\nof these geometries.   We will see later on how the\nconformal structure on ${\\mathbb{C}}$ agrees with that of $S$,\nand then how the simple hermitian metric (see Appendix~B)\n\\[\nds^2 =\n\\frac{dw d\\overline{w}}{\\left( 1 + {\\kappa_1} \\left| w \\right|^2 \\right)^2}\n\\]\ngives the main metric $g_1$ for $S$.  This metric can be\nused to help indicate the general character of the\nregion $\\varsigma$ for each of the nine types of Cayley--Klein geometries,\nas illustrated in Fig.~5.  Note that antipodal points\non the boundary of $\\varsigma$ (if there is a boundary)\nare to be identif\\\/ied.  For absolute-time\nspacetimes (when ${\\kappa_2} = 0$) the subsidiary metric $g_2$ is given by\n\\[\ng_2 = \\frac{dx^2}{\\left( 1 + {\\kappa_1} t_0^2 \\right)^2}\n\\]\nand is def\\\/ined on lines $w = t_0$ of simultaneous events.\nFor all spacetimes, with Here-Now at the origin, the set of zero-divisors gives the null cone for that event.\n\\begin{figure}[ht]\n\\begin{center}\n\\includegraphics[width=12.5cm]{McRae-fig5e}\n\\end{center}\n\\caption{The regions $\\varsigma$.}\n\\end{figure}\n\nVia this identif\\\/ication of points of $S$ with\npoints of $\\varsigma$, transformations of $S$ correspond\nto transformations of $\\varsigma$.  If the real\nparameters ${\\kappa_1}$ and ${\\kappa_2}$ are normalized to the values ${K_1}$ and ${K_2}$ so that\n\\begin{equation*}\nK_i =\n\\begin{cases}\n1,                   &\\text{if $\\kappa_i > 0$},   \\\\\n0,  &\\text{if $\\kappa_i = 0$} ,     \\\\\n-1,   &\\text{if $\\kappa_i < 0$}\n\\end{cases}\n\\end{equation*}\nthen Yaglom~\\cite{Y79} has shown that the linear isometries\nof ${\\mathbb{R}}^3$ (with metric $ds^2 = dz^2 + {K_1} dt^2 + {K_1}{K_2} dx^2$)\nacting on $\\bar{\\Sigma}$ project to those M\\\"{o}bius\ntransformations that preserve $\\varsigma$, and so these M\\\"{o}bius\ntransformations preserve\ncycles\\footnote{Yaglom projects from the point $(z, t, x) =\n (-1, 0, 0)$ onto the plane $z = 1$ whereas we project onto the plane $z = 0$.\n But this hardly matters as cycles are invariant under dilations of ${\\mathbb{C}}$.}:\n a cycle is a curve of constant curvature, corresponding to the\n intersection of a plane in ${\\mathbb{R}}^3$ with $\\bar{\\Sigma}$.\n We would like to show that elements of $SO(3)$ project to M\\\"{o}bius\n transformations if the parameters are not normalized, and then to f\\\/ind\n a realization of $so(3)$ as a real subalgebra of $M(2,{\\mathbb{C}})$.\n\nGiven ${\\kappa_1}$ and ${\\kappa_2}$ we may def\\\/ine a linear isomorphism of ${\\mathbb{R}}^3$ as indicated below.\n\\begin{table}[htdp]\n\\centering\n\\begin{tabular}{l | l | l} \\hline\n${\\kappa_1} \\neq 0, \\; {\\kappa_2} \\neq 0$ & ${\\kappa_1} \\neq 0, \\; {\\kappa_2} = 0$ & ${\\kappa_1} = {\\kappa_2} = 0$ \\\\ \\hline \\hline\n$z \\mapsto z^{\\prime} = z$                                          & $z \\mapsto z^{\\prime} = z$                                   & $z \\mapsto z^{\\prime} = z$ \\\\\n$t \\mapsto t^{\\prime} =  \\frac{1}{\\sqrt{ | {\\kappa_1} | }} t$        & $t \\mapsto t^{\\prime}\n=  \\frac{1    }{\\sqrt{ | {\\kappa_1} | }} t$ & $t \\mapsto t^{\\prime} = t$ \\\\\n\\bsep{1ex}$x \\mapsto x^{\\prime} = \\frac{1}{\\sqrt{ | {\\kappa_1} {\\kappa_2} | }} x$ & $x \\mapsto x^{\\prime} = x$\n                                 & $x \\mapsto x^{\\prime} = x$ \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\n\\noindent\nThis transformation preserves the projection point $(-1, 0, 0)$\nas well as the complex plane $z = 0$, and maps the projective\nquadric $\\bar{\\Sigma}$ for parameters ${K_1}$ and ${K_2}$ to that\nfor ${\\kappa_1}$ and ${\\kappa_2}$, and so gives a correspondence between\nelements of $SO_{{K_1},{K_2}}(3)$ with those of ${SO_{\\ka,\\kb}(3)}$ as well\nas the projections of these elements.   As the M\\\"{o}bius\ntransformations of ${\\mathbb{C}}$ are those transformations that preserve curves of the form\n\\[\n\\mbox{Im} \\frac{(w^{\\prime}_1 - w^{\\prime}_3)(w^{\\prime}_2\n- w^{\\prime})}{(w^{\\prime}_1 - w^{\\prime})(w^{\\prime}_2 - w^{\\prime}_3)} = 0\n\\]\n(where $w^{\\prime}_1$, $w^{\\prime}_2$, and $w^{\\prime}_3$ are three distinct\npoints lying on the cycle), then if this form is invariant under the\ninduced action of the linear isomorphism, then elements of $SO_{{\\kappa_1},{\\kappa_2}}(3)$\nproject to M\\\"{o}bius transformations of $\\varsigma$.  As a point $(z, t, x)$\nis projected to the point $\\left( 0, \\frac{t}{z + 1}, \\frac{x}{z + 1} \\right)$\ncorresponding to the complex number $w = \\frac{1}{z + 1}(t + \\mathcal{I}x) \\in {\\mathbb{C}}_{{K_2}}$,\nif the linear transformation sends $(z, t, x)$ to $(z^{\\prime}, t^{\\prime}, x^{\\prime})$,\nthen it sends $w = \\frac{1}{z + 1}(t + \\mathcal{I}x) \\in {\\mathbb{C}}_{{K_2}}$ to $w^{\\prime}\n= \\frac{1}{z^{\\prime} + 1}(t^{\\prime} + ix^{\\prime}) \\in {\\mathbb{C}}_{{\\kappa_2}} = {\\mathbb{C}}$,\nwhere $\\mathcal{I}^2 = -{K_2}$ and $i^2 = -{\\kappa_2}$.  We can then write that\n\\begin{table}[htdp]\n\\centering\n\\begin{tabular}{l | l | l} \\hline\n${\\kappa_1} \\neq 0, \\; {\\kappa_2} \\neq 0$ & ${\\kappa_1} \\neq 0, \\; {\\kappa_2} = 0$ & ${\\kappa_1} = {\\kappa_2} = 0$ \\\\ \\hline \\hline\n\\tsep{1ex} $w = \\frac{1}{z+1} \\left( t + \\mathcal{I}x \\right) \\mapsto$ &\n$w = \\frac{1}{z+1}  \\left( t + \\mathcal{I}x \\right) \\mapsto$ &\n$w = \\frac{1}{z+1} \\left( t + \\mathcal{I}x \\right) \\mapsto$  \\\\\n$w^{\\prime} = \\frac{1}{z+1} \\frac{1}{\\sqrt{| {\\kappa_1} |}} \\left( t + \\frac{\\mathcal{I}}{\\sqrt{| {\\kappa_2} |}} x  \\right)$ &\n$w^{\\prime} = \\frac{1}{z+1} \\left( \\frac{1}{\\sqrt{| {\\kappa_1} |}} t + \\mathcal{I}x  \\right)$ &\n$ w^{\\prime} = w$ \\bsep{1ex}\\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\n\\noindent And so\n\\[\n\\mbox{Im} \\frac{(w_1 - w_3)(w_2 - w)}{(w_1 - w)(w_2 - w_3)} = 0 \\iff\n\\mbox{Im} \\frac{(w^{\\prime}_1 - w^{\\prime}_3)(w^{\\prime}_2\n- w^{\\prime})}{(w^{\\prime}_1 - w^{\\prime})(w^{\\prime}_2 - w^{\\prime}_3)} = 0,\n\\]\nas can be checked directly, and we then have that elements of $SO(3)$\nproject to M\\\"{o}bius transformations of $\\varsigma$.\n\nThe rotations $e^{\\theta K}$ preserve the complex number plane\n$z = t + ix = 0$ and so correspond simply to the transformations of\n ${\\mathbb{C}}$ given by $w \\mapsto e^{i \\theta}w$, as $e^{i \\theta} = {C_{\\kb}}(\\theta) + i{S_{\\kb}}(\\theta)$,\n keeping in mind that $i^2 = -{\\kappa_2}$.   Now in order to express this rotation as a M\\\"{o}bius\n transformation, we can write\n\\[ w \\mapsto \\frac{e^{\\frac{\\theta}{2}i}w + 0}{0w + e^{-\\frac{\\theta}{2}i}} .\\]\nSince there is a group homomorphism from the subgroup of M\\\"{o}bius\ntransformations correspon\\-ding to $SO(3)$ to the group $M(2, {\\mathbb{C}})$ of\n $2 \\times 2$ matrices with entries in ${\\mathbb{C}}$, this transformation being def\\\/ined by\n\\[  \\frac{aw + b}{cw + d} \\mapsto\n \\left(\n   \\begin{matrix}\n   a & b \\\\\n   c &  d\n   \\end{matrix}\n   \\right),\n\\]\neach M\\\"{o}bius transformation is covered by two elements of $SL(2, {\\mathbb{C}})$.\nSo the rotations $e^{\\theta K}$ correspond to the matrices\n\\[\n   \\pm \\left(\n   \\begin{matrix}\n   e^{\\frac{\\theta}{2}i} & 0 \\\\\n   0 &  e^{-\\frac{\\theta}{2}i}\n   \\end{matrix}\n   \\right) =\n  \\pm e^{ \\frac{\\theta}{2}i\n    \\left(\n   \\begin{matrix}\n   1 & 0 \\\\\n   0 &  -1\n   \\end{matrix}\n   \\right) .\n   }\n\\]\nFor future reference let us now def\\\/ine {\\samepage\n\\[\n   {\\sigma_1} \\equiv\n    \\left(\n   \\begin{matrix}\n   1 & 0 \\\\\n   0 &  -1\n   \\end{matrix}\n   \\right) ,\n\\]\nwhere $\\frac{i}{2}{\\sigma_1}$ is then an element of the Lie algebra $so(3)$. }\n\nWe now wish to see which elements of $SL(2, {\\mathbb{C}})$ correspond to the\n motions $e^{\\alpha H}$ and $e^{\\beta P}$.   The $x$-axis,\n the $zt$-coordinate plane, and the unit sphere $\\Sigma$,\n are all preserved by $e^{\\alpha H}$.  So the $zt$-coordinate\n plane is given the complex structure ${\\C_{\\ka}} = \\{ {\\mathsf{w}} = z + it \\, | \\, i^2 = - {\\kappa_1} \\}$,\n for then the unit circle ${\\mathsf{w}} \\overline{{\\mathsf{w}}} = 1$ gives the\n intersection of $\\Sigma$ with ${\\C_{\\ka}}$, and the transformation\n induced on ${\\C_{\\ka}}$ by $e^{\\alpha H}$ is simply given by ${\\mathsf{w}} \\mapsto e^{i \\alpha} {\\mathsf{w}}$.\n Similarly the transformation induced by $e^{\\beta P}$ on\n ${\\C_{\\ka \\kb}} = \\{ {\\mathfrak{w}} = z + ix \\; | \\; i^2 = -{\\kappa_1} {\\kappa_2} \\}$ is given by ${\\mathfrak{w}} \\mapsto e^{i \\beta} {\\mathfrak{w}}$.\n\nIn order to explicitly determine the projection of the rotation\n${\\mathsf{w}} \\mapsto e^{i \\alpha} {\\mathsf{w}}$ of the unit circle in~${\\C_{\\ka}}$ and also\nthat of the rotation ${\\mathfrak{w}} \\mapsto e^{i \\beta} {\\mathfrak{w}}$ of the unit circle in ${\\C_{\\ka \\kb}}$,\nnote that the projection point $(z, t, x) = (-1, 0, 0)$ lies in either\nunit circle and that projection sends a point on the unit circle\n(save for the projection point itself) to a point on the imaginary axis as follows:\n\\[ {\\mathsf{w}} = e^{i \\phi}  \\mapsto i{T_{\\ka}}\\left(\\frac{\\phi}{2}\\right),\\qquad\n {\\mathfrak{w}} =  e^{i \\phi} \\mapsto i{T_{\\ka \\kb}}\\left(\\frac{\\phi}{2}\\right) \\]\n(where $T_{\\kappa}$ is the tangent function) for\n\\[  T_{\\kappa}\\left(\\frac{\\mu}{2}\\right) =   \\frac{S_{\\kappa}(\\mu)}{C_{\\kappa}(\\mu) + 1}  ,\\]\nnoting that a point $a + ib$ on the unit circle $w \\overline{w}$ of the\ncomplex plane $C_{\\kappa}$ can be written as $a + ib = e^{i \\psi} = {C_{\\kappa}}(\\psi) + i{S_{\\kappa}}(\\psi)$.\nSo the rotations $e^{\\alpha H}$ and $e^{\\beta P}$ induce the respective transformations\n\\[ i{T_{\\ka}}\\left(\\frac{\\phi}{2}\\right) \\mapsto i{T_{\\ka}}\\left(\\frac{\\phi + \\alpha}{2}\\right),\\qquad\ni{T_{\\ka \\kb}}\\left(\\frac{\\phi}{2}\\right) \\mapsto i{T_{\\ka \\kb}}\\left(\\frac{\\phi + \\beta}{2}\\right)  \\]\non the imaginary axes.  We know that such transformations of either imaginary or\nreal axes can be extended to M\\\"{o}bius transformations, and in fact uniquely\ndetermine such M\\\"{o}bius maps.\nFor example, if ${\\mathsf{w}} = i{T_{\\ka}} \\left( \\frac{\\phi}{2} \\right)$, then we have that\n\\[\n{\\mathsf{w}} \\mapsto \\frac{{\\mathsf{w}} + i {T_{\\ka}} \\left( \\frac{\\alpha}{2} \\right)}\n{1 - \\frac{ {\\kappa_1} {\\mathsf{w}}}{i} {T_{\\ka}} \\left( \\frac{\\alpha}{2} \\right)}\n\\]\nor\n\\[\n{\\mathsf{w}} \\mapsto \\frac{{C_{\\ka}} \\left( \\frac{\\alpha}{2} \\right) {\\mathsf{w}}\n+ i {S_{\\ka}} \\left( \\frac{\\alpha}{2} \\right)}{-\\frac{{\\kappa_1}}{i}\n{S_{\\ka}} \\left( \\frac{\\alpha}{2} \\right){\\mathsf{w}} + {C_{\\ka}} \\left( \\frac{\\alpha}{2} \\right)}\n\\]\nwith corresponding matrix representation\n\\[\n\\pm \\left(\n\\begin{matrix}\n{C_{\\ka}} \\left( \\frac{\\alpha}{2} \\right) &\ni {S_{\\ka}} \\left( \\frac{\\alpha}{2} \\right) \\vspace{1mm}\\\\\ni {S_{\\ka}} \\left( \\frac{\\alpha}{2} \\right) &\n{C_{\\ka}} \\left(  \\frac{\\alpha}{2} \\right)\n\\end{matrix}\n\\right)\n\\]\nin $SL(2, {C_{\\ka}})$, where we have applied the\ntrigonometric identity\\footnote{For Minkowski spacetimes\nthis trigonometric identity is the well-known formula for the addition of rapidities.}\n\\[ T_{\\kappa}(\\mu \\pm \\psi) = \\frac{T_{\\kappa}(\\mu) \\pm T_{\\kappa}(\\psi)}\n{1 \\mp \\kappa T_{\\kappa}(\\mu) T_{\\kappa}(\\psi)}. \\]\nHowever, it is not these M\\\"{o}bius transformations that we are after,\nbut those corresponding transformations of ${\\mathbb{C}}$.\n\nNow a transformation of the imaginary axis (the $x$-axis)\nof ${\\C_{\\ka \\kb}}$ corresponds to a transformation of the imaginary\naxis of ${\\mathbb{C}}$ (also the $x$-axis) while a transformation of\nthe imaginary axis of ${\\C_{\\ka}}$ (the $t$-axis) corresponds to\na transformation of the real axis of ${\\mathbb{C}}$ (also the $t$-axis).\nFor this reason, values on the $x$-axis, which are imaginary for\nboth the ${\\mathbb{C}}_{{\\kappa_1} {\\kappa_2}}$ as well as the ${\\mathbb{C}}$ plane, correspond as\n\\[\n i {T_{\\ka \\kb}} \\left( \\frac{\\phi}{2} \\right) = i \\frac{1}{\\sqrt{{\\kappa_1}}} {T_{\\kb}} \\left( \\sqrt{{\\kappa_1}} \\frac{\\phi}{2} \\right)\n\\]\nif ${\\kappa_1} > 0$,\n\\[\n i {T_{\\ka \\kb}} \\left( \\frac{\\phi}{2} \\right) =\n i \\frac{1}{\\sqrt{-{\\kappa_1}}} T_{-{\\kappa_2}} \\left( \\sqrt{-{\\kappa_1}} \\frac{\\phi}{2} \\right)\n\\]\nif ${\\kappa_1} < 0$, and\n\\[\n i {T_{\\ka \\kb}} \\left( \\frac{\\phi}{2} \\right) = i  \\left( \\frac{\\phi}{2} \\right)\n\\]\nif ${\\kappa_1} = 0$, as can be seen by examining the power series\nrepresentation for $T_{\\kappa}$.  The situation for the rotation $e^{i \\alpha}$ is similar.\nWe can then compute the elements of $SL(2,{\\mathbb{C}})$ corresponding\nto $e^{\\alpha H}$ and $e^{\\beta P}$ as given in tables $13$ and $14$\nin Appendix~C.  In all cases we have the simple result\nthat those elements of $SL(2, {\\mathbb{C}})$ corresponding to $e^{\\alpha H}$\ncan be written as $e^{\\frac{\\alpha}{2i} {\\sigma_3}}$ and those for $e^{\\beta P}$ as $e^{i \\frac{\\beta}{2} {\\sigma_2}}$, where\n\\[\n   {\\sigma_2} \\equiv\n    \\left(\n   \\begin{matrix}\n   0 & 1 \\\\\n   {\\kappa_1} &  0\n   \\end{matrix}\n   \\right) \\qquad \\mbox{and} \\qquad\n   {\\sigma_3} \\equiv\n    \\left(\n   \\begin{matrix}\n   0 & i \\\\\n   -{\\kappa_1} i &  0\n   \\end{matrix}\n   \\right) .\n\\]\nThus $\\frac{i}{2} {\\sigma_1}$, $\\frac{i}{2} {\\sigma_2}$, and $\\frac{1}{2i}{\\sigma_3}$\nare generators for the generalized Lie algebra $so(3)$, a subalgebra of the real matrix algebra $M(2,{\\mathbb{C}})$.\n\n\n\n\n\\section[The Clifford algebra $Cl_3$]{The Clif\\\/ford algebra $\\boldsymbol{Cl_3}$}\n\n\\begin{definition}\nLet $Cl_3$ be the 8-dimensional real Clif\\\/ford algebra that\nis identif\\\/ied with $M(2,{\\mathbb{C}})$ as indicated by Table~11,\nwhere ${\\mathbb{C}}$ denotes the generalized complex numbers ${\\mathbb{C}}_{{\\kappa_2}}$.\nHere we identify the scalar $1$ with the identity matrix and the\nvolume element $i$ with the $2 \\times 2$ identity matrix multiplied\nby the complex scalar $i$: in this case $\\frac{1}{i}{\\sigma_3}$ can be thought of as the $2 \\times 2$ matrix  $\\left(\n   \\begin{matrix}\n   0 & 1 \\\\\n   -{\\kappa_1}  &  0\n   \\end{matrix}\n   \\right)$.\nWe will also identify the generalized\nPaul spin matrices ${\\sigma_1}$, ${\\sigma_2}$, and ${\\sigma_3}$ with the\nvectors $\\hat{i} = \\langle 1, 0, 0 \\rangle$, $\\hat{j} = \\langle 0, 1, 0 \\rangle$,\nand $\\hat{k} = \\langle 0, 0, 1 \\rangle$ respectively of\nthe vector space ${\\mathbb{R}}^3 = \\{ (z, t, x) \\}$ given the Cayley--Klein\ninner product\\footnote{We will use the symbol $\\hat{v}$ to\ndenote a vector $v$ of length one under the standard inner product.}.\n\\end{definition}\n\\begin{table}[t]\n\\centering\n\\caption{The basis elements for $Cl_3$.}\n\\vspace{1mm}\n\n\\begin{tabular}{r c l} \\hline\nSubspace of & & with basis \\\\ \\hline \\hline\nscalars & ${\\mathbb{R}}$ & 1 \\\\\nvectors & ${\\mathbb{R}}^3$ & ${\\sigma_1}, {\\sigma_2}, {\\sigma_3}$ \\\\\nbivectors & $\\bigwedge^2 {\\mathbb{R}}^3$ & $i {\\sigma_1}, i {\\sigma_2}, \\frac{1}{i} {\\sigma_3}$ \\\\\nvolume elements & $\\bigwedge^3 {\\mathbb{R}}^3$ & $i$ \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\n\\begin{proposition}\nLet $Cl_3$ be the Clifford algebra given by Definition~{\\rm 3}.\n\\begin{itemize}\\itemsep=0pt\n\\item[(i)]  The Clifford product $\\sigma_i^2$ gives the square\nof the length of the vector $\\sigma_i$ under the Cayley--Klein inner product.\n\n\\item[(ii)]  The center Cen($Cl_3$) of $Cl_3$ is given by ${\\mathbb{R}} \\oplus \\bigwedge^3 {\\mathbb{R}}^3$,\nthe subspace of scalars and volume elements.\n\n\\item[(iii)]  The generalized Lie algebra $so(3)$ is isomorphic to the space of bivectors $\\bigwedge^2 {\\mathbb{R}}^3$, where\n\\[\nH =  \\frac{1}{2i} {\\sigma_3} , \\qquad P = \\frac{i}{2} {\\sigma_2} , \\qquad \\mbox{and} \\qquad K = \\frac{i}{2} {\\sigma_1} .\n\\]\n\\item[(iv)]  If $\\hat{n} = \\langle n^1, n^2, n^3 \\rangle$\nand $\\vec{\\sigma} = \\langle i{\\sigma_1}, i{\\sigma_2}, {\\sigma_3}\/i \\rangle$,\nthen we will let ${\\hat{n} \\cdot \\vec{\\sigma}}$ denote the bivector $n^1 i {\\sigma_1} + n^2 i {\\sigma_2} + n^3 \\frac{1}{i} {\\sigma_3}$.\nThis bivector is simple, and the parallel vectors $i{\\hat{n} \\cdot \\vec{\\sigma}}$ and $\\frac{1}{i} {\\hat{n} \\cdot \\vec{\\sigma}}$\nare perpendicular to any plane element represented by ${\\hat{n} \\cdot \\vec{\\sigma}}$.\nLet $\\eta$ denote the line through the origin that is determined by $i{\\hat{n} \\cdot \\vec{\\sigma}}$ or $\\frac{1}{i} {\\hat{n} \\cdot \\vec{\\sigma}}$.\n\\item[(v)] The generalized Lie group $SO(3)$ is also represented within $Cl_3$,\nfor if $a$ is the vector $a^1{\\sigma_1} + a^2{\\sigma_2} + a^3{\\sigma_3}$, then the linear\ntransformation of ${\\mathbb{R}}^3$ defined by the inner automorphism\n\\[\na \\mapsto e^{-\\frac{\\phi}{2} \\hat{n} \\cdot \\vec{\\sigma}} a \\, e^{\\frac{\\phi}{2} \\hat{n} \\cdot \\vec{\\sigma}}\n\\]\nfaithfully represents an element of $SO(3)$ as it preserves vector lengths given by the Cayley--Klein\ninner product, and is in fact a rotation, rotating the vector\n$\\langle a^1, a^2, a^3 \\rangle$ about the axis $\\eta$ through the angle $\\phi$.\nIn this way we see that the spin group is generated by the elements\n\\[\n e^{\\frac{\\theta}{2} i{\\sigma_1}}, \\qquad e^{\\frac{\\beta}{2} i{\\sigma_2}},\\qquad\n  \\mbox{and} \\qquad e^{\\frac{\\alpha}{2i} {\\sigma_3}}  .\n\\]\n\\item[(vi)]   Bivectors ${\\hat{n} \\cdot \\vec{\\sigma}}$ act as imaginary units\nas well as generators of rotations in the oriented planes they represent.\nLet $\\varkappa$ be the scalar $- \\left( {\\hat{n} \\cdot \\vec{\\sigma}} \\right)^2 $. Then if $a$ lies\nin an oriented plane determined by the bivector ${\\hat{n} \\cdot \\vec{\\sigma}}$,\nwhere this plane is given the complex structure of ${\\mathbb{C}}_{\\varkappa}$,\nthen $e^{-\\frac{\\phi}{2} {\\hat{n} \\cdot \\vec{\\sigma}}} a  e^{\\frac{\\phi}{2} {\\hat{n} \\cdot \\vec{\\sigma}}}$ is simply\nthe vector $\\langle a^1, a^2, a^3 \\rangle$ rotated by the angle $\\phi$\nin the complex plane ${\\mathbb{C}}_{\\varkappa}$, where $\\iota^2 = -\\varkappa$.\nSo this rotation is given by unit complex multiplication.\n\\end{itemize}\n\\end{proposition}\n\nThe goal of this section is to prove Proposition~1.\nWe can easily compute the following:\n\\begin{gather*}\n{\\sigma_1}^2 = 1, \\qquad {\\sigma_2}^2 = {\\kappa_1} , \\qquad {\\sigma_3}^2 = {\\kappa_1} {\\kappa_2} ,\n\\\\\n{\\sigma_3}{\\sigma_2}  =  -{\\sigma_2}{\\sigma_3}  =  {\\kappa_1} i {\\sigma_1}, \\qquad\n{\\sigma_1}{\\sigma_3}  =  -{\\sigma_3}{\\sigma_1}  =  i{\\sigma_2}, \\\\\n{\\sigma_1}{\\sigma_2}  =  -{\\sigma_2}{\\sigma_1}  =  \\frac{1}{i} {\\sigma_3} ,\n\\qquad\n{\\sigma_1} {\\sigma_2} {\\sigma_3} = -{\\kappa_1} i.\n\\end{gather*}\nRecalling that ${\\mathbb{R}}^3$ is given the Cayley--Klein inner product,\nwe see that $\\sigma_i^2$ gives the square of the length of the vector $\\sigma_i$.\nNote that when ${\\kappa_1} = 0$, $Cl_3$ is not generated by the vectors.\nCen($Cl_3$) of $Cl_3$ is given by ${\\mathbb{R}} \\oplus \\bigwedge^3 {\\mathbb{R}}^3$,\nand we can check directly that if\n\\[\nH \\equiv  \\frac{1}{2i} {\\sigma_3} , \\qquad\n P \\equiv \\frac{i}{2} {\\sigma_2} , \\qquad \\mbox{and} \\qquad K \\equiv \\frac{i}{2} {\\sigma_1} ,\n\\]\nthen we have the following commutators:\n\\begin{gather*}\n\\left[ H, P \\right]     =  HP - PH        =  \\frac{1}{4}\\left( {\\sigma_3}{\\sigma_2} - {\\sigma_2}{\\sigma_3} \\right)    =\n\\frac{{\\kappa_1} i {\\sigma_1}}{2}  =  {\\kappa_1} K ,\\\\\n\\left[ K, H \\right]  =  KH - HK  =  \\frac{1}{4}\\left( {\\sigma_1}{\\sigma_3} - {\\sigma_3}{\\sigma_1} \\right)    = \\frac{i {\\sigma_2}}{2}       =  P, \\\\\n\\left[ K, P \\right]  = KP - PK  =  \\frac{i^2}{4}\\left( {\\sigma_1}{\\sigma_2} - {\\sigma_2}{\\sigma_1} \\right)  =\n\\frac{i {\\sigma_3}}{2}       =  - {\\kappa_2} H.\n\\end{gather*}\nSo the Lie algebra $so(3)$ is isomorphic to the space of bivectors $\\bigwedge^2 {\\mathbb{R}}^3$.\n\nThe product of two vectors $a = a^1{\\sigma_1} + a^2{\\sigma_2}\n+ a^3{\\sigma_3}$ and $b = b^1{\\sigma_1} + b^2{\\sigma_2} + b^3{\\sigma_3}$ in $Cl_3$\ncan be expressed as $ab = a \\cdot b + a \\wedge b = \\frac{1}{2}(ab + ba)\n+ \\frac{1}{2}(ab - ba)$, where $a \\cdot b = \\frac{1}{2}(ab + ba)\n= a^1 b^1 + {\\kappa_1} a^2 b^2 + {\\kappa_1} {\\kappa_2} a^3 b^3$ is the Cayley--Klein inner product and the wedge product is given by\n\\[\na \\wedge b = \\frac{1}{2}(ab - ba) =\n\\left|\n   \\begin{matrix}\n   -{\\kappa_1} i {\\sigma_1} & -i {\\sigma_2} & \\frac{1}{i} {\\sigma_3} \\\\\n   a^1         & a^2    & a^3 \\\\\n   b^1         & b^2    & b^3\n   \\end{matrix}\n   \\right|,\n\\]\nso that $ab$ is the sum of a scalar and a bivector: here $| \\star |$ denotes the usual $3 \\times 3$ determinant.\n\nBy the properties of the determinant, if $e \\wedge f = g \\wedge h$\nand ${\\kappa_1} \\neq 0$, then the vectors $e$ and $f$\nspan the same oriented plane as the vectors $g$ and $h$.\nWhen ${\\kappa_1} = 0$ the bivector ${\\hat{n} \\cdot \\vec{\\sigma}}$ is no longer simple in the usual way.\nFor example, for the Galilean kinematical group (aka the Heisenberg group)\nwhere ${\\kappa_1} = 0$ and ${\\kappa_2} = 0$, we have that both ${\\sigma_1} \\wedge {\\sigma_3}\n= i {\\sigma_2}$ and $\\left( {\\sigma_1} + {\\sigma_2} \\right) \\wedge {\\sigma_3} = i {\\sigma_2}$,\nso that the bivector $i {\\sigma_2}$ represents plane elements that do\nno all lie in the same plane\\footnote{There is some interesting asymmetry for Galilean spacetime,\nin that the perpendicular to a timelike geodesic through a given point is uniquely\ndef\\\/ined as the lightlike geodesic that passes through that point,\nand this lightlike geodesic then has no unique perpendicular, since\nall timelike geodesics are perpendicular to it.}.  Recalling that ${\\sigma_1}$, ${\\sigma_2}$,\nand ${\\sigma_3}$ correspond to the vectors $\\hat{i}$, $\\hat{j}$, and $\\hat{k}$\nrespectively, we observe that the subgroup $\\mathcal{P}$ of the\nGalilean group f\\\/ixes the $t$-axis and preserves both of these planes,\ninducing the same kind of rotation upon each of them:  for the plane spanned by $\\hat{i}$ and $\\hat{k}$ we have that\n\\[\n e^{\\beta P}:\n     \\left(\n       \\begin{matrix}\n           \\hat{i} \\\\\n           \\hat{k}\n        \\end{matrix}\n      \\right)\n    \\mapsto\n      \\left(\n         \\begin{matrix}\n            \\hat{i} + \\beta \\hat{k} \\\\\n            \\hat{k}\n         \\end{matrix}\n       \\right)\n \\]\n while for the plane spanned by $\\hat{i} + \\hat{j}$ and $\\hat{k}$ we have that\n\\[\n e^{\\beta P}:\n     \\left(\n       \\begin{matrix}\n           \\hat{i} + \\hat{j} \\\\\n           \\hat{k}\n        \\end{matrix}\n      \\right)\n    \\mapsto\n      \\left(\n         \\begin{matrix}\n            \\hat{i} + \\hat{j} + \\beta \\hat{k} \\\\\n            \\hat{k}\n         \\end{matrix}\n       \\right).\n \\]\nIf we give either plane the complex structure of the dual\nnumbers so that $i^2 = 0$, then the rotation is given by\nsimply multiplying vectors in the plane by the unit complex number $e^{\\beta i}$.\nWe will see below that this kind of construction holds generally.\n\nWhat we need for our construction below is that any bivector\ncan be meaningfully expressed as $e \\wedge f$ for some vectors $e$ and $f$,\nso that the bivector represents at least one plane element: we will discuss\nthe meaning of the magnitude and orientation of the plane element at the end of the section.\nIf the bivector represents multiple plane elements spanning distinct planes, so much the better.\nIf $\\hat{n} = \\langle n^1, n^2, n^3 \\rangle$ and $\\vec{\\sigma} = \\langle i{\\sigma_1}, i{\\sigma_2}, {\\sigma_3}\/i \\rangle$,\nthen we will let ${\\hat{n} \\cdot \\vec{\\sigma}}$ denote the bivector $B = n^1 i {\\sigma_1} + n^2 i {\\sigma_2} + n^3 \\frac{1}{i} {\\sigma_3}$.  Now if\n\\begin{gather*}\na  = n^1 {\\sigma_3} + {\\kappa_1} n^3 {\\sigma_1} , \\qquad\nb  = -n^1 {\\sigma_2} + {\\kappa_1} n^2 {\\sigma_1},\\qquad\nc  = n^3 {\\sigma_2} + n^2 {\\sigma_3},\n\\end{gather*}\nthen\n\\begin{gather*}\na \\wedge c  = {\\kappa_1} n^3 {\\hat{n} \\cdot \\vec{\\sigma}}, \\qquad\nb \\wedge a  = {\\kappa_1} n^1 {\\hat{n} \\cdot \\vec{\\sigma}}, \\qquad\nb \\wedge c  = {\\kappa_1} n^2 {\\hat{n} \\cdot \\vec{\\sigma}},\n\\end{gather*}\nwhere at least one of the bivectors $n^i {\\hat{n} \\cdot \\vec{\\sigma}}$ is non-zero as ${\\hat{n} \\cdot \\vec{\\sigma}}$ is non-zero.\nIf ${\\kappa_1} = 0$ and $n^1 = 0$, then ${\\sigma_1} \\wedge c = {\\hat{n} \\cdot \\vec{\\sigma}}$.\nHowever, if both ${\\kappa_1} = 0$ and $n^1 \\neq 0$, then it is impossible\nto have $e \\wedge f = {\\hat{n} \\cdot \\vec{\\sigma}}$:  in this context we may simply replace\nthe expression ${\\hat{n} \\cdot \\vec{\\sigma}}$ with the expression ${\\sigma_3} \\wedge {\\sigma_2}$ whenever\n${\\kappa_1} = 0$ and $n^1 \\neq 0$ (as we will see at the end of this section,\nwe could just as well replace ${\\hat{n} \\cdot \\vec{\\sigma}}$ with any non-zero multiple of ${\\sigma_3} \\wedge {\\sigma_2}$).\nThe justif\\\/ication for this is given by letting ${\\kappa_1} \\rightarrow 0$, for then\n\\[\ne \\wedge f = \\left( \\sqrt{|{\\kappa_1}|} n^2 {\\sigma_1} - \\frac{n^1}{\\sqrt{|{\\kappa_1}|}} {\\sigma_2} \\right)\n\\wedge \\left( \\sqrt{|{\\kappa_1}|} \\frac{n^3}{n^1}{\\sigma_1} + \\frac{1}{\\sqrt{|{\\kappa_1}|}} {\\sigma_3} \\right) = {\\hat{n} \\cdot \\vec{\\sigma}}\n\\]\nshows that the plane spanned by the vectors $e$ and $f$ tends to the $xt$-coordinate plane.\n We will see below how each bivector ${\\hat{n} \\cdot \\vec{\\sigma}}$ corresponds to an element of $SO(3)$\n that preserves any oriented plane corresponding to ${\\hat{n} \\cdot \\vec{\\sigma}}$: in the case where\n ${\\kappa_1} = 0$ and $n^1 \\neq 0$, we will then have that this element preserves\n the $tx$-coordinate plane, which is all that we require.\n\nIt is interesting to note that the parallel vectors $i(a \\wedge b)$\nand $\\frac{1}{i} (a \\wedge b)$ (when def\\\/ined) are perpendicular to both $a$ and $b$\nwith respect to the Cayley--Klein inner product, as can be checked directly.\nHowever, due to the possible degeneracy of the Cayley--Klein inner product,\nthere may not be a unique direction that is perpendicular to any given plane.\nThe vector $i {\\hat{n} \\cdot \\vec{\\sigma}} = -{\\kappa_2} n^1 {\\sigma_1} - {\\kappa_2} n^2 {\\sigma_2} + n^3 {\\sigma_1}$ is non-zero\nand perpendicular to any plane element corresponding to ${\\hat{n} \\cdot \\vec{\\sigma}}$ except when\nboth ${\\kappa_2} = 0$ and $n^3 = 0$, in which case $i {\\hat{n} \\cdot \\vec{\\sigma}}$ is the zero vector.\nIn this last case the vector $\\frac{1}{i} {\\hat{n} \\cdot \\vec{\\sigma}} = n^1 {\\sigma_1} + n^2 {\\sigma_2}$ gives a non-zero\nnormal vector.  In either case, let $\\eta$ denote the axis through the origin that\ncontains either of these normal vectors.\n\nBefore we continue, let us reexamine those elements of $SO(3)$\nthat generate the subgroups $\\mathcal{K}$,~$\\mathcal{P}$, and $\\mathcal{H}$.\nHere the respective axes of rotation (parallel to ${\\sigma_1}$, ${\\sigma_2}$,\nand ${\\sigma_3}$) for the generators $e^{\\theta K}$, $e^{\\beta P}$,\nand $e^{\\alpha H}$ are given by $\\eta$, where ${\\hat{n} \\cdot \\vec{\\sigma}}$ is given\nby $i {\\sigma_1}$ (or ${\\sigma_3} \\wedge {\\sigma_2}$ by convention), $i {\\sigma_2} = {\\sigma_1} \\wedge {\\sigma_3}$,\nand $\\frac{1}{i} {\\sigma_3} = {\\sigma_1} \\wedge {\\sigma_2}$.\nThese plane elements are preserved under the respective rotations.\nIn fact, for each of these planes the rotations are given simply\nby multiplication by a~unit complex number, as the $zt$-coordinate\nplane is identif\\\/ied with ${\\C_{\\ka}}$, the $zx$-coordinate plane with ${\\C_{\\ka \\kb}}$,\nand the $tx$-coordinate plane with ${\\C_{\\kb}}$ as indicated in Fig.~4.\nNote that the basis bivectors act as imaginary units in $Cl_3$ since\n\\[\n\\left( \\frac{1}{i} {\\sigma_3} \\right)^2 = -{\\kappa_1}, \\qquad\n\\left( i {\\sigma_2} \\right)^2 = -{\\kappa_1} {\\kappa_2}, \\qquad \\mbox{and} \\qquad\n\\left( i {\\sigma_1} \\right)^2 = -{\\kappa_2}.\n\\]\n\nThe product of a vector $a$ and a bivector $B$ can be written as\n$aB = a \\dashv B + a \\wedge B = \\frac{1}{2}(aB - Ba) + \\frac{1}{2}(aB + Ba)$\nso that $aB$ is the sum of a vector $a \\dashv B$ (the left contraction of\n $a$ by~$B$) and a volume element $a \\wedge B$.  Let $B = b \\wedge c$ for some vectors $b$ and $c$.  Then\n\\[\n2a \\dashv (b \\wedge c)  = a (b \\wedge c) - (b \\wedge c)a\n                                      = \\frac{1}{2} a ( bc - cb) - \\frac{1}{2}(bc - cb)a\n\\]\nso that\n\\begin{gather*}\n4a \\dashv (b \\wedge c)  = cba + abc - acb - bca \\\\\n \\phantom{4a \\dashv (b \\wedge c)}{} = c(b \\cdot a + b \\wedge a) + (a \\cdot b + a \\wedge b)c -\n                                       (a \\cdot c + a \\wedge c)b - b(c \\cdot a + c \\wedge a) \\\\\n \\phantom{4a \\dashv (b \\wedge c)}{}  = 2(b \\cdot a)c - 2(c \\cdot a)b +\n                                      c(b \\wedge a) + (a \\wedge b)c - (a \\wedge c)b - b(c \\wedge a) \\\\\n\\phantom{4a \\dashv (b \\wedge c)}{} = 2(b \\cdot a)c - 2(c \\cdot a)b +\n                                      c(b \\wedge a) - (b \\wedge a)c + b(a \\wedge c) - (a \\wedge c)b \\\\\n\\phantom{4a \\dashv (b \\wedge c)}{} =  2(b \\cdot a)c - 2(c \\cdot a) b + 2 \\left[ c \\dashv (b \\wedge a) + b \\dashv (a\n                                             \\wedge c) \\right] \\\\\n\\phantom{4a \\dashv (b \\wedge c)}{} =  2(b \\cdot a)c - 2(c \\cdot a) b - 2 a \\dashv (c \\wedge b) \\\\\n\\phantom{4a \\dashv (b \\wedge c)}{} =  2(b \\cdot a)c - 2(c \\cdot a) b + 2 a \\dashv (b \\wedge c)\n\\end{gather*}\nwhere we have used the Jacobi identity\n\\[ c \\dashv (b \\wedge a) + b \\dashv (a \\wedge c) + a \\dashv (c \\wedge b) = 0, \\]\nrecalling that $M(2,{\\mathbb{C}})$ is a matrix algebra where the commutator is given by left contraction.  Thus\n\\begin{gather*}\n2a \\dashv (b \\wedge c)  = 2(b \\cdot a)c - 2(c \\cdot a)b\n\\end{gather*}\nand so\n\\[\na \\dashv (b \\wedge c) = (a \\cdot b) c - (a \\cdot c)b.\n\\]\nSo the vector $a \\dashv B$ lies in the plane determined by the plane element $b \\wedge c$.\nBecause of the possible degeneracy of the Cayley--Klein metric,\nit is possible for a non-zero vector $b$ that $b \\dashv (b \\wedge c) = 0$.\n\nWe will show that if $a$ is the vector\n$a^1{\\sigma_1} + a^2{\\sigma_2} + a^3{\\sigma_3}$, then the linear transformation of ${\\mathbb{R}}^3$ def\\\/ined by\n\\[\na \\mapsto e^{-\\frac{\\phi}{2} \\hat{n} \\cdot \\vec{\\sigma}} a \\, e^{\\frac{\\phi}{2} \\hat{n} \\cdot \\vec{\\sigma}}\n\\]\nfaithfully represents an element of $SO(3)$\n(and all elements are thus represented).\nIn this way we see that the spin group is generated by the elements\n\\[\n e^{\\frac{\\theta}{2} i{\\sigma_1}}, e^{\\frac{\\beta}{2} i{\\sigma_2}}, \\qquad \\mbox{and} \\qquad e^{\\frac{\\alpha}{2i} {\\sigma_3}}.\n\\]\n\n\nFirst, let us see how, using this construction, the vectors ${\\sigma_1}$, ${\\sigma_2}$,\nand ${\\sigma_3}$ (and hence the bivectors $i {\\sigma_1}$, $i {\\sigma_2}$, and $\\frac{1}{i} {\\sigma_3}$)\ncorrespond to rotations of the coordinate axes (and hence coordinate planes)\ngiven by $e^{\\theta K}$, $e^{\\beta P}$, and $e^{\\alpha H}$ respectively.\nSince\n\\begin{gather*}\ne^{\\frac{\\theta}{2} i {\\sigma_1}}  = {C_{\\kb}} \\left( \\frac{\\theta}{2} \\right) + i {S_{\\kb}} \\left( \\frac{\\theta}{2} \\right) {\\sigma_1},\n\\qquad\ne^{\\frac{\\beta}{2} i {\\sigma_2}}  = {C_{\\ka \\kb}} \\left( \\frac{\\beta}{2} \\right) + i {S_{\\ka \\kb}} \\left( \\frac{\\beta}{2} \\right) {\\sigma_2}, \\\\\ne^{\\frac{\\alpha}{2i}  {\\sigma_3}}  = {C_{\\ka}} \\left( \\frac{\\alpha}{2} \\right) + \\frac{1}{i} {S_{\\ka}} \\left( \\frac{\\alpha}{2} \\right) {\\sigma_3}\n\\end{gather*}\nand\n\\begin{gather*}\n2{C_{\\kappa}} \\left( \\frac{\\phi}{2} \\right) {S_{\\kappa}} \\left( \\frac{\\phi}{2} \\right)                      = {S_{\\kappa}} (\\phi),\\qquad\n{C_{\\kappa}}^2 \\left( \\frac{\\phi}{2} \\right) - \\kappa {S_{\\kappa}}^2 \\left( \\frac{\\phi}{2} \\right)   = {C_{\\kappa}} (\\phi), \\\\\n{C_{\\kappa}}^2 \\left( \\frac{\\phi}{2} \\right) + \\kappa {S_{\\kappa}}^2 \\left( \\frac{\\phi}{2} \\right)  = 1\n\\end{gather*}\n(noting that ${C_{\\kappa}}$ is an even function while ${S_{\\kappa}}$ is odd) it follows that\n\\begin{gather*}\ne^{-\\frac{\\theta}{2} i {\\sigma_1}} \\sigma_j e^{\\frac{\\theta}{2} i {\\sigma_1}}              =\n\\begin{cases}\n{\\sigma_1}                                                             & \\text{if $j = 1$}, \\\\\n{C_{\\kb}} (\\theta) {\\sigma_2} - {S_{\\kb}} (\\theta) {\\sigma_3} & \\text{if $j = 2$}, \\\\\n{C_{\\kb}} (\\theta) {\\sigma_3} + {\\kappa_2} {S_{\\kb}} (\\theta) {\\sigma_2}            & \\text{if $j = 3$},\n\\end{cases} \\\\\ne^{-\\frac{\\beta}{2} i {\\sigma_2}} \\sigma_j e^{\\frac{\\beta}{2} i {\\sigma_2}}              =\n\\begin{cases}\n{C_{\\ka \\kb}} (\\beta) {\\sigma_1} + {S_{\\ka \\kb}} (\\beta) {\\sigma_3} & \\text{if $j = 1$}, \\\\\n{\\sigma_2}                                                             & \\text{if $j = 2$}, \\\\\n{C_{\\ka \\kb}} (\\beta) {\\sigma_3} - {\\kappa_1} {\\kappa_2} {S_{\\ka \\kb}} (\\beta) {\\sigma_1}            & \\text{if $j = 3$},\n\\end{cases} \\\\\ne^{-\\frac{\\alpha}{2i}  {\\sigma_3}} \\sigma_j e^{\\frac{\\alpha}{2i}  {\\sigma_3}}           =\n\\begin{cases}\n{C_{\\ka}} (\\alpha) {\\sigma_1} + {S_{\\ka}} (\\alpha) {\\sigma_2}       & \\text{if $j = 1$}, \\\\\n{C_{\\ka}} (\\alpha) {\\sigma_2} - {\\kappa_1} {S_{\\ka}} (\\alpha) {\\sigma_1}            & \\text{if $j = 2$}, \\\\\n{\\sigma_3}                                                             & \\text{if $j = 3$}.\n\\end{cases}\n\\end{gather*}\nSo for each plane element, the $\\sigma_j$ transform as\nthe components of a vector under rotation in the clockwise direction,\ngiven the orientations of the respective plane elements:\n\\[\ni{\\sigma_1} \\; \\mbox{is represented by} \\; {\\sigma_3} \\wedge {\\sigma_2}, \\qquad\n i{\\sigma_2} = {\\sigma_1} \\wedge {\\sigma_3}, \\qquad \\mbox{and} \\qquad \\frac{1}{i} {\\sigma_3} = {\\sigma_1} \\wedge {\\sigma_2}.\n\\]\nNow we can write\n\\[\ne^{\\frac{\\phi}{2} {\\hat{n} \\cdot \\vec{\\sigma}}} =\n1 +\n\\frac{\\phi}{2} {\\hat{n} \\cdot \\vec{\\sigma}} +\n\\frac{1}{2!} \\left( \\frac{\\phi}{2} \\right)^2 \\left( {\\hat{n} \\cdot \\vec{\\sigma}} \\right)^2 +\n\\frac{1}{3!} \\left( \\frac{\\phi}{2} \\right)^3 \\left( {\\hat{n} \\cdot \\vec{\\sigma}} \\right)^3 + \\cdots.\n\\]\nIf $\\varkappa$ is the scalar $-\\left( {\\hat{n} \\cdot \\vec{\\sigma}} \\right)^2$, then\n\\begin{gather*}\ne^{\\frac{\\phi}{2} {\\hat{n} \\cdot \\vec{\\sigma}}} =  \\left(\n1 -\n\\frac{1}{2!} \\left( \\frac{\\phi}{2} \\right)^2 \\varkappa +\n\\frac{1}{4!} \\left( \\frac{\\phi}{2} \\right)^4 \\varkappa^2 - \\cdots\n\\right)\\\\\n\\phantom{e^{\\frac{\\phi}{2} {\\hat{n} \\cdot \\vec{\\sigma}}}=}{}\n +  {\\hat{n} \\cdot \\vec{\\sigma}} \\left(\n\\frac{\\phi}{2} -\n\\frac{1}{3!} \\left( \\frac{\\phi}{2} \\right)^3 \\varkappa +\n\\frac{1}{5!} \\left( \\frac{\\phi}{2} \\right)^5 \\varkappa^2 - \\cdots\n\\right) \\\\\n\\phantom{e^{\\frac{\\phi}{2} {\\hat{n} \\cdot \\vec{\\sigma}}}}{}  =  C_{\\varkappa} \\left( \\frac{\\phi}{2} \\right)\n+  {\\hat{n} \\cdot \\vec{\\sigma}} S_{\\varkappa} \\left( \\frac{\\phi}{2} \\right).\n\\end{gather*}\n\n\nAs $a = a^1 {\\sigma_1} + a^2 {\\sigma_2} + a^3 {\\sigma_3}$ is a vector, we can compute its length easily\nusing Clif\\\/ford multiplication as $a a = (a^1)^2 + {\\kappa_1} (a^2)^2 + {\\kappa_1} {\\kappa_2} (a^3)^2 = | a |^2$.\n We would like to show that $e^{-\\frac{\\phi}{2} {\\hat{n} \\cdot \\vec{\\sigma}}} a  e^{\\frac{\\phi}{2} {\\hat{n} \\cdot \\vec{\\sigma}}}$\n is also a vector with the same length as $a$.   If $g$ and $h$ are elements of a matrix Lie algebra,\n then so is $e^{-\\phi \\, {\\mbox{\\tiny ad}} \\, g} h = e^{-\\phi g} h e^{\\phi g}$ (see \\cite{SW86} for example).\n So if $B$ is a bivector $B^1 i{\\sigma_1} + B^2 i{\\sigma_2} + B^3\\frac{1}{i}{\\sigma_3}$,\n then $e^{-\\frac{\\phi}{2} {\\hat{n} \\cdot \\vec{\\sigma}}} B  e^{\\frac{\\phi}{2} {\\hat{n} \\cdot \\vec{\\sigma}}}$ is also a bivector.\n It follows that $e^{-\\frac{\\phi}{2} {\\hat{n} \\cdot \\vec{\\sigma}}} a  e^{\\frac{\\phi}{2} {\\hat{n} \\cdot \\vec{\\sigma}}}$ is a vector\n as the volume element $i$ lies in Cen($Cl_3$) so that $e^{-\\frac{\\phi}{2} {\\hat{n} \\cdot \\vec{\\sigma}}}\n {\\sigma_1}  e^{\\frac{\\phi}{2} {\\hat{n} \\cdot \\vec{\\sigma}}}$, $e^{-\\frac{\\phi}{2} {\\hat{n} \\cdot \\vec{\\sigma}}} {\\sigma_2}  e^{\\frac{\\phi}{2} {\\hat{n} \\cdot \\vec{\\sigma}}}$,\n and $e^{-\\frac{\\phi}{2} {\\hat{n} \\cdot \\vec{\\sigma}}} {\\sigma_3}  e^{\\frac{\\phi}{2} {\\hat{n} \\cdot \\vec{\\sigma}}}$ are all vectors.  Since\n \\[\n\\left(\ne^{-\\frac{\\phi}{2} {\\hat{n} \\cdot \\vec{\\sigma}}} a  e^{\\frac{\\phi}{2} {\\hat{n} \\cdot \\vec{\\sigma}}}\n\\right)\n\\left(\ne^{-\\frac{\\phi}{2} {\\hat{n} \\cdot \\vec{\\sigma}}} a  e^{\\frac{\\phi}{2} {\\hat{n} \\cdot \\vec{\\sigma}}}\n\\right) =\ne^{-\\frac{\\phi}{2} {\\hat{n} \\cdot \\vec{\\sigma}}} | a |^2  e^{\\frac{\\phi}{2} {\\hat{n} \\cdot \\vec{\\sigma}}}  =\n| a |^2  e^{-\\frac{\\phi}{2} {\\hat{n} \\cdot \\vec{\\sigma}}}  e^{\\frac{\\phi}{2} {\\hat{n} \\cdot \\vec{\\sigma}}}  =\n\\left| a \\right|^2\n\\]\nit follows that $e^{-\\frac{\\phi}{2} {\\hat{n} \\cdot \\vec{\\sigma}}} a  e^{\\frac{\\phi}{2} {\\hat{n} \\cdot \\vec{\\sigma}}}$\nhas the same length as $a$.  So the inner automorphism of ${\\mathbb{R}}^3$ given\nby $a \\mapsto e^{-\\frac{\\phi}{2} {\\hat{n} \\cdot \\vec{\\sigma}}} a  e^{\\frac{\\phi}{2} {\\hat{n} \\cdot \\vec{\\sigma}}}$ corresponds\nto an element of $SO(3)$.  We will see in the next section that all elements of $SO(3)$\nare represented by such inner automorphisms of ${\\mathbb{R}}^3$.\n\n\nFinally, note that $e^{-\\frac{\\phi}{2} {\\hat{n} \\cdot \\vec{\\sigma}}} \\left( {\\hat{n} \\cdot \\vec{\\sigma}} \\right) e^{\\frac{\\phi}{2} {\\hat{n} \\cdot \\vec{\\sigma}}} = {\\hat{n} \\cdot \\vec{\\sigma}}$\nas ${\\hat{n} \\cdot \\vec{\\sigma}}$ commutes with $e^{\\frac{\\phi}{2} {\\hat{n} \\cdot \\vec{\\sigma}}}$: so any plane element\nrepresented by ${\\hat{n} \\cdot \\vec{\\sigma}}$ is preserved by the corresponding element of $SO(3)$.\nIn fact, if ${\\hat{n} \\cdot \\vec{\\sigma}} = a \\wedge b$ for some vectors $a$ and $b$ and $\\varkappa$\nis the scalar $ - \\left( a \\wedge b \\right)^2$, then\n\\begin{gather*}\n     e^{-\\frac{\\phi}{2} a \\wedge b} ( a )  e^{\\frac{\\phi}{2} a \\wedge b}\n=  \\left[ C_{\\varkappa} \\left( \\frac{\\phi}{2} \\right) -  (a \\wedge b )S_{\\varkappa}\n\\left( \\frac{\\phi}{2} \\right) \\right] ( a )\n          \\left[ C_{\\varkappa} \\left( \\frac{\\phi}{2} \\right)\n          +  (a \\wedge b) S_{\\varkappa} \\left( \\frac{\\phi}{2} \\right) \\right] \\\\\n\\phantom{e^{-\\frac{\\phi}{2} a \\wedge b} ( a )  e^{\\frac{\\phi}{2} a \\wedge b}}{} =  C^2_{\\varkappa} \\left( \\frac{\\phi}{2} \\right) a\n          + C_{\\varkappa} \\left( \\frac{\\phi}{2} \\right) S_{\\varkappa}\n          \\left( \\frac{\\phi}{2} \\right)  (a \\wedge b) a (a \\wedge b) \\\\\n\\phantom{e^{-\\frac{\\phi}{2} a \\wedge b} ( a )  e^{\\frac{\\phi}{2} a \\wedge b}=}{}\n-  C_{\\varkappa} \\left( \\frac{\\phi}{2} \\right) S_{\\varkappa} \\left( \\frac{\\phi}{2} \\right) (a \\wedge b) a\n-       S^2_{\\varkappa} \\left( \\frac{\\phi}{2} \\right)  a(a \\wedge b).\n\\end{gather*}\nSince $a (a \\wedge b) = -(a \\wedge b) a$, then\n\\[\ne^{-\\frac{\\phi}{2} a \\wedge b} ( a )  e^{\\frac{\\phi}{2} a \\wedge b} =\n\\left[ C_{\\varkappa}(\\phi) - (a \\wedge b) S_{\\varkappa}(\\phi) \\right] a,\n\\]\nand so vectors lying in the plane determined by $a \\wedge b$ are simply\nrotated by an angle $-\\phi$, and this rotation is given by simple multiplication\nby a unit complex number $e^{-i \\phi}$ where $i^2 = -\\varkappa$.  Thus, the\nlinear combination $ua + vb$ is sent to $ue^{-i \\phi}a + ve^{-i \\phi}b$,\nand so the plane spanned by the vectors $a$ and $b$ is preserved.\n\n\nThe signif\\\/icance is that if $a$ lies in an oriented plane determined by\nthe bivector ${\\hat{n} \\cdot \\vec{\\sigma}}$ where this plane is given the complex structure of\n${\\mathbb{C}}_{\\varkappa}$, then $e^{-\\frac{\\phi}{2} {\\hat{n} \\cdot \\vec{\\sigma}}} a  e^{\\frac{\\phi}{2} {\\hat{n} \\cdot \\vec{\\sigma}}}$\nis simply the vector $a$ rotated by an angle of $-\\phi$ in the complex plane\n${\\mathbb{C}}_{\\varkappa}$, where $\\iota^2 = -\\varkappa$.  Furthermore, the axis of\nrotation is given by $\\eta$ as $\\eta$ is preserved (recall that $i$ lies\nin the center of $Cl_3$).  Since the covariant components $\\sigma_i$ of $a$\nare rotated clockwise, the contravariant components $a^j$ are rotated counterclockwise.\nSo $\\langle a^1, a^2, a^3 \\rangle$ is rotated by the angle $\\phi$ in the complex plane\n$C_{\\varkappa}$ determined by ${\\hat{n} \\cdot \\vec{\\sigma}}$.\n\nIf we use $b \\wedge a$ instead of $a \\wedge b$ to represent the plane element,\nthen $\\varkappa$ remains unchanged.  Note however that, if $c$ is a vector lying in this plane, then\n\\begin{gather*}\ne^{-\\frac{\\phi}{2} b \\wedge a} c e^{-\\frac{\\phi}{2} b \\wedge a}\n    = \\left[ C_{\\varkappa}(\\phi) - (b \\wedge a) S_{\\varkappa}(\\phi) \\right] c\n    = \\left[ C_{\\varkappa}(-\\phi) - (a \\wedge b) S_{\\varkappa}(-\\phi) \\right] c\n\\end{gather*}\nso that rotation by an angle of $\\phi$ in the plane oriented according\nto $b \\wedge a$ corresponds to a~rotation of angle $-\\phi$\nin the same plane under the opposite orientation as given by $a \\wedge b$.\n\n\nIt would be appropriate at this point to note two things:\none, the magnitude of ${\\hat{n} \\cdot \\vec{\\sigma}}$ appears to be important, since\n$\\varkappa = - \\left( {\\hat{n} \\cdot \\vec{\\sigma}} \\right)^2$, and two, the normalization\n$(n^1)^2 + (n^2)^2 + (n^3)^2 = 1$ of $\\hat{n}$ is somewhat\narbitrary\\footnote{Due to dimension requirements some kind of\nnormalization is needed as we cannot have $\\phi$, $n^1$, $n^2$, and $n^3$\nas independent variables, for $so(3)$ is 3-dimensional.}.\nThese two matters are one and the same.  We have chosen this normalization\nbecause it is a simple and natural choice.  This particular normalization\nis not essential, however.  For suppose that $\\varkappa = - (a \\wedge b)^2$\nwhile $\\varkappa^{\\prime} = - (n a \\wedge b)^2$, where $n$ is a positive constant.\n Let ${\\mathbb{C}}_{\\varkappa} = \\{ t + ix \\, | \\, i^2 = -\\varkappa \\}$\n with angle measure $\\phi$ and ${\\mathbb{C}}_{\\varkappa^{\\prime}} = \\{ t + \\iota x \\, | \\,\\iota^2\n = -\\varkappa^{\\prime} = -n^2 \\varkappa \\}$ with angle measure $\\theta$:\n  without loss of generality let $\\varkappa > 0$.  Then $\\phi = n\\theta$, for\n\\begin{gather*}       e^{i \\theta}\n = \\cos{\\left( \\sqrt{\\varkappa^{\\prime}} \\theta \\right)}\n - \\frac{\\iota}{\\sqrt{\\varkappa^{\\prime}}}\\sin{\\left( \\sqrt{\\varkappa^{\\prime}} \\theta \\right)}\n  = \\cos{\\left( n\\sqrt{\\varkappa} \\theta \\right)} - \\frac{\\iota}{n\\sqrt{\\varkappa}}\n  \\sin{\\left( n\\sqrt{\\varkappa} \\theta \\right)} \\\\\n \\phantom{e^{i \\theta}}{} = \\cos{\\left( \\sqrt{\\varkappa} \\phi \\right)} - \\frac{i}{\\sqrt{\\varkappa}}\\sin{\\left( \\sqrt{\\varkappa} \\phi \\right)}\n  = e^{i \\phi}.\n\\end{gather*}\nSo we see that $SO(3)$ is truly a rotation group, where each\nelement has a distinct axis of rotation as well as a well-def\\\/ined rotation angle.\n\n\n\\section[$SU(2)$]{$\\boldsymbol{SU(2)}$}\n\nSince the generators of the generalized Lie group $SO(3)$\ncan be represented by inner automorphisms of the subspace ${\\mathbb{R}}^3$\nof vectors of $Cl_3$ (see Def\\\/inition~3), then every element of $SO(3)$\ncan be represented by an inner automorphism, as the composition of\ninner automorphisms is an inner automorphism.  On the other hand,\nwe've seen that any inner automorphism represents an element of $SO(3)$.\nIn fact, each rotation belonging to $SO(3)$ is then represented by two elements\n$\\pm e^{\\frac{\\phi}{2} {\\hat{n} \\cdot \\vec{\\sigma}}}$ of $SL(2,{\\mathbb{C}})$, where as usual ${\\mathbb{C}}$ denotes the\ngeneralized complex number ${\\mathbb{C}}_{{\\kappa_2}}$: we will denote the subgroup of $SL(2,{\\mathbb{C}})$\nconsisting of elements of the form $\\pm e^{\\frac{\\phi}{2} {\\hat{n} \\cdot \\vec{\\sigma}}}$ by $SU(2)$.\n\n\\begin{definition}\nLet $A$ be the matrix\n\\[\nA =\n\\left(\n   \\begin{matrix}\n   {\\kappa_1} & 0 \\\\\n   0    & 1\n   \\end{matrix}\n\\right) .\n\\]\n\\end{definition}\n\nWe will now use Def\\\/inition 4 to show that $SU(2)$ is a subgroup of the subgroup $G$ of $SL(2,{\\mathbb{C}})$\nconsisting of those matrices $U$ where $U^{\\star} A U = A$: in fact, both these subgroups of\n$SL(2,{\\mathbb{C}})$ are one and the same, as we shall see.\nNow\n\\begin{gather*}\n\\big( e^{\\frac{\\phi}{2} {\\hat{n} \\cdot \\vec{\\sigma}}}\\big)^{\\star} A e^{\\frac{\\phi}{2} {\\hat{n} \\cdot \\vec{\\sigma}}}\n= \\left[ C_{\\varkappa}\\left( \\frac{\\phi}{2} \\right) +\n\\left( {{\\hat{n} \\cdot \\vec{\\sigma}}} \\right)^{\\star} S_{\\varkappa}\\left( \\frac{\\phi}{2} \\right)  \\right] A\n          \\left[ C_{\\varkappa}\\left( \\frac{\\phi}{2} \\right) + {\\hat{n} \\cdot \\vec{\\sigma}} S_{\\varkappa}\\left( \\frac{\\phi}{2} \\right) \\right] \\\\\n \\phantom{\\big( e^{\\frac{\\phi}{2} {\\hat{n} \\cdot \\vec{\\sigma}}}\\big)^{\\star} A e^{\\frac{\\phi}{2} {\\hat{n} \\cdot \\vec{\\sigma}}}}{}\n  = C_{\\varkappa}^2\\left( \\frac{\\phi}{2} \\right)A +\n\\left( {{\\hat{n} \\cdot \\vec{\\sigma}}} \\right)^{\\star} A \\left( {\\hat{n} \\cdot \\vec{\\sigma}} \\right) S_{\\varkappa}^2\\left( \\frac{\\phi}{2} \\right) \\\\\n\\phantom{\\big( e^{\\frac{\\phi}{2} {\\hat{n} \\cdot \\vec{\\sigma}}}\\big)^{\\star} A e^{\\frac{\\phi}{2} {\\hat{n} \\cdot \\vec{\\sigma}}}=}{}\n+ A \\left( {\\hat{n} \\cdot \\vec{\\sigma}} \\right) C_{\\varkappa}\\left( \\frac{\\phi}{2} \\right)S_{\\varkappa}\\left( \\frac{\\phi}{2} \\right) +\n\\left( {{\\hat{n} \\cdot \\vec{\\sigma}}} \\right)^{\\star} A C_{\\varkappa}\\left( \\frac{\\phi}{2} \\right)S_{\\varkappa}\\left( \\frac{\\phi}{2} \\right)\n = A\n\\end{gather*}\nbecause $A \\left( {\\hat{n} \\cdot \\vec{\\sigma}} \\right) = -\\left( {{\\hat{n} \\cdot \\vec{\\sigma}}} \\right)^{\\star} A$ implies that\n\\[\nA \\left( {\\hat{n} \\cdot \\vec{\\sigma}} \\right) C_{\\varkappa}\\left( \\frac{\\phi}{2} \\right)S_{\\varkappa}\\left( \\frac{\\phi}{2} \\right) +\n\\left( {{\\hat{n} \\cdot \\vec{\\sigma}}} \\right)^{\\star} A C_{\\varkappa}\\left( \\frac{\\phi}{2} \\right)S_{\\varkappa}\\left( \\frac{\\phi}{2} \\right)\n= 0\n\\]\nand $\\left( {{\\hat{n} \\cdot \\vec{\\sigma}}} \\right)^{\\star} A \\left( {\\hat{n} \\cdot \\vec{\\sigma}} \\right) = -A \\left( {\\hat{n} \\cdot \\vec{\\sigma}} \\right)^2 = \\varkappa A$ implies that\n\\begin{gather*}\n C_{\\varkappa}^2\\left( \\frac{\\phi}{2} \\right)A +\n\\left( {{\\hat{n} \\cdot \\vec{\\sigma}}} \\right)^{\\star} A \\left( {\\hat{n} \\cdot \\vec{\\sigma}} \\right) S_{\\varkappa}^2\\left( \\frac{\\phi}{2} \\right)\n = C_{\\varkappa}^2\\left( \\frac{\\phi}{2} \\right)A +\n\\varkappa S_{\\varkappa}^2\\left( \\frac{\\phi}{2} \\right) A   = A.\n\\end{gather*}\nSo $SU(2)$ is a subgroup of the subgroup $G$ of $SL(2,{\\mathbb{C}})$ consisting of those matrices $U$ where $U^{\\star} A U = A$.\n\nWe can characterize this subgroup $G$ as\n\\[\n\\left\\{\n\\left(\n   \\begin{matrix}\n   \\alpha               & \\beta \\\\\n   -{\\kappa_1} \\overline{\\beta} & \\overline{\\alpha}\n   \\end{matrix}\n\\right)\n | \\,\n\\alpha, \\beta \\in {\\mathbb{C}} \\; \\mbox{and} \\; \\alpha \\overline{\\alpha} + {\\kappa_1} \\beta \\overline{\\beta} = 1\n\\right\\}.\n\\]\nNow\n\\[\ne^{\\frac{\\phi}{2} {\\hat{n} \\cdot \\vec{\\sigma}}} =\n\\left(\n   \\begin{matrix}\n   C_{\\varkappa}^2\\left( \\frac{\\phi}{2} \\right) + n^1 i S_{\\varkappa}^2\\left( \\frac{\\phi}{2} \\right) &\n   n^2 i S_{\\varkappa}^2\\left( \\frac{\\phi}{2} \\right) + n^3 S_{\\varkappa}^2\\left( \\frac{\\phi}{2} \\right)\\vspace{1mm} \\\\\n   n^2 {\\kappa_1} i S_{\\varkappa}^2\\left( \\frac{\\phi}{2} \\right) - n^3 {\\kappa_1} S_{\\varkappa}^2\\left( \\frac{\\phi}{2} \\right) &\n   C_{\\varkappa}^2\\left( \\frac{\\phi}{2} \\right) - n^1 i S_{\\varkappa}^2\\left( \\frac{\\phi}{2} \\right)\n   \\end{matrix}\n\\right)\n\\]\nas can be checked directly, recalling that\n\\[\ne^{\\frac{\\phi}{2} {\\hat{n} \\cdot \\vec{\\sigma}}} = C_{\\varkappa}\\left( \\frac{\\phi}{2} \\right)\n+ \\left( {\\hat{n} \\cdot \\vec{\\sigma}} \\right) S_{\\varkappa}\\left( \\frac{\\phi}{2} \\right),\n\\]\nwhere\n\\[\n\\varkappa = - \\left( {\\hat{n} \\cdot \\vec{\\sigma}} \\right)^2 = \\left(n^1\\right)^2 {\\kappa_2}\n+ \\left( n^2 \\right)^2 {\\kappa_1} {\\kappa_2} + \\left( n^3 \\right)^2 {\\kappa_1}.\n\\]\nThus $\\det\\left( e^{\\frac{\\phi}{2} {\\hat{n} \\cdot \\vec{\\sigma}}} \\right) = 1$,\nand we see that any element of $G$ can be written in the form $e^{\\frac{\\phi}{2} {\\hat{n} \\cdot \\vec{\\sigma}}}$.\nSo the group $SU(2)$ can be characterized by\n\\[\nSU(2) =\n\\left\\{\n\\left(\n   \\begin{matrix}\n   \\alpha               & \\beta \\\\\n   -{\\kappa_1} \\overline{\\beta} & \\overline{\\alpha}\n   \\end{matrix}\n\\right)\n | \\,\n\\alpha, \\beta \\in {\\mathbb{C}} \\; \\mbox{and} \\; \\alpha \\overline{\\alpha} + {\\kappa_1} \\beta \\overline{\\beta} = 1\n\\right\\}.\n\\]\nNote that if $U(\\lambda)$ is a curve passing through the identity at $\\lambda = 0$, then\n\\[\n\\left. \\frac{d}{d\\lambda} \\right|_{\\lambda = 0} \\left( U^{\\star} A U = A \\right) \\ \\Longrightarrow \\\n  \\dot{U}^{\\star}A + A\\dot{U} = 0\n\\]\nso that $su(2)$ consists of those elements $B$ of $M(2,{\\mathbb{C}})$ such that $B^{\\star}A + AB = 0$.  Although $SU(2)$ is a double cover of $SO(3)$, it is not necessarily the universal cover for $SO(3)$, nor even connected, for sometimes $SO(3)$ is itself simply-connected.  Thus we have shown that:\n\n\\begin{theorem}\nThe Clifford algebra $Cl_3$ can be used to construct a double cover of the generalized Lie group $SO(3)$,\nfor a vector $a$ can be rotated by the inner automorphism\n\\[\n{\\mathbb{R}}^3 \\rightarrow {\\mathbb{R}}^3, \\qquad a \\mapsto \\mathit{s}^{-1} a \\mathit{s}\n\\]\nwhere $\\mathit{s}$ is an element of the group\n\\[\n{\\bf Spin}(3) =\n\\left\\{\n\\left(\n   \\begin{matrix}\n   \\alpha               & \\beta \\\\\n   -{\\kappa_1} \\overline{\\beta} & \\overline{\\alpha}\n   \\end{matrix}\n\\right)\n | \\,\n\\alpha, \\beta \\in {\\mathbb{C}} \\; \\mbox{and} \\; \\alpha \\overline{\\alpha} + {\\kappa_1} \\beta \\overline{\\beta} = 1\n\\right\\},\n\\]\nwhere ${\\mathbb{C}}$ denotes the generalized complex number ${\\mathbb{C}}_{{\\kappa_2}}$.\n\\end{theorem}\n\n\\begin{lemma}\nWe define the generalized special unitary group $SU(2)$ to be ${\\bf Spin}(3)$.  Then $su(2)$\nconsists of those matrices $B$ of $M(2,{\\mathbb{C}})$ such that $B^{\\star}A + AB = 0$.\n\\end{lemma}\n\n\n\\section[The conformal completion of $S$]{The conformal completion of $\\boldsymbol{S}$}\n\nYaglom~\\cite{Y79} has shown how the complex plane ${\\mathbb{C}}_{\\kappa}$ may be extended\nto a Riemann sphere $\\Gamma$ or inversive plane\\footnote{Yaglom did\nthis when $\\kappa \\in \\{-1, 0, 1 \\}$, but it is a simple matter to generalize his results.}\n(and so dividing by zero-divisors is allowed), upon which the entire set\nof M\\\"{o}bius transformations acts globally and so gives a group of conformal transformations.\nIn this last section we would like to take advantage of the simple structure of\nthis conformal group and give the conformal completion of $S$, where $S$ is conformally\nembedded simply by inclusion of the region $\\varsigma$ lying in ${\\mathbb{C}}$ and therefore lying in~$\\Gamma$.\nHerranz and Santander~\\cite{HS02b} found a conformal\ncompletion of $S$ by realizing the conformal group as a group\nof linear transformations acting on ${\\mathbb{R}}^4$, and then constructing the\nconformal completion as a homogeneous phase space of this conformal group.\nThe original Cayley--Klein geometry $S$ was then embedded\ninto its conformal completion by one of two methods, one a group-theoretical\none involving one-parameter subgroups and the other stereographic projection.\n\n\nThe 6-dimensional real Lie algebra for $SL(2,{\\mathbb{C}})$ consist of those matrices in $M(2,{\\mathbb{C}})$\nwith trace equal to zero.  In addition to the three generators $H$, $P$, and $K$\n\\[\nH = \\frac{1}{2i}{\\sigma_3} =\n\\left(\n   \\begin{matrix}\n   0 & \\frac{1}{2} \\\\\n   -\\frac{{\\kappa_1}}{2} & 0\n   \\end{matrix}\n\\right), \\qquad\nP = \\frac{i}{2}{\\sigma_2} =\n\\left(\n   \\begin{matrix}\n   0 & \\frac{i}{2} \\\\\n   \\frac{{\\kappa_1} i}{2} & 0\n   \\end{matrix}\n\\right), \\qquad\nK = \\frac{i}{2}{\\sigma_1} =\n\\left(\n   \\begin{matrix}\n   \\frac{i}{2} & 0 \\\\\n   0 & -\\frac{i}{2}\n   \\end{matrix}\n\\right)\n\\]\nthat come from the generalized Lie group $SO(3)$\nof isometries of $S$, we have three other generators for $SL(2, {\\mathbb{C}})$:\none, labeled $D$, for the subgroup of dilations centered at the origin and\ntwo others, labeled $G_1$ and $G_2$, for ``translations''.\n It is these transformations $D$, $G_1$, $G_2$, that necessitate extending $\\varsigma$\n to the entire Riemann sphere $\\Gamma$, upon which the set of M\\\"{o}bius\n transformations acts as a conformal group.  Note that the following correspondences\n for the M\\\"{o}bius transformations $w \\mapsto w + t$ and $w \\mapsto w + ti$\n (for real parameter $t$) are valid only if ${\\kappa_1} \\neq 0$, which explains why our\n ``translations\" $G_1$ and $G_2$ are not actually translations:\n\\begin{gather*}\n\\exp \\left[ t \\left(\n   \\begin{matrix}\n   0 & 1 \\\\\n   0 & 0\n   \\end{matrix}\n   \\right) \\right] =\n   \\left(\n   \\begin{matrix}\n   1 & t \\\\\n   0 & 1\n   \\end{matrix}\n   \\right)\n   \\hspace{0.25in} \\rightleftarrows \\hspace{0.25in}\n   w \\mapsto w + t,\n\\\\\n\\mbox{exp} \\left[ t \\left(\n   \\begin{matrix}\n   0 & i \\\\\n   0 & 0\n   \\end{matrix}\n   \\right) \\right] =\n   \\left(\n   \\begin{matrix}\n   1 & ti \\\\\n   0 & 1\n   \\end{matrix}\n   \\right)\n   \\hspace{0.25in} \\rightleftarrows \\hspace{0.25in}\n   w \\mapsto w + ti.\n\\end{gather*}\nPlease see Tables 15 and 16.\nThe structure constants $[\\star, \\star \\star]$ for this basis of $sl(2,{\\mathbb{C}})$\n(which is the same basis as that given in \\cite{HS02} save for a sign change in $G_2$) are given by Table~12.\n\n\\begin{table}[t]\n \\centering\n\\caption{Additional basis elements for $sl(2,{\\mathbb{C}})$.}\n\\vspace{1mm}\n \\begin{tabular}{ c ||  c   c   c   c   c   c   c  } \\hline\n$\\star \\diagdown \\star\\star $ & $H$ & $P$ & $K$ & $G_1$ & $G_2$ & $D$  \\\\ \\hline \\hline\n$H$     & 0                         & ${\\kappa_1} K$        & $-P$        &  $D$         & $K$     & $-H - {\\kappa_1} G_1$  \\\\\n$P$     & $-{\\kappa_1} K$      & 0                          & ${\\kappa_2} H$    & $K$  & $-{\\kappa_2} D$     &  $-P + {\\kappa_1} G_2$ \\\\\n$K$ & $P$                 &  $-{\\kappa_2} H$         & 0                   & $-S_2$    & ${\\kappa_2} G_2$   & $0$                        \\\\\n$G_1$    &  $-D$                   &  $-K$           &  $S_2$          & 0             & $0$              & $G_1$                   \\\\\n$G_2$    &  $-K$           &  ${\\kappa_2} D$               &  $-{\\kappa_2} G_2$  &  $0$         & 0                 & $G_2$                   \\\\\n$D$        & $H + {\\kappa_1} G_1$ &  $P - {\\kappa_1} G_2$ &   $0$             &  $-G_1$   &  $-G_2$      & 0                            \\\\\n\\hline\n \\end{tabular}\n \\end{table}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe Haagerup subfactor is a finite-depth subfactor with index $\\frac{5+\\sqrt{13}}{2} $; this is the smallest index above $4$ for any finite depth subfactor.  Its even parts are two fusion categories.  We will call the fusion category with four simple objects $\\mathscr{H}_1$ and the one with six simple objects $\\mathscr{H}_2$.   \n\n\nGiven a fusion category $\\mathscr{C}$, one important structural question is to understand all quantum subgroups of $\\mathscr{C}$ in the sense of Ocneanu \\cite{MR1907188}.  That is we wish to understand all simple module categories over  $\\mathscr{C}$ (by simple we mean semisimple and indecomposable).  The reason for the name ``quantum subgroup'' is that when $\\mathscr{C}$ is the category of $G$-modules for a finite group $G$ then the simple module categories correspond to  the subgroups $H \\subset G$ together with some additional cohomological data \\cite{MR1976233}.  When $\\mathscr{C}$ is a fusion category coming from quantum $\\mathfrak{su}_2$ at a root of unity, then the quantum subgroups are given by the ADE Dynkin diagrams.  (See \\cite{MR996454, OcnLect, MR1777347} for this result in subfactor language, and \\cite{MR1936496, MR1976459, MR2046203} for the translation of these results into the language of fusion categories and module categories.)  Ocneanu has announced the classification of quantum subgroups of the fusion categories coming from quantum $\\mathfrak{su}_3$ and $\\mathfrak{su}_4$ \\cite{MR1907188} (see \\cite{MR2545609,MR2553429} for details in the $\\mathfrak{su}_3$ case).  In this paper we find all quantum subgroups of $\\mathscr{H}_1$ and $\\mathscr{H}_2$.  One might think of the results of this paper as analogous to finding the subgroups of a sporadic finite simple group.\n\n\\begin{theorem}\nThere are exactly three quantum subgroups of each of $\\mathscr{H}_1$ and $\\mathscr{H}_2$.  The quantum subgroups of  $\\mathscr{H}_1$ have the following graphs for fusion with the object of dimension $\\frac{1+\\sqrt{13}}{2}$.\n\n\\hpic{m1} {.3in} \\quad \\hpic{m2} {.6in} \\quad \\hpic{m3} {.6in}\n\nThe quantum subgroups of $\\mathscr{H}_2$ have the following graphs for fusion with one of the objects of dimension $\\frac{3+\\sqrt{13}}{2}$.\n\n\\hpic{m4} {.6in} \\quad\n\\hpic{m5} {.6in} \\quad\n\\hpic{m6} {.4in}\n\\end{theorem}\n\nThe graphs in the above theorem are analogous to the ADE Dynkin diagrams for quantum subgroups of quantum $\\mathfrak{su}_2$.\n\nUnderstanding all quantum subgroups of $\\mathscr{C}$ also allows us to answer several other important structural questions about  $\\mathscr{C}$.  A subfactor whose principal even part is $\\mathscr{C}$ is roughly the same thing as a simple algebra object in $\\mathscr{C}$ (see Section \\ref{sec:subfactorvsalgebra} and \\cite{MR1257245, MR1444286, MR1966524, MR2075605}).  All simple algebra objects in $\\mathscr{C}$ can be realized as the internal endomorphisms of a simple object in some module category over $\\mathscr{C}$ \\cite{MR1976459}.  Hence we can use our classification of quantum subgroups to describe all subfactors whose even parts are $\\mathscr{H}_1$ or  $\\mathscr{H}_2$ (this generalizes the GHJ subfactors \\cite{MR999799} constructed from the quantum subgroups of quantum $\\mathfrak{su}_2$).\n\n\\begin{theorem}\nThere are exactly $7$ subfactors of the hyperfinite $II_1$ factor whose principal even part is $\\mathscr{H}_1$.  These subfactors have the following principal graphs, dual principal graphs, and indices.\n\n\\begin{tabular}{ccc}\n\\hpic{hu1} {0.6in} & \\hpic{hu8} {0.6in} &  $\\frac{5+\\sqrt{13}} {2}$\\\\\n\\hpic{hu2} {0.6in} & \\hpic{hu9} {0.6in} &  $12+3\\sqrt{13}$\\\\\n\\hpic{hu3} {0.6in} & \\hpic{hu10} {0.6in} &  $4+\\sqrt{13}$\\\\\n\\hpic{hu5} {0.6in} & \\hpic{hu12} {0.6in} &  $\\frac{15+3\\sqrt{13}}{2}$\\\\\n\\hpic{hu4} {0.6in} & \\hpic{hu4} {0.6in} &  $\\frac{11+3\\sqrt{13}}{2}$\\\\\n\\hpic{hu6} {0.6in} & \\hpic{hu6} {0.6in} &  $\\frac{19+5\\sqrt{13}}{2}$\\\\\n\\hpic{hu7} {0.3in} & \\hpic{hu7} {0.3in} &  $\\frac{7+\\sqrt{13}}{2}$\\\\\n\\end{tabular}\n\\end{theorem} \n\n\\begin{theorem}\nThere are exactly $4$ subfactors of the hyperfinite $II_1$ factor whose principal even part is $\\mathscr{H}_2$.  These subfactors have the following principal graphs, dual principal graphs, and indices.\n\n\\begin{tabular}{ccc}\n\\hpic{hu8} {0.6in} & \\hpic{hu1} {0.6in} &  $\\frac{5+\\sqrt{13}} {2}$\\\\\n\\hpic{hu9} {0.6in} & \\hpic{hu2} {0.6in} &  $12+3\\sqrt{13}$\\\\\n\\hpic{hu13} {0.6in} & \\hpic{hu13} {0.6in} &  $\\frac{33+9\\sqrt{13}}{2}$\\\\\n\\end{tabular}\n\n\\begin{tabular}{ccc}\\hpic{hu11} {0.6in} & \\hpic{hu11} {0.6in} &  $\\frac{11+3\\sqrt{13}}{2}$\\\\\n\\end{tabular}\n\\end{theorem}\n\nNote that for several of the subfactors in the above lists the dual subfactor does not appear on either list.  This is because for these subfactors the dual even part is not $\\mathscr{H}_1$ nor $\\mathscr{H}_2$.  Instead it is a new fusion category, which we call $\\mathscr{H}_3$.\n\n\\begin{theorem}\nThe (higher) Morita equivalence class of $\\mathscr{H}_1$ consists of exactly three fusion categories: $\\mathscr{H}_1$, $\\mathscr{H}_2$, and $\\mathscr{H}_3$.  The fusion category $\\mathscr{H}_3$ has six objects, three of dimension $1$ and three of dimension $\\frac{3+\\sqrt{13}}{2}$, and the same fusion ring as $\\mathscr{H}_2$.  The fusion category $\\mathscr{H}_3$ has exactly three quantum subgroups, the fusion graphs for these quantum subgroups (with respect to any of the $\\frac{3+\\sqrt{13}}{2}$ dimensional objects) are: \n\n\\hpic{m4} {.6in} \\quad\n\\hpic{m7} {.6in} \\quad\n\\hpic{m6} {.4in}\n\nFurthermore, $\\mathscr{H}_3$  appears as the even part of exactly four subfactors whose principal graphs, dual principal graphs, and indices are listed below.\n\n\\begin{tabular}{ccc}\n\\hpic{hu10} {0.6in} & \\hpic{hu3} {0.6in} &  $4+\\sqrt{13}$\\\\\n\\hpic{hu12} {0.6in} & \\hpic{hu5} {0.6in} &  $\\frac{15+3\\sqrt{13}}{2}$\\\\\n\\hpic{hu13} {0.6in} & \\hpic{hu13} {0.6in} &  $\\frac{33+9\\sqrt{13}}{2}$\\\\\n\\hpic{hu11} {0.6in} & \\hpic{hu11} {0.6in} &  $\\frac{11+3\\sqrt{13}}{2}$\\\\\n\\end{tabular}\n\\end{theorem}\n\nThe fusion category $\\mathscr{H}_3$ mentioned above is new.  It can be described most succinctly as the category of $(1+\\alpha+\\alpha^2)$-bimodule objects in $\\mathscr{H}_2$ where $\\alpha$ and $\\alpha^2$ are the nontrivial invertible objects in $\\mathscr{H}_2$.  It can also be described via an intermediate subfactor construction.  Namely, consider the Haagerup subfactor, and then look at the reduced subfactor constructed from the middle vertex of the principal graph.  This subfactor has three invertible objects at depth $2$ and thus has an intermediate of index $3$.  The other intermediate subfactor has $\\mathscr{H}_3$ as one of its even parts.\n\nNote that although the fusion categories $\\mathscr{H}_2$ and $\\mathscr{H}_3$ have the same Grothendieck rings, they are not the only fusion categories with this Grothendieck ring.  In particular both fusion categories have non-unitary Galois conjugates.  Furthermore according to \\cite{1006.1326} there may be unitary fusion categories which differ from $\\mathscr{H}_2$ by twisting by a class in $H^3(\\mathbb{Z}\/3\\mathbb{Z}, \\mathbb{C}^*)$ which are not Morita equivalent to $\\mathscr{H}_2$ (since their centers would have different modular invariants).  The classification of all fusion categories whose Grothendieck ring agrees with that of $\\mathscr{H}_2$ remains an interesting open question (which was suggested to us by Pavel Etingof).\n\n\nThe Brauer-Picard groupoid of a fusion category $\\mathscr{C}$ has points for every fusion category $\\mathscr{D}$ which is Morita equivalent to $\\mathscr{C}$ and an arrow for every Morita equivalence between $\\mathscr{C}$ and $\\mathscr{D}$ (up to equivalence of bimodule categories).  The Brauer-Picard groupoid is important in understanding the extension theory of $\\mathscr{C}$ \\cite{0909.3140}, and is essentially the same as Ocneanu's notion of ``maximal atlas.''  Morita equivalences between $\\mathscr{C}$ and $\\mathscr{D}$ correspond to a choice of module category over $\\mathscr{C}$ and a choice of isomorphism between $\\mathscr{D}$ and $\\mathscr{C}_\\mathscr{M}^*$ up to inner automorphism of $\\mathscr{D}$. Thus understanding Morita equivalences requires understanding the module categories and the outer automorphisms of $\\mathscr{C}$ and its Morita equivalent categories.\n\n\\begin{theorem}\nThe Brauer-Picard groupoid of $\\mathscr{H}_1$ has three objects $\\mathscr{H}_1$, $\\mathscr{H}_2$, $\\mathscr{H}_3$, and between any two (not necessarily distinct) of these there is exactly one Morita equivalence.  In particular, none of the $\\mathscr{H}_i$ has an outer automorphism.\n\\end{theorem}\n\nThe main technique of this paper is to move back-and-forth between algebra objects and module categories to exploit the richer combinatorial structure of the former and the richer algebraic structure of the latter.\n\nIn Section 2 we recall some background information on fusion categories, module categories, algebra objects, and subfactors. In Section 3 we find all fusion categories Morita equivalent to the Haagerup fusion categories, and classify all algebra objects in and module categories over these fusion categories. In Section 4 we find the full intermediate subfactor lattices for all subfactors arising from the quantum subgroups of the Haagerup fusion categories. In Section 5 we show that the outer automorphisms groups of the Haagerup fusion categories are trivial and compute the Brauer-Picard groupoid. In Section 6 we discuss how our results generalize to the Izumi subfactors, in particular the Izumi subfactor corresponding to $\\mathbb{Z} \/ 5\\mathbb{Z} $.\n\nWe would like to thank David Penneys who pointed out to us that we were both working on this problem.  Noah Snyder would like to thank Dmitri Nikshych who suggested this question at the Shanks workshop on Subfactors and Fusion Categories at Vanderbilt, and Dietmar Bisch and Richard Burstein for hosting that conference.  Pinhas Grossman would like to thank David Evans for helpful conversations, and Noah Snyder would like to thank David Jordan and Emily Peters.  Noah Snyder was supported by an NSF Postdoctoral Fellowship at Columbia University.  Pinhas Grossman was supported by a Marie Curie fellowship from the EU Noncommutative Geometry Network at Cardiff University, and later by a fellowship at IMPA in Brazil. He was also partially supported by NSF grant DMS-0801235. \n\n\\section{Background}\n\n\\subsection{Translating between fusion category language and subfactor language}\n\nThe goal of this subsection is to explain how to translate between fusion category language and subfactor language.  It is a cheat sheet for the rest of the background section, and we hope that it will enable people who are only familiar with one of the languages to understand the rest of the paper.  The key point is that if you have a subfactor $N \\subset M$ then you have a tensor category of all $N$-$N$ bimodules together with an algebra $M$ which can be thought of as an $N$-$N$ bimodule.  Thus subfactors roughly correspond to algebra objects in tensor categories.\n\nThe table below should be taken with three caveats, two of them technical and one of them important.  First, since factors are $C^*$ algebras, the corresponding tensor categories are always unitary and one may need to impose certain compatibility conditions with the $*$-structure.  Second, the word ``fusion'' means ``finite depth'' in the context of subfactors, so for infinite depth subfactors you should relax the adjective ``fusion.''  The important caveat will be explained below.\n\n\\begin{center}\\begin{tabular}{l|l}\nSubfactors & Fusion categories \\\\\n\\hline\n\\hline\n$N \\subset M$ & A fusion category $\\mathscr{C}$ with an algebra object $A$ \\\\\n\\hline\nThe standard invariant & The $2$-category consisting of all $1$-$1$, $1$-$A$,  $A$-$1$, \\\\& and $A$-$A$ bimodule objects in $\\mathscr{C}$ together with \\\\& a choice of $1$-morphism $A$ as an $A$-$1$ bimodule \\\\\n\\hline\nPrincipal even part & $\\mathscr{C}$ \\\\\n\\hline\nOdd part & The collection of $A$-modules (called $\\mathscr{M}$) \\\\ & as a right module category over $\\mathscr{C}$ \\\\\n\\hline\nDual even part & The dual fusion category $\\mathscr{C}_\\mathscr{M}^*$, or equivalently \\\\& the fusion category of $A$-$A$ bimodules. \\\\\n\\hline\nThe principal graphs & The fusion graphs for tensoring with $A$ \\\\\n\\hline\n$Q$-system & Algebra object \\\\\n\\hline\nTensoring odd objects & Internal Hom \\\\\n\\hline\n\\end{tabular} \\end{center}\n\n\\vspace{.1in}\n\nNow for the important caveat.  Notice that subfactors always correspond to a category theoretic construction {\\em together with} a choice of object $A$.  Furthermore, notice that the category of {\\em all} $N$-$N$ bimodules is quite large and unwieldy.  Thus, in subfactor theory, one always restricts to the subcategory tensor generated by $A$.  However, once you fix a fusion subcategory inside all $N$-$N$ bimodules (as we do throughout this paper), it then becomes unnatural to look only at the subcategory tensor generated by $A$.  Much of the power of this paper comes from using constructions which are natural on the fusion category side, but somewhat unnatural on the subfactor side because the $Q$-system doesn't tensor generate.  For example, if you look at the algebra object $1$ inside $\\mathscr{C}$, then the category of all $1$-$1$ bimodules is $\\mathscr{C}$ itself, but the corresponding subfactor is trivial.   We will often prove the uniqueness of a nontrivial $Q$-system by proving the uniqueness of a simpler $Q$-system which doesn't tensor generate.  To our knowledge, these arguments do not correspond to any arguments already appearing in the subfactor literature.\n\n\\subsection{Fusion categories, module categories, and bimodule categories}\n\nIn this subsection we sketch the definitions of fusion categories, module categories, bimodule categories, outer automorphisms and the Brauer-Picard groupoid.  For more details see \\cite{MR2183279, MR1976459, 0909.3140}.\n\n\\begin{definition} \\cite{MR2183279}\nA fusion category over  $\\mathbb{C}$ is a $\\mathbb{C}$-linear rigid semisimple monoidal category with finitely many simple objects (up to isomorphism) and finite-dimensional morphism spaces, such that the identity object is simple.  \n\\end{definition}\n\nRecall that a monoidal functor $\\mathscr{F}:\\mathscr{C} \\rightarrow \\mathscr{D}$ is a pair a functor $\\mathscr{F}$ together with a binatural transformation $\\sigma_{X,Y}: \\mathscr{F}(X \\otimes Y) \\rightarrow  \\mathscr{F}(X) \\otimes  \\mathscr{F}(Y)$ satisfying a certain naturality condition with the associator.  Somewhat nonstandardly we call an invertible monoidal functor an isomorphism (or an automorphism if it is an endofunctor) rather than an ``equivalence.\"  The main reason for this is that we want to avoid confusion with {\\bf Morita} equivalence, also it is harmless since the usual definition of ``isomorphism\" in category theory is well-known to be useless.  An automorphism is called inner if it is given by conjugation by an invertible object.  The group of outer automorphisms is the quotient of the group of all automorphisms by the subgroup of inner automorphisms.\n\nThe Grothendieck ring or fusion ring of a fusion category is the Grothendieck group of the fusion category (that is formal differences of objects modulo the natural relations) with multiplication by tensor product.\n\n\\begin{definition} \\cite{MR2183279}\nA dimension function is an assignment of a complex number to every object of $\\mathbb{C}$ which is multiplicative under tensor product and additive under direct sum.  There is a unique dimension function which sends all non-zero objects to a positive real number, called the Frobenius-Perron dimension.  It assigns to each object $X$ the Frobenius-Perron eigenvalue of the matrix of left multiplication by $[X]$ in the Grothendieck ring.\n\\end{definition}\n\n\\begin{definition} \\cite[Def. 6]{MR1976459}\nA left module category over $\\mathscr{C}$ is a $\\mathbb{C}$-linear category $\\mathscr{M}$ together with a biexact bifunctor $\\otimes: \\mathscr{C} \\times \\mathscr{M} \\rightarrow \\mathscr{M}$ and natural associtivity and unit isomorphisms satisfying certain coherence conditions.  A right module category over $\\mathscr{C}$ and a $\\mathscr{C}$-$\\mathscr{D}$ bimodule category are defined similarly.\n\\end{definition}\n\nAs with fusion categories, a functor of module categories is a functor of the underlying category together with a binatural transformation $c_{X,M}: F(X \\otimes M) \\rightarrow F(X) \\otimes F(M)$ satisfying certain compatibility relations.  See \\cite[Def. 7]{MR1976459}.\n\n\\begin{definition}\nA module category is called indecomposable if it cannot be written as the direct sum of two module categories and simple if it is semisimple and indecomposable.\n\\end{definition}\n\n\\begin{definition} \\cite[\\S3.2]{MR1976459}\nLet $\\mathscr{M}$ be a semisimple module category over a fusion category $\\mathscr{C}$.\nLet $M_1$ and $M_2$ be two objects of $\\mathscr{M}$.  Their internal Hom, $\\underline{Hom}(M_1, M_2)$ is the (unique up to unique isomoprhism) object of $\\mathscr{C}$ which represents the functor $X \\mapsto Hom(X \\otimes M_1, M_2)$.\n\\end{definition}\n\nAs you might expect given the name, you can compose internal Homs: $$\\underline{Hom}(M_2 , M_3)\\otimes \\underline{Hom}(M_1, M_2) \\rightarrow \\underline{Hom}(M_1, M_3).$$\n\n\\begin{definition}\nGiven a semisimple fusion category $\\mathscr{C}$ and a semisimple module category $\\mathscr{M}$, we can define the Frobenius-Perron dimension of objects in $\\mathscr{M}$ by $\\dim (M) = \\sqrt{\\dim(\\underline{Hom}(M,M))}$.\n\\end{definition}\n\nOne important notion in the theory of rings and modules is that of Morita equivalence.  Two rings are Morita equivalent if there is an invertible bimodule between them.  Since fusion categories categorify rings and module categories categorify modules we should have a categorified version of Morita equivalence of fusion categories.\n\n\\begin{definition}\nA Morita equivalence between $\\mathscr{C}$ and $\\mathscr{D}$ is an invertible $\\mathscr{C}$-$\\mathscr{D}$ bimodule category, i.e. $\\mathscr{C}$-$\\mathscr{D}$ bimodule category $\\mathscr{M}$ such that there exists $\\mathscr{M}'$ a $\\mathscr{D}$-$\\mathscr{C}$ such that $\\mathscr{M} \\otimes_\\mathscr{D} \\mathscr{M}'$ is equivalent to $\\mathscr{C}$ as a $\\mathscr{C}$-$\\mathscr{C}$ bimodule category and $\\mathscr{M}' \\otimes_\\mathscr{C} \\mathscr{M}$ is equivalent to $\\mathscr{D}$ as a $\\mathscr{D}$-$\\mathscr{D}$ bimodule category.  See \\cite{MR1966524,0909.3140} for more details.\n\\end{definition}\n\nThe collection of all Morita equivalences naturally forms a $3$-groupoid.  We can think of it as just a groupoid by modding out by equivalences of bimodule categories.\n\n\\begin{definition}\nThe Brauer-Picard groupoid of a fusion category $\\mathscr{C}$ has points for every $\\mathscr{D}$ which is Morita equivalent to $\\mathscr{C}$ and an arrow for every Morita equivalence between $\\mathscr{C}$ and $\\mathscr{D}$ (up to equivalence of bimodule categories).  \n\\end{definition}\n\nThe Brauer-Picard groupoid is important in understanding the extension theory of $\\mathscr{C}$ \\cite{0909.3140}.\n\nIf $R$ is a ring and $M$ is an $R$-module, then putting an $R$-$S$ bimodule structure on $M$ is the same as giving a map of rings $S \\rightarrow \\mathrm{End}_R(M)$.  Furthermore, $M$ is invertible if and only if the corresponding map is an isomorphism.  Two such bimodules are equivalent if and only if the two maps $S \\rightarrow \\mathrm{End}_R(M)$ differ by conjugation by an invertible element in $S$.  Hence a Morita equivalence between $R$ and $S$ corresponds (non-canonically) to a pair consisting of an $R$-module $M$ whose commutant is $S$ and an outer automorphism of $S$.\n\nThe same story holds in the categorified setting as well.  If $\\mathscr{M}$ is a left $\\mathscr{C}$ module category, then giving $\\mathscr{M}$ the structure of a $\\mathscr{C}$--$\\mathscr{D}$ bimodule category is the same thing as giving a monoidal functor $\\mathscr{D} \\rightarrow \\mathscr{C}_\\mathscr{M}^*$ (where $\\mathscr{C}_\\mathscr{M}^*$ is the category of $\\mathscr{C}$ module category endofunctors of $\\mathscr{M}$).  Furthermore, a $\\mathscr{C}$--$\\mathscr{D}$ bimodule category is invertible if and only if the monoidal functor $\\mathscr{D} \\rightarrow \\mathscr{C}_\\mathscr{M}^*$ is an equivalence of monoidal categories.    Finally, two $\\mathscr{C}$--$\\mathscr{D}$ bimodule categories are equivalent, if and only if the corresponding functors $\\mathscr{D} \\rightarrow \\mathscr{C}_\\mathscr{M}^*$ differ by an inner automorphism (that is by conjugation by an invertible object in $\\mathscr{D}$).  Hence, Morita equivalences between $\\mathscr{C}$ and $\\mathscr{D}$ correspond (non-canonically) to a pair of a $\\mathscr{C}$ module category $\\mathscr{M}$ and an outer automorphism of $\\mathscr{D}$.\n\nThe above result is important because it allows us to easily count the number of Morita equivalences between two categories.  In particular, we will prove that $\\mathscr{H}_2$ has exactly three module categories and that their duals are $\\mathscr{H}_1$, $\\mathscr{H}_2$, and $\\mathscr{H}_3$.  It follows that the number of Morita equivalences between $\\mathscr{H}_2$ and $\\mathscr{H}_i$ is just the number of outer automorphisms of $\\mathscr{H}_i$.  Thus, the number of outer automorphisms is the same for each of the $\\mathscr{H}_i$.  In particular, we will prove that there are no nontrivial outer automorphisms of $\\mathscr{H}_2$ and thus that there are no nontrivial outer automorphisms of $\\mathscr{H}_1$ and $\\mathscr{H}_3$.  Hence, using the same counting argument again, there is exactly one module category over each of the $\\mathscr{H}_i$ whose dual is each of the $\\mathscr{H}_j$.\n\n\\subsection{Algebra objects}\n\nIn this subsection we sketch the definition of an algebra object in a fusion category and the relationship between algebra objects and module categories.  For more details see \\cite[\\S 3]{MR1976459}.\n\n\\begin{definition}\nAn algebra object in a fusion category  $\\mathscr{C}$ is an object $A$ together with a unit morphism $\\mathbf{1} \\rightarrow A$ and a multiplication morphism $A \\otimes A \\rightarrow A$ satisfying the usual compatibility relations (associativity, left unit, right unit).\n\\end{definition}\n\nA left module object over $A$ is defined in the obvious way, namely it is an object $M$ together with a morphism $A \\otimes M \\rightarrow M$ satisfying the usual compatibility relations.  Similarly we can define a right module object over an algebra $A$, and an $A$-$B$ bimodule object over two algebras.\n\nA particularly important example of an algebra object is the internal endomorphisms $\\underline{Hom}(M, M)$ with composition as the algebra structure.  The unit structure comes from the identity morphism from $M$ to $M$ via the definition of the internal hom.\n\nNote that if $A$ is an algebra object in $\\mathscr{C}$ then the category of left $A$-modules in $\\mathscr{C}$ is a \\emph{right} $\\mathscr{C}$-module category.  Similarly the category of right $A$-modules in $\\mathscr{C}$ is a left $\\mathscr{C}$-module category.  A key theorem of Ostrik's gives a converse to this fact.\n\n\\begin{theorem} \\cite[Theorem 1]{MR1976459}\nLet $\\mathscr{M}$ be a simple left (resp. right) module category over a fusion category $\\mathscr{C}$ and let $X$ be any simple object in $\\mathscr{M}$, then $\\mathscr{M}$ is equivalent as a module category to the category of right (resp. left) $\\underline{Hom}(X, X)$ modules in $\\mathscr{C}$.\n\\end{theorem}\n\nNote that a simple algebra $A$ is the internal endomorphisms of itself as an $A$-module, so the simple algebras in $\\mathscr{C}$ are precisely the $\\underline{Hom}(X,X)$ for $X$ in some simple module category over $\\mathscr{C}$.\n\n\\begin{definition}\nWe call a simple algebra object $A$ in $\\mathscr{C}$ \\emph{minimal} if its Frobenius-Perron dimension is minimal among all $\\underline{Hom}(X, X)$ where $X$ varies over simple objects in the category of right $A$-modules.\n\\end{definition}\n\n\\begin{corollary}\nEvery simple module category can be realized as the category of modules over a minimal algebra object.\n\\end{corollary}\n\nIf $X$ is an invertible object in a fusion category $\\mathscr{C}$, then conjugation by $X$ gives an automorphism of the fusion category $\\mathscr{C}$.  Thus if $A$ is an algebra object, then so is $X \\otimes A \\otimes X^*$ (here the multiplication is just given by contracting the middle $X^* \\otimes X$ and then multiplying in $A$).\n\n\\begin{lemma} \\label{lem:inneraut}\nIf $\\mathscr{C}$ is a fusion category, $A$ is an algebra object, and $X$ is an invertible object, then the category of left (resp. right) $A$-modules and the category of left (resp. right) $X \\otimes A \\otimes X^*$-modules are isomorphic as right (resp. left) module categories.\n\\end{lemma}\n\\begin{proof}\nThe functor is given by $V \\mapsto X \\otimes V$ (resp. $V \\mapsto V \\otimes X^*$) where the action of $X\\otimes A \\otimes X^*$ on $X \\otimes V$ is given by contracting the middle factor and then acting $A$ on $V$.  The binatural transformation is just given by the associator $(X\\otimes V) \\otimes W \\rightarrow X\\otimes (V \\otimes W)$.  The inverse functor is given by $V \\mapsto X^* \\otimes V$.  \n\\end{proof}\n\nThus in order to classify all simple module categories it is enough to classify minimal algebra objects $A$ up to inner automorphism.  This result is useful because $\\mathscr{H}_2$ has several invertible objects.\n\nAs pointed out to us by the referee, Lemma \\ref{lem:inneraut} does not hold for arbitrary automorphisms of $\\mathscr{C}$.  For example, if $\\mathscr{C}$ is the category of $G$-graded vector spaces and $A$ is the group ring of $H \\subset G$, then any subgroup conjugate to $H$ gives the same module category (namely $\\mathrm{Vec}(G\/H)$), while outer automorphisms (either coming from outer automorphisms of $G$ itself, or coming from a nontrivial $2$-cocycle) typically give a different module category.  In the special case where $G$ is abelian (so $\\mathrm{Vec}(G) \\cong \\mathrm{Rep}(\\hat{G})$) see  \\cite[Theorem 2]{MR1976459}.\n\n\\subsection{Subfactors} \\label{sec:subfactorvsalgebra}\n\nIn this subsection we recall the relationship between subfactors and algebra objects.  For more details see \\cite{MR1257245, MR1444286, MR1966524, MR1976459}.\n\nA subfactor is a unital inclusion $N \\subset M$ of von Neumann algebras with trivial centers.  The index measures the dimension of $M$ as an $N$-module.  We will only consider subfactors of a Type II$_1$ factor with finite index.  We call $N \\subset M$ irreducible if $M$ is irreducible as an $M$-$N$ bimodule.  Given a subfactor $N \\subset M$ its \\emph{principal even part} is the monoidal category of $N$-$N$ bimodules which appear as summands of tensor powers of ${}_NM_N$, the \\emph{dual even part} is the monoidal category of $M$-$M$ bimodules which appear as summands of tensor powers of ${}_M M\\otimes_N M_{M}$.   These two categories have the structure of $C^*$-tensor categories; the subfactor is said to have \\emph{finite depth} if they are fusion categories, i.e. if there are only finitely many simple objects up to isomorphism. A $C^*$-fusion category is also called a unitary fusion category.\n\nFurthermore, given a subfactor $N \\subset M$ with even parts $\\mathscr{C}$ and $\\mathscr{D}$ we have a Morita equivalence between $\\mathscr{C}$ and $\\mathscr{D}$ given by the category of $N-M$ bimodules generated by ${}_N M_M $. \n\nLet $\\kappa$ denote the $M-N$ bimodule ${}_M M _N$. We will often use sector notation: objects are labeled by Greek letters, square brackets denote isomorphism classes,  and tensor symbols are omitted, so that e.g. $\\bar{\\kappa} \\kappa $ means ${}_N M _M \\otimes_M {}_M M _N $. Then $\\bar{\\kappa} \\kappa $ is an algebra object in $\\mathscr{C}$, and the index of the subfactor is the dimension of the algebra object. Conversely, any rigid $C^*$ tensor category with countably many simple objects and simple identity object arises as a category of finite index $N$-$N$ bimodules over a factor \\cite{MR1444286, MR1960417}. Moreover, any algebra object whose multiplication is a scalar multiple of a coisometry can be realized as $\\bar{\\kappa} \\kappa $ for some subfactor $N \\subseteq M$; such algebra objects are called Q-systems \\cite{MR1257245, MR1444286}.\n\nIn general, it is not clear whether every simple algebra object in a $C^*$ fusion category gives a Q-system (as multiplication may not be a multiple of a coisometry).  However, for the Haagerup fusion categories that we consider every simple algebra object does indeed give a Q-system; thus we often elide the distinction in the statements of the major theorems.   Nonetheless we do need to explicitly check that certain algebra objects give Q-systems.  For these results the following lemma will be useful.\n\n\\begin{lemma} \\label{lem:QTransfer}\nSuppose that $\\mathscr{C}$ is a $C^*$ fusion category and that $A$ is a Q-system in $\\mathscr{C}$.  Let $\\mathscr{M}$ be the left $\\mathscr{C}$-module category of right $A$ modules, and let $X$ be a simple object in $\\mathscr{C}$.  Then $\\underline{\\mathrm{End}}(X)$ is a $Q$-system in $\\mathscr{C}$.\n\\end{lemma}\n\\begin{proof}\nSince $A$ is a $Q$-system, there's a $2$-$C^*$-category (in the sense of \\cite[\\S 7]{MR1444286}) whose objects are $1$ and $A$, whose $1$-morphisms are the bimodule objects over those algebras, and whose $2$-morphisms are maps of bimodules.  In this context $\\underline{\\mathrm{End}}(X)$ becomes $\\overline{X} X$ where we think of $X$ as a $1$-morphism between $1$ and $A$.  Hence, by \\cite[\\S 7]{MR1444286}, $\\underline{\\mathrm{End}}(X)$ is a $Q$-system.\n\\end{proof}\n\nThe right way to think of the above result is that it says that there's a good notion of $C^*$ module categories over $C^*$ fusion category $\\mathscr{C}$, and that the $Q$-system condition just says that the corresponding module category is a $C^*$ module category.  Thus we need only check the $Q$-system condition once per module category.\n\n\\begin{remark}\nNote that in the literature the generalization of $Q$-systems to the nonalgebraic context is typically that of a Frobenius Algebra \\cite{MR1966524, MR2075605}.  However, in context of simple algebras, the Frobenius trace just comes from the (unique up to rescaling) splitting of the unit morphism.\n\\end{remark}\n\nWe recall the definition of the principal graphs of a subfactor. For any two objects $\\rho$ and $\\sigma$ in a fusion category or $C^*$ tensor category $\\mathscr{C}$, let $(\\rho, \\sigma )= dim(Hom( \\rho, \\sigma ))$. \n\nLet $N \\subset M$ be a finite index subfactor, with principal and dual even parts $\\mathscr{N}$ and $\\mathscr{M}$. Let $\\kappa= {}_M M_N$, and let $\\mathscr{K} $ be the category of $M-N$ bimodules generated by $\\kappa \\mathscr{N}$. The principal graph of the subfactor is the bipartite graph with even vertices indexed by the simple objects of $\\mathscr{N}$, odd vertices indexed by the simple objects of $\\mathscr{K}$, and for any pair of simple objects $\\xi \\in \\mathscr{N}, \\eta \\in \\mathscr{K}$, $(\\kappa \\xi,\\eta ) $ edges connecting the corresponding vertices. It can be made into a pointed graph by distinguishing the even vertex corresponding to the identity object in $\\mathscr{N}$, which is denoted by $*$. The dual graph is defined the same way but with $\\mathscr{M}$ replacing $\\mathscr{N}$ and $\\bar{\\mathscr{K}}= \\mathscr{M} \\kappa $. \n\nIf the subfactor has finite depth, then the Frobenius-Perron dimensions of the objects are given by the Frobenius-Perron weights of the graphs, normalized to be $1$ at $*$, and the index of the subfactor is the squared norm of the graph.\n\nMoreover, if $\\kappa$ is any object in a semisimple module category over a fusion category, we can define the principal graph the same way.\n\n\\subsection{The Haagerup subfactor}\n\nThe Haagerup subfactor \\cite{MR1686551} is a finite-depth subfactor with index $\\frac{5+\\sqrt{13}}{2} $; this is the smallest index above $4$ for any finite depth subfactor \\cite{MR1317352}. The Haagerup subfactor is unique, up to duality. It has the following principal and dual graphs:\n\n$$\\hpic{hgraphs} {2in} $$\n\nWe will call the fusion category with four simple objects $\\mathscr{H}_1$ and the one with six simple objects $\\mathscr{H}_2 $. The Frobenius-Perron dimensions of the simple objects are $d(\\alpha)=1$, $d = d(\\xi)=d(\\eta)=d(\\mu)+1=d(\\nu)-1=\\frac{3+\\sqrt{13}}{2}$, $d(\\kappa)=\\sqrt{d+1} $, $d(\\lambda)=\\sqrt{(d+1)(d+2)} $. \n\nThe fusion ring for $\\mathscr{H}_2$ will be called $H_6$; it satisfies the relations $[\\alpha^3]=[1], [\\alpha \\xi] = [\\xi \\alpha^2]$, and $[\\xi^2]=[ 1 ]\\oplus[\\xi ] + [\\alpha \\xi ] + [ \\alpha^2 \\xi]$. The fusion ring for $\\mathscr{H}_1 $ will be called $H_4$; it is commutative and the fusion rules are determined by the ring homomorphism property of dimension, self-duality of all simple objects, and Frobenius reciprocity.\n\n\\begin{table}\n\\begin{tabular}{ c || c | c | c | c | c | c }\n\n                 & $1$             & $\\alpha$       & $\\alpha^2 $ & $\\xi$          & $\\alpha \\xi$     & $\\alpha^2 \\xi $  \\\\ \\hline \\hline\n$1$              & $1$             & $\\alpha$       & $\\alpha^2 $ & $\\xi$          & $\\alpha \\xi$     & $\\alpha^2 \\xi $ \\\\ \\hline\n$\\alpha$         & $\\alpha$        & $\\alpha^2$     & $1 $        & $\\alpha \\xi$   & $ \\alpha^2 \\xi$  & $\\xi $ \\\\ \\hline\n$\\alpha^2 $      & $\\alpha^2$      & $1$            & $\\alpha $   & $\\alpha^2 \\xi$ & $\\xi$            & $\\alpha \\xi $ \\\\ \\hline\n$\\xi$            & $\\xi$           & $\\alpha^2 \\xi$ & $\\alpha \\xi$& $1+Z$ & $\\alpha^2+Z$ & $\\alpha+Z$ \\\\ \\hline\n$\\alpha \\xi $    & $\\alpha \\xi$    & $\\xi$           & $\\alpha^2 \\xi $ & $\\alpha+Z$ & $1+Z$ & $\\alpha^2+Z$ \\\\ \\hline\n$\\alpha^2 \\xi  $ & $\\alpha^2 \\xi$  & $\\alpha \\xi$   & $\\xi $ & $\\alpha^2+Z$ & $\\alpha+Z$ & $1+Z$ \\\\\n\\end{tabular}\n\\caption{ $H_6$ multiplication table.  We use the abbreviation $Z = \\xi+\\alpha\\xi+\\alpha^2\\xi$}\n\\end{table}\n\\begin{table} \n\\begin{tabular}{ c || c | c | c | c}\n\n            & $1$   & $\\nu$                  & $\\eta $ & $\\mu$    \\\\ \\hline \\hline\n $1$  \t    & $1$   & $\\nu$                  & $\\eta $ & $\\mu$    \\\\ \\hline \n  $\\nu$     & $\\nu$ & $1+2\\nu+2\\eta+\\mu$     & $2\\nu+\\eta+\\mu $ & $\\nu+\\eta+\\mu$    \\\\ \\hline \n   $\\eta $  & $\\eta$ & $2\\nu+\\eta+\\mu$                 & $1+\\nu+\\eta+\\mu$ & $\\nu+\\eta$    \\\\ \\hline\n  $\\mu$     & $\\mu$  & $\\nu+ \\eta+\\mu$       & $\\nu+\\eta $ & $1+\\nu$    \\\\ \n\\end{tabular}\n\\caption{ $H_4$ multiplication table}\n\\label{h4mult}\n\\end{table}\n\n\n\n\nThe fusion category $\\mathscr{H}_2$ has two non-trivial invertible objects: $\\alpha$ and $\\alpha^2$.  The inner automorphism given by conjugation by $\\alpha$ cyclically permutes $\\xi$, $\\alpha \\xi$, and $\\alpha^2 \\xi$.  \n\nThe full subcategory generated by $\\alpha$ has three invertible objects and is thus monoidally equivalent to $\\mathrm{Vec}(\\mathbb{Z}\/3\\mathbb{Z}, \\omega)$ for some associator $\\omega \\in H^3(\\mathbb{Z}\/3, \\mathbb{C}^*)$.  In fact, as was pointed out to us by David Jordan, this cocycle must be trivial.\n\n\\begin{lemma} \\label{lem:cocycle}\nThe full subcategory generated by $\\alpha$ is equivalent as a fusion category to the category of $\\mathbb{Z}\/3\\mathbb{Z}$-graded vector spaces.  \n\\end{lemma}\n\\begin{proof}\nNotice that since $\\alpha \\lambda = \\lambda$, the category of vector spaces (thought of as sums of $\\lambda$) is a module category over $\\mathscr{D}$.  Hence $\\mathscr{D}$ has trivial associator.\n\\end{proof}\n\n\n\n\n\n\\section{Algebra objects, principal graphs and subfactors in the Haagerup categories}\n\nThe goal of this section is to classify all simple algebra objects in each of the $\\mathscr{H}_i$, and to classify all indecomposable module categories over each of the $\\mathscr{H}_i$.  The outline of the argument is as follows.  We use combinatorics to describe the possible objects which could have a simple algebra structure.  However, this list is somewhat large, and since some of the objects are relatively complex it is difficult to determine how many algebra structures each such object admits. Fortunately, in order to classify all module categories it is enough to only consider the algebra objects whose dimensions are minimal among all algebras which yield the same module category.  Furthermore we need only consider these algebra objects up to inner automorphisms of the category.\n\nThere are many fewer candidates for these minimal algebra objects and we are able to easily determine when the algebra structure exists and that it is unique when it does exist.  Thus we obtain a complete list of all indecomposable module categories, and using the internal Hom construction we are able to read off the full list of (not necessarily minimal) simple algebra objects.   We do this first for $\\mathscr{H}_2$, and then use this classification to read off the same classification for the other $\\mathscr{H}_i$.\n\nIn essence what we are doing is moving back-and-forth between algebra objects and module categories in order to exploit the more accessible combinatorial structure of algebra objects and the more rich algebraic structure of module categories.  This interplay allows us to avoid computations which would otherwise be extremely difficult.  To illustrate the general technique we prove the following result which was proved with considerable effort in the appendix of \\cite{MR2418197}.\n\n\\begin{lemma}\nThere exists a unique simple algebra structure on $1 + \\nu$ in $\\mathscr{H}_1$.  This algebra is also a Q-system.\n\\end{lemma}\n\\begin{proof}\nSince $1 + \\nu \\cong \\mu \\bar{\\mu}$, it has at least one algebra structure given by contraction.  Furthermore, since the algebra $1$ is a $Q$-system and $\\mu$ is an object in the category of $1$-$1$ bimodules, by Lemma \\ref{lem:QTransfer} this algebra is a $Q$-system.\n\nNow suppose that we have any algebra structure on $1 + \\nu$.  A simple combinatorial calculation (see Example \\ref{ex}) shows that the principal graph of the corresponding subfactor must be \\hpic{h7} {.3in}\n\nThe vertex all the way on the right is an odd vertex of dimension $1$ which we call $ \\theta$.  Note that as an odd vertex, $\\theta$ is a simple object in the category of $(1 + \\nu)$-modules.  A dimension count shows that $\\underline{\\mathrm{Hom}}(\\theta,\\theta) \\cong 1$, and so the category of $(1+ \\nu)$-modules is equivalent to the category of $1$-modules which is just  $\\mathscr{H}_1$ itself with the usual module action.  Hence, the algebra structure on $1 + \\nu$ can be realized as the internal endomorphisms of some object in $\\mathscr{H}_1$.  A dimension count shows that this object must be $\\mu$, hence we have that $1 + \\nu \\cong \\underline{\\mathrm{Hom}}(\\mu,\\mu) \\cong \\mu \\bar{\\mu}$ as algebra objects.\n\\end{proof}\n\nWe now turn to the general question of classifying all simple algebra objects in and all simple module categories over $\\mathscr{H}_1$ and $\\mathscr{H}_2$.\n\nLet $\\mathscr{C}$ be a fusion category, and let $ \\mathscr{K}_{\\mathscr{C}}$ be a module category over $\\mathscr{C}$. Let $\\xi_0=1,\\xi_1,...\\xi_n$ be an enumeration of the simple objects in $\\mathscr{C}$, and let $\\kappa_0, \\kappa_1, ... \\kappa_m$ be an enumeration of the simple objects in $\\mathscr{K}$. Let $\\kappa$ be an object in $\\mathscr{K}$. \n\n\\begin{definition}\n The fusion matrix of $\\kappa$ is the matrix $(F^{\\kappa}_{ij} )_{0 \\leq i \\leq n, 0 \\leq j \\leq m}$ given by $F^{\\kappa}_{ij}=( \\kappa \\xi_i, \\eta_j) $.\n\\end{definition}\n\nNote that the fusion matrix is only defined up to a choice of ordering of the simple objects.\n\n\\begin{example}\n Let $N \\subset M$ an irreducible, finite-index, finite depth subfactor, and let $\\kappa= {}_M M_N$. Then the fusion matrix $A=F^{\\kappa}$ is an adjacency matrix of the principal graph, with the first row corresponding to $*$. In this case we take the convention $\\kappa_0=\\kappa$, so that the first column of the matrix gives the edges emanating from $\\kappa$.\n\\end{example}\n\nWe would like to figure out which objects in a fusion category can admit a simple algebra structure. By an abuse of notation, we will often omit reference to the algebra structure and refer to an object $\\gamma$, or even its isomorphism class $[\\gamma]$, as an algebra.\n\n\n\\begin{lemma}\nLet $\\gamma= 1 + \\sum_{i=1}^n \\limits a_i \\xi_i  $ be a simple algebra object in a fusion category $\\mathscr{\\mathscr{C}}$. Let $\\kappa$ be a simple right $\\gamma$-module such that $\\gamma \\cong \\underline{\\mathrm{Hom}}(\\kappa,\\kappa)$. Then $F^{\\gamma}=AA^T $, where $A=F^{\\kappa}$ is the adjacency matrix of the principal graph of $\\kappa $ (with the same ordering of the simple objects of $\\mathscr{C}$).\n\\end{lemma}\n\\begin{proof}\nFix $\\kappa$ such that $\\gamma \\cong \\bar{\\kappa} \\kappa $. Then we have $F^{\\gamma}_{ij}=(\\gamma \\xi_i, \\xi_j ) =( \\bar{\\kappa} \\kappa \\xi_i ,\\xi_j )= (\\kappa \\xi_i, \\kappa \\xi_j)$. On the other hand, $(AA^T)_{ij}= \\sum_{l}(\\kappa \\xi_i, \\kappa_l )(\\kappa \\xi_j, \\kappa_l )=(\\kappa \\xi_i, \\kappa \\xi_j )$. \n\\end{proof}\n\nWe will call the graph given by $A$ a principal graph of the algebra object. Note that it is not uniquely determined by the object $\\gamma$, although it is uniquely determined by the algebra structure. Nevertheless in many cases there is at most one possible choice for the graph given an object $\\gamma$. If $\\gamma = 1 + \\sum_{i=1}^n \\limits a_i \\xi_i  $ is a simple algebra object, then as we have seen the first column of $A$ is given by the cofficients $1, (a_i)$ of $\\gamma$, and the rest of the first row is $0$. It is therefore sometimes convenient to lop off the first row and column when doing computations.\n\\begin{definition}\n The reduced fusion matrix of $\\gamma=1 + \\sum_{i=1}^n \\limits a_i \\xi_i $,\\\\ $F^{\\gamma,r}$,  is given by $F^{\\gamma, r}_{ij}=F^{\\gamma}_{ij} - a_i a_j=\\sum_{k=0}^n \\limits a_k (\\xi_k \\xi_i, \\xi_j) -a_i a_j $ for $1 \\leq i,j \\leq n$. A reduced principal graph of $\\gamma $ is a graph given by the matrix $A^r$, defined by $ A^r_{ij}=A_{ij}$ for $i,j \\geq 1$, where $A$ is the adjacency matrix of a principal graph of $\\gamma$. \n\\end{definition}\n\\begin{lemma} \\label{findgraphs}\nWe have $F^{\\gamma, r}=A^r(A^r)^T $ (with the same ordering of simple objects of $\\mathscr{C}$).\n\\end{lemma}\n\\begin{proof}\n By definition, we have $F^{\\gamma, r}_{ij}=F^{\\gamma}_{ij} - a_i a_j=(AA^T)_{ij}-a_ia_j=(A^r(A^r)^T)_{ij}$.\n\\end{proof}\nThis allows us to quickly find possible algebra objects in a fusion category by checking which objects have reduced fusion matrices that decompose as $AA^T$ for some matrix $A$ all of whose entries are nonnegative integers. Then the reduced principal graph is given by such an $A$, and the full principal graph is obtained by adding the vertices corresponding to $1$ and $\\kappa$. \n\nWe now apply this analysis to the Haagerup fusion categories. \n\n\\begin{example} \\label{ex}\n\nWe explain the method for a few objects in $\\mathscr{H}_1$. We always use the following ordering of simple objects: $1, \\nu,  \\eta, \\mu $.\n\n(a) Let $\\gamma=1 + \\nu $. Then the reduced fusion matrix can be computed using \\ref{h4mult}; it is: \n$\\begin{pmatrix}\n 2 & 2 & 1\\\\\n2 & 2 & 1\\\\\n 1 & 1 & 2\\\\ \n\\end{pmatrix}$.\nIt is easy to see that up to graph equivalence, the only candidate for $A$ is: \n$\\begin{pmatrix}\n 1 & 1 & 0\\\\\n1 & 1 & 0\\\\\n 1 & 0 & 1\\\\\n\\end{pmatrix}$.\nAdding a column with the coefficients of the simple objects in $\\gamma$,  $1,1,0,0$, along with a row of zeros on top gives\n$\\begin{pmatrix}\n1 & 0 & 0 & 0\\\\\n 1 & 1 & 1 & 0\\\\\n0 & 1 & 1 & 0\\\\\n0 & 1 & 0 & 1\\\\ \n\\end{pmatrix}$. Therefore the only graph compatible with an algebra structure on $\\gamma$ is \\hpic{h7} {0.4in} .\n \n(b) $\\gamma=1 + 2\\nu$. Then the reduced fusion matrix is \n$\\begin{pmatrix}\n 1 & 4 & 2\\\\\n4 & 3 & 2\\\\\n 2 & 2 & 3\\\\ \n\\end{pmatrix}$.\nSince this matrix is not positive semi-definite, it does not decompose as $AA^T$ and $\\gamma $ does not admit an algebra structure.\n\n(c) Let $\\gamma=1 + 4\\nu + 3\\eta + 2\\mu$. Then the reduced fusion matrix is \n$\\begin{pmatrix}\n 1 & 1 & 1\\\\\n1 & 1 & 1\\\\\n 1 & 1 & 1\\\\ \n\\end{pmatrix}$,\nwhich decomposes as $AA^T$ only when $A$ is the matrix:\n $\\begin{pmatrix}\n 1\\\\\n1\\\\\n 1\\\\ \n\\end{pmatrix}$;\nthe principal graph is then given by\n$\\begin{pmatrix}\n1 & 0 \\\\\n 4 & 1 \\\\\n3 & 1 \\\\\n2 & 1 \\\\ \n\\end{pmatrix}$.\nThe Frobenius-Perron weight corresponding to the second column is $\\sqrt{3}$. Suppose such an algebra object exists, and let $\\kappa'$  be the simple object with dimension $\\sqrt{3} $. Then $[\\kappa' \\bar{\\kappa'}]=[1]+[ \\sigma] $, where $[\\sigma] $ is a nonnegative integral linear combination of $[\\nu], [\\eta], [\\mu] $. Since $d(\\sigma )=3-1=2$, this is impossible. Therefore $\\gamma $ does not admit an algebra structure.\n\\end{example}\n\nIn order to turn the classification of simple algebra objects in a fusion category into a finite problem we need a bound on the size of possible simple algebra objects.\n\n\\begin{lemma}\\label{lem:algebra-bound}\nIf $\\gamma$ is a simple algebra object and $\\xi$ is any simple object, then $(\\gamma, \\xi ) \\leq \\dim(\\xi)$.\n\\end{lemma}\n\\begin{proof}\nThis result is well-known in the subfactor context, but we quickly prove it in order to see that it works in the fusion category context.  Recall that $\\gamma = \\eta \\bar{\\eta}$ for some simple object $\\eta$ in a module category (namely $\\eta$ is just $\\gamma$ as a $\\gamma$-module).  By Frobenius reciprocity $(\\gamma, \\xi) = (\\eta, \\xi \\eta)$.  Since $\\eta$ is simple, $(\\eta, \\xi \\eta)$ just measures the number of copies in $\\xi \\eta$.  Since Frobenius-Perron dimensions are always positive, $(\\eta, \\xi \\eta)$ is at most $\\dim \\xi \\eta\/\\dim \\eta = \\dim \\xi$.\n\\end{proof}\n\n\\begin{lemma}\nLet $\\gamma $ be a nontrivial simple algebra object in a fusion category with fusion ring isomorphic to $H_4$. Then any principal graph of $\\gamma$ is one of the following seven graphs, with the indicated indices and indicated even objects:\\\\ \n\n(1) \\hpic{h1} {0.6in}  $\\displaystyle \\frac{5+\\sqrt{13}} {2}$, (2) \\hpic{h2} {0.6in} $12+3\\sqrt{13}$ \\\\\n\n\n(3) \\hpic{h3} {0.6in}  $4+\\sqrt{13}$, (4) \\hpic{h4} {1.1in}  $\\displaystyle \\frac{11+3\\sqrt{13}} {2}$ \\\\\n\n\n(5) \\hpic{h5} {0.6in} $\\displaystyle \\frac{15+3\\sqrt{13}} {2}$, (6) \\hpic{h6} {0.6in} $\\displaystyle \\frac{19+5\\sqrt{13}} {2}$ \\\\\n\n\n(7) \\hpic{h7} {0.4in} $\\displaystyle \\frac{7+\\sqrt{13}} {2}$ \\\\\n\nFurthermore, of the above algebra objects the only minimal ones are (1) and (3).\n\\end{lemma}\n\n\n\n\n\n\\begin{lemma} \\label{lem:H6}\nLet $\\gamma$ be a nontrivial simple algebra object in a fusion category with fusion ring isomorphic to $H_6$. Then up to inner automorphism of the category any principal graph of $\\gamma$ is one of the following seven graphs, with the indicated indices and even objects:\\\\\n\n(1') \\hpic{h8} {0.6in}  $\\displaystyle \\frac{5+\\sqrt{13}} {2}$, (2') \\hpic{h9} {0.6in}  $12+3\\sqrt{13}$ \\\\\n\n\n(3') \\hpic{h10} {1in}  $4+\\sqrt{13}$, (4') \\hpic{h11} {1.1in}  $\\displaystyle \\frac{11+3\\sqrt{13}} {2}$ \\\\\n\n\n(5') \\hpic{h12} {0.5in}  $\\displaystyle \\frac{15+3\\sqrt{13}} {2}$, (6') \\hpic{h13} {0.6in}  $\\displaystyle \\frac{33+9\\sqrt{13}} {2}$ \\\\\n\n\n(7') \\hpic{h14} {0.4in}  $3$  \\\\\n\n\nFurthermore, of the above algebra objects the only minimal ones are (1'), (3'), and (7').\n\\end{lemma}\n\n\\begin{proof}\nUsing Lemma \\ref{lem:algebra-bound} and Lemma \\ref{findgraphs} this is a tedious calculation. The only other admissible graphs encountered are two graphs each for $[1]+[\\mu]+[\\nu] $ and $[1] + 2[\\mu] + [\\nu] + [\\eta] $, which are eliminated due to having odd vertices violating the Jones index restriction; and the graph mentioned in Example \\ref{ex}, (c), which was eliminated there.\n\\end{proof}\n\n\\begin{corollary}\nIf $\\mathscr{M}$ is a left module category over $\\mathscr{H}_2$ then $\\mathscr{M}$ is the category of right $A$ modules for some algebra structure on one of the objects $1$, $1+ \\xi$, $1+\\alpha+\\alpha^2$, or $1+\\xi + \\alpha \\xi$.\n\\end{corollary}\n\\begin{proof}\nFrom the above lemma any minimal algebra object is of one of these forms. \n\\end{proof}\n\n\\begin{lemma}\nThere exists a unique algebra object structure on the object $1$ in $\\mathscr{H}_2$.  This algebra object is a $Q$-system.\n\\end{lemma}\n\\begin{proof}\nThe proof is immediate.\n\\end{proof}\n\n\\begin{lemma}\nThere exists a unique algebra object structure on the object $1+\\xi$ in $\\mathscr{H}_2$.  This algebra object is a $Q$-system.\n\\end{lemma}\n\\begin{proof}\nExistence of a $Q$-system follows from the existence of the Haagerup subfactor.  Uniqueness of the algebra follows from $3$-supertransitivity.  Namely, since $\\xi \\xi$ only contains one copy of $1$ and only one copy of $\\xi$, the multiplication and unit morphisms must lie inside Temperley-Lieb.  The uniqueness of the algebra structure inside Temperley-Lieb is a straightforward well-known calculation.\n\\end{proof}\n\n\\begin{lemma}\nThere exists a unique algebra object structure on the object $1+\\alpha+\\alpha^2$ in $\\mathscr{H}_2$.  This algebra object is a $Q$-system.\n\\end{lemma}\n\\begin{proof}\n$\\mathscr{H}_2$ has a fusion full subcategory $\\mathscr{D}$ consisting of sums of $1$, $\\alpha$, and $\\alpha^2$.  By Lemma \\ref{lem:cocycle} this category is equivalent to $\\mathbb{Z}\/3\\mathbb{Z}$-graded vector spaces.    Now existence and uniqueness follow from the same fact about the category of $\\mathbb{Z}\/3\\mathbb{Z}$-graded vector spaces (which is essentially just the existence and uniqueness of the $D_4$ subfactor).\n\\end{proof}\n\n\\begin{lemma}\\label{noalg}\nThere is no algebra object structure on  $1+ \\xi + \\alpha \\xi$.\n\\end{lemma}\n\\begin{proof}\nSuppose $1+\\xi+\\alpha \\xi$ had an algebra object structure. Let $\\kappa$ be $1+\\xi+\\alpha \\xi$ as a left module over itself (so that $1+\\xi+\\alpha\\xi \\cong \\kappa \\bar{\\kappa} $).  Let $\\iota$ be $1+\\alpha^2 \\xi$ as a right module over itself (so that $1+\\alpha^2 \\xi \\cong \\bar{\\iota } \\iota$). Then $\\iota \\kappa $ is a $(1+\\xi+\\alpha\\xi)-(1+\\alpha^2 \\xi)  $ bimodule. Moreover, by Frobenius reciprocity, $(\\iota \\kappa, \\iota \\kappa )=(\\bar{\\iota} \\iota, \\kappa \\bar{\\kappa} ) $, so $\\iota \\kappa $ is irreducible. Then  $\\iota \\kappa \\bar{\\kappa} \\bar{\\iota}$ must be a simple algebra object in the category of $(1+\\alpha^2 \\xi)-(1+\\alpha^2 \\xi)$-bimodules, which is $\\mathscr{H}_1$.  The algebra $\\iota \\kappa \\bar{\\kappa} \\bar{\\iota}$ has dimension $\\mathrm{dim}(1+\\alpha^2 \\xi )\\cdot \\mathrm{dim}(1+\\xi+\\alpha\\xi )=\\frac{33+9\\sqrt{13}}{2} $. But there is no admissible principal graph in $\\mathscr{H}_1 $ with that index. \n\n\\end{proof}\n\n\\begin{corollary}\nThere are exactly three simple module categories over $\\mathscr{H}_2$, namely the category of $1$-modules, the category of $(1+\\xi)$-modules and the category of $(1+\\alpha+\\alpha^2)$-modules for each of the above algebra structures.  These module categories have the following graphs for fusion with any of the objects of dimension $\\frac{3+\\sqrt{13}}{2}$.\n\n\\hpic{m4} {.6in} \\quad\n\\hpic{m5} {.6in} \\quad\n\\hpic{m6} {.4in}\n\\end{corollary}\n\\begin{proof}\nWe have classified all minimal algebra objects up to inner automorphism of the fusion category; thus we have classified all simple module categories.  The fusion graphs can be read off directly from the principal graphs.\n\\end{proof}\n\nFrom the above it is also easy to read off the complete list of simple algebra objects in $\\mathscr{H}_2$, and thus all of the subfactors whose principal even part is $\\mathscr{H}_2$.  Furthermore, their principal graphs can be easily identified on the list in Lemma \\ref{lem:H6}. However, it is a bit more work to identify the dual even parts and the dual principal graphs.\n\n\\begin{lemma} \\label{lem:dualspart1}\nThe category of $1$-$1$ bimodules in $\\mathscr{H}_2$ is $\\mathscr{H}_2$.\n\\end{lemma}\n\\begin{proof}\nObvious.\n\\end{proof}\n\n\\begin{lemma}\\label{lem:dualspart2}\nThe category of $(1+\\xi)$-$(1+\\xi)$ bimodules in $\\mathscr{H}_2$ is $\\mathscr{H}_1$.\n\\end{lemma}\n\\begin{proof}\nThis follows from the definitions of $\\mathscr{H}_2$ and $\\mathscr{H}_1$ as the even parts of the Haagerup subfactor.\n\\end{proof}\n\n\\begin{definition} \\label{lem:dualspart3}\nLet $\\mathscr{H}_3$ be the category of $(1+\\alpha+\\alpha^2)$-$(1+\\alpha+\\alpha^2)$ bimodules in $\\mathscr{H}_2$.\n\\end{definition}\n\nNote that we do not yet know that $\\mathscr{H}_3$ is distinct from $\\mathscr{H}_1$ and $\\mathscr{H}_2$.\n\n\n\\begin{definition}\n If $\\mathscr{C}$ and $\\mathscr{D}$ are fusion categories, a subfactor $N \\subset M$ will be called a $\\mathscr{C}-\\mathscr{D}$ subfactor if its principal even part is  $\\mathscr{C}$ and its dual even part is $\\mathscr{D}$.  Wild cards will be used when one of the categories is unknown or unspecified.\n\\end{definition}\n\n\n\n\n\n\n\n\n\n\\begin{lemma}\\label{existence}\nThe object $1+\\mu+\\nu$ in $\\mathscr{H}_1$ has a $Q$-system structure.  The corresponding subfactor is an $\\mathscr{H}_1$-$\\mathscr{H}_3$ subfactor and its principal graphs are (3) and (3').\n\\end{lemma}\n\\begin{proof}\n Let $N \\subset M$ be an $\\mathscr{H}_1-\\mathscr{H}_2 $ subfactor with principal graphs (2) and (2'). From the graph (2') we see that there is an intermediate subfactor $N \\subset P \\subset M$ with $[M:P] = 3$.  Note that, by definition of $\\mathscr{H}_3$, we have that $P \\subset M$ induces a Morita equivalence between $\\mathscr{H}_3$ and $\\mathscr{H}_2$.  Thus, $N \\subset P $ is an $\\mathscr{H}_1-\\mathscr{H}_3 $ subfactor which must have principal graph (3), since that is the only $H_4$ compatible graph with the right index. \n\n\\begin{figure} \n\\begin{center}\n\\hpic{part_graph} {1.2in}  $\\rightarrow$ \\hpic{part2_graph} {1.2in} $\\rightarrow$ \\hpic{part3_graph} {1.2in}\n\\caption{Schematic representation of the graph computation} \\label{buildinggraph}\n\\end{center}\n\\end{figure}\n\n\nTo compute the dual graph, let $\\iota={}_N P_P $ and $\\kappa={}_P M_M $. Note that $P \\subset M $ is an $*-\\mathscr{H}_2$ subfactor, so it must have dual graph (7'), which implies that is also has principal graph (7'). Let $\\beta$ be a dimension one $P-P$ bimodule such that $[\\kappa \\bar{\\kappa}]=[1] + [\\beta] + [\\beta^2]$. Then by Frobenius reciprocity, $(\\bar{\\iota} {\\iota}, \\kappa \\bar{\\kappa} )=(\\iota \\kappa, \\iota \\kappa ) =1$. In a similar way we find $([(\\bar{\\iota} {\\iota})^2], [\\beta] + [\\beta^2])=2 $. This implies that there are two other vertices in the dual graph (labeled by $\\beta$ and $\\beta^2$ sharing an order three symmetry with $*$ (the vertex labeled by $1$), such that there are no length two paths but there are two length four paths from $*$ to these vertices. Moreover, by Frobenius reciprocity, the vertex corresponding to the fundamental object in the dual graph is trivalent, and therefore so are the (unique) vertices adjacent to the other two dimension one vertices. (See Figure \\ref{buildinggraph}.) \nTaken together, this implies that the graph contains a triangle of odd vertices, with one even vertex in the middle of each side (which we label by $\\sigma, \\beta \\sigma, \\beta^2 \\sigma$) and one even vertex (labeled by $1,\\beta,\\beta^2$) attached by an edge to each corner of the triangle (labeled by $\\bar{\\iota}, \\beta \\bar{\\iota}, \\beta^2 \\bar{\\iota}$). There is one additional odd vertex (which we label by $\\zeta$), which must be connected to the vertices labeled by $\\sigma, \\beta \\sigma, \\beta^2 \\sigma $. By symmetry, it must be connected to all three of them by at least one edge. Adding these three edges to the graph gives the correct index, so the graph can't be any larger. \n\\end{proof}\n\n\\begin{corollary}\n $\\mathscr{H}_3 $ is not equivalent to $\\mathscr{H}_1 $ or $\\mathscr{H}_2$.\n\\end{corollary}\n\n\\begin{proof}\n Clearly, $\\mathscr{H}_3 $ is not equivalent to $\\mathscr{H}_1 $ (they are already different at the level of fusion rings). Suppose $\\mathscr{H}_3 $ were equivalent to $\\mathscr{H}_2$. Then from the dual graph (3'), we see that $1+\\xi+\\alpha \\xi $ would have to have an algebra structure. But by Lemma \\ref{noalg}, that is not the case.\n\\end{proof}\n\n\n\\begin{lemma}\\label{fring}\n Let $N \\subset M $ be a subfactor with principal graphs (3) and (3'). Then its dual even part has fusion ring isomorphic to $H_6$.\n\\end{lemma}\n\\begin{proof}\n From the graph (3'), we see that the $M-M$ fusion ring contains three dimension one objects (as before we will label them by $1, \\beta, \\beta^2$), which generate a subring isomorphic to $\\mathbb{Z} \/ 3\\mathbb{Z}$. Then there are there are three objects of dimension $\\frac{3+\\sqrt{13}}{2} $ (as before labeled $\\sigma, \\beta \\sigma, \\beta^2 \\sigma $), with $\\beta \\sigma$ and  $\\beta^2 \\sigma$ the two new objects at depth two. Then either $[\\beta \\sigma]=[ \\bar{ \\beta \\sigma}] $ or $[\\beta \\sigma] =[ \\bar{\\beta^2 \\sigma}]$. Since the two new objects in depth two in graph (3) have different dimensions, they are each self-conjugate, so the former holds by \\cite[\\S 3.3]{1007.1730}  (using an annular tangles argument \\cite{MR1929335}). Then $[\\beta \\sigma] = [\\bar{\\beta \\sigma}]=[ \\sigma \\beta^2 ]$, and the isomorphism to $H_6$ is easy to deduce.   \n\\end{proof}\n\n\\begin{corollary}\n There exists a unitary fusion category with fusion ring $H_6$ which is Morita equivalent to $\\mathscr{H}_2 $ but not isomorphic to it.\n\\end{corollary}\n\n\n\n\n\n\\begin{theorem} \\label{thm:H1H3}\nThere are exactly three simple module categories over each of $\\mathscr{H}_1$ and $\\mathscr{H}_3$.  In each case the three dual fusion categories are the three $\\mathscr{H}_i$.  The simple module categories over $\\mathscr{H}_1$ have the following graphs for fusion with the object of dimension $\\frac{1+\\sqrt{13}}{2}$.\n\n\\hpic{m1} {.3in} \\quad \\hpic{m2} {.6in} \\quad \\hpic{m3} {.6in}\n\nThe fusion graphs for the module categories over $\\mathscr{H}_3$ (with respect to any of the $\\frac{3+\\sqrt{13}}{2}$ dimensional objects) are: \n\n\\hpic{m4} {.6in} \\quad\n\\hpic{m7} {.6in} \\quad\n\\hpic{m6} {.4in}\n\\end{theorem}\n\\begin{proof}\nLet $M$ be the number of Morita equivalences between $\\mathscr{H}_i$ and $\\mathscr{H}_j$ (this is clearly independent of $i$ and $j$), let $O_i$ be the number of outer automorphisms of $\\mathscr{H}_i$, and let $B_{ij}$ be the number of simple module categories over $\\mathscr{H}_i$ whose dual is isomorphic to $\\mathscr{H}_j$.  We know that $M = O_j B_{ij}$ and that $B_{2j} = 1$ for any $i$ and $j$.  Hence, we have that $M = O_j B_{2j} = O_j$ for all $j$.  Thus we conclude that $B_{ij} = M\/O_j = 1$.  Hence there are exactly three simple module categories over each $\\mathscr{H}_i$.\n\nThe $\\mathscr{H}_1$ module category whose dual is  $\\mathscr{H}_1$ is trivial, and so its fusion graph can be read off from the fusion rules for $\\mathscr{H}_1$.  The $\\mathscr{H}_1$ module category whose dual is $\\mathscr{H}_2$ comes from the Haagerup subfactor, and its fusion graph can be read off from fusion rules for the Haagerup subfactor.  Finally, the $\\mathscr{H}_1$ module category whose dual is $\\mathscr{H}_3$ can be read off from the dual principal graph (3').\n\nThe calculations for $\\mathscr{H}_3$ are similar.\n\\end{proof}\n\\begin{corollary}\nAny fusion category Morita equivalent to the principal even parts of the Haagerup subfactor must be isomorphic to $\\mathscr{H}_1, \\mathscr{H}_2 $ or $\\mathscr{H}_3 $.\n\\end{corollary}\n\n\n\n\n\n\\begin{theorem} \\label{subthm}\n The complete list of irreducible subfactors whose even parts are Morita equivalent to $\\mathscr{H}_i$ is as follows; each exists and is unique up to isomorphism of the planar algebra. Note that the $\\mathscr{H}_i-\\mathscr{H}_i $ subfactors are self-dual while the $\\mathscr{H}_i-\\mathscr{H}_j $ subfactors for $i \\neq j $ come in non-self-dual pairs; we only list each pair once.\n\n(a) $\\mathscr{H}_1-\\mathscr{H}_2$: One with principal graphs (1)-(1') and one with principal graphs (2)-(2').\n\n(b) $\\mathscr{H}_1-\\mathscr{H}_3$: One subfactor with principal graphs (3)-(3') and one with principal graphs (5)-(5').\n\n(c) $\\mathscr{H}_2-\\mathscr{H}_3$: One subfactor with principal graphs (6')-(6'). \n\n(d) $\\mathscr{H}_1-\\mathscr{H}_1$: One subfactor each with principal graphs (4)-(4), (6)-(6), and (7)-(7).\n\n(e) $\\mathscr{H}_2-\\mathscr{H}_2$: One subfactor with principal graph (4')-(4').\n\n(f) $\\mathscr{H}_3-\\mathscr{H}_3$: One subfactor with principal graph (4')-(4').\n\\end{theorem}\n\n\\begin{proof}\n For each choice of $\\mathscr{H}_i-\\mathscr{H}_j$, we have a unique bimodule category up to isomorphism, so all the irreducible subfactors come from taking internal endomorphisms of simple objects in that bimodule category. The odd bimodules in the previously constructed $\\mathscr{H}_1-\\mathscr{H}_2$ and $\\mathscr{H}_1-\\mathscr{H}_3$ subfactors give us the list of simple objects in those two categories; note that in each case the three odd bimodules of the same dimension give the same subfactor, since they are just twists by the $\\mathbb{Z}\/3$ action. Similarly, looking at the even bimodules in those subfactors give the $\\mathscr{H}_i-\\mathscr{H}_i$ type subfactors. That leaves $\\mathscr{H}_2-\\mathscr{H}_3$; an $\\mathscr{H}_2-\\mathscr{H}_3$ subfactor with index $\\frac{33+\\sqrt{13}}{2} $ can be constructed as in the proof of Lemma \\ref{noalg}. It must have principal graphs (6) and (6'), from which the list of simple objects and subfactors may be obtained.    \n\\end{proof}\n\n\\section{The intermediate subfactor lattices}\n\nAnother important analogue of ``subgroups'' which appears in subfactor theory is the lattice of intermediate subfactors.  For the $M^G \\subset M$ the lattice of intermediate subfactors is the subgroup lattice for $G$.  From the point of view of fusion categories, a subfactor is an algebra object $A$ in a fusion category $\\mathscr{C}$, and the intermediate subfactors are just subalgebras of $A$.  This lattice has been studied much more on the subfactor side (e.g. \\cite{MR1409040, MR1437496, MR2257402, MR2418197, MR2670925, MR2493615, MR2393428}), while it has not been a major topic in the study of fusion categories.  Nonetheless we believe that this topic should be equally interesting in both fields.  We will use subfactor language in this section so as to be able to use results from \\cite{MR2418197} without modification; however none of the results in this section rely seriously on subfactor techniques and could easily be translated into fusion category language.\n\nRecall that the Galois group (written $Gal(M \/ N)$) of a subfactor $N \\subset M$ is the group of automorphisms of $M$ which fix $N$ pointwise.  (In terms of algebra objects, this is just the group of algebra automorphisms of $A$ which restrict to the identity morphism on the unit subobject.)  For an irreducible subfactor, the Galois group is given by the group of invertible objects which are located at a distance of at most $2$ from the identity object (``*'') on the dual principal graph. The invertible objects at depth $2$ are recognizable on the graph as the $1$-valent vertices at depth $2$. The Galois group acts on the lattice of intermediate subfactors $\\{ P | N \\subseteq P \\subseteq M\\}$. Since the lattice of intermediate subfactors can be naturally identified with the dual lattice of the intermediate subfactor lattice of the dual subfactor via the basic construction, the Galois group of the dual subfactor also acts on the intermediate subfactor lattice of $N \\subset M$. In general these two groups and actions are different.  (From the fusion category perspective, the ``basic construction'' replaces $A \\in \\mathscr{C}$ with the algebra object $A \\otimes A$ in the fusion category of $A$-$A$ bimodule objects in $\\mathscr{C}$.)\n\n We will use the labeling convention $\\iota={}_P P_N $ and $\\kappa={}_M M_P $ when discussing an intermediate subfactor $N \\subset P \\subset M$. We will only consider irreducible subfactors. When dealing with multiple intermediate subfactors, we will label the objects $\\iota_P, \\kappa_P $, etc. If $\\alpha$ is an element of the Galois group of $N \\subset M$, we will freely identify it with the corresponding $M-M$ bimodule ${}_M L^2(M)_{\\alpha(M)}  $.\n\n\\begin{lemma} \\label{gal}\nLet $N \\subset P \\subset M $ be an intermediate subfactor and let $\\alpha$ be a nontrivial element of the Galois group. Then $\\alpha(P) \\cong P $ as $N-N$ bimodules. If $N \\subset P $ and $P \\subset M $ have trivial Galois groups, then $\\alpha(P) \\neq P $.   \n\\end{lemma}\n\\begin{proof}\nThe first statement is obvious. For the second, suppose that $N \\subset P $ and $P \\subset M $ have trivial Galois groups and $\\alpha(P)=P $. By triviality of $Gal(P\/N) $, we have $\\alpha$ restricted to $P$ is the identity.  Thus $\\alpha$ is an element of $Gal(M\/P)$ and hence is trivial.\n\\end{proof}\n\n\\begin{lemma} \\label{biglattice}\nThe $\\mathscr{H}_2-\\mathscr{H}_3$ subfactor $N \\subset M$ with graphs (6')-(6') has exactly nine index $4+\\sqrt{13}$ intermediate subfactors.\n\\end{lemma}\n\\begin{proof}\nLet $\\xi$ be a noninvertible object in $\\mathscr{H}_3 $ and let $\\alpha $ be a nontrivial invertible object in $\\mathscr{H}_3$, such that $\\alpha $ is an element of the Galois group of $N \\subset M$. Similarly, let $\\rho$ be a noninvertible object in $\\mathscr{H}_2 $ and let $\\beta $ be a nontrivial invertible object in $\\mathscr{H}_3$, such that $\\beta $ is an element of the dual Galois group (i.e. the Galois group of the downward basic construction $N_{-1} \\subset N \\subset M$). \n\nBy the construction in the proof of Lemma \\ref{existence} there is at least one intermediate subfactor $N \\subset P \\subset M$ with index $[M:P]=4+\\sqrt{13}$. Without loss of generality we may assume that $[\\bar{\\iota} \\iota] = [1] + [\\rho] $. Then by Lemma \\ref{gal}, $P, \\alpha(P), \\alpha^2(P)$ are distinct, and all isomorphic to $[1] + [\\rho] $ as $N-N$ bimodules.     \n\n  Let $\\bar{P} $ be the dual intermediate subfactor to $P$ in the downward basic construction $N_{-1} \\subset N \\subset M$. Let $\\zeta = {}_N N {}_{\\bar{P}} $. Then $[\\zeta \\bar{\\zeta}]=[\\bar{\\iota}\\iota ]=[ 1] + [\\rho ] $. Consider the subfactor $\\bar{Q}=\\beta(\\bar{P})$; let $\\zeta_Q = {}_N N {}_{\\bar{Q}} $. Let $N \\subset Q \\subset M$ be the dual intermediate subfactor of $\\bar{Q}$. Then $[\\bar{\\iota_Q} \\iota_Q]=[\\zeta_Q \\bar{\\zeta_Q}] = [\\beta \\zeta \\bar{\\zeta} \\beta^{-1} ] =[1]+ [\\beta \\rho \\beta^{-1} ]= [1 ] + [\\beta^2 \\rho]$. Similarly, the dual subfactor of $\\beta^2(\\bar{P}) $ is an intermediate subfactor of $N \\subset M$ which is isomorphic to $[1] + [\\beta \\rho] $ as an $N-N$ bimodule. Looking at the actions of $\\alpha $ and $\\alpha^2 $ on these subfactors shows that there must be three of each.\n\nSo we have found nine intermediate subfactors of $N \\subset M$ with index $4+\\sqrt{13}$. There is a set of three for each $N-N$ bimodule class $[1] + [\\rho], [1]+ [\\beta \\rho ], [ 1] + [\\beta^2 \\rho] $. The Galois group cyclically permutes each set, and the dual Galois group cyclically permutes the three sets. It remains to show that there are no others.\n\nOnce again let $P$ be an intermediate subfactor with $[\\bar{\\iota} \\iota]=[1 ] + [\\rho ] $, and let $Q$ be another intermediate with $[\\bar{\\iota_Q} \\iota_Q ]=[1 ]+[\\rho ] $. By \\cite[Lemma 6.1]{MR1932664} there is an isomorphism $\\pi: P \\rightarrow Q $ which fixes $N$ pointwise; we will also use $\\pi $ to refer to the corresponding $Q-P $ bimodule, $_Q L^2(Q)_{\\pi(P)} $. Note that $\\pi \\iota = \\iota_Q $. By a dimension comparison, the $M-M$ sector $[\\kappa_Q \\pi \\bar{\\kappa}]$ must contain a sector of dimension $1$, i.e. one of $[1],[\\alpha], [\\alpha^2]$. Call this one dimensional sector $[\\theta] $. Then as in the proof of \\cite[Theorem 4.3]{MR2418197}, we have by Frobenius reciprocity $[\\theta \\kappa]=[\\kappa_Q \\pi] $, and if we choose $\\theta $ in the Galois group then $\\theta(P)=Q $. This shows that any intermediate subfactor whose $N-N$ bimodule structure is $[1 ]+[\\rho ]$ is in the orbit of $P$ under the action of the Galois group, so there can be only three of them. Similarly, there are only three each for $[1 ]+[\\beta \\rho ] $ and $[1 ]+[\\beta^2 \\rho ]$.\n\n\n\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\\begin{theorem}\n The subfactors in the list in Theorem \\ref{subthm} have the following intermediate subfactor lattices: The subfactor with graphs (5)-(5') has a single proper intermediate subfactor, with index $3$. The subfactor with graphs (2)-(2') has four proper intermediate subfactors, one with index $3$ and the other three with index $\\frac{5+\\sqrt{13}}{2}$. The subfactor with graphs (6)-(6) has two proper intermediate subfactors, both with index $\\frac{5+\\sqrt{13}}{2}$. The ($\\mathscr{H}_2-\\mathscr{H}_3$) subfactor with graphs (6')-(6') has five proper intermediate subfactors: one with index $3$, one with index $\\frac{11+3\\sqrt{13}}{2}$, and nine with index $4+\\sqrt{13}$. All other $\\mathscr{H}_i-\\mathscr{H}_j$ subfactors have no proper intermediate subfactors.\n\\end{theorem}\n\n\\begin{proof}\n Note that if $N \\subset M$ is a $\\mathscr{H}_i-\\mathscr{H}_j$ subfactor and $N \\subset P \\subset M$ is a proper intermediate subfactor, then $[M:P],[P:N]$ must be indices of the graphs (1)-(7),(1')-(7'). This immediately implies the last statement.\n\nFor the first statement, we see from the graph of (5') that there is a unique index $3$ intermediate subfactor. From the graph (5) we see that there is no co-index $3$ intermediate subfactor. From the list of indices there cannot be any other intermediate subfactors.\n\nIn \\cite[Theorem 5.19]{MR2418197}, a (2)-(2') subfactor with three index $\\frac{5+\\sqrt{13}}{2}$ subfactors was constructed. From the graph (2') we see that there is also an index $3$ intermediate subfactor. Let $P$ and $Q$ be distinct index $\\frac{5+\\sqrt{13}}{2}$ intermediate subfactors. From the graph (2) and \\cite[Theorem 3.10]{MR2418197}  we find that the quadrilateral generated by $P$ and $Q$ is noncommuting, and then it follows from \\cite[Theorem 5.19]{MR2418197} that $P$ is in the orbit of $Q$ under the action of the Galois group of $N \\subset M$ on the intermediate subfactor lattice. Since the Galois group is $\\mathbb{Z}\/3\\mathbb{Z} $, it follows that there are only three intermediate subfactors of index $\\frac{5+\\sqrt{13}}{2}$. From the graph (2') we see that there is no intermediate subfactor of co-index $\\frac{5+\\sqrt{13}}{2}$. From the list of indices there cannot be any other intermediate subfactors. \n\nIn \\cite[Theorem 5.2]{MR2418197} and the following discussion, a (6)-(6) subfactor $N \\subset M$ with two index $\\frac{5+\\sqrt{13}}{2}$ intermediate subfactors was constructed; by uniqueness it is the same (6)-(6) subfactor which appears on the list in Theorem \\ref{subthm}. By \\cite[ Lemma 3.14]{MR2418197} there are no other index $\\frac{5+\\sqrt{13}}{2}$ intermediate subfactors. From the list of indices there cannot be any other intermediate subfactors.\n\n\nFrom the graph (6'), we see that the $\\mathscr{H}_2-\\mathscr{H}_3$ subfactor with graphs (6')-(6') has a unique index $3$ intermediate subfactor and unique co-index $3$ intermediate subfactor. From Lemma \\ref{noalg} there is no index $\\frac{5+\\sqrt{13}}{2}$ intermediate subfactor, and by Lemma \\ref{biglattice} there are exactly nine intermediate subfactors with index $4+\\sqrt{13}$. From the list of indices there are no other intermediate subfactors.\n\n\n\n\n\\end{proof}\n\n\\begin{figure}\n  \n  \\begin{center}\n    \\hpic{lattice} {4in}\n  \\end{center}\n  \\caption{The intermediate subfactor lattice of the $\\displaystyle \\frac{33+9\\sqrt{13}}{2} $ subfactor: $G = Gal(M\/N)=\\{1, \\alpha, \\alpha^2 \\}$, $H=Gal(N\/N_{-1})=\\{1, \\beta, \\beta^2 \\} $, $S=N \\ltimes H  $, $T= M^G $, $[P;N]=[Q:N]=[R:N]=\\frac{5+\\sqrt{13}}{2} $ .}\n\\end{figure}\n\n\n\n\\section{Outer automorphisms and the Brauer-Picard Groupoid}\n\nThe goal of this section is to prove that the outer automorphism group of each of the $\\mathscr{H}_i$ is trivial.  This completes the calculation of the Brauer-Picard groupoid.  The argument we give uses Emily Peters's description of the Haagerup subfactor planar algebra \\cite{0902.1294}.\n\n\\begin{lemma}\n$\\mathrm{Out}(\\mathscr{H}_1) \\cong \\mathrm{Out}(\\mathscr{H}_2) \\cong \\mathrm{Out}(\\mathscr{H}_3)$.\n\\end{lemma}\n\\begin{proof}\nFor each $i$ there is only one $\\mathscr{H}_i$ module category whose dual is  $\\mathscr{H}_i$ (see Lemmas \\ref{lem:dualspart1}, \\ref{lem:dualspart2}, Definition \\ref{lem:dualspart3}, and Theorem \\ref{thm:H1H3}).  Thus, the group of Morita autoequivalences of $\\mathscr{H}_i$ is isomorphic to the group of outer automorphisms of $\\mathscr{H}_i$.   But since all three fusion categories are Morita equivalent, their groups of Morita autoequivalences are all isomorphic to each other.\n\\end{proof}\n\nThus, it is enough to show that the outer automorphism group of $\\mathscr{H}_1$ is trivial.  We concentrate on $\\mathscr{H}_1$ because it is one of the even parts of the Haagerup subfactor (and thus can be described via Peters's construction) and because it has no inner automorphisms.  Essentially the same argument applies directly to $\\mathscr{H}_2$ with a little extra care.\n\nIn general an automorphism of a fusion category does not give an automorphism of the corresponding planar algebra.  This is because the chosen algebra object is built into the planar algebra formalism.  Only automorphisms which act trivially on the algebra object correspond to automorphisms of the planar algebra.  That is, we must have that $\\mathscr{F}(A) \\cong A$ and furthermore that the multiplication map is also acted on trivially.  Explicitly this means that the composition $$A \\otimes A = \\mathscr{A} \\otimes \\mathscr{A} \\rightarrow \\mathscr{F}(A \\otimes A) \\rightarrow \\mathscr{F}(A) = A$$ should agree with multiplication, where the first map is part of the data of a tensor functor and the second map is the functor applied to the multiplication morphism.\n\n\\begin{lemma}\nAny automorphism of $\\mathscr{H}_1$ is naturally isomorphic to an automorphism which acts trivially on the algebra object $1+\\eta$.\n\\end{lemma}\n\\begin{proof}\nFirst note that no other simple object in $\\mathscr{H}_1$ has the same dimension as $1$ or $\\eta$, hence the automrophism must send $1+\\eta$ to itself.  Now the $3$-supertransitivity of the Haagerup subfactor guarantees that up to algebra isomorphism, there is only one algebra structure on $1+\\eta$.  Since $1+\\eta$ is multiplicity free, we can extend this algebra isomorphism to a natural transformation which sends our original automorphism into an automorphism which acts trivially on $1+\\eta$.\n\\end{proof}\n\n\\begin{lemma}\nThe Haagerup planar algebra (constructed in \\cite{0902.1294}) has no nontrivial automorphisms.\n\\end{lemma}\n\\begin{proof}\nThe subspace spanned by the generator $T$ is characterized by being the low weight space for the action of annular Temperley-Lieb on the $4$-box space.  Thus any automorphism of the planar algebra must send $T$ to a multiple of itself.  Since $T^2 = \\frac{1}{2} f^{(4)}$ we see that $T$ needs to be sent to $\\pm T$.  However, the third twisted moments of $T$ and $-T$ are not equal to each other \\cite[Lemma 4.1]{0902.1294}, hence the automorphism must send $T$ to $T$.  Since $T$ generates the planar algebra we see that the automorphism is automatically trivial.\n\\end{proof}\n\n\\begin{remark}\nNote that the Haagerup planar algebra does have an anti-linear automorphism interchanging $T$ and $-T$.\n\\end{remark}\n\n\\begin{theorem}\n$\\mathrm{Aut}(\\mathscr{H}_1)$ is trivial.\n\\end{theorem}\n\\begin{proof}\nBy the above lemmas, any automorphism of $\\mathscr{H}_1$ is naturally isomorphic to one which acts trivially on the algebra object $1+\\eta$ and thus corresponds to an automorphism of the Haagerup planar algebra, which must therefore be trivial.\n\\end{proof}\n\n\\begin{remark}\nWe have also checked that $\\mathrm{Out}(\\mathscr{H}_2)$ is trivial in a completely independent way using 6j-symbols instead of planar algebras.  In this other approach $\\mathscr{H}_2$ is more convenient because its tensor product rules are multiplicity free, thereby simplifying the theory of 6j-symbols.  First, after possibly applying an inner automorphism, we may assume that the automorphism fixes the object $\\xi$.  Furthermore, by looking at the connection (or equivalently the 6j-symbols) any linear automorphism which fixes $\\xi$ actually fixes all the objects.  Any such automorphism is a gauge automorphism in the sense of \\cite{Liptrap}.  All gauge automorphisms can be found following \\cite{Liptrap} using  nothing more than highschool algebra.  However, our argument, elementary as it was, was also extremely tedious.  Since we were unable to find a good way to shorten or clarify this argument we have chosen not to inflict it on the reader.\n\\end{remark}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{The Izumi subfactors}\nIn \\cite[Section 7]{MR1832764} Izumi listed a system of equations associated to a finite Abelian group $G$ of odd order such that any solution gives a unitary fusion category with sectors $[\\alpha_{g}], [\\alpha_{g} \\xi], g \\in G$ satisfying $[\\alpha_g \\xi ]= [\\xi \\alpha_{-g}] $ and $[\\xi^2 ]=\\sum_{g \\in G} \\limits [\\alpha_g \\xi ] $. He showed that $[1] + [\\xi ] $ admits a Q-system.  The sector $[\\xi]$ has Forbenius-Perron dimension $d=d(\\xi)=\\displaystyle \\frac{n+\\sqrt{n^2+4}}{2}$, where $n=|G|$, so the corresponding subfactor has index $1+d$. Note the relation $d^2=1+nd$. For $\\mathbb{Z} \/ 3 \\mathbb{Z}$ one recovers the Haagerup subfactor, and he showed that there is unique solution up to equivalence for $\\mathbb{Z} \/ 5 \\mathbb{Z} $.   The goal of this section is to discuss how our results generalize to other Izumi subfactors.\n\nIzumi's equations were solved for $\\mathbb{Z}\/n\\mathbb{Z} $ for $n=7,9$ (in the latter case there are two non-equivalent solutions) by Evans and Gannon \\cite[Theorem 5]{1006.1326}, who also found numerical evidence for solutions for the cases $n=11,13,15,17,19$. They also computed the fusion rings of the dual fusion categories of the subfactors subject to conditions on the modular data \\cite[Theorem 7]{1006.1326}; these conditions are satisfied for at least one solution for each $n$ for which solutions are known. \n\nThe dual fusion ring is then commutative and is described as follows: there is the identity $1$; there is a simple object $\\eta$ with $d(\\eta)=d $; there are $\\frac{n-1}{2} $ simple objects $\\nu_j$ with $d(\\nu_j)=d+1$; and there are $\\frac{n-1}{2} $ simple objects $\\mu_j$ with $d(\\mu_j)=d-1$.\n\nFrom now on we will fix $n$ and let $\\mathscr{I}_1 $ and $\\mathscr{I}_2$ denote, respectively, the commutative and noncommutative fusion categories of an Izumi subfactor associated to $\\mathbb{Z} \/ n \\mathbb{Z} $ and satisfying the Evans-Gannon conditions. Let $I_1 $ be the fusion ring of $\\mathscr{I}_1  $ and let $I_2 $ be the fusion ring of $\\mathscr{I}_2 $. Let $\\alpha=\\alpha_g$ be an object corresponding to a generator of the group, so that the invertible objects are given by $1,\\alpha^i, 1 \\leq i \\leq n-1$.\n \nIf $\\gamma$ is a simple algebra object in a fusion category, then we have $(\\gamma,\\rho) \\leq d(\\rho ) $ for every simple object $\\rho $ in $\\mathscr{C} $.\n\n\\begin{definition}\n A simple algebra object $\\gamma$ in a fusion category $\\mathscr{C}$ will be called saturated if $(\\gamma,\\rho )=\\lfloor d(\\rho) \\rfloor $ for every simple object $\\rho$ in $\\mathscr{C}$. \n\\end{definition}\n\nIf $[\\gamma]$ is a saturated simple algebra object in $\\mathscr{C} $, then the vertex in the principal graph corresponding to the fundamental object is connected to the vertex corresponding to $\\rho$ by $\\lfloor d(\\rho) \\rfloor$ edges, for every simple object $\\rho$ in $\\mathscr{C}$. Let $G_{\\mathscr{C}}$ be the graph obtained by adding one more odd vertex, along with one edge from the new vertex to each non-invertible simple object in $\\mathscr{C}$. (For examples, see Example \\ref{ex}, (c), and Haagerup graph (6')).\n\n\\begin{lemma}\n Let $\\mathscr{C}$ be a fusion category with fusion ring $I_1$ or $I_2$. Then the only possible principal graph of a saturated simple algebra is $G_{\\mathscr{C}} $. In either case, the Frobenius-Perron weight of the second odd vertex is $\\sqrt{n}$ and the dimension of the algebra object is $n+n^2d$.  \n\\end{lemma}\n\\begin{proof}\n For $I_2$, one can check using the Evans-Gannon fusion rules that all entries of the reduced fusion matrix of the object $\\sum_{i=0}^{n} \\limits  \\lfloor d(\\rho) \\rfloor \\rho_i $ are $1$; the Frobenius-Perron weights are easy to compute. For $I_1$, the proof is the same except that only the entries corresponding to two non-invertible objects are $1$. \n\\end{proof}\n\n\\begin{corollary} \\label{imp}\n Let $\\mathscr{C}$ be a fusion category with fusion ring $I_2$. Then $\\mathscr{C} $ does not have a saturated algebra object.\n\\end{corollary}\n\n\\begin{proof}\n As in Example \\ref{ex}, (c), let $\\kappa'$ be the object corresponding to the second odd vertex. Then $[\\kappa' \\bar{\\kappa'}]=[1]+[ \\sigma] $, where $[\\sigma] $ is a nonnegative integral linear combination of $[\\nu_j], [\\eta], [\\mu_j] $. Since $d(\\sigma )=n-1$, this is impossible.\n\\end{proof}\n\n\n\n\\begin{theorem} \\label{I3}\nThere exists a unitary fusion category $\\mathscr{I}_3 $ which is Morita equivalent to $\\mathscr{I}_1$ or $\\mathscr{I}_2$ but not isomorphic to either of them.\n\\end{theorem}\n\\begin{proof}\n Let $\\lambda$ denote the simple object corresponding to the middle vertex of the dual graph of the Izumi subfactor. Let $\\gamma=\\lambda \\bar{\\lambda}$ be the correponding algebra object, which is also a Q-system. Then $\\gamma$ is the $M-M$ Q-system for an $\\mathscr{I}_1-\\mathscr{I}_2$ subfactor $N \\subset M$ with index $d(\\gamma)=\\frac{(nd)^2}{d+1}$. For any $0 \\leq k \\leq n$, $(\\gamma, \\alpha^k)=(\\lambda, \\alpha^k \\lambda )=(\\lambda, \\lambda)=1 $, so there is an intermediate $\\mathscr{I}_1-*$ subfactor $N \\subset P \\subset M$ with index $[P:N]=d(\\gamma )\/n = \\frac{nd^2}{d+1}=1+(n-1)d$. Let $\\mathscr{I}_3$ be the $P-P$ even part of $N \\subset P$.\n\nThen $\\mathscr{I}_3 $ contains a nontrivial invertible object, so $\\mathscr{I}_3 \\ncong \\mathscr{I}_1$. Suppose $\\mathscr{I}_3 \\cong \\mathscr{I}_2$. Then as in the proof of Lemma \\ref{noalg}, there would have to exist an $\\mathscr{I}_1-* $ subfactor $R \\subset S$ of index $(1+d)(1+(n-1)d )=n(1+nd)$. But this would imply the existence of a saturated algebra object in $\\mathscr{I}_1$, which is impossible by Lemma \\ref{imp}. \n\\end{proof}\n\n\n\nAs we have seen, in the case $n=3$, $\\mathscr{I}_3$ has fusion ring $I_2$ and there is a unique simple bimodule category between each pair $\\mathscr{I}_i, \\mathscr{I}_j, 1 \\leq i,j \\leq 3 $ up to isomorphism. It is natural to wonder whether this holds true for other values of $n$. The first question that needs to be resolved is finding the principal graphs of the $\\mathscr{I}_1-\\mathscr{I}_3$ subfactor constructed in Theorem \\ref{I3}. There are natural candidates for these graphs based on the $n=3$ and $n=5$ (below) cases, but we cannot verify the following conjecture for arbitrary $n$ at this time.\n\n\n\n\\begin{conjecture}\n Let $N \\subset M$ be the  $\\mathscr{I}_1-\\mathscr{I}_3$ subfactor constructed above, and let $\\gamma$ be the corresponding algebra object in $\\mathscr{I}_1 $. Then $\\gamma \\cong 1 + \\sum_{j=1}^{\\frac{n-1}{2}} \\limits (\\nu_j + \\mu_j) $. \n\n\n\\end{conjecture}\n\n\n\n\n\n\\subsection{The Izumi subfactor for $\\mathbb{Z} \/ 5 \\mathbb{Z}$}\n\nThere is a unique Izumi subfactor corresponding to $n=5$. We consider a list of graphs analogous to that obtained in the Haagerup ($n=3$) case. As before, the unprimed series are consistent with the $I_1$ fusion rules and the primed series is consistent with the $I_2 $ fusion rules.\n\n(1) \\hpic{i14} {1.2in} , $\\displaystyle \\frac{7+\\sqrt{29}} {2}$, (2) \\hpic{i13} {1.2in} , $55+10\\sqrt{29}$ \\\\\n\n\n(3) \\hpic{i9} {1.2in} , $11+2\\sqrt{29}$, (4) \\hpic{i10} {1.4in} , $\\displaystyle \\frac{27+5\\sqrt{29}} {2}$ \\\\\n\n\n(5) \\hpic{i12} {1.2in} , $\\displaystyle \\frac{35+5\\sqrt{29}} {2}$, (6) \\hpic{i11} {1.2in} , $\\displaystyle \\frac{39+7\\sqrt{29}} {2}$ \\\\\n\n\n(7) \\hpic{i8} {1.2in} , $\\displaystyle \\frac{19+3\\sqrt{29}} {2}$ \\\\\n\n\n(1') \\hpic{i4} {1.2in} , $\\displaystyle \\frac{7+\\sqrt{29}} {2}$, (2') \\hpic{i6} {1.2in} , $55+10\\sqrt{29}$ \\\\\n\n\n(3') \\hpic{i5} {1.2in} , $11+2\\sqrt{29}$, (4') \\hpic{i1} {1.2in} , $\\displaystyle \\frac{27+5\\sqrt{29}} {2}$ \\\\\n\n\n(5') \\hpic{i7} {1.2in} , $\\displaystyle \\frac{35+5\\sqrt{29}} {2}$, (6') \\hpic{i2} {1.2in} , $\\displaystyle \\frac{135+25\\sqrt{29}} {2}$ \\\\\n\n\n(7') \\hpic{i3} {0.8in} , $5$ \\\\\n\n\n\\begin{lemma}\\label{i10graphs}\n Let $[\\gamma] $ be a nontrivial simple algebra object in a fusion category with fusion ring isomorphic to $I_2$. Then the principal graph of $[\\gamma]$ is one of the seven graphs (1')-(7') listed above.\n\\end{lemma}\n\n\\begin{proof}\n Besides (1')-(7'), there are six other irreducible graphs that are compatible with the $I_2$ fusion rules. These are: the graphs obtained from (2') and (5') by in each case replacing four of the five symmetric odd vertices by a single odd vertex with two edges to each of the five even vertices; and two pairs of graphs each at the indices $\\displaystyle \\frac{85+15\\sqrt{29}} {2}$ and $30+5\\sqrt{29}$.  In each case, the square of the Frobenius-Perron dimension of one of the other odd vertices gives an index which does not admit a graph consistent with $I_2$.\n\\end{proof}\n\nThe proofs of the following results are now essentially the same as in the $n=3$ case.\n\n\\begin{lemma}\n There exist unique algebra structures in $\\mathscr{I}_2 $ on $1$, $1+\\xi$, $1+\\alpha+\\alpha^2+\\alpha^3+\\alpha^4$; there does not exist an algebra structure on $1+\\alpha^k \\xi+\\alpha^{k+1} \\xi+\\alpha^{k+2} \\xi + \\alpha^{k+3} \\xi$ for any $k$.\n\\end{lemma}\n\n\\begin{lemma}\n The $\\mathscr{I}_1-\\mathscr{I}_3 $ subfactor constructed in \\ref{I3} has principal and dual graphs (3) and (3').\n\\end{lemma}\n\n\\begin{lemma}\n The fusion category $\\mathscr{I}_3 $ has fusion ring $I_2$.\n\\end{lemma}\n\n\n\\begin{theorem}\nEach $\\mathscr{I}_i, i=1,2,3 $ has exactly three module categories, whose dual categories in each case are again the three $\\mathscr{I}_i$. The complete list of subfactors whose fusion categories are Morita equivalent to $\\mathscr{I}_i$ is as follows; each exists and is unique up to isomorphism of the planar algebra. The $\\mathscr{I}_i-\\mathscr{I}_i $ subfactors are self-dual while the $\\mathscr{I}_i-\\mathscr{I}_j $ subfactors for $i \\neq j $ come in non-self-dual pairs; we only list each pair once.\n\n(a) $\\mathscr{I}_1-\\mathscr{I}_2$: One with principal graphs (1)-(1') and one with principal graphs (2)-(2').\n\n(b) $\\mathscr{I}_1-\\mathscr{I}_3$: One subfactor with principal graphs (3)-(3') and one with principal graphs (5)-(5').\n\n(c) $\\mathscr{I}_2-\\mathscr{I}_3$: One subfactor with principal graphs (6')-(6'). \n\n(d) $\\mathscr{I}_1-\\mathscr{I}_1$: One subfactor each with principal graphs (4)-(4), (6)-(6), and (7)-(7).\n\n(e) $\\mathscr{I}_2-\\mathscr{I}_2$: One subfactor with principal graph (4')-(4').\n\n(f) $\\mathscr{I}_3-\\mathscr{I}_3$: One subfactor with principal graph (4')-(4').\n\\end{theorem}\n\n\n\n\n\n\n\n\\newcommand{}{}\n\\bibliographystyle{alpha}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}