# Collins diffraction formula and the Wigner function in entangled state
representation

Hong-Yi Fan1,2 and Li-yun Hu${}^{1\text{*}}$
1Department of Physics, Shanghai Jiao Tong University, Shanghai 200030, China
2Department of Material Science and Engineering,
University of Science and Technology of China, Hefei, Anhui 230026, China
Corresponding author. E-mail address<EMAIL_ADDRESS>
###### Abstract

Based on the correspondence between Collins diffraction formula (optical
Fresnel transform) and the transformation matrix element of a three-parameters
two-mode squeezing operator in the entangled state representation (Opt. Lett.
31 (2006) 2622) we further explore the relationship between output field
intensity determined by the Collins formula and the input field’s probability
distribution along an infinitely thin phase space strip both in spacial domain
and frequency domain. The entangled Wigner function is introduced for
recapitulating the result.

OCIS codes: 070.2590, 270.6570

In a preceding Letter [1] we have reported that the Collins diffraction
formula in cylindrical coordinates is just the transformation matrix element
of a three-parameter ($k$ and $t$ are complex and satisfy the unimodularity
condition $kk^{\ast}-tt^{\ast}=1)$ two-mode squeezing operator [2, 3]

$F^{\left(t,k\right)}=\exp\left(\frac{t}{k^{\ast}}a_{1}^{\dagger}a_{2}^{\dagger}\right)\exp\left[\left(a_{1}^{\dagger}a_{1}+a_{2}^{\dagger}a_{2}+1\right)\ln\left(k^{\ast}\right)^{-1}\right]\exp\left(-\frac{t^{\ast}}{k^{\ast}}a_{1}a_{2}\right),$
(1)

in the deduced entangled state representation $\left\langle
s,r^{\prime}\right|,$

$\displaystyle\phi_{s}\left(r^{\prime}\right)$ $\displaystyle\equiv$
$\displaystyle\left\langle
s,r^{\prime}\right|\left.\phi\right\rangle=\left\langle
s,r^{\prime}\right|F^{\left(t,k\right)}\left|\psi\right\rangle$ (2)
$\displaystyle=$
$\displaystyle\frac{\mathtt{i}^{s}}{2\mathtt{i}B}\int_{0}^{\infty}\mathtt{d}\left(r^{2}\right)\exp\left[\frac{\mathtt{i}}{2B}\left(Ar^{2}+Dr^{\prime
2}\right)\right]J_{s}\left(-\frac{rr^{\prime}}{B}\right)\psi_{s}\left(r\right),$

where $\psi_{s}$ and $\phi_{s}$ denote the incoming and output light,
respectively, $[a_{i},a_{j}^{\dagger}]=\delta_{i,j},$

$k=\frac{1}{2}\left[A+D-\mathtt{i}\left(B-C\right)\right],\text{\ \
}t=\frac{1}{2}\left[A-D+\mathtt{i}\left(B+C\right)\right],$ (3)

we see that the relation $kk^{\ast}-tt^{\ast}=1$ becomes $AD-BC=1$, $J_{s}$ is
the $s$th Bessel function, and

$\left\langle
s,r^{\prime}\right|=\frac{1}{2\pi}\int_{0}^{2\pi}\mathtt{d}\theta
e^{\mathtt{i}s\theta}\left\langle\eta=r^{\prime}e^{\mathtt{i}\theta}\right|,$
(4)

here $\left|\eta\right\rangle$ is the entangled states in two-mode Fock space
[4, 5, 6, 7] named after Einstein-Podolsky-Rosen (EPR)’s [8] concept of
quantum entanglement,

$\left|\eta\right\rangle=\exp\\{-\frac{1}{2}\left|\eta\right|^{2}+\eta
a_{1}^{\dagger}-\eta^{\ast}a_{2}^{\dagger}+a_{1}^{\dagger}a_{2}^{\dagger}\\}\left|00\right\rangle.$
(5)

Thus $\left\langle
s,r^{\prime}\right|F^{\left(t,k\right)}\left|\psi\right\rangle$ is the quantum
optics version of the Collins formula (generalized Hankel transformation). In
[9] we have also found

$\mathcal{K}^{\left(t,k\right)}\left(\eta^{\prime},\eta\right)=\frac{1}{\pi}\left\langle\eta^{\prime}\right|F^{\left(t,k\right)}\left|\eta\right\rangle=\frac{1}{2\mathtt{i}B\pi}\exp\left\\{\frac{\mathtt{i}}{2B}\left[A\left|\eta\right|^{2}-\left(\eta\eta^{\prime\ast}+\eta^{\ast}\eta^{\prime}\right)+D\left|\eta^{\prime}\right|^{2}\right]\right\\}.$
(6)

Comparing with the integral kernel of usual Fresnel transform which describes
how a general beam $\psi\left(x^{\prime}\right),$ propagating through an
$\left(ABCD\right)$ optical paraxial system, becomes output field
$\phi\left(x\right)$[10, 11]

$\phi\left(x\right)=\int_{-\infty}^{\infty}\mathcal{K}\left(x,x^{\prime}\right)\psi\left(x^{\prime}\right)\mathtt{d}x^{\prime},$
(7)

where $AD-BC=1,$

$\mathcal{K}\left(x,x^{\prime}\right)=\frac{1}{\sqrt{2\pi\mathtt{i}B}}\exp\left[\frac{\mathtt{i}}{2B}\left(Ax^{\prime
2}-2x^{\prime}x+Dx^{2}\right)\right],$ (8)

we see that $\mathcal{K}^{\left(t,k\right)}\left(\eta^{\prime},\eta\right)$
can be considered as the integration kernel of 2-dimensional entangled optical
Fresnel transform,

$\Psi\left(\eta^{\prime}\right)=\int\mathcal{K}^{\left(t,k\right)}\left(\eta^{\prime},\eta\right)\Phi\left(\eta\right)\mathtt{d}^{2}\eta,$
(9)

in this sense $F^{\left(t,k\right)}$ can be named entangled Fresnel operator
(EFO), here
$\Phi\left(\eta\right)=\left\langle\eta\right|\left.\Phi\right\rangle,$
$\Psi\left(\eta^{\prime}\right)=\left\langle\eta^{\prime}\right|\left.\Psi\right\rangle,$
and we have used the completeness relation
$\int\frac{\mathtt{d}^{2}\eta}{\pi}\left|\eta\right\rangle\left\langle\eta\right|=1.$

Clearly, if the $\left[ABCD\right]$ system is changed to
$\left[D\left(-B\right)\left(-C\right)A\right]$ system, then Eq. (9) should
read

$\Psi\left(\eta^{\prime}\right)=\int\mathcal{K}_{2}^{\left(D,-B,-C\right)}\left(\eta^{\prime},\eta\right)\Phi\left(\eta\right)\mathtt{d}^{2}\eta,$
(10)

where $\mathcal{K}_{2}^{\left(D,-B,-C\right)}$ is

$\mathcal{K}_{2}^{\left(D,-B,-C\right)}\left(\eta^{\prime},\eta\right)=\frac{1}{-2\mathtt{i}B\pi}\exp\left\\{\frac{\mathtt{i}}{-2B}\left[D\left|\eta\right|^{2}-\left(\eta\eta^{\prime\ast}+\eta^{\ast}\eta^{\prime}\right)+A\left|\eta^{\prime}\right|^{2}\right]\right\\}.$
(11)

On the other hand, signals or images in optical information theory may be
described directly or indirectly by the Wigner distribution function (WDF)
[12]. In one-dimensional (1D) case, the WDF of an optical signal field
$\psi\left(x\right)$ is defined as

$W_{\psi}(\nu,x)=\int_{-\infty}^{+\infty}\frac{\mathtt{d}u}{2\pi}e^{\mathtt{i}\nu
u}\psi^{\ast}\left(x+\frac{u}{2}\right)\psi\left(x-\frac{u}{2}\right).$ (12)

$W_{\psi}(\nu,x)$ involves both spatial distribution information and space-
frequency distribution information of the signal, $\nu$ is named space
frequency. Now, let us consider the entangled case. Like Eq. (12), it is
natural to introduce the 2-D complex Wigner transform as

$W\left(\sigma,\gamma\right)=\int\frac{\mathtt{d}^{2}\eta}{\pi^{3}}\psi\left(\sigma+\eta\right)\psi^{\ast}\left(\sigma-\eta\right)e^{\eta\gamma^{\ast}-\eta^{\ast}\gamma},$
(13)

where $\sigma,$ $\gamma,$ $\eta$ are all complex variables. To see its
physical meaning, using the integration formula of Dirac $\delta-$function, we
perform the following integration,

$\int\mathtt{d}^{2}\gamma
W\left(\sigma,\gamma\right)=\int\frac{\mathtt{d}^{2}\eta}{\pi}\psi\left(\sigma+\eta\right)\psi^{\ast}\left(\sigma-\eta\right)\delta\left(\eta\right)\delta\left(\eta^{\ast}\right)=\frac{1}{\pi}\left|\psi\left(\sigma\right)\right|^{2},$
(14)

which is just the probability distribution of the complex function
$\psi$($\sigma$). Further, let the ordinary Fourier transforms of
$\psi\left(\sigma\right)$ be $j\left(\zeta\right),$

$\psi\left(\sigma\right)=\int\frac{\mathtt{d}^{2}\zeta}{2\pi}j\left(-\zeta\right)e^{\left(\zeta^{\ast}\sigma-\zeta\sigma^{\ast}\right)/2},$
(15)

then substituting (15) into (13) leads to

$\displaystyle W\left(\sigma,\gamma\right)$ $\displaystyle=$
$\displaystyle\int\frac{\mathtt{d}^{2}\eta}{\pi^{3}}\frac{\mathtt{d}^{2}\zeta}{2\pi}\frac{\mathtt{d}^{2}\zeta^{{}^{\prime}}}{2\pi}j\left(-\zeta\right)j^{\ast}\left(-\zeta^{{}^{\prime}}\right)e^{\frac{\left(\zeta^{\ast}-\zeta^{/\ast}\right)\sigma-\left(\zeta-\zeta^{/}\right)\sigma^{\ast}}{2}}e^{\eta\left(\gamma^{\ast}+\frac{\zeta^{\ast}+\zeta^{/\ast}}{2}\right)-\eta^{\ast}\left(\gamma+\frac{\zeta+\zeta^{{}^{\prime}}}{2}\right)}$
(16) $\displaystyle=$
$\displaystyle\int\frac{\mathtt{d}^{2}\zeta}{\pi^{3}}j\left(-\zeta\right)j^{\ast}\left(2\gamma+\zeta\right)e^{\left(\zeta^{\ast}+\gamma^{\ast}\right)\sigma-\left(\zeta+\gamma\right)\sigma^{\ast}}=\int\frac{\mathtt{d}^{2}\zeta}{\pi^{3}}j\left(\gamma-\zeta\right)j^{\ast}\left(\gamma+\zeta\right)e^{\zeta^{\ast}\sigma-\zeta\sigma^{\ast}}.$

It then follows from (16) that

$\displaystyle\int\mathtt{d}^{2}\sigma W\left(\sigma,\gamma\right)$
$\displaystyle=$
$\displaystyle\int\frac{\mathtt{d}^{2}\zeta}{\pi^{3}}j\left(\gamma-\zeta\right)j^{\ast}\left(\zeta+\gamma\right)\int\mathtt{d}^{2}\sigma
e^{\zeta^{\ast}\sigma-\zeta\sigma^{\ast}}$ (17) $\displaystyle=$
$\displaystyle\int\frac{\mathtt{d}^{2}\zeta}{\pi}j\left(\gamma-\zeta\right)j^{\ast}\left(\zeta+\gamma\right)\delta\left(\zeta\right)\delta\left(\zeta^{\ast}\right)=\frac{1}{\pi}\left|j\left(\gamma\right)\right|^{2},$

which is the probability distribution of the complex function
$j\left(\gamma\right)$. Thus our definition in (13) leads to two marginal
distributions in $\sigma$ and $\gamma$ phase space, respectively. Hence
$W\left(\sigma,\gamma\right)$ is indeed the correct complex 2-D Wigner
function (Wigner transform) of complex function $\psi\left(\sigma\right)$ or
$j\left(\gamma\right)$. If one wants to reconstruct the Wigner function by
using various probability distributions, obviously the “position density”
$\left|\left\langle\sigma\right|\left.\psi\right\rangle\right|^{2}$ and the
space-frequency density
$\left|\left\langle\gamma\right|\left.\psi\right\rangle\right|^{2}$ are not
enough, so we extend
$\delta\left(\eta\right)\delta\left(\eta^{\ast}\right)\equiv\delta\left(\eta_{1}\right)\delta\left(\eta_{2}\right)$
to
$\delta\left(\eta_{1}-D\sigma_{1}-B\gamma_{2}\right)\delta\left(\eta_{2}-D\sigma_{2}+B\gamma_{1}\right)$
and generalize (14) to,

$R_{2}\left(\eta_{1},\eta_{2}\right)\equiv\pi\int\delta\left(\eta_{1}-D\sigma_{1}-B\gamma_{2}\right)\delta\left(\eta_{2}-D\sigma_{2}+B\gamma_{1}\right)W\left(\sigma,\gamma\right)\mathtt{d}^{2}\sigma\mathtt{d}^{2}\gamma,$
(18)

$R_{2}\left(\eta_{1},\eta_{2}\right)$ is also a probability distribution along
an infinitely thin phase space strip denoted by the real parameters $B,D$,
which is a generalized entangled Radon transform [13, 14] of the two-mode
Wigner function (in the entangled form) [15, 16],

Then an interesting question naturally arises: what is the relation between
the generalized Fresnel transform and the WDF in entangled state
representation?

We begin with rewriting the 2-D WF (13) as

$\displaystyle W\left(\sigma,\gamma\right)$ $\displaystyle=$
$\displaystyle\int\mathtt{d}^{2}\sigma^{\prime}\mathtt{d}^{2}\sigma^{\prime\prime}\int\frac{\mathtt{d}^{2}\eta}{\pi^{3}}\psi\left(\sigma^{\prime}\right)\psi^{\ast}\left(\sigma^{\prime\prime}\right)\delta^{(2)}\left(\sigma^{\prime}-\sigma-\eta\right)\delta^{(2)}\left(\sigma-\eta-\sigma^{\prime\prime}\right)e^{\eta\gamma^{\ast}-\eta^{\ast}\gamma}$
(19) $\displaystyle=$
$\displaystyle\int\frac{\mathtt{d}^{2}\sigma^{\prime}\mathtt{d}^{2}\sigma^{\prime\prime}}{\pi^{3}}\psi\left(\sigma^{\prime}\right)\psi^{\ast}\left(\sigma^{\prime\prime}\right)\delta^{\left(2\right)}\left(2\sigma-\sigma^{\prime}-\sigma^{\prime\prime}\right)e^{\left(\sigma^{\prime}-\sigma\right)\gamma^{\ast}-\left(\sigma^{\prime}-\sigma\right)^{\ast}\gamma}.$

Substituting (19) into (18) we rewrite the Radon transform of
$W\left(\sigma,\gamma\right)$ as
($\mathtt{d}^{2}\sigma=\mathtt{d}\sigma_{1}\mathtt{d}\sigma_{2},$
$\mathtt{d}^{2}\gamma=\mathtt{d}\gamma_{1}\mathtt{d}\gamma_{2}$)

$\displaystyle R_{2}\left(\eta_{1},\eta_{2}\right)$ $\displaystyle=$
$\displaystyle\int\frac{\mathtt{d}^{2}\sigma^{\prime}\mathtt{d}^{2}\sigma^{\prime\prime}}{\pi^{2}}\psi\left(\sigma^{\prime}\right)\psi^{\ast}\left(\sigma^{\prime\prime}\right)\int\mathtt{d}^{2}\sigma\mathtt{d}^{2}\gamma\delta\left(\eta_{2}-D\sigma_{2}+B\gamma_{1}\right)$
(20)
$\displaystyle\times\delta\left(\eta_{1}-D\sigma_{1}-B\gamma_{2}\right)\delta^{\left(2\right)}\left(2\sigma-\sigma^{\prime}-\sigma^{\prime\prime}\right)e^{\left(\sigma^{\prime}-\sigma\right)\gamma^{\ast}-\left(\sigma^{\prime}-\sigma\right)^{\ast}\gamma}$
$\displaystyle=$
$\displaystyle\int\frac{\mathtt{d}^{2}\sigma^{\prime}\mathtt{d}^{2}\sigma^{\prime\prime}}{4\pi^{2}}\psi\left(\sigma^{\prime}\right)\psi^{\ast}\left(\sigma^{\prime\prime}\right)\int\mathtt{d}^{2}\gamma\delta\left(\eta_{2}-D\frac{\sigma_{2}^{\prime}+\sigma_{2}^{\prime\prime}}{2}+B\gamma_{1}\right)$
$\displaystyle\times\delta\left(\eta_{1}-D\frac{\sigma_{1}^{\prime}+\sigma_{1}^{\prime\prime}}{2}-B\gamma_{2}\right)\exp\left\\{\allowbreak
i\left[\left(\sigma_{2}^{\prime}-\sigma_{2}^{\prime\prime}\right)\gamma_{1}-\left(\sigma_{1}^{\prime}-\sigma_{1}^{\prime\prime}\right)\gamma_{2}\right]\right\\}$
$\displaystyle=$
$\displaystyle\int\frac{\mathtt{d}^{2}\sigma^{\prime}\mathtt{d}^{2}\sigma^{\prime\prime}}{4B^{2}\pi^{2}}\psi\left(\sigma^{\prime}\right)\psi^{\ast}\left(\sigma^{\prime\prime}\right)$
$\displaystyle\times\exp\left\\{\allowbreak\frac{i}{B}\left[\left(\sigma_{2}^{\prime}-\sigma_{2}^{\prime\prime}\right)\left(-\eta_{2}+D\frac{\sigma_{2}^{\prime}+\sigma_{2}^{\prime\prime}}{2}\right)-\left(\sigma_{1}^{\prime}-\sigma_{1}^{\prime\prime}\right)\left(\eta_{1}-D\frac{\sigma_{1}^{\prime}+\sigma_{1}^{\prime\prime}}{2}\right)\right]\right\\}$
$\displaystyle=$
$\displaystyle\int\frac{\mathtt{d}^{2}\sigma^{\prime}\mathtt{d}^{2}\sigma^{\prime\prime}}{4B^{2}\pi^{2}}\psi\left(\sigma^{\prime}\right)\psi^{\ast}\left(\sigma^{\prime\prime}\right)\exp\left\\{\frac{i}{2B}\left[D\left(\left|\sigma^{\prime}\right|^{2}-\left|\sigma^{\prime\prime}\right|^{2}\right)-2\eta_{1}\left(\sigma_{1}^{\prime}-\allowbreak\sigma_{1}^{\prime\prime}\right)-2\eta_{2}\left(\sigma_{2}^{\prime}-\sigma_{2}^{\prime\prime}\right)\right]\right\\}.$

On the other hand, when the beam $\Phi\left(\eta\right)$ propagates through
the $\left[D\left(-B\right)\left(-C\right)A\right]$ optical system, according
to the Fresnel integration (10)-(11), we have

$\displaystyle\left|\Psi\left(\eta^{\prime}\right)\right|^{2}$
$\displaystyle=$
$\displaystyle\int\frac{\mathtt{d}^{2}\eta}{\pi}\mathcal{K}_{2}^{\left(D,-B,-C\right)}\left(\eta^{\prime},\eta\right)\Phi\left(\eta\right)\int\frac{\mathtt{d}^{2}\eta^{\prime\prime}}{\pi}\mathcal{K}_{2}^{\ast\left(D,-B,-C\right)}\left(\eta^{\prime},\eta^{\prime\prime}\right)\Phi^{\ast}\left(\eta^{\prime\prime}\right)$
(21) $\displaystyle=$
$\displaystyle\frac{1}{4B^{2}}\int\frac{\mathtt{d}^{2}\eta}{\pi}\exp\left\\{\frac{\mathtt{i}}{2B}\left[-D\left|\eta\right|^{2}+\left(\eta\eta^{\prime\ast}+\eta^{\ast}\eta^{\prime}\right)-A\left|\eta^{\prime}\right|^{2}\right]\right\\}\Phi\left(\eta\right)$
$\displaystyle\times\int\frac{\mathtt{d}^{2}\eta^{\prime\prime}}{\pi}\exp\left\\{\frac{\mathtt{i}}{2B}\left[D\left|\eta^{\prime\prime}\right|^{2}-\left(\eta^{\prime\prime\ast}\eta^{\prime}+\eta^{\prime\prime}\eta^{\prime\ast}\right)+A\left|\eta^{\prime}\right|^{2}\right]\right\\}\Phi^{\ast}\left(\eta^{\prime\prime}\right)$
$\displaystyle=$
$\displaystyle\frac{1}{4B^{2}\pi^{2}}\int\frac{\mathtt{d}^{2}\eta}{\pi}\Phi\left(\eta\right)\Phi^{\ast}\left(\eta^{\prime\prime}\right)\exp\left\\{\frac{\mathtt{i}}{2B}\left[D\left(\left|\eta^{\prime\prime}\right|^{2}-\left|\eta\right|^{2}\right)-2\eta_{1}^{\prime}\left(\eta_{1}^{\prime\prime}-\eta_{1}\right)-2\eta_{2}^{\prime}\left(\eta_{2}^{\prime\prime}-\eta_{2}\right)\right]\right\\},$

which is the same as $R_{2}\left(\eta_{1},\eta_{2}\right)$ in (20). So
combining (20), (10)-(11) and (21) we reach the conclusion

$\displaystyle\left|\frac{1}{-2\mathtt{i}B}\int\frac{\mathtt{d}^{2}\eta}{\pi}\exp\left\\{\frac{\mathtt{i}}{-2B}\left[D\left|\eta\right|^{2}-\left(\eta\eta^{\prime\ast}+\eta^{\ast}\eta^{\prime}\right)+A\left|\eta^{\prime}\right|^{2}\right]\right\\}\Phi\left(\eta\right)\right|^{2}$
(22) $\displaystyle=$
$\displaystyle\pi\int\delta\left(\eta_{1}^{\prime}-D\sigma_{1}-B\gamma_{2}\right)\delta\left(\eta_{2}^{\prime}-D\sigma_{2}+B\gamma_{1}\right)W\left(\sigma,\gamma\right)\mathtt{d}^{2}\sigma\mathtt{d}^{2}\gamma,$

where $AD-BC=1$. The physical meaning of Eq. (22) is: when an input field
propagates through an optical $\left[D\left(-B\right)\left(-C\right)A\right]$
system, the energy density of the output field is equal to the Radon transform
of the two-mode entangled Wigner function of the input field. So far as our
knowledge is concerned, this conclusion seems new. Eq. (22) is the
relationship between the input amplitude and output one in spatial-domain.
Next we turn to the frequency domain.

If taking the matrix element of $F^{\left(t,k\right)}$ in the
$\left|\xi\right\rangle$ representation which is conjugate to
$\left|\eta\right\rangle$, where the overlap
$\left\langle\eta\right|\left.\xi\right\rangle$ is
$\left\langle\eta\right|\left.\xi\right\rangle=\frac{1}{2}\exp[(\xi\eta^{\ast}-\xi^{\ast}\eta)/2],$
we obtain the 2-dimensional GFT in its ‘frequency domain’, i.e.,

$\displaystyle\frac{1}{\pi}\left\langle\xi^{\prime}\right|F^{\left(t,k\right)}\left|\xi\right\rangle=\int\frac{\mathtt{d}^{2}\eta\mathtt{d}^{2}\eta^{\prime}}{\pi^{3}}\left\langle\xi^{\prime}\right|\left.\eta^{\prime}\right\rangle\left\langle\eta^{\prime}\right|F^{\left(t,k\right)}\left|\eta\right\rangle\left\langle\eta\right|\left.\xi\right\rangle$
(23) $\displaystyle=$
$\displaystyle\frac{1}{4}\int\frac{\mathtt{d}^{2}\eta\mathtt{d}^{2}\eta^{\prime}}{\pi^{2}}\exp\left(\frac{\xi^{\prime\ast}\eta^{\prime}-\xi^{\prime}\eta^{\prime\ast}+\xi\eta^{\ast}-\xi^{\ast}\eta}{2}\right)\mathcal{K}^{\left(t,k\right)}\left(\eta^{\prime},\eta\right)$
$\displaystyle=$
$\displaystyle\frac{1}{2\mathtt{i}\left(-C\right)\pi}\exp\left[\frac{\mathtt{i}}{2\left(-C\right)}\left(D\left|\xi\right|^{2}+A\left|\xi^{\prime}\right|^{2}-\xi^{\prime\ast}\xi-\xi^{\prime}\xi^{\ast}\right)\right]\equiv\mathcal{K}_{2}^{N}\left(\xi^{\prime},\xi\right),$

where the superscript $N$ of $\mathcal{K}_{2}^{N}$ means that this transform
kernel corresponds to the parameter matrix $N=\left[D,-C,-B,A\right].$ Thus if
the $\left[D,-C,-B,A\right]$ system is changed to
$\tilde{N}=\left[A,C,B,D\right]$ system, the GFT in its ‘frequency domain’ is
given by

$\Psi\left(\xi^{\prime}\right)=\int\mathcal{K}_{2}^{\tilde{N}}\left(\xi^{\prime},\xi\right)\Phi\left(\xi\right)\mathtt{d}^{2}\xi,$
(24)

where $\mathcal{K}_{2}^{\tilde{N}}\left(\xi^{\prime},\xi\right)$ is

$\mathcal{K}_{2}^{N}\left(\xi^{\prime},\xi\right)=\frac{1}{2\mathtt{i}C\pi}\exp\left[\frac{\mathtt{i}}{2C}\left(A\left|\xi\right|^{2}+D\left|\xi^{\prime}\right|^{2}-\xi^{\prime\ast}\xi-\xi^{\prime}\xi^{\ast}\right)\right].$
(25)

It then follows from Eqs.(24) and (25) that

$\displaystyle\left|\Psi\left(\xi^{\prime}\right)\right|^{2}$ $\displaystyle=$
$\displaystyle\int\mathcal{K}_{2}^{\tilde{N}}\left(\xi^{\prime},\xi\right)\Phi\left(\xi\right)\mathtt{d}^{2}\xi\int\mathcal{K}_{2}^{\ast\tilde{N}}\left(\xi^{\prime},\xi^{\prime\prime}\right)\Phi^{\ast}\left(\xi^{\prime\prime}\right)\mathtt{d}^{2}\xi^{\prime\prime}$
(26) $\displaystyle=$
$\displaystyle\frac{1}{4\pi^{2}C^{2}}\int\mathtt{d}^{2}\xi\mathtt{d}^{2}\xi^{\prime\prime}\Phi\left(\xi\right)\Phi^{\ast}\left(\xi^{\prime\prime}\right)$
$\displaystyle\times\exp\left\\{\frac{\mathtt{i}}{2C}\left[A\left(\left|\xi\right|^{2}-\left|\xi^{\prime\prime}\right|^{2}\right)+2\xi_{1}^{\prime}\left(\xi_{1}^{\prime\prime}-\xi_{1}\right)+2\xi_{2}^{\prime}\left(\xi_{2}^{\prime\prime}-\xi_{2}\right)\right]\right\\}.$

On the other hand, in similar to (18), we consider the integration transform,

$R_{2}\left(\xi_{1},\xi_{2}\right)=\pi\int\delta\left(\xi_{1}-A\sigma_{1}-C\gamma_{2}\right)\delta\left(\xi_{2}-A\sigma_{2}+C\gamma_{1}\right)W\left(\sigma,\gamma\right)\mathtt{d}^{2}\sigma\mathtt{d}^{2}\gamma,$
(27)

$R_{2}\left(\xi_{1},\xi_{2}\right)$ is also a probability distribution along
an infinitely thin phase space strip denoted by the real parameters $A,C$.
Substituting (19) into (27) yields

$\displaystyle R_{2}\left(\xi_{1},\xi_{2}\right)$ $\displaystyle=$
$\displaystyle\int\frac{\mathtt{d}^{2}\sigma^{\prime}\mathtt{d}^{2}\sigma^{\prime\prime}}{\pi^{2}}\psi\left(\sigma^{\prime}\right)\psi^{\ast}\left(\sigma^{\prime\prime}\right)\int\delta^{\left(2\right)}\left(2\sigma-\sigma^{\prime}-\sigma^{\prime\prime}\right)\mathtt{d}^{2}\sigma\mathtt{d}^{2}\gamma$
(28)
$\displaystyle\times\delta\left(\xi_{2}-A\sigma_{2}+C\gamma_{1}\right)\delta\left(\xi_{1}-A\sigma_{1}-C\gamma_{2}\right)e^{\left(\sigma^{\prime}-\sigma\right)\gamma^{\ast}-\left(\sigma^{\prime}-\sigma\right)^{\ast}\gamma}$
$\displaystyle=$
$\displaystyle\int\frac{\mathtt{d}^{2}\sigma^{\prime}\mathtt{d}^{2}\sigma^{\prime\prime}}{4\pi^{2}}\psi\left(\sigma^{\prime}\right)\psi^{\ast}\left(\sigma^{\prime\prime}\right)\int\mathtt{d}^{2}\gamma\delta\left(\xi_{2}-A\frac{\sigma_{2}^{\prime}+\sigma_{2}^{\prime\prime}}{2}+C\gamma_{1}\right)$
$\displaystyle\times\delta\left(\xi_{1}-A\frac{\sigma_{1}^{\prime}+\sigma_{1}^{\prime\prime}}{2}-C\gamma_{2}\right)\exp\left[\frac{\sigma^{\prime}-\sigma^{\prime\prime}}{2}\gamma^{\ast}-\frac{\sigma^{\prime\ast}-\sigma^{\prime\prime\ast}}{2}\gamma\right]$
$\displaystyle=$
$\displaystyle\int\frac{\mathtt{d}^{2}\sigma^{\prime}\mathtt{d}^{2}\sigma^{\prime\prime}}{4\pi^{2}C^{2}}\psi\left(\sigma^{\prime}\right)\psi^{\ast}\left(\sigma^{\prime\prime}\right)$
$\displaystyle\times\exp\left\\{\frac{\mathtt{i}}{2C}\left[A\left(\left|\sigma^{\prime}\right|^{2}-\left|\sigma^{\prime\prime}\right|^{2}\right)+2\xi_{1}\left(\sigma_{1}^{\prime\prime}-\sigma_{1}^{\prime}\right)+2\xi_{2}\left(\sigma_{2}^{\prime\prime}-\sigma_{2}^{\prime}\right)\right]\right\\},$

which is the same as $\left|\Psi\left(\xi^{\prime}\right)\right|^{2}$ in (26).
So combining (28), (24)-(25), and (26) we can draw the conclusion

$\displaystyle\left|\frac{1}{2\mathtt{i}C\pi}\int\exp\left[\frac{\mathtt{i}}{2C}\left(A\left|\xi\right|^{2}+D\left|\xi^{\prime}\right|^{2}-\xi^{\prime\ast}\xi-\xi^{\prime}\xi^{\ast}\right)\right]\Phi\left(\xi\right)\mathtt{d}^{2}\xi\right|^{2}$
(29) $\displaystyle=$
$\displaystyle\pi\int\delta\left(\xi_{1}-A\sigma_{1}-C\gamma_{2}\right)\delta\left(\xi_{2}-A\sigma_{2}+C\gamma_{1}\right)W\left(\sigma,\gamma\right)\mathtt{d}^{2}\sigma\mathtt{d}^{2}\gamma.$

This is the relationship between the output amplitude and input one’s
entangled Wigner function in ‘frequency domain’.

In sum, based on the correspondence between Collins diffraction formula
(optical Fresnel transform) and the transformation matrix element of a three-
parameters two-mode squeezing operator in the entangled state representation,
we have explored the relationship between output field intensity determined by
the Collins formula and the input field’s probability distribution along an
infinitely thin phase space strip. The entangled Wigner function is introduced
for recapitulating the result.

Work supported by the National Natural Science Foundation of China (Grant Nos
10775097 and 10874174).

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# On the center of near-group fusion category of type $\mathbb{Z}_{3}+6$

Zhiqiang Yu

###### Abstract

Let $\mathcal{A}$ be a near-group fusion category of type $\mathbb{Z}_{3}+6$.
We show that there is a modular tensor equivalence
$\mathcal{Z}(\mathcal{A})\cong\mathcal{C}(\mathbb{Z}_{3},\eta)\boxtimes\mathcal{C}(\mathfrak{sl}_{3},9)_{\mathbb{Z}_{3}}^{0}$.
Moreover, we construct two non-trivial faithful extensions of $\mathcal{A}$
explicitly, whose Drinfeld centers can also be obtained from representation
categories quantum groups at root of unity.

Keywords: Drinfeld center; near-group fusion category

Mathematics Subject Classification 2020: 18M20 $\cdot$ 81R50

##  1 Introduction

A fusion category $\mathcal{C}$ over the complex field $\mathbb{C}$ is a
semisimple finite tensor category. Let $G$ be a finite group. A fusion
category $\mathcal{C}$ is called pointed if its simple objects are all
invertible [7]. Equivalently, $\mathcal{C}$ is tensor equivalent to the
category $\text{Vec}_{G}^{\omega}$ of finite-dimensional $G$-graded vector
spaces over $\mathbb{C}$, where $\omega\in Z^{3}(G,\mathbb{C}^{*})$ is a
(normalized) $3$-cocycle, which provides the associator of the monoidal
structure of $\text{Vec}_{G}^{\omega}$ [6, 7].

$\mathcal{C}$ is called a near-group fusion category if all of the isomorphism
classes of its simple objects except one are invertible [21]. Specifically,
let $n$ be a non-negative integer and let $G$ be the finite group of
isomorphism classes of $\mathcal{C}$, and $X$ be the unique non-invertible
simple object of $\mathcal{C}$. Then the Grothendieck ring
$\text{Gr}(\mathcal{C})$ is determined by the following fusion relations

$\displaystyle g\cdot h=gh,g\cdot X=X\cdot g=X,X\cdot X=\sum_{g\in
G}g+nX,\forall g,h\in G.$

We will say $\mathcal{C}$ is a near-group fusion category of type $G+n$. It
turns out that the group $G$ and integer $n$ have to meet strict restrictions
to guarantee that there exists a near-group fusion category, see [9, 11, 21]
and the references therein for details. For example, if $\text{FPdim}(X)$ is
irrational, then $n$ is a multiple of the order of $|G|$ [17, Appendix A].

Let $G=\mathbb{Z}_{3}$, it was proved in [13, Theorem 4.11] that if there is a
near-group fusion category of type $\mathbb{Z}_{3}+n$, then $n=0,2,3,6$. The
existences of fusion categories in the first three cases was clear then, and
the existence of near-group fusion category of type $\mathbb{Z}_{3}+6$ was
later proved in [11, Theorem 10.19] 111Indeed, it was pointed by D. Penneys
that the existence of this near-group fusion category dates back to 2000, as
the even half of the subfactor whose principal graph was described in Figure 5
[2, P. 28]., who attributes the result to Zhengwei Liu and Noah Snyder. In
this paper, we pay our attention to consider the Drinfeld center of near-group
fusion category $\mathcal{A}$ of type $\mathbb{Z}_{3}+6$. We show that
$\mathcal{Z}(\mathcal{A})$ is factored in to Deligne tensor product of a
pointed modular fusion category and
$\mathcal{C}(\mathfrak{sl}_{3},9)_{\mathbb{Z}_{3}}^{0}$ (see Theorem 3.6, also
[5, Theorem 5.2]). Notice that both of these fusion categories can be obtained
from representation categories of quantum groups of root of unity, hence the
near-group fusion categories of type $\mathbb{Z}_{3}+6$ is not exotic. And we
construct two faithful $G$-graded extensions of $\mathcal{A}$ by using Witt
equivalence and non-degenerate fusion categories developed in [3].

$\mathbf{Note:}$ Recently, the author was informed by S-H. Ng and Y. Wang that
Theorem 3.6 was obtained in [5, Theorem 5.2] already, the author claims that
this is worked out independently.

This paper is organized as follows. In Section 2, we recall some basic notions
and notations of fusion categories, such as Frobenius-Perron dimension,
spherical structure, étale algebras, etc. In Section 3, we consider the
Drinfeld centers and two faithful graded extensions of near-group fusion
category $\mathcal{A}$ of type $\mathbb{Z}_{3}+6$. In Subsection 3.1, we first
determine the fusion rules of
$\mathcal{C}(\mathfrak{sl}_{3},9)_{\mathbb{Z}_{3}}^{0}$, then we obtain the
$S$-matrix with the help of Verlinde formula in Proposition 3.5, in the last,
we determine the structure of $\mathcal{Z}(\mathcal{A})$ in Theorem 3.6. In
Subsection 3.2, we construct two faithful graded extension of $\mathcal{A}$,
and we determine the fusion relations of them explicitly in Theorem 3.8 and
Theorem 3.10, respectively.

##  2 Preliminaries

In this section, we will recall some most used definitions and properties
about fusion categories and modular fusion categories, we refer the readers to
[4, 6, 7, 15] to standard conclusions for fusion categories and braided fusion
categories.

Let $\mathcal{C}$ be a fusion category, $\mathcal{O}(\mathcal{C})$ the set of
isomorphism classes of simple objects of $\mathcal{C}$. Then there is a unique
ring homomorphism $\text{FPdim}(-)$, called the Frobenius-Perron homomorphism,
from the Grothendieck ring $\text{Gr}(\mathcal{C})$ to $\mathbb{C}$ such that
$\text{FPdim}(X)$ is a positive algebraic integer for all non-zero object $X$
of $\mathcal{C}$ [6, 7]. The sum

$\displaystyle\text{FPdim}(\mathcal{C}):=\sum_{X\in\mathcal{O}(\mathcal{C})}\text{FPdim}(X)^{2}$

is called the Frobenius-Perron dimension of fusion category $\mathcal{C}$.

A fusion category $\mathcal{C}$ is spherical if $\mathcal{C}$ admits a
spherical structure $j$. Specifically, $j$ is a natural isomorphism from
identity tensor functor $\text{id}_{\mathcal{C}}$ to the double dual tensor
functor $(-)^{**}$ and $\dim_{j}(X)=\dim_{j}(X^{*})$ for all objects $X$ of
$\mathcal{C}$, where $\dim_{j}(X)$ is the quantum dimension of an object $X$
is defined by the (categorical) trace of $\text{id}_{X}$, that is,

$\displaystyle\dim_{j}(X)=\text{Tr}(\text{id}_{X}):=\text{ev}_{X}\circ(j_{X}\circ\text{id}_{X})\otimes\text{id}_{X^{*}}\circ\text{coev}_{X},$

here we suppress the associativity and unit constraints of $\mathcal{C}$. Then
the global (or categorical) dimension of the spherical fusion category
$\mathcal{C}$ is defined as

$\displaystyle\dim(\mathcal{C}):=\sum_{X\in\mathcal{O}(\mathcal{C})}\dim_{j}(X)^{2}.$

The global dimension $\dim(\mathcal{C})$ is independent of the choice of the
spherical structure of $\mathcal{C}$. In this following, we will fix a
spherical structure $j$ on $\mathcal{C}$ and denote $\dim_{j}(-)$ by
$\dim(-)$. A fusion category $\mathcal{C}$ is pseudo-unitary if
$\text{FPdim}(\mathcal{C})=\dim(\mathcal{C})$.

Let $\mathcal{C}$ be a braided fusion category with braiding $c$. Then
$\mathcal{C}$ is called a pre-modular fusion category if $\mathcal{C}$ is
spherical. Let $\theta$ be the corresponding ribbon structure of
$\mathcal{C}$. The two matrices $S=(S_{X,Y})$ and $T=(\delta_{X,Y}\theta_{X})$
are called the $S$-matrix and $T$-matrix of $\mathcal{C}$, respectively, where
$S_{X,Y}:=\text{Tr}(c_{Y,X}c_{X,Y})$, $X,Y\in\mathcal{O}(\mathcal{C})$. A
modular fusion category $\mathcal{C}$ is a pre-modular fusion category such
that the $S$-matrix $S$ is non-degenerate [4, 6]. Drinfeld centers of
spherical fusion categories are always modular [15], for example. It is well-
known that a braided pointed fusion category $\text{Vec}_{G}^{\omega}$ is
modular if and only if $G$ is a metric group equipped with a non-degenerate
quadratic form $\eta$ [4, 6]. In this paper, we will use $\mathcal{C}(G,\eta)$
to denote the modular fusion category determined by metric group $(G,\eta)$.

Recall that a commutative algebra $A$ in a braided fusion category
$\mathcal{C}$ is said to be a connected étale algebra if the category
$\mathcal{C}_{A}$ of right $A$-modules in $\mathcal{C}$ is semisimple and
$\text{Hom}_{\mathcal{C}}(I,A)=\mathbb{C}$ [3, Definition 3.1]. For example,
let $\text{Rep}(G)\subseteq\mathcal{C}$ be a Tannakian fusion subcategory,
then the function (or regular) algebra $\text{Fun}(G)\in\text{Rep}(G)$ is a
connected étale algebra [3, 4].

Let $A$ be a connected étale algebra, and let $(M,\mu_{M})\in\mathcal{C}_{A}$,
the fusion category of right $A$-modules in $\mathcal{C}$, where
$\mu_{M}:M\otimes A\to M$ is the right $A$-module morphism of $M$. Then $M$ is
called a local (or dyslectic) module if $\mu_{M}=\mu_{M}\circ(c_{A,M}c_{M,A})$
[3, 12, 18]. The category of local modules over $A$ is a braided fusion
category [12, 18], which will be denoted by $\mathcal{C}_{A}^{0}$ below. In
addition, if $\mathcal{C}$ is modular, so is $\mathcal{C}_{A}^{0}$ [3, 4, 12].

##  3 Center and extensions of near-group fusion categories

###  3.1 Modular data of
$\mathcal{C}(\mathfrak{sl}_{3},9)_{\mathbb{Z}_{3}}^{0}$ and Drinfeld center

In this subsection, we determine the fusion rules and modular data of
$\mathcal{C}(\mathfrak{sl}_{3},9)_{\mathbb{Z}_{3}}^{0}$, and we show that the
Drinfeld center of the near-group fusion category of type $\mathbb{Z}_{3}+6$
is modular tensor equivalent to
$\mathcal{C}\boxtimes\mathcal{C}(\mathfrak{sl}_{3},9)_{\mathbb{Z}_{3}}^{0}$,
where $\mathcal{C}$ is a pointed modular fusion category of Frobenius-Perron
dimension $3$.

Let $k$ be a positive integer. Let $\mathcal{C}(\mathfrak{sl}_{3},k)$ be the
modular fusion category obtained from semisimplification of the category
$\text{Rep}(U_{q}(\mathfrak{sl}_{3}))$ of finite-dimensional representation of
$U_{q}(\mathfrak{sl}_{3})$ at root of unity $q=e^{\frac{2\pi i}{3(k+3)}}$, we
refer the readers to [1, 6] for a detailed construction of modular fusion
categories $\mathcal{C}(\mathfrak{g},k)$, where $\mathfrak{g}$ is a semisimple
Lie algebra over $\mathbb{C}$.

Let $\alpha_{1},\alpha_{2}$ be the simple roots of $\mathfrak{sl}_{3}$, and
let $\lambda_{1}=\frac{2\alpha_{1}+\alpha_{2}}{3}$ and
$\lambda_{2}=\frac{\alpha_{1}+2\alpha_{2}}{3}$ be the fundamental dominant
weights of $\mathfrak{sl}_{3}$ [1]. Notice that simple objects of
$\mathcal{C}(\mathfrak{sl}_{3},k)$ are in bijection with pairs of non-negative
integers $(m_{1},m_{2})$ such that $m_{1}+m_{2}\leq k$ [1, 20], each
$(m_{1},m_{2})$ corresponds to a dominant weight
$m_{1}\lambda_{1}+m_{2}\lambda_{2}$, indeed. Hence we will use $(m_{1},m_{2})$
to denote simple objects of $\mathcal{C}(\mathfrak{sl}_{3},k)$. Moreover, for
an arbitrary simple object $(m_{1},m_{2})$, we have

(1)
$\displaystyle\dim((m_{1},m_{2}))=\frac{\sin\left(\frac{(m_{1}+1)\pi}{k+3}\right)\sin\left(\frac{(m_{2}+1)\pi}{k+3}\right)\sin\left(\frac{(m_{1}+m_{2}+2)\pi}{k+3}\right)}{\sin\left(\frac{2\pi}{k+3}\right)\sin^{2}\left(\frac{\pi}{k+3}\right)},$

which is a positive algebraic integer, so it is the Frobenius-Perron dimension
of $(m_{1},m_{2})$, and then $\mathcal{C}(\mathfrak{sl}_{3},k)$ is pseudo-
unitary [6].

If $3\mid k$, then $\mathcal{C}(\mathfrak{sl}_{3},k)$ contains a maximal
Tannakian fusion subcategory $\text{Rep}(\mathbb{Z}_{3})$, whose isomorphism
classes of simple objects are $\\{(0,0),(k,0),(0,k)\\}$. Let
$A:=(0,0)\oplus(k,0)\oplus(0,k)$ be the corresponding connected étale algebra
of $\text{Rep}(\mathbb{Z}_{3})$, and let

$\displaystyle
F_{A}:\mathcal{C}(\mathfrak{sl}_{3},k)\to\mathcal{C}(\mathfrak{sl}_{3},k)_{A},M\mapsto
M\otimes A,$

be the free right $A$-module functor, where
$\mathcal{C}(\mathfrak{sl}_{3},k)_{A}$ is the fusion category of right
$A$-modules in $\mathcal{C}(\mathfrak{sl}_{3},k)$ with unit object $A$.

######  Lemma 3.1.

The isomorphism classes of simple objects of
$\mathcal{C}(\mathfrak{sl}_{3},9)_{\mathbb{Z}_{3}}^{0}$ are

$\displaystyle\\{I,X_{1},X_{2},X_{3},Y_{1},Y_{2},Y_{3},Y_{4},Y_{5}\\},$

$\text{FPdim}(X_{i})=\text{FPdim}(Y_{j})=d$ for $1\leq i,j\leq 3$,
$\text{FPdim}(Y_{4})=2d+2$ and $\text{FPdim}(Y_{5})=2d+1$, where
$d:=3+2\sqrt{3}$.

###### Proof.

It was characterized in [20, subsection 3.5] that
$\mathcal{C}(\mathfrak{sl}_{3},9)_{\mathbb{Z}_{3}}^{0}$ contains two types of
simple objects. The first classes are the orbits of simple objects
$(m_{1},m_{2})$ under action of the algebra $A=(0,0)\oplus(9,0)\oplus(0,9)$,
where the weight $m_{1}\lambda_{1}+m_{2}\lambda_{2}$ determined by
$(m_{1},m_{2})$ belongs to root lattice of $\mathfrak{sl}_{3}$. A direct
computation shows that these simple objects are

$\displaystyle I:$
$\displaystyle=F_{A}((0,0))=(0,0)\oplus(0,9)\oplus(9,0),Y_{1}:=F_{A}((1,1))=(1,1)\oplus(1,7)\oplus(7,1),$
$\displaystyle Y_{2}:$
$\displaystyle=F_{A}((3,0))=(3,0)\oplus(0,6)\oplus(6,3),Y_{3}:=F_{A}((0,3))=(0,3)\oplus(3,6)\oplus(6,0),$
$\displaystyle Y_{4}:$
$\displaystyle=F_{A}((2,2))=(2,2)\oplus(2,5)\oplus(5,2),Y_{5}:=F_{A}((4,1))=(4,1)\oplus(1,4)\oplus(4,4).$

The other type is three simple objects $X_{1},X_{2},X_{3}$, which are
isomorphic to $(3,3)$ as objects but non-isomorphic as right $A$-modules. The
Frobenius-Perron dimensions of simple objects of
$\mathcal{C}(\mathfrak{sl}_{3},9)_{\mathbb{Z}_{3}}^{0}$ follows from [12,
Theorem 1.18] and equation (1). ∎

Next, we determine the fusion rules of
$\mathcal{C}(\mathfrak{sl}_{3},9)_{\mathbb{Z}_{3}}^{0}$.

######  Proposition 3.2.

The fusion rules between $Y_{j}$ ($1\leq j\leq 5$) are the following equations

$\displaystyle Y_{1}\otimes Y_{1}=I\oplus Y_{1}\oplus f_{1}\oplus
Y_{4},Y_{1}\otimes Y_{2}=Y_{1}\oplus Y_{2}\oplus f_{2},Y_{1}\otimes
Y_{3}=Y_{1}\oplus Y_{3}\oplus f_{2},$ $\displaystyle Y_{1}\otimes
Y_{4}=f_{1}\oplus 2f_{2}\oplus f_{3},Y_{1}\otimes Y_{5}=Y_{2}\oplus
Y_{3}\oplus 2f_{2}\oplus f_{3},Y_{2}\otimes Y_{2}=2Y_{3}\oplus f_{2},$
$\displaystyle Y_{2}\otimes Y_{3}=I\oplus Y_{1}\oplus Y_{4}\oplus
f_{3},Y_{2}\otimes Y_{4}=f_{1}\oplus 2f_{2}\oplus f_{3},$ $\displaystyle
Y_{2}\otimes Y_{5}=Y_{1}\oplus Y_{3}\oplus 2f_{2}\oplus f_{3},Y_{3}\otimes
Y_{3}=2Y_{2}\oplus f_{2},Y_{3}\otimes Y_{4}=f_{1}\oplus 2f_{2}\oplus f_{3},$
$\displaystyle Y_{3}\otimes Y_{5}=Y_{1}\oplus Y_{2}\oplus 2f_{2}\oplus
f_{3},Y_{4}\otimes Y_{4}=I\oplus 2f_{1}\oplus 5f_{2}\oplus 2f_{3},$
$\displaystyle Y_{4}\otimes Y_{5}=2f_{1}\oplus Y_{4}\oplus 4f_{2}\oplus
2f_{3},Y_{5}\otimes Y_{5}=I\oplus 2f_{1}\oplus 4f_{2}\oplus 2f_{3},$

where $f_{1}:=Y_{1}\oplus Y_{2}\oplus Y_{3}$, $f_{2}:=Y_{4}\oplus Y_{5}$ and
$f_{3}:=X_{1}\oplus X_{2}\oplus X_{3}$.

###### Proof.

We compute these fusion rules by using program SageMath [19]. For example,

$\displaystyle Y_{1}\otimes Y_{1}$
$\displaystyle=F_{A}((1,1))\otimes_{A}F_{A}((1,1))=F_{A}((1,1)\otimes(1,1))$
$\displaystyle=F_{A}((0,0)\oplus(0,3)\oplus(3,0)\oplus 2(1,1)\oplus(2,2))$
$\displaystyle=F_{A}((0,0))\oplus F_{A}((0,3))\oplus F_{A}((3,0))\oplus
2F_{A}((1,1))\oplus F_{A}((2,2))$ $\displaystyle=I\oplus 2Y_{1}\oplus
Y_{2}\oplus Y_{3}\oplus Y_{4}.$

The rest fusion rules can be computed in the same way. ∎

######  Proposition 3.3.

For simple objects $X_{i}$ ($1\leq i\leq 3$) and $Y_{l}$ ($1\leq l\leq 5$),
the fusion rules are

$\displaystyle X_{i}\otimes Y_{1}=f_{2}\oplus X_{j}\oplus X_{k},X_{i}\otimes
Y_{2}=Y_{2}\oplus f_{2}\oplus X_{i},X_{i}\otimes Y_{3}=Y_{3}\oplus f_{2}\oplus
X_{i},$ $\displaystyle X_{i}\otimes Y_{4}=f_{1}\oplus 2f_{2}\oplus
f_{3},X_{i}\otimes Y_{5}=f_{1}\oplus 2f_{2}\oplus X_{j}\oplus X_{k},$
$\displaystyle X_{i}\otimes X_{i}=I\oplus Y_{2}\oplus Y_{3}\oplus Y_{4}\oplus
2X_{i},X_{i}\otimes X_{j}=Y_{1}\oplus f_{2}\oplus X_{k}.$

where $1\leq i,j,k\leq 3$ are distinct and $f_{2}=Y_{4}\oplus Y_{5}$ .

###### Proof.

Notice that simple objects $X_{j}$ are self-dual, $1\leq j\leq 3$. For $1\leq
i\neq j\leq 3$, by decomposing $\text{FPdim}(X_{i}\otimes X_{j})$ into a sum
of Frobenius-Perron dimensions of simple objects of
$\mathcal{C}(\mathfrak{sl}_{3},9)_{\mathbb{Z}_{3}}^{0}$, and together with
Proposition 3.2, we find

$\displaystyle X_{i}\otimes Y_{5}$ $\displaystyle=Y_{1}\oplus Y_{2}\oplus
Y_{3}\oplus 2Y_{4}\oplus 2Y_{5}\oplus X_{j}\oplus X_{k},$ $\displaystyle
X_{i}\otimes Y_{4}$ $\displaystyle=Y_{1}\oplus Y_{2}\oplus Y_{3}\oplus
2Y_{4}\oplus 2Y_{5}\oplus X_{1}\oplus X_{2}\oplus X_{3},$

where $i,j,k$ are distinct. By Proposition 3.2, $X_{1}\otimes
Y_{2}=Y_{2}\otimes Y_{4}\oplus Y_{5}\oplus X_{j}$ for some $j$, let
$X_{j}\otimes Y_{2}=Y_{2}\oplus Y_{4}\oplus Y_{5}\oplus X_{k}$. Since
$(X_{1}\otimes Y_{2})\otimes Y_{2}=X_{1}\otimes(Y_{2}\otimes Y_{2})$, we find
$X_{j}=X_{k}=X_{1}$. By symmetry of this computation, we have $X_{j}\otimes
Y_{2}=Y_{2}\oplus Y_{4}\oplus Y_{5}\oplus X_{j}$ for $1\leq j\leq 3$.
Similarly, for $1\leq i\leq 3$, it follows the decomposition of $(X_{i}\otimes
Y_{1})\otimes Y_{1}=X_{i}\otimes(Y_{1}\otimes Y_{1})$ that $X_{i}\otimes
Y_{1}=Y_{4}\oplus Y_{5}\oplus X_{j}\oplus X_{k}$ with $i,j,k$ being distinct.

By Proposition 3.2, for $1\leq i\neq j\leq 3$, we know $X_{i}\otimes
X_{j}=Y_{1}\oplus Y_{4}\oplus Y_{5}\oplus X_{ij}$ for some
$X_{ij}\in\\{X_{1},X_{2},X_{3}\\}$, so
$\dim_{\mathbb{C}}\text{Hom}_{\mathcal{C}(\mathfrak{sl}_{3},9)_{\mathbb{Z}_{3}}^{0}}(X_{i}\otimes
X_{j},X_{i}\otimes X_{j})=4$. Meanwhile,

$\displaystyle\dim_{\mathbb{C}}\text{Hom}_{\mathcal{C}(\mathfrak{sl}_{3},9)_{\mathbb{Z}_{3}}^{0}}(X_{i}\otimes
X_{j},X_{i}\otimes
X_{j})=\dim_{\mathbb{C}}\text{Hom}_{\mathcal{C}(\mathfrak{sl}_{3},9)_{\mathbb{Z}_{3}}^{0}}(X_{i}\otimes
X_{i},X_{j}\otimes X_{j}),$

hence $X_{i}\otimes X_{i}=I\oplus Y_{2}\oplus Y_{3}\oplus Y_{4}\oplus 2X_{ii}$
for some $X_{ii}\in\\{X_{1},X_{2},X_{3}\\}$, and $X_{ii}\neq X_{jj}$ if $i\neq
j$. In summary, we have

$\displaystyle X_{i}\otimes X_{j}=\left\\{\begin{array}[]{ll}Y_{1}\oplus
Y_{4}\oplus Y_{5}\oplus X_{ij},&\hbox{ if $i\neq j$;}\\\ I\oplus Y_{2}\oplus
Y_{3}\oplus 2X_{ii},&\hbox{if $i=j$.}\end{array}\right.$

If $X_{i}\neq X_{ii}$, then it follows from the decomposition of
$(X_{i}\otimes X_{i})\otimes Y_{1}=X_{i}\otimes(X_{i}\otimes Y_{1})$ and the
decomposition of $X_{i}\otimes Y_{1}$ that

$\displaystyle
4=\dim_{\mathbb{C}}\text{Hom}_{\mathcal{C}(\mathfrak{sl}_{3},9)_{\mathbb{Z}_{3}}^{0}}(X_{i}\otimes
Y_{1},X_{ii}\otimes
Y_{1})=\dim_{\mathbb{C}}\text{Hom}_{\mathcal{C}(\mathfrak{sl}_{3},9)_{\mathbb{Z}_{3}}^{0}}(X_{i}\otimes
X_{ii},Y_{1}\otimes Y_{1})=3,$

it is a contradiction. Therefore, $X_{i}=X_{ii}$ for all $i$; consequently,
$X_{i}\otimes X_{j}=Y_{1}\oplus Y_{4}\oplus Y_{5}\oplus X_{k}$ for $i,j,k$
being distinct. ∎

In the last, we determine the modular data of
$\mathcal{C}(\mathfrak{sl}_{3},9)_{\mathbb{Z}_{3}}^{0}$. Let
$\zeta_{n}:=e^{\frac{2\pi i}{n}}$.

######  Lemma 3.4.

The ribbon structures of simple objects of
$\mathcal{C}(\mathfrak{sl}_{3},9)_{\mathbb{Z}_{3}}^{0}$ are given by

$\displaystyle\theta_{Y_{1}}=\theta_{X_{i}}=\zeta_{4},\theta_{Y_{2}}=\theta_{Y_{3}}=-1,\theta_{Y_{4}}=\zeta_{3}^{2},\theta_{Y_{5}}=1,1\leq
i\leq 3,$

###### Proof.

It follows from [1] that the ribbon structure of simple object $(m_{1},m_{2})$
of $\mathcal{C}(\mathfrak{sl}_{3},k)$ is

$\displaystyle\theta_{(m_{1},m_{2})}=e^{\frac{(m_{1}^{2}+3m_{1}+m_{1}m_{2}+3m_{2}+m_{2}^{2})}{3(k+3)}2\pi
i},$

meanwhile, [12, Theorem 1.17] states that ribbon structures of objects of
$\mathcal{C}(\mathfrak{sl}_{3},9)_{\mathbb{Z}_{3}}^{0}$ is inherited from
$\mathcal{C}(\mathfrak{sl}_{3},9)$, then the lemma follows. ∎

Given a modular fusion category $\mathcal{C}$, the balancing equation [6,
Proposition 8.13.8] states that

$\displaystyle
S_{X,Y}=\theta_{X}^{-1}\theta_{Y}^{-1}\sum_{Z\in\mathcal{O}(\mathcal{C})}N_{X,Y}^{Z}\dim(Z)\theta_{Z},\quad\forall
X,Y\in\mathcal{O}(\mathcal{C}),$

where $N_{X,Y}^{Z}:=\dim_{\mathbb{C}}\dim_{\mathcal{C}}(X\otimes Y,Z)$. A
direct computation with the balancing equation, we obtain the $S$-matrix of
$\mathcal{C}(\mathfrak{sl}_{3},9)_{\mathbb{Z}_{3}}^{0}$.

######  Proposition 3.5.

The $S$-matrix of $\mathcal{C}(\mathfrak{sl}_{3},9)_{\mathbb{Z}_{3}}^{0}$ is

$\displaystyle\left(\begin{array}[]{ccccccccc}1&d&d&d&2(1+d)&1+2d&d&d&d\\\
d&3d&d&d&0&-d&-d&-d&-d\\\ d&d&-(1+2\zeta_{4})d&-(1-2\zeta_{4})d&0&-d&d&d&d\\\
d&d&-(1-2\zeta_{4})d&-(1+2\zeta_{4})d&0&-d&d&d&d\\\
2(1+d)&0&0&0&-2(1+d)&2(1+d)&0&0&0\\\ 1+2d&-d&-d&-d&2(1+d)&1&-d&-d&-d\\\
d&-d&d&d&0&-d&3d&-d&-d\\\ d&-d&d&d&0&-d&-d&3d&-d\\\ d&-d&d&d&0&-d&-d&-d&3d\\\
\end{array}\right),$

where $d=3+2\sqrt{3}$.

We are ready to prove the following theorem

######  Theorem 3.6.

([5, Theorem 5.2]) There is a modular tensor equivalence
$\mathcal{C}(\mathbb{Z}_{3},\eta)\boxtimes\mathcal{C}(\mathfrak{sl}_{3},9)_{\mathbb{Z}_{3}}^{0}\cong\mathcal{Z}(\mathcal{A})$,
where $\mathcal{A}$ is a near-group fusion category of type
$\mathbb{Z}_{3}+6$.

###### Proof.

It was shown in [10, 20] that
$\mathcal{C}(\mathfrak{sl}_{3},9)_{\mathbb{Z}_{3}}^{0}$ contains a non-trivial
étale algebra $A$ such that

$\displaystyle\mathcal{C}(\mathbb{Z}_{3},\eta)\cong(\mathcal{C}(\mathfrak{sl}_{3},9)_{\mathbb{Z}_{3}}^{0})_{A}^{0}$

is a modular tensor equivalence, where $\mathbb{Z}_{3}=\langle g\rangle$ and
$\eta(g)=\zeta_{3}^{2}$. Hence, it follows from [3, Corollary 3.30] that there
is a modular tensor equivalence

$\displaystyle\mathcal{C}(\mathbb{Z}_{3},\eta)^{\text{rev}}\boxtimes\mathcal{C}(\mathfrak{sl}_{3},9)_{\mathbb{Z}_{3}}^{0}=\mathcal{C}(\mathbb{Z}_{3},\eta^{-1})\boxtimes\mathcal{C}(\mathfrak{sl}_{3},9)_{\mathbb{Z}_{3}}^{0}\cong\mathcal{Z}((\mathcal{C}(\mathfrak{sl}_{3},9)_{\mathbb{Z}_{3}}^{0})_{A}).$

Let
$\mathcal{A}:=(\mathcal{C}(\mathfrak{sl}_{3},9)_{\mathbb{Z}_{3}}^{0})_{A}$.
Then

$\displaystyle\text{FPdim}(\mathcal{A})^{2}=3\text{FPdim}(\mathcal{C}(\mathfrak{sl}_{3},9)_{\mathbb{Z}_{3}}^{0})=12^{2}(2+\sqrt{3})^{2}.$

Notice that connected étale subalgebras $B$ of $\mathcal{I}(I)$ are in
bijective correspondence with fusion subcategories $\mathcal{B}$ of
$\mathcal{A}$ [3, Theorem 4.10] satisfying
$\text{FPdim}(\mathcal{B})\text{FPdim}(B)=\text{FPdim}(\mathcal{A})$, where
$\mathcal{I}:\mathcal{A}\to\mathcal{Z}(\mathcal{A})$ is the adjoint functor to
the forgetful functor $F:\mathcal{Z}(\mathcal{A})\to\mathcal{A}$. Since étale
subalgebras of $\mathcal{I}(I)$ have trivial ribbon structures [16, Theorem
4.1], the image of $\mathcal{C}(\mathbb{Z}_{3},\eta^{-1})$ is a non-trivial
pointed fusion subcategory of $\mathcal{A}$. Up to isomorphism, a direct
computation shows that
$\mathcal{C}(\mathbb{Z}_{3},\eta^{-1})\boxtimes\mathcal{C}(\mathfrak{sl}_{3},9)_{\mathbb{Z}_{3}}^{0}$
contains exactly four such simple objects

$\displaystyle I\boxtimes I,I\boxtimes Y_{5},g\boxtimes Y_{4},g^{2}\boxtimes
Y_{4}.$

By comparing their Frobenius-Perron dimensions, we find that there exist only
two possible non-trivial étale subalgebras of $\mathcal{I}(I)$: $I\boxtimes
I\oplus I\boxtimes Y_{5}$ and $\mathcal{I}(I)$ itself. Therefore,
$\mathcal{A}$ only contains a unique non-trivial fusion subcategory, which is
also pointed.

By decomposing $\text{FPdim}(\mathcal{A})=24+12\sqrt{3}$ into sums of squares
of Frobenius-Perron dimensions of simple objects of $\mathcal{A}$ over the
field $\mathbb{Q}(\sqrt{3})$, there exist only two solutions:

$\displaystyle 24+12\sqrt{3}=3+(3+2\sqrt{3})^{2}=3+3(2+\sqrt{3})^{2}.$

In the later case, $\mathcal{A}$ contains exactly three simple objects
$\\{Z_{1},Z_{2},Z_{3}\\}$ (up to isomorphism) of Frobenius-Perron dimension
$2+\sqrt{3}$, so $Z_{i}\otimes Z_{i}^{*}$ is a direct sum of simple objects of
$\mathcal{A}$. However, it is easy to see that there does not exist such a
decomposition. Hence, $\mathcal{O}(\mathcal{A})$ contains a unique non-
invertible object $X$ of Frobenius-Perron dimension $3+\sqrt{3}$, thus we have

$\displaystyle X\otimes X=I\oplus g\oplus g^{2}\oplus 6X.$

That is, $\mathcal{A}$ is a near-group fusion category of type
$\mathbb{Z}_{3}+6$. ∎

It would be interesting to show whether other near-group fusion categories
(generally, quadratic fusion categories) are coming from quantum groups at
roots of unity. For example, by using the same method, we also have the
following result.

######  Proposition 3.7.

Let $\mathcal{C}$ be a near-group fusion category of type $\mathbb{Z}_{4}+4$,
then there exists a modular tensor equivalence
$\mathcal{Z}(\mathcal{C})_{\mathbb{Z}_{2}}^{0}\cong\mathcal{C}(\mathbb{Z}_{2},\eta)\boxtimes\mathcal{C}(\mathfrak{sl}_{3},5)_{\text{ad}}$.

###### Proof.

Notice that the simple objects of
$\mathcal{C}(\mathfrak{sl}_{3},5)_{\text{ad}}$ are

$\displaystyle\\{(0,0),(0,3),(3,0),(1,1),(2,2),(1,4),(4,1)\\}.$

It is well-known that $\mathcal{C}(\mathfrak{sl}_{3},5)_{\text{ad}}$ contains
a connected étale algebra $A=(0,0)\oplus(2,2)$ such that
$\mathcal{C}(\mathbb{Z}_{2},\eta)=(\mathcal{C}(\mathfrak{sl}_{3},5)_{\text{ad}})_{A}^{0}$
[10, 20]. Hence,
$\mathcal{C}(\mathbb{Z}_{2},\eta)^{\text{rev}}\boxtimes\mathcal{C}(\mathfrak{sl}_{3},5)_{\text{ad}}\cong\mathcal{Z}(\mathcal{B})$
for fusion category
$\mathcal{B}:=(\mathcal{C}(\mathfrak{sl}_{3},5)_{\text{ad}})_{A}$ [3]. Then
$\mathcal{B}$ is a fusion category of rank $4$ and fusion rules are

$\displaystyle X\otimes X^{*}=I\oplus X\oplus X^{*},g\otimes X=X^{*}=X\otimes
g,$ $\displaystyle g\otimes g=I,X\otimes X=X^{*}\otimes X^{*}=g\oplus X\oplus
X^{*}.$

Indeed, let $X=F(I\boxtimes(1,4))$, where
$F:\mathcal{Z}(\mathcal{B})\to\mathcal{B}$ is the forgetful functor, then $X$
is a simple object of Frobenius-Perron dimension $1+\sqrt{2}$, hence

$\displaystyle X\otimes X$ $\displaystyle=F(I\boxtimes(1,4)\otimes
I\boxtimes(1,4))$ $\displaystyle=F(I\boxtimes(3,0)\oplus I\boxtimes(4,1))$
$\displaystyle=F(I\boxtimes(3,0))\oplus F(I\boxtimes(4,1)).$

Since $\text{FPdim}((3,0))=2+\sqrt{2}$, it is not simple by counting the
Frobenius-Perron dimension of $\mathcal{B}$. Hence, $F(I\boxtimes(3,0))$ must
contains an invertible object $g$ as a direct summand. However,
$\theta_{(3,0)}=\zeta_{4}^{3}$ by Lemma 3.4,
$I\boxtimes(3,0)\nsubseteq\mathcal{I}(I)$, where $\mathcal{I}$ is the adjoint
functor to $F$, so $g\neq I$. Thus $g\subseteq F(I\boxtimes(3,0))$ and
$X^{*}=F(I\boxtimes(4,1))\subseteq X\otimes X^{*}$, which implies

$\displaystyle X\otimes X=g\oplus X\oplus X^{*}=X^{*}\otimes X^{*},$

then the rest fusion rules are easy. And it was proved that the
$\mathbb{Z}_{2}$-equivariantization of $\mathcal{B}$ is a near-group fusion
category of type $\mathbb{Z}_{4}+4$ [14][11]. ∎

###  3.2 Faithful extensions of $\mathcal{A}$

In this subsection, we construct two faithful graded extension of the near-
group fusion categories of type $\mathbb{Z}_{3}+6$ by using Witt equivalences
of non-degenerate fusion categories [3].

######  Theorem 3.8.

The near-group fusion category $\mathcal{A}$ admits a faithful
$\mathbb{Z}_{2}$-extension $\mathcal{B}$, which is a fusion category of rank
$8$. Explicitly,
$\mathcal{O}(\mathcal{B})=\\{I,g,g^{2},X,Y,Z_{1},Z_{2},Z_{3}\\}$, and

$\displaystyle X\otimes X=I\oplus g\oplus g^{2}\oplus 6X,g\otimes X=X\otimes
g=X,X\otimes Y=Y\otimes X=Z_{1}\oplus Z_{2}\oplus Z_{3},$ $\displaystyle
g\otimes Z_{3}=Z_{1}=Z_{3}\otimes g^{2},g^{2}\otimes Z_{3}=Z_{2}=Z_{3}\otimes
g,Z_{i}\otimes Z_{j}=g^{i+2j}\oplus 2X,$ $\displaystyle Y\otimes Y=I\oplus
g\oplus g^{2},Y\otimes Z_{i}=Z_{i}\otimes Y=X,Z_{i}\otimes X=X\otimes
Z_{i}=Y\oplus 2Z_{1}\oplus 2Z_{2}\oplus 2Z_{3}.$

###### Proof.

Consider the modular fusion category
$\mathcal{C}(\mathfrak{sl}_{3},9)_{\mathbb{Z}_{3}}^{0}\boxtimes\mathcal{C}(\mathfrak{sl}_{2},4)$.
Since $\mathcal{C}(\mathfrak{sl}_{2},4)$ contains a Tannakian fusion
subcategory $\text{Rep}(\mathbb{Z}_{2})$ such that
$\mathcal{C}(\mathfrak{sl}_{2},4)_{\mathbb{Z}_{2}}^{0}\cong\mathcal{C}(\mathbb{Z}_{3},\eta^{-1})$,
we have the following braided tensor equivalences

$\displaystyle\left(\mathcal{C}(\mathfrak{sl}_{3},9)_{\mathbb{Z}_{3}}^{0}\boxtimes\mathcal{C}(\mathfrak{sl}_{2},4)\right)_{\mathbb{Z}_{2}}^{0}\cong\mathcal{C}(\mathfrak{sl}_{3},9)_{\mathbb{Z}_{3}}^{0}\boxtimes\mathcal{C}(\mathfrak{sl}_{2},4)_{\mathbb{Z}_{2}}^{0}\cong\mathcal{Z}(\mathcal{A}),$

by Theorem 3.6. Then it follows from [8, Theorem 1.3] that
$\mathcal{C}(\mathfrak{sl}_{3},9)_{\mathbb{Z}_{3}}^{0}\boxtimes\mathcal{C}(\mathfrak{sl}_{2},4)\cong\mathcal{Z}(\mathcal{B})$
for a fusion category $\mathcal{B}$ with $\mathcal{B}$ being a faithful
$\mathbb{Z}_{2}$-extension of $\mathcal{A}$. Let
$\mathcal{B}=\oplus_{h\in\mathbb{Z}_{2}}\mathcal{B}_{g}$ be the corresponding
grading of $\mathcal{B}$ with $\mathcal{B}_{e}=\mathcal{A}$.

Let $F:\mathcal{Z}(\mathcal{B})\to\mathcal{B}$ be the forgetful functor.
Notice that the above tensor equivalence implies that
$\mathcal{Z}(\mathcal{B})$ contains a simple object of Frobenius-Perron
dimension $\sqrt{3}$, the image of this simple object in $\mathcal{B}$,
denoted by $Y$, must be simple. Obviously, $Y\in\mathcal{B}_{h}$ and $Y\otimes
Y=I\oplus g\oplus g^{2}$. Since

$\displaystyle\text{Hom}_{\mathcal{B}}(X\otimes Y,X\otimes
Y)\cong\text{Hom}_{\mathcal{B}}(X,X\otimes Y\otimes Y)=\mathbb{C}^{3},$

$X\otimes Y=Z_{1}\oplus Z_{2}\oplus Z_{3}$, where $Z_{j}$ are distinct simple
objects. It follows from

$\displaystyle\text{Hom}_{\mathcal{B}}(Y\otimes Y,X)=\text{Hom}(g^{i}\otimes
Y,Z_{j})\cong\text{Hom}(g^{i},Z_{j}\otimes Y)=0$

that $Z_{j}\otimes Y=n_{j}X$ for some integers $n_{j}\geq 1$, $1\leq j\leq 3$.
Therefore,

$\displaystyle 3X=X\otimes(Y\otimes Y)=(X\otimes Y)\otimes
Y=(n_{1}+n_{2}+n_{3})X,$

thus $Z_{j}\otimes Y=X$, similarly, $Y\otimes Z_{j}=X$. By comparing the
Frobenius-Perron dimensions, we obtain
$\mathcal{O}(\mathcal{B}_{h})=\\{Y,Z_{1},Z_{2},Z_{3}\\}$.

Notice that at least one of $Z_{j}$ are self-dual. Without loss of generality,
assume $Z_{3}\otimes Z_{3}=I\oplus 2X$, $g\otimes Z_{3}=Z_{1}=Z_{3}\otimes
g^{k}$ and $g\otimes Z_{1}=Z_{2}$ for some $k$. Then $X\otimes Z_{i}=X\otimes
Z_{3}=Y\oplus 2Z_{1}\oplus 2Z_{2}\oplus 2Z_{3}$ and $Z_{i}\otimes
Z_{j}=g^{i+kj}\otimes 2X$. Meanwhile, ribbon structure of simple objects of
$\mathcal{C}(\mathfrak{sl}_{2},4)$ are explicitly
$1,1,\zeta_{3},\zeta_{8},\zeta_{8}^{5}$, so
$\mathcal{C}(\mathfrak{sl}_{2},4)\boxtimes\mathcal{C}(\mathfrak{sl}_{3},9)_{\mathbb{Z}_{3}}^{0}$
has exactly five simple objects of trivial twists by Lemma 3.4, then it
follows from [17, Corollary 2.16] that $\mathcal{I}(I)$ can’t be multiplicity-
free. Hence, $\text{Gr}(\mathcal{B})$ is noncommutative, so $g\otimes
Z_{3}=Z_{3}\otimes g^{2}$. ∎

Notice that we also have modular equivalence

$\displaystyle(\mathcal{C}(\mathbb{Z}_{3},\eta^{-1})\boxtimes\mathcal{C}(\mathfrak{sl}_{3},9))_{\mathbb{Z}_{3}}^{0}\cong\mathcal{C}(\mathbb{Z}_{3},\eta^{-1})\boxtimes\mathcal{C}(\mathfrak{sl}_{3},9))_{\mathbb{Z}_{3}}^{0}\cong\mathcal{Z}(\mathcal{A})$

hence, there
$\mathcal{C}(\mathbb{Z}_{3},\eta)\boxtimes\mathcal{C}(\mathfrak{sl}_{3},9)\cong\mathcal{Z}(\mathcal{D})$
as modular fusion category by [8, Theorem 1.3], where $\mathcal{D}$ is a
faithful $\mathbb{Z}_{3}$-extension of $\mathcal{A}$.

######  Lemma 3.9.

Fusion category $\mathcal{D}$ contains a simple object of Frobenius-Perron
dimension $1+\sqrt{3}$.

###### Proof.

By considering the Frobenius-Perron dimensions of simple objects of
$\mathcal{C}(\mathfrak{sl}_{3},9)$, we know it contains a simple object $W$
(the simple object $(1,0)$, for example) with non-trivial twist of Frobenius-
Perron dimension $1+\sqrt{3}$. Let $F:\mathcal{Z}(\mathcal{D})\to\mathcal{D}$
be the forgetful functor, we claim that $F(I\boxtimes W)$ must be simple
Indeed, if $F(I\boxtimes W)$ is not simple, then $F(I\boxtimes W)=W_{1}\oplus
W_{2}$, where $W_{j}$ are simple. Then
$\text{FPdim}(W_{j})=2\cos\frac{\pi}{n_{j}}$ for some integer $n_{j}\geq 3$
[6], which means $n_{1}=3$ and $n_{2}=6$. Therefore, each grading component of
$\mathcal{C}$ contains at least three invertible simple objects up to
isomorphism and the non-trivial components contains simple objects of
Frobenius-Perron dimension $\sqrt{3}$. Let $\mathcal{B}$ be the strictly
weakly integral fusion subcategory of $\mathcal{D}$, so
$\text{FPdim}(\mathcal{B}_{\text{pt}})=3+6a$ and
$\text{FPdim}(\mathcal{D})=3+6a+6b$. Notice that

$\displaystyle\text{FPdim}(\mathcal{B}_{\text{pt}})\mid\text{FPdim}(\mathcal{B})\mid\text{FPdim}(\mathcal{D})=36(2+\sqrt{3})$

by [7, Proposition 8.15], which then means $a=1$ and $b=0$, it is impossible.
Hence object $F(I\boxtimes W)$ must be simple. ∎

######  Theorem 3.10.

The faithful $\mathbb{Z}_{3}$-extension $\mathcal{D}$ is a fusion category of
rank $12$, which has commutative fusion rules. Explicitly,
$\mathcal{O}(\mathcal{D})=\\{I,g,g^{2},X,V,W_{1},W_{2},W_{3},V^{*},W_{1}^{*},W_{2}^{*},W_{3}^{*}\\}$,
and

$\displaystyle X\otimes X=I\oplus g\oplus g^{2}\oplus 6X,g\otimes X=X,g\otimes
W_{1}=W_{2},g\otimes W_{2}=W_{3},V^{*}\otimes W_{j}=2X,$ $\displaystyle
W_{i}\otimes W_{j}=g^{i+j-2}\otimes W_{1}^{*}\oplus V^{*},V\otimes
V^{*}=I\oplus g\oplus g^{2}\oplus 3X,W_{i}\otimes W_{j}^{*}=g^{i-j}\oplus X,$
$\displaystyle V\otimes W_{i}=W_{1}^{*}\oplus W_{2}^{*}\oplus W_{3}^{*}\oplus
V^{*},V\otimes V=W_{1}^{*}\oplus W_{2}^{*}\oplus W_{3}^{*}\oplus 3V^{*},$
$\displaystyle W_{i}\otimes X=W_{1}\oplus W_{2}\oplus W_{3}\oplus 2V,X\otimes
V=V\otimes X=2W_{1}\oplus 2W_{2}\oplus 2W_{3}\oplus 3V.$

###### Proof.

Let $\mathcal{D}=\oplus_{h\in\mathbb{Z}_{3}}\mathcal{D}_{h}$ with
$\mathcal{D}_{e}=\mathcal{A}$. To simplify notations, we use
$\mathcal{C}(\mathbb{Z}_{3},\eta^{-1})\boxtimes\mathcal{C}(\mathfrak{sl}_{3},9)$
to denote the Drinfeld center $\mathcal{Z}(\mathcal{D})$. By Lemma 3.9, both
of the non-trivial components $\mathcal{D}_{h}$ and $\mathcal{D}_{h^{2}}$
contain at least three simple objects $W_{i}$ of Frobenius-Perron dimension
$1+\sqrt{3}$, and obviously $W_{i}\otimes W_{i}^{*}=I\oplus X$. In fact, we
can regard $F(I\boxtimes(0,1))$ as one of the simple objects $W_{j}$. Assume
$g\otimes W_{1}=W_{2}$ and $g\otimes W_{2}=W_{3}$. Hence $W_{i}\otimes
X=X\otimes W_{i}=W_{1}\oplus W_{2}\oplus W_{3}\oplus 2V$ with
$\text{FPdim}(V)=3+\sqrt{3}$. Indeed, since

$\displaystyle\dim_{\mathbb{C}}\text{Hom}(X\otimes W_{i},X\otimes
W_{i})=\dim_{\mathbb{C}}\text{Hom}(X,X\otimes W_{i}\otimes W_{i})=7,$

$W_{i}\otimes X=W_{1}\oplus W_{2}\oplus W_{3}\oplus M$ with $M$ not being
simple. If $M$ is multiplicity-free, then $M=M_{1}\oplus M_{2}\oplus
M_{3}\oplus M_{4}$, where $M_{j}$ ($1\leq j\leq 4$) are simple. Let
$\text{FPdim}(M_{4})$ being the smallest among $\text{FPdim}(W_{j})$, so
$\text{FPdim}(M_{4})\leq\frac{3+\sqrt{3}}{2}$. Meanwhile,
$N_{X,W_{i}}^{M_{4}}=N_{W_{i},M_{4}^{*}}^{X}=1$, thus
$\text{FPdim}(W_{i})\text{FPdim}(M_{4})\geq\text{FPdim}(X)$, which means
$\text{FPdim}(M_{4})\geq\frac{3+\sqrt{3}}{2}$. However, $\frac{3+\sqrt{3}}{2}$
is not an algebraic integer, it is a contradiction. So $M=2V$ with $V$ being
simple. By comparing the Frobenius-Perron dimensions of simple objects, we
have $\mathcal{O}(\mathcal{D}_{h})=\\{W_{1},W_{2},W_{3},V\\}$ and
$\mathcal{O}(\mathcal{D}_{h^{2}})=\\{W_{1}^{*},W_{2}^{*},W_{3}^{*},V^{*}\\}$,
so $\text{rank}(\mathcal{D})=12$.

Obviously, $V\otimes V^{*}=I\oplus g\oplus g^{2}\oplus 3X$. Since

$\displaystyle\dim_{\mathbb{C}}\text{Hom}(V\otimes W_{i},V\otimes
W_{i})=\dim_{\mathbb{C}}\text{Hom}(V,V\otimes(I\oplus X))=4,$

we also have $V\otimes V=W_{1}^{*}\oplus W_{2}^{*}\oplus W_{3}^{*}\oplus
3V^{*}$ and $V\otimes W_{i}=W_{i}\otimes V=W_{1}^{*}\oplus W_{2}^{*}\oplus
W_{3}^{*}\oplus V^{*}$. Therefore, $X\otimes V=V\otimes X=2W_{1}\oplus
2W_{2}\oplus 2W_{3}\oplus 3V$, $V^{*}\otimes W_{j}=W_{j}^{*}\otimes V=2X$.

Notice that $\text{Vec}_{\mathbb{Z}_{3}}$ is exactly the image of
$\mathcal{C}(\mathbb{Z}_{3},\eta^{-1})$ under the forgetful functor $F$, hence
$g\otimes W_{i}=W_{i}\otimes g$ for all $1\leq i\leq 3$. Consequently, we know
$\mathcal{D}$ has commutative fusion rules. Assume $W_{1}\otimes
W_{1}=V^{*}\oplus W_{l}^{*}$ for some simple object $W_{l}$, then

$\displaystyle W_{i}\otimes W_{j}=W_{1}\otimes W_{1}\otimes
g^{i+j-2}=V^{*}\oplus W_{l}^{*}\otimes g^{i+j-2}$

Thus, we only need to determine $W_{1}\otimes W_{1}$. Meanwhile,

$\displaystyle F(I\boxtimes(0,1))\otimes
F(I\boxtimes(0,1))=F(I\boxtimes(0,1)\otimes(0,1))=F(I\boxtimes(1,0)\oplus
I\boxtimes(0,2)),$

which implies $W_{j}^{*}\subseteq W_{j}\otimes W_{j}$ for all $1\leq j\leq 3$,
as $I\boxtimes(1,0)$ is the dual object of $I\boxtimes(0,1)$. This finishes
the proof of the theorem. ∎

A direct computations shows that the following simple objects

$\displaystyle
I\boxtimes(0,0),I\boxtimes(0,9),I\boxtimes(0,9),I\boxtimes(4,1),I\boxtimes(4,4),I\boxtimes(1,4),$
$\displaystyle
g\boxtimes(2,2),g\boxtimes(2,5),g\boxtimes(5,2),g^{2}\boxtimes(2,2),g^{2}\boxtimes(2,5),g^{2}\boxtimes(5,2)$

are the simple objects of
$\mathcal{C}(\mathbb{Z}_{3},\eta^{-1})\boxtimes\mathcal{C}(\mathfrak{sl}_{3},9)$
with trivial ribbon. Since
$\text{Vec}_{\mathbb{Z}_{3}}\subseteq\mathcal{A}\subseteq\mathcal{D}$, we know
$\mathcal{I}(I)$ contains two non-trivial connected étale subalgebras
$A_{1}\subseteq A_{2}$ such that
$\mathcal{Z}(\mathcal{D})_{A_{1}}^{0}\cong\mathcal{Z}(\mathcal{A})$ and
$\mathcal{Z}(\mathcal{D})_{A_{2}}^{0}\cong\mathcal{Z}(\text{Vec}_{\mathbb{Z}_{3}})$
[3], where $A_{1}:=\text{Fun}(\mathbb{Z}_{3})=I\boxtimes(0,0)\oplus
I\boxtimes(0,9)\oplus I\boxtimes(0,9)$. Notice that $A_{2}$ is an étale
algebra over $A_{1}$ and $\text{FPdim}(A_{2})=24+12\sqrt{3}$, we must have

$\displaystyle A_{2}=A_{1}\oplus I\boxtimes(4,1)\oplus I\boxtimes(4,4)\oplus
I\boxtimes(1,4).$

And $\text{Gr}(\mathcal{D})$ is commutative, so $\mathcal{I}(I)$ is
multiplicity-free by [17, Corollary 2.16]. Then

$\displaystyle\mathcal{I}(I)=A_{2}\oplus\oplus_{j=1}^{2}(g^{j}\boxtimes(2,2)\oplus
g^{j}\boxtimes(2,5)\oplus g^{j}\boxtimes(5,2)).$

######  Remark 3.11.

In addition,
$\mathcal{C}(\mathfrak{sl}_{2},4)\boxtimes\mathcal{C}(\mathfrak{sl}_{3},9)\cong\mathcal{Z}(\mathcal{C})$,
where $\mathcal{C}$ is a $\mathbb{Z}_{6}$-extension of $\mathcal{A}$,
$\mathcal{C}$ is also a $\mathbb{Z}_{2}$-extension of $\mathcal{D}$ and
$\mathbb{Z}_{3}$-extension of $\mathcal{B}$. Notice that

$\displaystyle\text{Hom}(W_{i}\otimes Y,W_{i}\otimes
Y)=\text{Hom}(W_{i},W_{i}\otimes Y\otimes Y)=\mathbb{C},$

so $U:=W_{i}\otimes Y=W_{1}\otimes Y$ must be simple, and $U\otimes
U^{*}=I\oplus g\oplus g^{2}\oplus 3X$. Notice that

$\displaystyle\text{Hom}_{\mathcal{C}}(V\otimes Y,V\otimes
Y)=\text{Hom}_{\mathcal{C}}(V,V\otimes Y\otimes Y)=\mathbb{C}^{3},$

Hence $V\otimes Y=T_{1}\oplus T_{2}\oplus T_{3}$, where $T_{j}$ are simple.
Assume $\text{FPdim}(T_{3})\leq\text{FPdim}(T_{j})$ for all $j$, then
$\text{FPdim}(T_{3})\leq 1+\sqrt{3}$, meanwhile,
$N_{V,Y}^{T_{3}}=N_{Y,T_{3}^{*}}^{V^{*}}=1$, so

$\displaystyle\text{FPdim}(Y)\text{FPdim}(T_{3})\geq\text{FPdim}(V)=3+\sqrt{3}.$

Therefore, $\text{FPdim}(T_{j})=1+\sqrt{3}$ for all $1\leq j\leq 3$.

Consequently, $\mathcal{C}=\oplus_{h\in\mathbb{Z}_{6}}\mathcal{C}_{h}$ with
$\text{rank}(\mathcal{C})=24$, and

$\displaystyle\mathcal{O}(\mathcal{C}_{e})$
$\displaystyle=\\{I,g,g^{2},X\\},\mathcal{O}(\mathcal{C}_{h})=\\{U^{*},T_{1}^{*},T_{2}^{*},T_{3}^{*}\\},\mathcal{O}(\mathcal{C}_{h^{2}})=\\{W_{1},W_{2},W_{3},V\\},$
$\displaystyle\mathcal{O}(\mathcal{C}_{h^{3}})$
$\displaystyle=\\{Z_{1},Z_{2},Z_{3},Y\\},\mathcal{O}(\mathcal{C}_{h^{4}})=\\{W_{1}^{*},W_{2}^{*},W_{3}^{*},V^{*}\\},\mathcal{O}(\mathcal{C}_{h^{5}})=\\{U,T_{1},T_{2},T_{3}\\}.$

The fusion rules can be obtained by in a similar method, we omit it here.

## Acknowledgements

The author thanks D. Penneys for clarification on the existence of near-group
fusion category of type $\mathbb{Z}_{3}+6$ and for providing reference [2].
The author thanks C. Edie-Michell, S-H. Ng and Y. Wang for comments and
communications. The author is supported by the National Natural Science
Foundation of China (no.12101541), the Natural Science Foundation of Jiangsu
Province (no.BK20210785), and the Natural Science Foundation of Jiangsu Higher
Institutions of China (no.21KJB110006).

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Zhiqiang Yu
Email<EMAIL_ADDRESS>
School of Mathematical Science, Yangzhou University, Yangzhou 225002, China