Affine vertex operator superalgebra $L_{\widehat{osp(1|2)}}(\mathcal{l},0)$ at
admissible level

Huaimin Li, Qing Wang 111Supported by China NSF grants No.12071385 and
No.12161141001.

School of Mathematical Sciences, Xiamen University, Xiamen, China 361005

###### Abstract

Let $L_{\widehat{osp(1|2)}}(\mathcal{l},0)$ be the simple affine vertex
operator superalgebra with admissible level $\mathcal{l}$. We prove that the
category of weak $L_{\widehat{osp(1|2)}}(\mathcal{l},0)$-modules on which the
positive part of $\widehat{osp(1|2)}$ acts locally nilpotent is semisimple.
Then we prove that $\mathbb{Q}$-graded vertex operator superalgebras
$(L_{\widehat{osp(1|2)}}(\mathcal{l},0),\omega_{\xi})$ with new Virasoro
elements $\omega_{\xi}$ are rational and the irreducible modules are exactly
the admissible modules for $\widehat{osp(1|2)}$, where $0<\xi<1$ is a rational
number. Furthermore, we determine the Zhu’s algebras
$A(L_{\widehat{osp(1|2)}}(\mathcal{l},0))$ and their bimodules
$A(L(\mathcal{l},\mathcal{j}))$ for
$(L_{\widehat{osp(1|2)}}(\mathcal{l},0),\omega_{\xi})$, where $\mathcal{j}$ is
the admissible weight. As an application, we calculate the fusion rules among
the irreducible ordinary modules of
$(L_{\widehat{osp(1|2)}}(\mathcal{l},0),\omega_{\xi})$.

##  1 Introduction

Kac and Wakimoto introduced the notion of admissible highest weight
representations for affine Lie (super)algebras to study the modular invariant
representations of affine Lie (super)algebras in [17], then they classified
the admissible weights for all affine Lie algebras in [18]. Admissible modules
also studied in the context of vertex operator algebras, in [3], Adamović and
Milas proved the simple affine vertex operator algebra
$L_{\widehat{sl_{2}}}(\mathcal{l},0)$ at admissible level is rational in the
category $\mathcal{O}$, and conjectured this holds for simple affine vertex
operator algebras $L_{\widehat{\mathfrak{g}}}(\mathcal{l},0)$ associated to
any simple finite-dimensional Lie algebra $\mathfrak{g}$. This conjecture has
been proved by Arakawa in [1]. However, simple affine vertex operator algebras
$L_{\widehat{\mathfrak{g}}}(\mathcal{l},0)$ at admissible level $\mathcal{l}$
may not be rational if $\mathcal{l}$ is not a positive integer. In [7], Dong,
Li and Mason showed that $L_{\widehat{sl_{2}}}(\mathcal{l},0)$ is a rational
$\mathbb{Q}$-graded vertex operator algebra under a new Virasoro element and
that the irreducible modules are exactly the admissible modules for
$\widehat{sl_{2}}$. Furthermore, these results has been proved by Lin for
$L_{\widehat{\mathfrak{g}}}(\mathcal{l},0)$ associated to any simple finite-
dimensional Lie algebra $\mathfrak{g}$ in [23]. For superalgebra case, Wood
classified the irreducible relaxed highest-weight modules for the simple
affine vertex operator superalgebra $L_{\widehat{osp(1|2)}}(\mathcal{l},0)$ at
admissible level, and also proved that $L_{\widehat{osp(1|2)}}(\mathcal{l},0)$
is rational in the category $\mathcal{O}$ in [25]. Gorelik and Serganova later
showed the rationality in the category $\mathcal{O}$ for
$L_{\widehat{osp(1|2n)}}(\mathcal{l},0)$ in [13].

Let $\mathfrak{g}={osp}(1|2)=\mbox{span}_{\mathbb{C}}\\{h,e,f,x,y\\}$, where
$\\{h,e,f,x,y\\}$ is the standard Chevalley basis of $\mathfrak{g}$. In our
paper we study the simple affine vertex operator superalgebra
$L_{\widehat{\mathfrak{g}}}(\mathcal{l},0)$ with admissible level
$\mathcal{l}$. Let $V_{\widehat{\mathfrak{g}}}(\mathcal{l},\mathbb{C})$ be the
universal affine vertex operator superalgebra, for a $\mathbb{C}h$-module $U$,
we construct the weak
$V_{\widehat{\mathfrak{g}}}(\mathcal{l},\mathbb{C})$-module $L(\mathcal{l},U)$
in Subsection 3.1 and determine the condition that $L(\mathcal{l},U)$ is a
weak $L_{\widehat{\mathfrak{g}}}(\mathcal{l},0)$-module. We prove that the
irreducible highest weight $\widehat{\mathfrak{g}}$-module
$L(\mathcal{l},\mathcal{j})$ is a weak
$L_{\widehat{\mathfrak{g}}}(\mathcal{l},0)$-module if and only if
$\mathcal{j}$ is admissible, this result implies that the weak
$L_{\widehat{\mathfrak{g}}}(\mathcal{l},0)$-modules in category $\mathcal{O}$
are complete reducible and the irreducible modules are those
$L(\mathcal{l},\mathcal{j})$ for $\mathcal{j}$ being admissible. This also
obtained in [25] by computing explicit presentation for the Zhu’s algebra
associated to $L_{\widehat{\mathfrak{g}}}(\mathcal{l},0)$ and in [13] by
establishing the correspondence between the weak
$L_{\widehat{osp(1|2n)}}(\mathcal{l},0)$-modules and the weak
$L_{\widehat{sp_{2n}}}(\mathcal{l},0)$-modules. Furthermore, we consider the
category $\mathcal{C}_{\mathcal{l}}$ in Subsection 3.2, where
$\mathcal{C}_{\mathcal{l}}$ is the subcategory of the weak
$L_{\widehat{\mathfrak{g}}}(\mathcal{l},0)$-module category such that $M$ is
an object in $\mathcal{C}_{\mathcal{l}}$ if and only if the sum of all
positive root spaces of $\widehat{\mathfrak{g}}$ acts locally nilpotently on
$M$. We prove that the category $\mathcal{C}_{\mathcal{l}}$ is semisimple and
there are finitely many irreducible weak
$L_{\widehat{\mathfrak{g}}}(\mathcal{l},0)$-modules belonging to
$\mathcal{C}_{\mathcal{l}}$ up to isomorphism. It implies that any ordinary
$L_{\widehat{\mathfrak{g}}}(\mathcal{l},0)$-module is completely reducible and
there are finitely many irreducible ordinary
$L_{\widehat{\mathfrak{g}}}(\mathcal{l},0)$-modules up to isomorphism, this
result also established for $\mathfrak{g}=osp(1|2n)$ in [4] by using the
theory of vertex superalgebra extensions. Moreover, we consider
$\mathbb{Q}$-graded vertex operator superalgebras
$(L_{\widehat{\mathfrak{g}}}(\mathcal{l},0),\omega_{\xi})$ associated to a
family of new Virasoro elements $\omega_{\xi}$ in Subsection 3.3, where
$0<\xi<1$ is a rational number. We apply the semisimplicity of category
$\mathcal{C}_{\mathcal{l}}$ to prove that
$(L_{\widehat{\mathfrak{g}}}(\mathcal{l},0),\omega_{\xi})$ are rational
$\mathbb{Q}$-graded vertex operator superalgebras. In [9], the $A(V)$-theory
for $\mathbb{Q}$-graded vertex operator superalgebras has been studied. In
this paper we determine the Zhu’s algebras
$A(L_{\widehat{\mathfrak{g}}}(\mathcal{l},0))$ and their bimodules
$A(L(\mathcal{l},\mathcal{j}))$ for
$(L_{\widehat{\mathfrak{g}}}(\mathcal{l},0),\omega_{\xi})$ in Subsection 4.2.
Then we apply the Zhu’s algebras and their bimodules associated to
$(L_{\widehat{\mathfrak{g}}}(\mathcal{l},0),\omega_{\xi})$ to calculate the
fusion rules among the irreducible ordinary modules of
$(L_{\widehat{\mathfrak{g}}}(\mathcal{l},0),\omega_{\xi})$, the fusion rules
we obtained are also those with respect to the old vertex operator
superalgebra $L_{\widehat{\mathfrak{g}}}(\mathcal{l},0)$. The fusion rules
among the irreducible relaxed highest-weight modules over
$L_{\widehat{\mathfrak{g}}}(\mathcal{l},0)$ have been calculated in [5] by
viewing $L_{\widehat{\mathfrak{g}}}(\mathcal{l},0)$ as extensions of the
tensor product of certain simple Virasoro vertex operator algebra and simple
affine vertex operator algebra associated to $sl_{2}$.

This paper is organized as follows. In Section 2, we recall some concepts
about $\mathbb{Z}$-graded vertex operator superalgebras and some facts about
admissible modules of $\widetilde{\mathfrak{g}}$. In Section 3, we construct
the weak $V_{\widehat{\mathfrak{g}}}(\mathcal{l},\mathbb{C})$-module
$L(\mathcal{l},U)$ and give the condition for $L(\mathcal{l},U)$ being a weak
$L_{\widehat{\mathfrak{g}}}(\mathcal{l},0)$-module. Then we prove that
$L_{\widehat{\mathfrak{g}}}(\mathcal{l},0)$ at admissible level is rational in
category $\mathcal{C}_{\mathcal{l}}$ and
$(L_{\widehat{\mathfrak{g}}}(\mathcal{l},0),\omega_{\xi})$ are rational
$\mathbb{Q}$-graded vertex operator superalgebras with respect to a family of
Virasoro elements $\omega_{\xi}$. In Section 4, we determine the Zhu’s algebra
$A(L_{\widehat{\mathfrak{g}}}(\mathcal{l},0))$ and their bimodules
$A(L(\mathcal{l},\mathcal{j}))$ for
$(L_{\widehat{\mathfrak{g}}}(\mathcal{l},0),\omega_{\xi})$. As an application,
we calculate the fusion rules among the irreducible ordinary modules of
$(L_{\widehat{\mathfrak{g}}}(\mathcal{l},0),\omega_{\xi})$. Throughout the
paper, $\mathbb{Z}$, $\mathbb{N}$, $\mathbb{Z}_{+}$, $\mathbb{Q}$ and
$\mathbb{C}$ are the sets of integers, nonnegative integers, positive
integers, rational numbers and complex numbers respectively.

##  2 Preliminaries

In this section, we recall some concepts about $\mathbb{Z}$-graded vertex
operator superalgebra and some facts about admissible modules of
$\widetilde{{osp}(1|2)}$.

###  2.1 Vertex operator superalgebras and their modules

Let $V=V_{\bar{0}}\oplus V_{\bar{1}}$ be a vector superspace, the elements in
$V_{\bar{0}}$ (resp. $V_{\bar{1}}$) are called even (resp. odd). For any $v\in
V_{\bar{i}}$ with $i=0,1$, define $|v|=i$. First we recall the definitions of
$\mathbb{Z}$-graded vertex operator superalgebra and their various module
categories following [8, 20].

######  Definition 2.1.

A vertex superalgebra is a quadruple $(V,\textbf{1},D,Y)$, where
$V=V_{\bar{0}}\oplus V_{\bar{1}}$ is a vector superspace, $D$ is an
endomorphism of $V$, 1 is a specified even vector called the vacuum vector of
$V$, and $Y$ is a linear map

$\displaystyle Y(\cdot,z):$ $\displaystyle
V\rightarrow(\mbox{End}V)[[z,z^{-1}]]$ $\displaystyle a\mapsto
Y(a,z)=\sum_{n\in\mathbb{Z}}a_{n}z^{-n-1}~{}~{}(a_{n}\in\mbox{End}V)$

such that
(1) For any $a,b\in V$, $a_{n}b=0$ for $n$ sufficiently large;
(2) $[D,Y(a,z)]=Y(D(a),z)=\frac{d}{dz}Y(a,z)$ for any $a\in V$;
(3) $Y(\textbf{1},z)=\mbox{Id}_{V}$;
(4) $Y(a,z)\textbf{1}\in(\mbox{End}V)[[z]]$ and $\mbox{lim}_{z\rightarrow
0}Y(a,z)\textbf{1}=a$ for any $a\in V$;
(5) For $\mathbb{Z}_{2}$-homogeneous elements $a,b\in V$, the following Jacobi
identity holds:

$\displaystyle
z_{0}^{-1}\delta(\frac{z_{1}-z_{2}}{z_{0}})Y(a,z_{1})Y(b,z_{2})-(-1)^{|a||b|}z_{0}^{-1}\delta(\frac{z_{2}-z_{1}}{-z_{0}})Y(b,z_{2})Y(a,z_{1})$
$\displaystyle=z_{2}^{-1}\delta(\frac{z_{1}-z_{0}}{z_{2}})Y(Y(a,z_{0})b,z_{2}).$

A vertex superalgebra $V$ is called a $\mathbb{Z}$-graded vertex operator
superalgebra if there is an even vector $\omega$ called the Virasoro element
of $V$ such that
(6) Set $Y(\omega,z)=\sum_{n\in\mathbb{Z}}L(n)z^{-n-2}$, for any
$m,n\in\mathbb{Z}$,

$[L(m),L(n)]=(m-n)L(m+n)+\frac{m^{3}-m}{12}\delta_{m+n,0}c,$

where $c\in\mathbb{C}$ is called the central charge;
(7) $L(-1)=D$;
(8) $V$ is $\mathbb{Z}$-graded such that
$V=\bigoplus_{n\in\mathbb{Z}}V_{(n)}$,
$L(0)\mid_{V_{(n)}}=n\mbox{Id}_{V_{(n)}}$, $\mbox{dim}~{}V_{(n)}<\infty$ for
all $n\in\mathbb{Z}$ and $V_{(n)}=0$ for $n$ sufficiently small. For $v\in
V_{(n)}$, the conformal weight of $v$ is defined to be $\mbox{wt}~{}v=n$.

######  Definition 2.2.

Let $V$ be a $\mathbb{Z}$-graded vertex operator superalgebra. A weak
$V$-module is a triple $(M,d,Y_{M})$ where $M=M_{\bar{0}}\oplus M_{\bar{1}}$
is a vector supersapce, $d$ is an endomorphism of $M$ and $Y_{M}$ is a linear
map

$\displaystyle Y_{M}(\cdot,z):$ $\displaystyle
V\rightarrow(\mbox{End}M)[[z,z^{-1}]]$ $\displaystyle a\mapsto
Y_{M}(a,z)=\sum_{n\in\mathbb{Z}}a_{n}z^{-n-1}~{}~{}(a_{n}\in\mbox{End}M)$

such that
(1) For any $a\in V,u\in M$, $a_{n}u=0$ for $n$ sufficiently large;
(2) $[d,Y_{M}(a,z)]=Y_{M}(D(a),z)=\frac{d}{dz}Y_{M}(a,z)$ for any $a\in V$;
(3) $Y_{M}(\textbf{1},z)=\mbox{Id}_{M}$;
(4) For $\mathbb{Z}_{2}$-homogeneous elements $a,b\in V$, the following Jacobi
identity holds:

$\displaystyle
z_{0}^{-1}\delta(\frac{z_{1}-z_{2}}{z_{0}})Y_{M}(a,z_{1})Y_{M}(b,z_{2})-(-1)^{|a||b|}z_{0}^{-1}\delta(\frac{z_{2}-z_{1}}{-z_{0}})Y_{M}(b,z_{2})Y_{M}(a,z_{1})$
$\displaystyle=z_{2}^{-1}\delta(\frac{z_{1}-z_{0}}{z_{2}})Y_{M}(Y(a,z_{0})b,z_{2}).$

A weak $V$-module $M$ is called an admissible module ($\mathbb{Z}_{+}$-graded
weak module) if $M$ has a $\mathbb{Z}_{+}$-gradation
$M=\bigoplus_{n\in\mathbb{Z}_{+}}M(n)$ such that

$a_{m}M(n)\subseteq M(\mbox{wt}~{}a+n-m-1)$

for any $\mathbb{Z}$-homogeneous element $a\in
V,m\in\mathbb{Z},n\in\mathbb{Z}_{+}$.

A weak $V$-module $M$ is called an ordinary module if
$M=\bigoplus_{\lambda\in\mathbb{C}}M_{\lambda}$, where $M_{\lambda}=\\{w\in
M\mid L(0)w=\lambda w\\}$ such that $\mbox{dim}~{}M_{\lambda}<\infty$ for all
$\lambda\in\mathbb{C}$ and $M_{\lambda}=0$ for the real part of $\lambda$
sufficiently small.

######  Definition 2.3.

Let $V$ be a $\mathbb{Z}$-graded vertex operator superalgebra and
$(W_{i},Y_{i}),(W_{j},Y_{j})$, $(W_{k},Y_{k})$ be three weak $V$-modules. An
intertwining operator of type $\begin{pmatrix}W_{i}\\\
W_{j}~{}~{}W_{k}\end{pmatrix}$ is a linear map

$\displaystyle\mathcal{Y}(\cdot,z):$ $\displaystyle
W_{j}\rightarrow(\mbox{Hom}(W_{k},W_{i}))\\{z\\}$ $\displaystyle
w\mapsto\mathcal{Y}(w,z)=\sum_{n\in\mathbb{Q}}w_{n}z^{-n-1}~{}~{}(w_{n}\in\mbox{Hom}(W_{k},W_{i}))$

such that for any $v\in V,w^{j}\in W_{j},w^{k}\in W_{k}$,
(1) $w_{n}^{j}w^{k}=0$ for $n$ sufficiently large;
(2) $\mathcal{Y}(L(-1)w^{j},z)=\frac{d}{dz}\mathcal{Y}(w^{j},z)$, where
$L(-1)$ is the operator acting on $W_{j}$;
(3) for $\mathbb{Z}_{2}$-homogeneous elements $v,w^{j}$, the following Jacobi
identity holds for the operators acting on the element $w^{k}$:

$\displaystyle
z_{0}^{-1}\delta(\frac{z_{1}-z_{2}}{z_{0}})Y_{i}(v,z_{1})\mathcal{Y}(w^{j},z_{2})w^{k}-(-1)^{|v||w^{j}|}z_{0}^{-1}\delta(\frac{z_{2}-z_{1}}{-z_{0}})\mathcal{Y}(w^{j},z_{2})Y_{k}(v,z_{1})w^{k}$
$\displaystyle=z_{2}^{-1}\delta(\frac{z_{1}-z_{0}}{z_{2}})\mathcal{Y}(Y_{j}(v,z_{0})w^{j},z_{2})w^{k}.$

For any irreducible weak $V$-modules $W_{i},W_{j},W_{k}$, the space of all
intertwining operators of type $\begin{pmatrix}W_{i}\\\
W_{j}~{}~{}W_{k}\end{pmatrix}$ is denoted by $I_{V}\begin{pmatrix}W_{i}\\\
W_{j}~{}~{}W_{k}\end{pmatrix}$. Let
$N_{W_{j},W_{k}}^{W_{i}}=\mbox{dim}~{}I_{V}\begin{pmatrix}W_{i}\\\
W_{j}~{}~{}W_{k}\end{pmatrix}$. These integers $N_{W_{j},W_{k}}^{W_{i}}$ are
usually called the fusion rules.

###  2.2 Admissible modules of $\widetilde{osp(1|2)}$

Let $\mathfrak{g}={osp}(1|2)$ be the finite dimensional simple Lie
superalgebra with the basis $\\{h,e,f,x,y\\}$ such that the even part
$\mathfrak{g}_{0}=\mbox{span}_{\mathbb{C}}\\{h,e,f\\}\cong sl_{2}$ and the odd
part $\mathfrak{g}_{1}=\mbox{span}_{\mathbb{C}}\\{x,y\\}$. The anticommutation
relations are given by

$\displaystyle[e,f]=h,~{}[h,e]=2e,~{}[h,f]=-2f,$
$\displaystyle[h,x]=x,~{}[e,x]=0,~{}[f,x]=-y,$
$\displaystyle[h,y]=-y,~{}[e,y]=-x,~{}[f,y]=0,$
$\displaystyle\\{x,x\\}=2e,\\{x,y\\}=h,\\{y,y\\}=-2f.$

The invariant nondegenerate even supersymmetric bilinear form $(\cdot,\cdot)$
on $\mathfrak{g}$ such that non-trivial products are given by

$(e,f)=(f,e)=1,(h,h)=2,(x,y)=-(y,x)=2.$

Let
$\tilde{\mathfrak{g}}=\mathfrak{g}\otimes\mathbb{C}[t,t^{-1}]\oplus\mathbb{C}k$
be the affine Lie superalgebra of $osp(1|2)$ with the
$\mathbb{Z}_{2}$-gradation

$|a\otimes t^{n}|=|a|~{}(a\in\mathfrak{g}),~{}|k|=0,$

and anticommutation relations

$[a\otimes t^{n},b\otimes t^{m}]=[a,b]\otimes
t^{m+n}+m\delta_{m+n,0}(a,b)k~{}(a,b\in\mathfrak{g},m,n\in\mathbb{Z}),~{}[k,\tilde{\mathfrak{g}}]=0.$

We identify $\mathfrak{g}$ with $\mathfrak{g}\otimes t^{0}$ and set
$a(n)=a\otimes t^{n}$ for $a\in\mathfrak{g}$ and $n\in\mathbb{Z}$ for
convenience. Define subalgebras

$\displaystyle N_{+}=\mathbb{C}e\oplus\mathbb{C}x\oplus\mathfrak{g}\otimes
t\mathbb{C}[t],$ (2.1) $\displaystyle
N_{-}=\mathbb{C}f\oplus\mathbb{C}y\oplus\mathfrak{g}\otimes
t^{-1}\mathbb{C}[t^{-1}],$ $\displaystyle
B=N_{+}\oplus\mathbb{C}h\oplus\mathbb{C}k,~{}P=\mathfrak{g}\otimes\mathbb{C}[t]\oplus\mathbb{C}k.$

Let $\hat{\mathfrak{g}}=\tilde{\mathfrak{g}}\oplus\mathbb{C}d$ be the extended
affine Lie superalgebra with $|d|=0$ and

$[d,a\otimes t^{n}]=na\otimes
t^{n}~{}(a\in\mathfrak{g},n\in\mathbb{Z}),~{}[d,k]=0.$

Let $H=\mathbb{C}h\oplus\mathbb{C}k\oplus\mathbb{C}d$ be the Cartan subalgebra
of $\hat{\mathfrak{g}}$. For any $\lambda\in H^{*}$, denote by
$M(\lambda)$(resp. $L(\lambda)$) the Verma (resp. the irreducible highest
weight) $\hat{\mathfrak{g}}$-module. It is clear that $L(\lambda)$ is an
irreducible $\tilde{\mathfrak{g}}$-module, denote by
$L(\mathcal{l},\mathcal{j})$ the $\tilde{\mathfrak{g}}$-module $L(\lambda)$,
where
$\mathcal{l}=\langle\lambda,k\rangle,\mathcal{j}=\langle\lambda,h\rangle$.
Conversely, for any restricted $\tilde{\mathfrak{g}}$-module $M$ of level
$\mathcal{l}\neq-\frac{3}{2}$, by Sugawara construction we can extend $M$ to a
$\hat{\mathfrak{g}}$-module by letting $d$ act on $M$ as $-L(0)$. In this
paper we shall consider any restricted $\tilde{\mathfrak{g}}$-module as a
$\hat{\mathfrak{g}}$-module in this way.

Let $\mathcal{l}\in\mathbb{C}$ and $U$ be a $\mathbb{C}h$-module, $U$ can be
regarded as a $B$-module by setting $N_{+}$ acting trivially and $k$ acting as
$\mathcal{l}$, let $M(\mathcal{l},U)=U(\tilde{\mathfrak{g}})\otimes_{U(B)}U$.
If $U=\mathbb{C}$ is a one-dimensional $\mathbb{C}h$-module on which $h$ acts
as a fixed complex number $\mathcal{j}$, the corresponding module is a Verma
module denoted by $M(\mathcal{l},\mathcal{j})$. Then
$M(\mathcal{l},\mathcal{j})$ has a unique maximal submodule and
$L(\mathcal{l},\mathcal{j})$ is isomorphic to the corresponding irreducible
quotient. Similarly we can define the generalized Verma
$\tilde{\mathfrak{g}}$-module
$V(\mathcal{l},U)=U(\tilde{\mathfrak{g}})\otimes_{U(P)}U$ for any
$\mathfrak{g}$-module $U$ which can be extend to a $P$-module by setting
$\mathfrak{g}\otimes t\mathbb{C}[t]$ acting trivially and $k$ acting as
$\mathcal{l}$. Note that if $U=\mathbb{C}$ is the trivial
$\mathfrak{g}$-module, then $V(\mathcal{l},\mathbb{C})$ is a quotient of
$M(\mathcal{l},0)$ and $L(\mathcal{l},0)$ is isomorphic to the irreducible
quotient of $V(\mathcal{l},\mathbb{C})$.

We recall the following reducibility criterion in [14] which generalized the
Kac-Kazhdan reducibility criterion [16].

######  Proposition 2.4.

[14]. The Verma module
$M(\mathcal{l},\mathcal{j})(\mathcal{l}\neq-\frac{3}{2})$ is reducible if and
only if

$\mathcal{j}=\frac{m-1}{2}-s(\mathcal{l}+\frac{3}{2}),$

where $m,s\in\mathbb{Z}$ such that $m+s\in 2\mathbb{Z}+1$ and either
$m>0,s\geq 0$ or $m<0,s<0$.

In [17], Kac and Wakimoto gave the definition of admissible weight for Kac-
Moody (super)algebras and charactered the admissible weight of
$\hat{\mathfrak{g}}$ as an example. There is an equivalent characterization of
the admissible weight of $\hat{\mathfrak{g}}$ in [14].

######  Proposition 2.5.

[14]. For any $\lambda\in H^{*}$, let
$\mathcal{l}=\langle\lambda,k\rangle,\mathcal{j}=\langle\lambda,h\rangle$.
(1) $\mathcal{l}$ is an admissible level if and only if
$\mathcal{l}+\frac{3}{2}=\frac{p}{2q}$, where $p,q$ are positive integers such
that $p\geq 2,p\equiv q(\mbox{mod}~{}2)$ and $(\frac{p-q}{2},q)=1$.
(2) For an admissible level $\mathcal{l}$, $\mathcal{j}$ is an admissible
weight if and only if $\mathcal{j}=\frac{m-1}{2}-ls$, where
$l=\mathcal{l}+\frac{3}{2}=\frac{p}{2q},m,s\in\mathbb{Z}$ such that $m+s\equiv
1(\mbox{mod}~{}2)$ and $1\leq m\leq p-1,0\leq s\leq q-1$.

From now on we will assume that $l=\mathcal{l}+\frac{3}{2}=\frac{p}{2q}$,
where $p,q$ are positive integers such that $p\geq 2,p\equiv
q(\mbox{mod}~{}2)$ and $(\frac{p-q}{2},q)=1$.

######  Remark 2.6.

Let $\mathcal{j}=\frac{m-1}{2}-ls=\frac{m^{\prime}-1}{2}-ls^{\prime}$ be an
admissible weight such that

$\displaystyle m,m^{\prime},s,s^{\prime}\in\mathbb{Z},1\leq m\leq p-1,1\leq
m^{\prime}\leq p-1,0\leq s^{\prime}\leq s\leq q-1,$ $\displaystyle m+s\equiv
1(\mbox{mod}~{}2),m^{\prime}+s^{\prime}\equiv 1(\mbox{mod}~{}2).$

Suppose $s>s^{\prime}$, then $\frac{m-m^{\prime}}{s-s^{\prime}}=\frac{p}{q}$.
Since $p\equiv q(\mbox{mod}~{}2)$, we have $p,q\in 2\mathbb{Z}$ or $p,q\in
2\mathbb{Z}+1$. If $p,q\in 2\mathbb{Z}+1$, by $(\frac{p-q}{2},q)=1$ we have
$(p,q)=1$, it is a contradiction. If $p,q\in 2\mathbb{Z}$, by
$(\frac{p-q}{2},q)=1$ we have
$s-s^{\prime}=\frac{1}{2}q,m-m^{\prime}=\frac{1}{2}p$, then
$(m-m^{\prime},s-s^{\prime})=1$ and $m-m^{\prime}+s-s^{\prime}\equiv
1(\mbox{mod}~{}2)$, it contradicts to $m+s\equiv 1(\mbox{mod}~{}2)$ and
$m^{\prime}+s^{\prime}\equiv 1(\mbox{mod}~{}2)$. Hence
$s=s^{\prime},m=m^{\prime}$, it shows that the expression
$\mathcal{j}=\frac{m-1}{2}-ls$ of an admissible weight $\mathcal{j}$ with
$m,s\in\mathbb{Z},m+s\equiv 1(\mbox{mod}~{}2)$ and $1\leq m\leq p-1,0\leq
s\leq q-1$ is unique.

A vector $w$ in a highest weight module $M$ for $\hat{\mathfrak{g}}$ is called
a singular vector if $w$ as a highest weight vector generates a proper
submodule of $M$. Note that $l=\frac{p}{2q}$ and
$\mathcal{j}=\frac{m-1}{2}-ls=\frac{m-p-1}{2}-l(s-q)$, then from Remark 2.6,
together with the proof of Theorem 4.2 in [14] and Corollary 1 in [17], we
have the following proposition.

######  Proposition 2.7.

Let $\mathcal{j}=\frac{m-1}{2}-ls$, where $m,s\in\mathbb{Z}$ with $m+s\equiv
1(\mbox{mod}~{}2)$ and $1\leq m\leq p-1,0\leq s\leq q-1$ and let $v$ be a
highest weight vector which generates the Verma module
$M(\mathcal{l},\mathcal{j})$. Set

$\displaystyle\begin{aligned}
F_{1}(m,s)=&y(0)(y(0)^{2})^{\frac{m-1}{2}+ls}e(-1)^{\frac{m}{2}+l(s-1)}y(0)(y(0)^{2})^{\frac{m-1}{2}+l(s-2)}\cdots\\\
&e(-1)^{\frac{m}{2}-l(s-1)}y(0)(y(0)^{2})^{\frac{m-1}{2}-ls},\end{aligned}$
(2.2) $\displaystyle\begin{aligned}
F_{2}(m,s)=&e(-1)^{\frac{p-m}{2}+l(q-s-1)}y(0)(y(0)^{2})^{\frac{p-m-1}{2}+l(q-s-2)}e(-1)^{\frac{p-m}{2}+l(q-s-3)}\cdots\\\
&y(0)(y(0)^{2})^{\frac{p-m-1}{2}-l(q-s-2)}e(-1)^{\frac{p-m}{2}-l(q-s-1)}.\end{aligned}$

Then $v_{j,1}=F_{1}(m,s)v,v_{j,2}=F_{2}(m,s)v$ are singular vectors of
$M(\mathcal{l},\mathcal{j})$. Moreover, the maximal proper submodule of
$M(\mathcal{l},\mathcal{j})$ is generated by $v_{\mathcal{j},1}$ and
$v_{\mathcal{j},2}$.

We regard $(y(0)^{2})^{\gamma}$ as an even element for any
$\gamma\in\mathbb{C}$. It follows from [24] that $F_{1}(m,s)$ and $F_{2}(m,s)$
make sense as an element of $U(\tilde{\mathfrak{g}})$, and we have

$a^{\gamma}b=ba^{\gamma}+\sum_{j>0}\begin{pmatrix}\gamma\\\
j\end{pmatrix}(\mbox{ad}a)^{j}(b)a^{\gamma-j}$ (2.3)

for $a\in\tilde{\mathfrak{g}}_{\bar{0}},b\in
U(\tilde{\mathfrak{g}}),\gamma\in\mathbb{C}$.

######  Remark 2.8.

For $\mathcal{j}=0$, from Remark 2.6, we have $m=1,s=0$. By Proposition 2.7,
$v_{0,1}=F_{1}(1,0)\textbf{1},v_{0,2}=F_{2}(1,0)\textbf{1}$ are singular
vectors of $M(\mathcal{l},0)$. Since $v_{0,1}=F_{1}(1,0)\textbf{1}=0$ in
$V(\mathcal{l},\mathbb{C})$, we have $v_{0,2}=F_{2}(1,0)\textbf{1}$ generates
the maximal proper submodule of $V(\mathcal{l},\mathbb{C})$.

Set $P(\alpha)=xy+\alpha$ and $Q(\alpha)=yx-\alpha$ for $\alpha\in\mathbb{C}$.
We have the following commutation relations in $U(\mathfrak{g})$ (cf. [14]).

######  Proposition 2.9.

For any $\alpha,\beta,\gamma\in\mathbb{C}$, we have

$\displaystyle[P(\alpha),P(\beta)]=0,[P(\alpha),Q(\beta)]=0,[Q(\alpha),Q(\beta)]=0,$
(2.4) $\displaystyle
e^{\gamma}P(\alpha)=P(\alpha-\gamma)e^{\gamma},e^{\gamma}Q(\alpha)=Q(\alpha+\gamma)e^{\gamma},$
$\displaystyle
f^{\gamma}P(\alpha)=P(\alpha+\gamma)f^{\gamma},f^{\gamma}Q(\alpha)=P(\alpha-\gamma)f^{\gamma},$
$\displaystyle
xP(\alpha)=Q(1-\alpha)x,xQ(\alpha)=P(-\alpha)x,yP(\alpha)=Q(-\alpha)y,yQ(\alpha)=P(1-\alpha)y,$
$\displaystyle
e^{\gamma}y=Q(\gamma)xe^{\gamma-1},xe^{\gamma}y=P(-\gamma)e^{\gamma},f^{\gamma}x=-P(\gamma)yf^{\gamma-1},yf^{\gamma}x=Q(-\gamma)f^{\gamma}.$

Let $\sigma$ be the anti-automorphism of superalgebra $U(\mathfrak{g})$ such
that $\sigma(a)=-a$ for any $a\in\mathfrak{g}$. Then
$\sigma(P(\alpha))=-Q(\alpha)$ and $\sigma(Q(\alpha))=-P(\alpha)$ for any
$\alpha\in\mathbb{C}$. Let $\pi$ be the projection $\tilde{\mathfrak{g}}$ onto
$\mathfrak{g}$ such that $\pi(a\otimes t^{n})=a$ for any $a\in\mathfrak{g}$
and $\pi(c)=0$. From the Lemma 4.2 in [14], we have the following proposition.

######  Proposition 2.10.

[14].The following projection formulas hold:

$\displaystyle\pi(F_{1}(m,s))=\prod_{j=1}^{s}\prod_{i=1}^{m}\\{\prod_{i+j\in
2\mathbb{Z}}P(\frac{i}{2}+jl)\prod_{i+j\in
2\mathbb{Z}+1}Q(-\frac{i-1}{2}-jl)\\}y^{m},$ (2.5)
$\displaystyle\pi(F_{2}(m,s))=\prod_{j=1}^{q-s-1}\prod_{i=1}^{p-m}\\{\prod_{i+j\in
2\mathbb{Z}}Q(\frac{i}{2}+jl)\prod_{i+j\in
2\mathbb{Z}+1}P(-\frac{i-1}{2}-jl)\\}x^{p-m}.$

##  3 Rationality of
$(L_{\widehat{\mathfrak{g}}}(\mathcal{l},0),\omega_{\xi})$

In this section, we first construct the weak
$V_{\widehat{\mathfrak{g}}}(\mathcal{l},\mathbb{C})$-module $L(\mathcal{l},U)$
and determine the condition that $L(\mathcal{l},U)$ is a weak
$L_{\widehat{\mathfrak{g}}}(\mathcal{l},0)$-module. Then we prove that
$L_{\widehat{\mathfrak{g}}}(\mathcal{l},0)$ at admissible level is rational in
category $\mathcal{C}_{\mathcal{l}}$ and
$(L_{\widehat{\mathfrak{g}}}(\mathcal{l},0),\omega_{\xi})$ are rational
$\mathbb{Q}$-graded vertex operator superalgebras respect to a family of
Virasoro elements $\omega_{\xi}$, where $0<\xi<1$ is a rational number.

###  3.1 The weak $L_{\widehat{\mathfrak{g}}}(\mathcal{l},0)$-module
$L(\mathcal{l},U)$

We know that for $\mathcal{l}\neq-\frac{3}{2}$, $V(\mathcal{l},\mathbb{C})$
and $L(\mathcal{l},0)$ have natural $\mathbb{Z}$-graded vertex operator
superalgebra structures and any $M(\mathcal{l},U)$ is a weak module for
$V(\mathcal{l},\mathbb{C})$. We denote by
$L_{\widehat{\mathfrak{g}}}(\mathcal{l},0)$ the $\mathbb{Z}$-graded vertex
operator superalgebra $L(\mathcal{l},0)$. For a $\mathbb{C}h$-module $U$, we
define a linear function on $U^{*}\otimes M(\mathcal{l},U)$ as follows:

$\langle
u^{\prime},u\rangle=u^{\prime}(\pi^{\prime}(u)),~{}\mbox{for}~{}~{}u^{\prime}\in
U^{*},u\in M(\mathcal{l},U),$

where $\pi^{\prime}$ is the projection of $M(\mathcal{l},U)$ onto the subspace
$U$. Define

$I=\\{u\in M(\mathcal{l},U)\mid\langle
u^{\prime},au\rangle=0~{}~{}\mbox{for~{}~{}any}~{}~{}u^{\prime}\in U^{*},a\in
U(\tilde{\mathfrak{g}})\\}.$ (3.1)

######  Lemma 3.1.

$I$ is unique and maximal in the set of submodules of $M(\mathcal{l},U)$ which
intersects with $U$ trivially.

###### Proof.

Let $I^{\prime}$ is a submodule of $M(\mathcal{l},U)$ which intersects with
$U$ trivially. Suppose $I^{\prime}\nsubseteq I$, we have there exists $u\in
I^{\prime}$ such that $\langle u^{\prime},au\rangle\neq 0$ for some
$u^{\prime}\in U^{*}$, then $\pi^{\prime}(au)\neq 0$. Since $I^{\prime}$ is a
submodule, we have $au\in I^{\prime}$. From the natural gradation of
$M(\mathcal{l},U)$ as a $\hat{\mathfrak{g}}$-module, we have
$\pi^{\prime}(au)\in I^{\prime}$, it contradicts to that $I^{\prime}$
intersects with $U$ trivially, hence $I^{\prime}\subseteq I$. Therefore $I$ is
unique and maximal in the set of submodules of $M(\mathcal{l},U)$ which
intersects with $U$ trivially. ∎

Set $L(\mathcal{l},U)=M(\mathcal{l},U)/I$ and we regard $U$ as a subspace of
$L(\mathcal{l},U)$. Then $\pi^{\prime}$ induces a projection of
$L(\mathcal{l},U)$ to $U$, which we still denote it by $\pi^{\prime}$. It is
clear that $L(\mathcal{l},U)$ is a weak module for
$V(\mathcal{l},\mathbb{C})$. Let $Y(\cdot,z)$ be the vertex operator map
defining the module structure on $L(\mathcal{l},U)$, then $Y(\cdot,z)$ is an
intertwining operator of type $\begin{pmatrix}L(\mathcal{l},U)\\\
V(\mathcal{l},\mathbb{C})L(\mathcal{l},U)\end{pmatrix}$. Let

$\mathcal{Y}(u,z)v=(-1)^{|u||v|}e^{zL(-1)}Y(v,-z)u$

for any homogeneous element $u\in L(\mathcal{l},U),v\in
V(\mathcal{l},\mathbb{C})$, then $\mathcal{Y}(\cdot,z)$ is an intertwining
operator of type $\begin{pmatrix}L(\mathcal{l},U)\\\
L(\mathcal{l},U)V(\mathcal{l},\mathbb{C})\end{pmatrix}$ (cf. [11]).

######  Lemma 3.2.

The $\tilde{\mathfrak{g}}$-module $L(\mathcal{l},U)$ is a weak module for
$L_{\widehat{\mathfrak{g}}}(\mathcal{l},0)$ if and only if

$\langle
u^{\prime},\mathcal{Y}(u,z)v_{0,2}\rangle=0~{}~{}\mbox{for~{}~{}any}~{}~{}u^{\prime}\in
U^{*},u\in U(\mathfrak{g})U\subset L(\mathcal{l},U).$ (3.2)

###### Proof.

Let $J$ be the maximal submodule of $V(\mathcal{l},\mathbb{C})$. From Remark
2.8, we have $J=U(N_{-})v_{0,2}$. Since
$L_{\widehat{\mathfrak{g}}}(\mathcal{l},0)=V(\mathcal{l},\mathbb{C})/J$ is the
quotient vertex operator superalgebra of $V(\mathcal{l},\mathbb{C})$ and
$L(\mathcal{l},U)$ is a weak module for the vertex operator superalgebra
$L_{\widehat{\mathfrak{g}}}(\mathcal{l},0)$, then for any $v\in J$ we have
$Y(v,z)L(\mathcal{l},U)=0$. Hence

$\mathcal{Y}(u,z)v_{0,2}=(-1)^{|u||v_{0,2}|}e^{zL(-1)}Y(v_{0,2},-z)u=0$

for any homogeneous element $u\in L(\mathcal{l},U)$, then $\langle
u^{\prime},\mathcal{Y}(u,z)v_{0,2}\rangle=0~{}~{}\mbox{for~{}~{}any}~{}~{}u^{\prime}\in
U^{*},u\in U(\mathfrak{g})U\subset L(\mathcal{l},U)$.

For the other hand, if $a\in N_{-}U(N_{-})$, we have
$\pi^{\prime}(a\mathcal{Y}(u,z)w)=0$. Then

$\langle
u^{\prime},a\mathcal{Y}(u,z)w\rangle=0~{}~{}\mbox{for~{}~{}any}~{}~{}u^{\prime}\in
U^{*},u\in L(\mathcal{l},U),a\in N_{-}U(N_{-}),w\in
V(\mathcal{l},\mathbb{C}).$

Let $\Gamma=\\{w\in V(\mathcal{l},\mathbb{C})\mid\langle
u^{\prime},\mathcal{Y}(u,z)w\rangle=0~{}~{}\mbox{for~{}~{}any}~{}~{}u^{\prime}\in
U^{*},u\in U(\mathfrak{g})U\subset L(\mathcal{l},U)\\}$, then
$v_{0,2}\in\Gamma$. From the Jacobi identity for the intertwining operator we
can get the following commutator formula

$[a(m),\mathcal{Y}(u,z)]=\sum_{j\geq 0}\begin{pmatrix}m\\\
j\end{pmatrix}\mathcal{Y}(a(j)u,z)z^{m-j}$ (3.3)

for any $a\in\mathfrak{g},m\in\mathbb{Z}$ and $u\in L(\mathcal{l},U)$. If
$w\in\Gamma,a(m)\in N_{-},u\in U(\mathfrak{g})U\subset L(\mathcal{l},U)$,

$\displaystyle\mathcal{Y}(u,z)a(m)w$
$\displaystyle=(-1)^{|u||a|}(a(m)\mathcal{Y}(u,z)w-[a(m),\mathcal{Y}(u,z)]w)$
$\displaystyle=(-1)^{|u||a|}(a(m)\mathcal{Y}(u,z)w-\sum_{j\geq
0}\begin{pmatrix}m\\\ j\end{pmatrix}\mathcal{Y}(a(j)u,z)z^{m-j}w),$

then by $j\geq 0$ we have $a(j)u\in U(\mathfrak{g})U\subset L(\mathcal{l},U)$
and $\langle u^{\prime},\mathcal{Y}(u,z)a(m)w\rangle=0$, hence
$a(m)w\in\Gamma$. Then we have $J\subseteq\Gamma$. From the Jacobi identity we
can also get

$\mathcal{Y}(a(n)u,z)=\sum_{j\geq 0}\begin{pmatrix}n\\\
j\end{pmatrix}a(n-j)\mathcal{Y}(u,z)z^{j}-(-1)^{n+|u||a|}\sum_{j\geq
0}\begin{pmatrix}n\\\ j\end{pmatrix}\mathcal{Y}(u,z)a(j)z^{n-j}$ (3.4)

for any $u\in L(\mathcal{l},U),a\in\mathfrak{g},n\in\mathbb{Z}$. Since
$L(\mathcal{l},U)$ is generated by $U$ as $\tilde{\mathfrak{g}}$-module, then
we have $\langle
u^{\prime},\mathcal{Y}(u,z)J\rangle=0~{}~{}\mbox{for~{}~{}any}~{}~{}u^{\prime}\in
U^{*},u\in L(\mathcal{l},U)$. From (3.3), we have

$\langle
u^{\prime},x\mathcal{Y}(u,z)J\rangle=0~{}~{}\mbox{for~{}~{}any}~{}~{}u^{\prime}\in
U^{*},x\in U(\tilde{\mathfrak{g}}),u\in L(\mathcal{l},U).$

Then $\mathcal{Y}(u,z)J\subseteq I$ for any $u\in L(\mathcal{l},U)$, i.e.,
$\mathcal{Y}(u,z)J=0$ in $L(\mathcal{l},U)$. It implies that
$\mathcal{Y}(\cdot,z)$ induces an intertwining operator of type
$\begin{pmatrix}L(\mathcal{l},U)\\\
L(\mathcal{l},U)L_{\widehat{\mathfrak{g}}}(\mathcal{l},0)\end{pmatrix}$. Hence
$L(\mathcal{l},U)$ is a weak module for the vertex operator superalgebra
$L_{\widehat{\mathfrak{g}}}(\mathcal{l},0)$. ∎

######  Theorem 3.3.

$L(\mathcal{l},U)$ is a weak
$L_{\widehat{\mathfrak{g}}}(\mathcal{l},0)$-module if and only if $f(h)U=0$,
where

$f(h)=\prod_{i+j\in 2\mathbb{Z}+1,0\leq j\leq q-1,1\leq i\leq
p-1}(h-\frac{i-1}{2}+jl).$ (3.5)

###### Proof.

For $n\in\mathbb{Z},a\in\mathfrak{g}$, we define $\mbox{deg}(a\otimes
t^{n})=n$. For any $u^{\prime}\in U^{*},a(m)\in N_{-},u\in
U(\mathfrak{g})U\subset L(\mathcal{l},U),w\in V(\mathcal{l},\mathbb{C})$, by
the commutator formula (3.3) we have

$\displaystyle\langle u^{\prime},\mathcal{Y}(u,z)a(m)w\rangle$
$\displaystyle=(-1)^{|u||a|}(\langle
u^{\prime},a(m)\mathcal{Y}(u,z)w\rangle-\langle
u^{\prime},[a(m),\mathcal{Y}(u,z)]w\rangle)$
$\displaystyle=(-1)^{|u||a|+1}\langle u^{\prime},\sum_{j\geq
0}\begin{pmatrix}m\\\ j\end{pmatrix}\mathcal{Y}(a(j)u,z)z^{m-j}w\rangle$
$\displaystyle=(-1)^{|u||a|+1}\langle
u^{\prime},\mathcal{Y}(a(0)u,z)z^{m}w\rangle$ $\displaystyle=\langle
u^{\prime},(-1)^{|u||a|+1}z^{\mbox{deg}(a(m))}\mathcal{Y}(\pi(a(m))u,z)w\rangle.$

Hence for any $u^{\prime}\in U^{*},a\in U(N_{-}),u\in U(\mathfrak{g})U\subset
L(\mathcal{l},U),w\in V(\mathcal{l},\mathbb{C})$ we have

$\langle u^{\prime},\mathcal{Y}(u,z)aw\rangle=\langle u^{\prime},\xi
z^{\mbox{deg}(a)}\mathcal{Y}(\sigma\pi(a)u,z)w\rangle,$

where $\xi=1$ or $-1$. Let $a=F_{2}(1,0)$, then $v_{0,2}=a\textbf{1}$. From
Lemma 3.2, $L(\mathcal{l},U)$ is a weak
$L_{\widehat{\mathfrak{g}}}(\mathcal{l},0)$-module if and only if

$\langle
u^{\prime},\mathcal{Y}(\sigma\pi(a)u,z)\textbf{1}\rangle=0~{}~{}\mbox{for~{}~{}any}~{}~{}u^{\prime}\in
U^{*},u\in U(\mathfrak{g})U\subset L(\mathcal{l},U).$

By Proposition 2.9 and Proposition 2.10 we have

$\displaystyle\sigma\pi(a)=\sigma\prod_{j=1}^{q-1}\prod_{i=1}^{p-1}\\{\prod_{i+j\in
2\mathbb{Z}}Q(\frac{i}{2}+jl)\prod_{i+j\in
2\mathbb{Z}+1}P(-\frac{i-1}{2}-jl)\\}x^{p-1}$
$\displaystyle=(-1)^{q(p-1)}x^{p-1}\prod_{j=1}^{q-1}\prod_{i=1}^{p-1}\\{\prod_{i+j\in
2\mathbb{Z}}P(\frac{i}{2}+jl)\prod_{i+j\in
2\mathbb{Z}+1}Q(-\frac{i-1}{2}-jl)\\}$
$\displaystyle=\begin{cases}(-1)^{q(p-1)}\prod_{j=1}^{q-1}\prod_{i=1}^{p-1}\\{\prod_{i+j\in
2\mathbb{Z}}P(-\frac{p-i-1}{2}+jl)\prod_{i+j\in
2\mathbb{Z}+1}Q(\frac{p-i}{2}-jl)\\}x^{p-1},&p\notin 2\mathbb{Z}\\\
(-1)^{q(p-1)}\prod_{j=1}^{q-1}\prod_{i=1}^{p-1}\\{\prod_{i+j\in
2\mathbb{Z}}Q(\frac{p-i}{2}-jl)\prod_{i+j\in
2\mathbb{Z}+1}P(-\frac{p-i-1}{2}+jl)\\}x^{p-1},&p\in 2\mathbb{Z}\end{cases}$
$\displaystyle=(-1)^{q(p-1)}\prod_{j=1}^{q-1}\prod_{i=1}^{p-1}\\{\prod_{i+j\in
2\mathbb{Z}+1}P(-\frac{i-1}{2}+jl)\prod_{i+j\in
2\mathbb{Z}}Q(\frac{i}{2}-jl)\\}x^{p-1}.$

Note that
$\mathcal{Y}(\sigma\pi(a)u,z)\textbf{1}=e^{zL(-1)}Y(\textbf{1},-z)\sigma\pi(a)u=e^{zL(-1)}\sigma\pi(a)u$.
Then for any $u^{\prime}\in U^{*},u\in U(\mathfrak{g})U\subset
L(\mathcal{l},U)$,

$\langle u^{\prime},\mathcal{Y}(\sigma\pi(a)u,z)\textbf{1}\rangle=\langle
u^{\prime},e^{zL(-1)}\sigma\pi(a)u\rangle=\langle
u^{\prime},\sigma\pi(a)u\rangle,$

thus $L(\mathcal{l},U)$ is a weak
$L_{\widehat{\mathfrak{g}}}(\mathcal{l},0)$-module if and only if for any
$u^{\prime}\in U^{*}$,

$\langle u^{\prime},\prod_{j=1}^{q-1}\prod_{i=1}^{p-1}\\{\prod_{i+j\in
2\mathbb{Z}+1}P(-\frac{i-1}{2}+jl)\prod_{i+j\in
2\mathbb{Z}}Q(\frac{i}{2}-jl)\\}x^{p-1}U(\mathfrak{g})U\rangle=0.$

From the grading restriction on the bilinear pair, it is equivalent to

$\prod_{j=1}^{q-1}\prod_{i=1}^{p-1}\\{\prod_{i+j\in
2\mathbb{Z}+1}P(-\frac{i-1}{2}+jl)\prod_{i+j\in
2\mathbb{Z}}Q(\frac{i}{2}-jl)\\}x^{p-1}y^{p-1}U=0.$

By Proposition 2.9, we have

$\displaystyle x^{p-1}y^{p-1}$
$\displaystyle=\begin{cases}\prod_{i=1}^{\frac{p-1}{2}}Q(\frac{p+1}{2}-i)P(-\frac{p-1}{2}+i),&p\notin
2\mathbb{Z}\\\
P(0)\prod_{i=1}^{\frac{p-2}{2}}P(-\frac{p}{2}+i)Q(\frac{p}{2}-i),&p\in
2\mathbb{Z}\end{cases}$ $\displaystyle=\prod_{i\in 2\mathbb{Z}+1,1\leq i\leq
p-1}P(-\frac{i-1}{2})\prod_{i\in 2\mathbb{Z},1\leq i\leq p-1}Q(\frac{i}{2}).$

Since $xU=0$, we have for any $\alpha\in\mathbb{C}$,

$P(\alpha)U=xyU+\alpha U=(h+\alpha)U,Q(\alpha)U=-\alpha U.$

Then $L(\mathcal{l},U)$ is a weak
$L_{\widehat{\mathfrak{g}}}(\mathcal{l},0)$-module if and only if

$\prod_{i+j\in 2\mathbb{Z}+1,1\leq j\leq q-1,1\leq i\leq
p-1}(h-\frac{i-1}{2}+jl)\prod_{i\in 2\mathbb{Z}+1,1\leq i\leq
p-1}(h-\frac{i-1}{2})U=0.$

It is equivalent to

$f(h)U=\prod_{i+j\in 2\mathbb{Z}+1,0\leq j\leq q-1,1\leq i\leq
p-1}(h-\frac{i-1}{2}+jl)U=0.$

∎

From Proposition 2.5 and Theorem 3.3, we obtain the following corollary.

######  Corollary 3.4.

The $\tilde{\mathfrak{g}}$-module $L(\mathcal{l},\mathcal{j})$ is a weak
$L_{\widehat{\mathfrak{g}}}(\mathcal{l},0)$-module if and only if
$\mathcal{j}$ is admissible.

Let $\mathcal{O}_{\mathcal{l}}$ be the subcategory of the weak
$L_{\widehat{\mathfrak{g}}}(\mathcal{l},0)$-module category such that $M$ is
an object in $\mathcal{O}_{\mathcal{l}}$ if and only if $M\in\mathcal{O}$ as
$\hat{\mathfrak{g}}$-module. In [25], Wood showed that
$\mathcal{O}_{\mathcal{l}}$ is semisimple.

######  Theorem 3.5.

[25]. The weak $L_{\widehat{\mathfrak{g}}}(\mathcal{l},0)$-modules in the
category $\mathcal{O}$ are completely reducible, i.e.,
$\mathcal{O}_{\mathcal{l}}$ is semisimple. Moreover, the irreducible modules
correspond to $L(\mathcal{l},\mathcal{j})$, where $\mathcal{j}$ is admissible.

Corollary 3.4 can also be obtained by Theorem 3.5. Conversely, by Corollary
3.4 and the proof of Theorem 4.1 in [18], we can obtain Theorem 3.5 in a
different way. From Theorem 3.5, for any $M\in\mathcal{O}_{\mathcal{l}}$, let
$U=\\{w\in M\mid N_{+}w=0\\}$, it is clear that $U$ is a $\mathbb{C}h$-module
and $M\cong L(\mathcal{l},U)$.

###  3.2 Category $\mathcal{C}_{\mathcal{l}}$

######  Definition 3.6.

The category $\mathcal{C}_{\mathcal{l}}$ is the subcategory of the weak
$L_{\widehat{\mathfrak{g}}}(\mathcal{l},0)$-module category such that $M$ is
an object in $\mathcal{C}_{\mathcal{l}}$ if and only if $a$ acts locally
nilpotently on $M$ for all $a\in N_{+}$.

It is clear that the object in $\mathcal{O}_{\mathcal{l}}$ is also an object
in $\mathcal{C}_{\mathcal{l}}$. We will show that $\mathcal{C}_{\mathcal{l}}$
is semisimple. Let $\Omega(M)=\\{v\in M\mid(\mathfrak{g}\otimes
t\mathbb{C}[t]).v=0\\}$ for a weak
$L_{\widehat{\mathfrak{g}}}(\mathcal{l},0)$-module $M$. We can prove the
following proposition.

######  Proposition 3.7.

Let $M$ be a weak $L_{\widehat{\mathfrak{g}}}(\mathcal{l},0)$-module belonging
to $\mathcal{C}_{\mathcal{l}}$, then $M$ has a highest weight vector.

###### Proof.

Similar to the Proposition 3.6 of [23] or the Theorem 3.7 of [6], we have
$\Omega(M)\neq 0$. Since $M$ is an object of $\mathcal{C}_{\mathcal{l}}$, we
have $e$ and $x$ act locally nilpotent on $\Omega(M)$. Let
$\mathfrak{n}_{+}=\mathbb{C}e\oplus\mathbb{C}x$, then $U(\mathfrak{n}_{+})v$
is finite dimensional for any $v\in\Omega(M)$. Since $\mathfrak{n}_{+}$ is a
nilpotent Lie superalgebra, then by the Engel theorem for Lie superalgebra
(cf. [15]), there exists a nonzero $w\in U(\mathfrak{n}_{+})v$ such that
$\mathfrak{n}_{+}.w=0$. Since $\mathfrak{n}_{+}.\Omega(M)\subseteq\Omega(M)$,
we have $w\in\Omega(M)$, then $N_{+}.w=0$. Furthermore, let
$U=U(\mathfrak{g})w$ and $W$ be the weak
$L_{\widehat{\mathfrak{g}}}(\mathcal{l},0)$-submodule generated by $U$. Then
$W$ is isomorphic to a quotient module of $M(\mathcal{l},U)$ as
$\tilde{\mathfrak{g}}$-module. From Lemma 3.1, $L(\mathcal{l},U)$ is
isomorphic to some quotient module of $W$ as $\tilde{\mathfrak{g}}$-module,
then $L(\mathcal{l},U)$ is a quotient of $W$ as weak
$V(\mathcal{l},\mathbb{C})$-module. Therefore $L(\mathcal{l},U)$ is a weak
$L_{\widehat{\mathfrak{g}}}(\mathcal{l},0)$-module. From Theorem 3.3, we have
$f(h)U=0$, then $h$ semisimply acts on $U$, i.e., $w$ is a highest weight
vector. ∎

We now classify simple objects in the category $\mathcal{C}_{\mathcal{l}}$.

######  Proposition 3.8.

Any irreducible weak $L_{\widehat{\mathfrak{g}}}(\mathcal{l},0)$-module
belonging to $\mathcal{C}_{\mathcal{l}}$ is of the form
$L(\mathcal{l},\mathcal{j})$ with admissible weight $\mathcal{j}$. Moreover,
up to isomorphism, there are finitely many irreducible weak
$L_{\widehat{\mathfrak{g}}}(\mathcal{l},0)$-modules belonging to
$\mathcal{C}_{\mathcal{l}}$.

###### Proof.

Let $M$ be an irreducible weak
$L_{\widehat{\mathfrak{g}}}(\mathcal{l},0)$-module belonging to
$\mathcal{C}_{\mathcal{l}}$, then by Proposition 3.7, $M$ contains a highest
weight vector $w$. Since $M$ is an irreducible weak
$L_{\widehat{\mathfrak{g}}}(\mathcal{l},0)$-module, we have $M$ is an
irreducible weak $V(\mathcal{l},\mathbb{C})$-module, then $M$ is an
irreducible $\tilde{\mathfrak{g}}$-module generated by $w$. Therefore $M$ is
an irreducible highest weight module of $\tilde{\mathfrak{g}}$. By Corollary
3.4, $M$ is of the form $L(\mathcal{l},\mathcal{j})$ for some admissible
weight $\mathcal{j}$. ∎

Then we can prove the category $\mathcal{C}_{\mathcal{l}}$ is semisimple.

######  Theorem 3.9.

The category $\mathcal{C}_{\mathcal{l}}$ is semisimple. Moreover, any object
$M$ in $\mathcal{C}_{\mathcal{l}}$ is a direct sum of
$L(\mathcal{l},\mathcal{j})$ with admissible weight $\mathcal{j}$.

###### Proof.

Let $M$ be a weak $L_{\widehat{\mathfrak{g}}}(\mathcal{l},0)$-module belonging
to $\mathcal{C}_{\mathcal{l}}$. From Proposition 3.7, $M$ contains a highest
weight vector $w$. Let $W$ be the weak
$L_{\widehat{\mathfrak{g}}}(\mathcal{l},0)$-submodule generated by $w$, then
$W$ is a quotient of certain Verma module
$M(\mathcal{l},\mathcal{j}^{\prime})$ as $\tilde{\mathfrak{g}}$-module. Hence
$W$ is an object in $\mathcal{O}_{\mathcal{l}}$. From Theorem 3.5, it is
completely reducible and $W=\bigoplus_{\mathcal{j}}L(\mathcal{l},\mathcal{j})$
for admissible weights $\mathcal{j}$. But the highest weight subspace of
$M(\mathcal{l},\mathcal{j}^{\prime})$ is one dimensional, then
$W=L(\mathcal{l},\mathcal{j}^{\prime})$ with admissible weight
$\mathcal{j}^{\prime}$. Let $W^{\prime}$ be the sum of irreducible weak
$L_{\widehat{\mathfrak{g}}}(\mathcal{l},0)$-module of $M$, then $W^{\prime}$
is a direct sum of $L(\mathcal{l},\mathcal{j})$ for admissible weights
$\mathcal{j}$.

If $W^{\prime}$ is a proper submodule of $M$, then $M/W^{\prime}$ is a weak
$L_{\widehat{\mathfrak{g}}}(\mathcal{l},0)$-module belonging to
$\mathcal{C}_{\mathcal{l}}$, thus it contains a highest weight vector
$\bar{w^{\prime}}$ by Proposition 3.7. Let $w^{\prime}$ be a preimage of
$\bar{w^{\prime}}$, then $N_{+}.w^{\prime}\subseteq W^{\prime}$. Since $N_{+}$
is finitely generated, there exist submodules
$L(\mathcal{l},\mathcal{j}_{1}),L(\mathcal{l},\mathcal{j}_{2}),\cdots,L(\mathcal{l},\mathcal{j}_{s})$
of $W^{\prime}$ such that

$N_{+}.w^{\prime}\subseteq L(\mathcal{l},\mathcal{j}_{1})\oplus
L(\mathcal{l},\mathcal{j}_{2})\oplus\cdots\oplus
L(\mathcal{l},\mathcal{j}_{s}).$

From Theorem 3.5, the submodule of $M$ generated by $w^{\prime}$ and
$L(\mathcal{l},\mathcal{j}_{1})\oplus
L(\mathcal{l},\mathcal{j}_{2})\oplus\cdots\oplus
L(\mathcal{l},\mathcal{j}_{s})$ is completely reducible. Then the submodule of
$M$ generated by $w^{\prime}$ is a direct sum of certain
$L(\mathcal{l},\mathcal{j})$ with admissible weights $\mathcal{j}$, it is a
contradiction. Therefore $W^{\prime}=M$, i.e., $M$ is a direct sum of
$L(\mathcal{l},\mathcal{j})$ with admissible weight $\mathcal{j}$. Then we
have that the category $\mathcal{C}_{\mathcal{l}}$ is semisimple. ∎

As an application of the semisimplicity of $\mathcal{C}_{\mathcal{l}}$, we
prove the category of ordinary modules for
$L_{\widehat{\mathfrak{g}}}(\mathcal{l},0)$ is semisimple.

######  Proposition 3.10.

Any ordinary $L_{\widehat{\mathfrak{g}}}(\mathcal{l},0)$-module $M$ is
completely reducible.

###### Proof.

Let $M$ be an ordinary $L_{\widehat{\mathfrak{g}}}(\mathcal{l},0)$-module,
then $M=\bigoplus_{h\in\mathbb{C}}M_{h}$ such that $\mbox{dim}~{}M_{h}<\infty$
for any $h\in\mathbb{C}$ and $M_{h+n}=0$ for $n\in\mathbb{Z}$ sufficiently
small. For any $a(n)\in\mathfrak{g}\otimes t\mathbb{C}[t]$, we have
$\mbox{wt}~{}a(n)w=\mbox{wt}~{}w-n<\mbox{wt}~{}w$ for all $w\in M$, then
$a(n)$ acts locally nilpotently on $M$. For any $w\in
M_{h}~{}(h\in\mathbb{C})$, since $M$ is a $\tilde{\mathfrak{g}}$-module and
$\mbox{wt}~{}aw=\mbox{wt}~{}w$ for any $a\in\mathfrak{g}$, we have $M_{h}$ is
a finite dimensional $\mathfrak{g}$-module, then $x$ and $e$ act locally
nilpotently on $M_{h}$. Therefore $x$ acts locally nilpotently on $M$ for all
$x\in N_{+}$, i.e., $M$ is an object of $\mathcal{C}_{\mathcal{l}}$, then $M$
is completely reducible by Theorem 3.9. ∎

######  Proposition 3.11.

Any irreducible ordinary $L_{\widehat{\mathfrak{g}}}(\mathcal{l},0)$-module is
of the form $L(\mathcal{l},\mathcal{j})$ such that
$0\leq\mathcal{j}\leq\frac{p-2}{2},\mathcal{j}\in\mathbb{Z}$. Moreover, up to
isomorphism, there are finitely many irreducible ordinary
$L_{\widehat{\mathfrak{g}}}(\mathcal{l},0)$-modules.

###### Proof.

Let $U$ be an irreducible finite dimensional $\mathfrak{g}$-module, then the
irreducible quotient $L(U)$ of
$\mbox{Ind}_{\mathfrak{g}}^{\hat{\mathfrak{g}}}(U)$ is an irreducible ordinary
$V(\mathcal{l},\mathbb{C})$-module (cf. [20]). It is similar to the proof of
Theorem 6.2.23 of [22], we have the modules $L(U)$ for irreducible finite
dimensional $\mathfrak{g}$-module $U$ exhaust the irreducible ordinary
$V(\mathcal{l},\mathbb{C})$-modules up to equivalence. Let $M$ be an
irreducible ordinary $L_{\widehat{\mathfrak{g}}}(\mathcal{l},0)$-module, then
from Proposition 3.8 and Proposition 3.10, $M$ is of the form
$L(\mathcal{l},\mathcal{j})$ with admissible weight $\mathcal{j}$. Since $M$
is an irreducible ordinary $V(\mathcal{l},\mathbb{C})$-module, we have $M$ is
of the form $L(U)$ for irreducible finite dimensional $\mathfrak{g}$-module
$U$. Hence $M$ is of the form $L(\mathcal{l},\mathcal{j})$ with admissible
weight $\mathcal{j}\in\mathbb{Z}$.

Since $\mathcal{j}$ is admissible, $\mathcal{j}=\frac{m-1}{2}-ls$, where
$l=\frac{p}{2q},m,s\in\mathbb{Z}$ such that $m+s\equiv 1(\mbox{mod}~{}2)$ and
$1\leq m\leq p-1,0\leq s\leq q-1$, $p,q$ are positive integers such that
$p\geq 2,p\equiv q(\mbox{mod}~{}2)$ and $(\frac{p-q}{2},q)=1$. If $p,q\in
2\mathbb{Z}+1$, we have $(p,q)=1$, $l\notin\mathbb{Z}$, then $s=0$,
$\mathcal{j}=\frac{m-1}{2}$ for $1\leq m\leq p-1,m\in 2\mathbb{Z}+1$. If
$p,q\in 2\mathbb{Z}$, we have $\frac{p-q}{2}\in 2\mathbb{Z}+1$ and
$(\frac{p}{2},\frac{q}{2})=1$, for $\frac{p}{2}\in 2\mathbb{Z}+1$,
$l=\frac{p/2}{q}\notin\mathbb{Z}$ and $(\frac{p}{2},q)=1$, then $s=0$,
$\mathcal{j}=\frac{m-1}{2}$ for $1\leq m\leq p-1,m\in 2\mathbb{Z}+1$; for
$\frac{p}{2}\in 2\mathbb{Z}$, $l=\frac{p/4}{q/2}$ and
$(\frac{p}{4},\frac{q}{2})=1$, by $m+s\equiv 1(\mbox{mod}~{}2)$,
$\mathcal{j}=\frac{m-1}{2}-\frac{p}{4}\notin\mathbb{Z}$ for $s=\frac{q}{2}$,
then $s=0$, $\mathcal{j}=\frac{m-1}{2}$ for $1\leq m\leq p-1,m\in
2\mathbb{Z}+1$. Therefore we have
$0\leq\mathcal{j}\leq\frac{p-2}{2},\mathcal{j}\in\mathbb{Z}$. ∎

Proposition 3.10 and Proposition 3.11 are also obtained in [4] by using the
theory of vertex superalgebra extensions.

###  3.3 $\mathbb{Q}$-graded vertex operator superalgebras
$(L_{\widehat{\mathfrak{g}}}(\mathcal{l},0),\omega_{\xi})$

A $\mathbb{Q}$-graded vertex operator superalgebra $V$ is $\mathbb{Q}$-graded
by weights instead of $\mathbb{Z}$-graded (cf. [10]). Similarly we can define
the weak module, $\mathbb{Q}_{+}$-graded weak $V$-module and ordinary module
for $\mathbb{Q}$-graded vertex operator superalgebra.

######  Definition 3.12.

A $\mathbb{Q}$-graded vertex operator superalgebra $V$ is called rational if
any $\mathbb{Q}_{+}$-graded weak $V$-module is a direct sum of irreducible
$\mathbb{Q}_{+}$-graded weak $V$-modules.

Let $(V,Y,\textbf{1},\omega)$ be a $\mathbb{Z}$-graded vertex operator
superalgebra, $h\in V_{(1)}$ be a vector satisfying the following conditions:

$\displaystyle[L(m),h_{n}]=-nh_{m+n}-\frac{m^{2}+m}{2}\delta_{m+n,0}\kappa_{1}\mbox{Id}_{V},$
(3.6) $\displaystyle[h_{m},h_{n}]=2m\delta_{m+n,0}\kappa_{2}\mbox{Id}_{V},$

where $\kappa_{1},\kappa_{2}\in\mathbb{C}$. Assume that $h_{0}$ acts
semisimply on $V$ and the eigenvalues of $h_{0}$ are rational numbers. For a
rational number $\xi$, set $\omega_{\xi}=\omega+\frac{\xi}{2}L(-1)h$ and
$Y(\omega_{\xi},z)=\sum_{n\in\mathbb{Z}}L^{\prime}(n)z^{-n-2}$. Then we have

$L^{\prime}(n)=L(n)-\frac{\xi}{2}(n+1)h_{n}~{}~{}\mbox{for~{}~{}any}~{}~{}n\in\mathbb{Z},$

hence the component operators of $\omega_{\xi}$ satisfy the Virasoro relations
and $L^{\prime}(-1)=L(-1)=D$. Since $L(0)$ and $h_{0}$ are commutative, each
$V_{(n)}$ is a direct sum of eigenspaces of $h_{0}$. Set $V_{(m,n)}=\\{v\in
V_{(m)}\mid h_{0}v=nv\\}$ for $m\in\mathbb{Z},n\in\mathbb{Q}$. Let

$V^{\prime}_{(m)}=\\{v\in V\mid
L^{\prime}(0)v=mv\\}=\coprod_{s\in\mathbb{Z},t\in\mathbb{Q},s-\frac{\xi
t}{2}=m}V_{(s,t)}.$ (3.7)

Then we have the following proposition (cf. [2, 7, 23]).

######  Proposition 3.13.

Let $(V,Y,\textbf{1},\omega)$ be a $\mathbb{Z}$-graded vertex operator
superalgebra of central charge $c$, $h\in V_{(1)}$ be a vector satisfying the
conditions (3.6) and $h_{0}$ acts semisimply on $V$, the eigenvalues of
$h_{0}$ are rational numbers. Suppose that
$\mbox{dim}~{}V^{\prime}_{(m)}<\infty$ for any $m\in\mathbb{Q}$ and
$V^{\prime}_{(m)}=0$ for $m$ sufficiently small. Then
$(V,Y,\textbf{1},\omega_{\xi})$ is a $\mathbb{Q}$-graded vertex operator
superalgebra of central charge $c-6\xi(\kappa_{1}+\xi\kappa_{2})$.

For the $\mathbb{Z}$-graded vertex operator superalgebra
$L_{\widehat{\mathfrak{g}}}(\mathcal{l},0)$, by the Sugawara construction,

$\displaystyle L(n)=$
$\displaystyle\frac{1}{2\mathcal{l}+3}\sum_{m\in\mathbb{Z}}(\frac{1}{2}:h(m)h(n-m):+:e(m)f(n-m):+:f(m)e(n-m):$
$\displaystyle-\frac{1}{2}:x(m)y(n-m):+\frac{1}{2}:y(m)x(n-m):).$

A direct calculation shows that $[L(m),h(n)]=-nh(m+n)$, then $h$ satisfies
conditions (3.6) with $\kappa_{1}=0,\kappa_{2}=\mathcal{l}$. Hence
$L_{\widehat{\mathfrak{g}}}(\mathcal{l},0)$ have a $\mathbb{Q}$-gradation by
weights with respect to $L^{\prime}(0)$. From the construction of
$V(\mathcal{l},\mathbb{C})$, for any $m\in\mathbb{Z}_{+}$, if
$V(\mathcal{l},\mathbb{C})_{(m,n)}\neq 0$, we have $-2m\leq n\leq 2m$. Since
$L_{\widehat{\mathfrak{g}}}(\mathcal{l},0)$ is a quotient of
$V(\mathcal{l},\mathbb{C})$, for any $m\in\mathbb{Z}_{+}$, if
$L_{\widehat{\mathfrak{g}}}(\mathcal{l},0)_{(m,n)}\neq 0$, we also have
$-2m\leq n\leq 2m$. Let $0<\xi<1$, for any $m\in\mathbb{Q}$, since
$L_{\widehat{\mathfrak{g}}}(\mathcal{l},0)^{\prime}_{(m)}=\coprod_{s\in\mathbb{Z},t\in\mathbb{Q},s-\frac{\xi
t}{2}=m}L_{\widehat{\mathfrak{g}}}(\mathcal{l},0)_{(s,t)}$, we have

$\frac{m}{1+\xi}\leq s\leq\frac{m}{1-\xi}$

for $L_{\widehat{\mathfrak{g}}}(\mathcal{l},0)_{(s,t)}\neq 0$, then
$\mbox{dim}~{}L_{\widehat{\mathfrak{g}}}(\mathcal{l},0)^{\prime}_{(m)}<\infty$.
Note that $\frac{m}{1+\xi}\leq\frac{m}{1-\xi}$, we have
$L_{\widehat{\mathfrak{g}}}(\mathcal{l},0)^{\prime}_{(m)}=0$ for $m<0$.
Therefore, by Proposition 3.13,
$(L_{\widehat{\mathfrak{g}}}(\mathcal{l},0),\omega_{\xi})$ is a
$\mathbb{Q}$-graded vertex operator superalgebra of central charge
$c-6\xi^{2}\mathcal{l}$.

######  Theorem 3.14.

Let $0<\xi<1$, $(L_{\widehat{\mathfrak{g}}}(\mathcal{l},0),\omega_{\xi})$ is a
rational $\mathbb{Q}$-graded vertex operator superalgebra. Moreover, any
$\mathbb{Q}_{+}$-graded weak
$(L_{\widehat{\mathfrak{g}}}(\mathcal{l},0),\omega_{\xi})$-module is a direct
sum of $L(\mathcal{l},\mathcal{j})$ with admissible weight $\mathcal{j}$.

###### Proof.

Let $M$ be a $\mathbb{Q}_{+}$-graded weak
$(L_{\widehat{\mathfrak{g}}}(\mathcal{l},0),\omega_{\xi})$-module, we denote
by $\mbox{deg}~{}w$ the degree of homogeneous element $w\in W$. For any
$\alpha\in\Delta_{\mathfrak{g}}^{+}$ and the corresponding root vector
$x_{\alpha}$,

$L^{\prime}(0)x_{\alpha}=L(0)x_{\alpha}-\frac{\xi}{2}h_{0}x_{\alpha}=(1-\frac{\xi}{2}\alpha(h))x_{\alpha}.$

Since $0<\xi<1$, we have
$\mbox{deg}~{}x_{\alpha}w=\mbox{wt}~{}x_{\alpha}+\mbox{deg}~{}w-1<\mbox{deg}~{}w$
for all $w\in M$, then $x$ and $e$ act locally nilpotently on $M$. For any
$a(n)\in\mathfrak{g}\otimes t\mathbb{C}[t]$, we have

$\mbox{deg}~{}a(n)w=\mbox{deg}~{}w+\mbox{wt}~{}a-n-1<\mbox{deg}~{}w$

for all $w\in M$, then $a(n)$ acts locally nilpotently on $M$. Therefore $a$
acts locally nilpotently on $M$ for all $a\in N_{+}$, i.e., $M$ is an object
of $\mathcal{C}_{\mathcal{l}}$. Then $M$ is completely reducible by Theorem
3.9.

Next we prove that $L(\mathcal{l},\mathcal{j})$ is a $\mathbb{Q}_{+}$-graded
weak $(L_{\widehat{\mathfrak{g}}}(\mathcal{l},0),\omega_{\xi})$-module for any
admissible weight $j$. Similar to the proof that
$(L_{\widehat{\mathfrak{g}}}(\mathcal{l},0),\omega_{\xi})$ is a
$\mathbb{Q}$-graded vertex operator superalgebra, $L(\mathcal{l},\mathcal{j})$
with admissible weight $\mathcal{j}$ is an ordinary
$(L_{\widehat{\mathfrak{g}}}(\mathcal{l},0),\omega_{\xi})$-module, hence
$L(\mathcal{l},\mathcal{j})$ is a $\mathbb{Q}_{+}$-graded weak
$(L_{\widehat{\mathfrak{g}}}(\mathcal{l},0),\omega_{\xi})$-module. ∎

##  4 $A(V)$-theory for
$(L_{\widehat{\mathfrak{g}}}(\mathcal{l},0),\omega_{\xi})$

In this section, we first recall the $A(V)$-theory for $\mathbb{Q}$-graded
vertex operator superalgebras. Then we determine the Zhu’s algebras
$A(L_{\widehat{\mathfrak{g}}}(\mathcal{l},0))$ and their bimodules
$A(L(\mathcal{l},\mathcal{j}))$ for $\mathbb{Q}$-graded vertex operator
superalgebras $(L_{\widehat{\mathfrak{g}}}(\mathcal{l},0),\omega_{\xi})$.
Furthermore, we apply the Zhu’s algebras and their bimodules associated to
$(L_{\widehat{\mathfrak{g}}}(\mathcal{l},0),\omega_{\xi})$ to calculate the
fusion rules among the irreducible ordinary modules of
$(L_{\widehat{\mathfrak{g}}}(\mathcal{l},0),\omega_{\xi})$.

###  4.1 $A(V)$-theory for $\mathbb{Q}$-graded vertex operator superalgebra

Let $V$ be a $\mathbb{Q}$-graded vertex operator superalgebra. Define a
function $\varepsilon$ for all homogeneous elements of $V$ as follows:

$\varepsilon(a)=\begin{cases}1,&\mbox{wt}~{}a\in\mathbb{Z},\\\
0,&\mbox{wt}~{}a\notin\mathbb{Z}.\\\ \end{cases}$

For any homogeneous element $a\in V$, we define

$a\ast b=\varepsilon(a)\mbox{Res}_{z}\frac{(1+z)^{[{\rm wt}~{}a]}}{z}Y(a,z)b$
(4.1)

for any $b\in V$, where $[\cdot]$ denotes the greatest-integer function. Then
we can extend $\ast$ on $V$. Let $O(V)$ be the subspace of $V$ linearly
spanned by

$\mbox{Res}_{z}\frac{(1+z)^{[{\rm wt}~{}a]}}{z^{1+\varepsilon(a)}}Y(a,z)b$
(4.2)

for any homogeneous element $a\in V$ and for any $b\in V$. We have

$\mbox{Res}_{z}\frac{(1+z)^{[{\rm
wt}~{}a]+m}}{z^{1+\varepsilon(a)+n}}Y(a,z)b\in O(V)$ (4.3)

for $n\geq m\geq 0$ (cf. [12]). Let $M$ be any weak $V$-module, we define

$\Omega(M)=\\{u\in M\mid a_{m}u=0~{}~{}\mbox{for}~{}~{}a\in
V,m>\mbox{wt}~{}a-1\\},$

and $o$ to be the linear map from $V$ to $\mbox{End}~{}\Omega(M)$ such that
$o(a)=\varepsilon(a)a_{[{\rm wt}~{}a]-1}$ for any homogeneous element $a\in
V$. The following theorem were established in [9].

######  Theorem 4.1.

(a) The subspace $O(V)$ is a two-sided ideal of $V$ with respect to the
product $\ast$ and $A(V)=V/O(V)$ is an associative superalgebra with identity
$\textbf{1}+O(V)$. Moreover, $\omega+O(V)$ lies in the center of $A(V)$.
(b) For any weak $V$-module $M$, $\Omega(M)$ is an $A(V)$-module with $a$ acts
as $o(a)$ for $a\in V$.
(c) There is an induction functor $L$ from the category of $A(V)$-modules to
the category of $\mathbb{Q}_{+}$-graded weak $V$-modules. Moreover,
$\Omega(L(U))=U$ for any $A(V)$-module $U$.
(d) $\Omega$ and $L$ are inverse bijections between the sets of irreducible
modules in each category.

As a consequence, we have the following proposition.

######  Proposition 4.2.

If $V$ is rational, then $A(V)$ is a semisimple associative superalgebra.

###### Proof.

Let $U$ be a $A(V)$-module, from Theorem 4.1 $L(U)$ is a
$\mathbb{Q}_{+}$-graded weak $V$-modules with $\Omega(L(U))=L(U)(0)=U$. Since
$V$ is rational, we have $L(U)$ is complete reducible. Then $U$ is complete
reducible as $A(V)$-module. Hence $A(V)$ is semisimple. ∎

For any weak $V$-module $M$, let $O(M)$ be the subspace of $M$ linearly
spanned by

$\mbox{Res}_{z}\frac{(1+z)^{[{\rm wt}~{}a]}}{z^{1+\varepsilon(a)}}Y_{M}(a,z)u$
(4.4)

for any homogeneous element $a\in V$ and for any $u\in M$. Similar to the
Theorem 2.11 in [21] (see also [12, 19]), we have the following theorem.

######  Theorem 4.3.

(a) The quotient space $A(M)=M/O(M)$ is an $A(V)$-bimodule with the following
left and right actions:

$\displaystyle a\ast u=\varepsilon(a){\rm Res}_{z}\frac{(1+z)^{[{\rm
wt}~{}a]}}{z}Y_{M}(a,z)u,$ (4.5) $\displaystyle u\ast
a=(-1)^{|a||u|}\varepsilon(a){\rm Res}_{z}\frac{(1+z)^{[{\rm
wt}~{}a]-1}}{z}Y_{M}(a,z)u$

for any homogeneous element $a\in V,u\in M$.
(b) Let $W_{1},W_{2},W_{3}$ be irreducible $V$-modules and suppose $V$ is
rational. Then there is a linear isomorphism from the space ${\rm
Hom}_{A(V)}(A(W_{1})\otimes_{A(V)}W_{2}(0),W_{3}(0))$ to the space of
intertwining operators of type $\begin{pmatrix}W_{3}\\\
W_{1}~{}~{}W_{2}\end{pmatrix}$.

We also obtain the following lemma which is similar to the Proposition 1.5.4
in [12].

######  Lemma 4.4.

Let $V$ be a $\mathbb{Q}$-graded vertex operator superalgebra and let $M$ be a
weak $V$-module with a weak submodule $W$, $I$ be an ideal of $V$.
(a) As $A(V)$-bimodule $A(M/W)\cong M/(O(M)+W)$.
(b) $(I+O(V))/O(V)$ is a two sided ideal of $A(V)$ and $A(V/I)\cong
A(V)/((I+O(V))/O(V))$.
(c) If $I\cdot M\subset W$, then $I\ast A(M)\subset(W+O(M))/O(M)$, $A(M)\ast
I\subset(W+O(M))/O(M)$ and $A(M/W)\cong A(M)/((W+O(M))/O(M))$ as
$A(V/I)$-bimodules.

###  4.2 Zhu’s algebras $A(L_{\widehat{\mathfrak{g}}}(\mathcal{l},0))$ and
their bimodules $A(L(\mathcal{l},\mathcal{j}))$

Take $0<\xi<1$, we have
$(L_{\widehat{\mathfrak{g}}}(\mathcal{l},0),\omega_{\xi})$ and
$(V(\mathcal{l},\mathbb{C}),\omega_{\xi})$ are $\mathbb{Q}$-graded vertex
operator superalgebras. First we determine the Zhu’s algebra
$A(V(\mathcal{l},\mathbb{C}))$ and their bimodules
$A(M(\mathcal{l},\mathcal{j}))$ for
$(V(\mathcal{l},\mathbb{C}),\omega_{\xi})$.

######  Proposition 4.5.

Let $M$ be any weak $(V(\mathcal{l},\mathbb{C}),\omega_{\xi})$-module. Then
$O(M)=CM$, where

$C=\mathbb{C}[t^{-1}](t^{-1}+1)\otimes f+\mathbb{C}[t^{-1}](t^{-1}+1)\otimes
y+\mathbb{C}[t^{-1}]t^{-1}\otimes e+\mathbb{C}[t^{-1}]t^{-1}\otimes
x+\mathbb{C}[t^{-1}](t^{-2}+t^{-1})\otimes h.$

###### Proof.

Since

$\displaystyle\mbox{wt}~{}h(-1)\textbf{1}=1,\mbox{wt}~{}e(-1)\textbf{1}=1-\xi,\mbox{wt}~{}f(-1)\textbf{1}=1+\xi,$
$\displaystyle\mbox{wt}~{}x(-1)\textbf{1}=1-\frac{1}{2}\xi,\mbox{wt}~{}y(-1)\textbf{1}=1+\frac{1}{2}\xi,$

we have

$\displaystyle\mbox{Res}_{z}\frac{(1+z)^{[{\rm
wt}~{}f]}}{z^{m}}Y_{M}(f,z)u=(f(-m)+f(1-m))u;$ (4.6)
$\displaystyle\mbox{Res}_{z}\frac{(1+z)^{[{\rm
wt}~{}y]}}{z^{m}}Y_{M}(y,z)u=(y(-m)+y(1-m))u;$
$\displaystyle\mbox{Res}_{z}\frac{(1+z)^{[{\rm
wt}~{}e]}}{z^{m}}Y_{M}(e,z)u=e(-m)u;$
$\displaystyle\mbox{Res}_{z}\frac{(1+z)^{[{\rm
wt}~{}x]}}{z^{m}}Y_{M}(x,z)u=x(-m)u;$
$\displaystyle\mbox{Res}_{z}\frac{(1+z)^{[{\rm
wt}~{}h]}}{z^{m+1}}Y_{M}(h,z)u=(h(-m-1)+h(-m))u$

for any positive integer $m$ and for $u\in M$. It is clear that all those
elements in (4.6) are in $O(M)$. Let $W$ be the subspace linearly spanned by
the elements in (4.6), then $W=CM$.

Let $L$ be the linear span of homogeneous elements $a$ of
$(V(\mathcal{l},\mathbb{C}),\omega_{\xi})$ such that for any positive integer
$n$,

$\mbox{Res}_{z}\frac{(1+z)^{[{\rm
wt}~{}a]}}{z^{n+\varepsilon(a)}}Y_{M}(a,z)M\subseteq W.$

Let $a$ be any homogeneous element of $L$, similar to the proof of the
Proposition 4.1 in [7], we have $h(-m)a,e(-m)a,f(-m)a,x(-m)a,y(-m)a\in L$ for
any positive integer $m$. Since $\textbf{1}\in L$ and
$V(\mathcal{l},\mathbb{C})=U(t^{-1}\mathbb{C}[t^{-1}]\otimes\mathfrak{g})\textbf{1}$,
we have $L=V(\mathcal{l},\mathbb{C})$. Therefore $O(M)=W=CM$. ∎

######  Proposition 4.6.

The associative superalgebra $A(V(\mathcal{l},\mathbb{C}))$ for
$(V(\mathcal{l},\mathbb{C}),\omega_{\xi})$ is isomorphic to the polynomial
algebra $\mathbb{C}[t]$.

###### Proof.

Define a linear map

$\displaystyle\psi:$ $\displaystyle\mathbb{C}[t]\rightarrow
A(V(\mathcal{l},\mathbb{C}))$ $\displaystyle g(t)\mapsto
g(h(-1))\textbf{1}+O(V(\mathcal{l},\mathbb{C}))$

for any $g(t)\in\mathbb{C}[t]$. Since $[h(-1),h(0)]=0$ and $h(0)\textbf{1}=0$,
we have $g(h(-1))\textbf{1}=g(h(-1)+h(0))\textbf{1}$ for any
$g(t)\in\mathbb{C}[t]$. Since $h(-1)\textbf{1}\ast u=(h(-1)+h(0))u$ for any
$u\in V(\mathcal{l},\mathbb{C})$, we have

$\displaystyle\psi(f(t)g(t))$
$\displaystyle=f(h(-1))g(h(-1))\textbf{1}+O(V(\mathcal{l},\mathbb{C}))$
$\displaystyle=f(h(-1)+h(0))g(h(-1))\textbf{1}+O(V(\mathcal{l},\mathbb{C}))$
$\displaystyle=f(h(-1))\textbf{1}\ast
g(h(-1))\textbf{1}+O(V(\mathcal{l},\mathbb{C}))=\psi(f(t))\ast\psi(g(t)),$

thus $\psi$ is an superalgebra homomorphism. By
$N_{-}=C\oplus\mathbb{C}f(0)\oplus\mathbb{C}y(0)\oplus\mathbb{C}h(-1)$,

$U(N_{-})=U(C)U(\mathbb{C}h(-1))U(\mathbb{C}f(0))U(\mathbb{C}y(0)).$

Then by Proposition 4.5, we have

$O(V(\mathcal{l},\mathbb{C}))=CV(\mathcal{l},\mathbb{C})=CU(N_{-})\textbf{1}\cong
CU(C)U(\mathbb{C}h(-1)).$

Hence
$A(V(\mathcal{l},\mathbb{C}))\cong(U(C)U(\mathbb{C}h(-1)))/(CU(C)U(\mathbb{C}h(-1)))\cong
U(\mathbb{C}h(-1))\cong\mathbb{C}[t]$. ∎

######  Proposition 4.7.

The $A(V(\mathcal{l},\mathbb{C}))$-bimodule $A(M(\mathcal{l},\mathcal{j}))$ is
isomorphic to $\mathbb{C}[t_{1},t_{2}]$ with the biaction as follows:

$t\ast f(t_{1},t_{2})=(t_{1}+\mathcal{j}-t_{2}\frac{\partial}{\partial
t_{2}})f(t_{1},t_{2}),f(t_{1},t_{2})\ast t=t_{1}f(t_{1},t_{2})$ (4.7)

for any $f(t_{1},t_{2})\in\mathbb{C}[t_{1},t_{2}]$.

###### Proof.

Let $v$ be a highest weight vector of $M(\mathcal{l},\mathcal{j})$. Then by
Proposition 4.5, we have

$O(M(\mathcal{l},\mathcal{j}))=CM(\mathcal{l},\mathcal{j})=CU(N_{-})v\cong
CU(C)U(\mathbb{C}h(-1))U(\mathbb{C}f(0))U(\mathbb{C}y(0)).$

Then

$A(M(\mathcal{l},\mathcal{j}))=\bigoplus_{m_{1},m_{2}\in\mathbb{Z}_{+}}\mathbb{C}(h(-1)^{m_{1}}y(0)^{m_{2}}+O(M(\mathcal{l},\mathcal{j}))).$

From Theorem 4.3, we have

$\displaystyle\begin{aligned}
h(-1)\textbf{1}\ast(h(-1)^{m_{1}}y(0)^{m_{2}}v)&=(h(-1)+h(0))h(-1)^{m_{1}}y(0)^{m_{2}}v\\\
&=(h(-1)+\mathcal{j}-m_{2})h(-1)^{m_{1}}y(0)^{m_{2}}v,\\\ \end{aligned}$
$\displaystyle(h(-1)^{m_{1}}y(0)^{m_{2}}v)\ast
h(-1)\textbf{1}=h(-1)h(-1)^{m_{1}}y(0)^{m_{2}}v,$

for any $m_{1},m_{2}\in\mathbb{Z}_{+}$. Define a linear map

$\displaystyle\varphi:$ $\displaystyle\mathbb{C}[t_{1},t_{2}]\rightarrow
A(M(\mathcal{l},\mathcal{j}))$ $\displaystyle
t_{1}^{m_{1}}t_{2}^{m_{2}}\mapsto
h(-1)^{m_{1}}y(0)^{m_{2}}+O(M(\mathcal{l},\mathcal{j}))~{}~{}~{}~{}\forall
m_{1},m_{2}\in\mathbb{Z}_{+},$

it is clear $\varphi$ is a bimodule isomorphic. ∎

Now we determine the Zhu’s algebras
$A(L_{\widehat{\mathfrak{g}}}(\mathcal{l},0))$ and their bimodules
$A(L(\mathcal{l},\mathcal{j}))$ for
$(L_{\widehat{\mathfrak{g}}}(\mathcal{l},0),\omega_{\xi})$.

######  Theorem 4.8.

The associative superalgebra $A(L_{\widehat{\mathfrak{g}}}(\mathcal{l},0))$
for $(L_{\widehat{\mathfrak{g}}}(\mathcal{l},0),\omega_{\xi})$ is semisimple
and isomorphic to the quotient algebra $\mathbb{C}[t]/\langle f(t)\rangle$ of
the polynomial algebra $\mathbb{C}[t]$, where

$f(t)=\prod_{i+j\in 2\mathbb{Z}+1,0\leq j\leq q-1,1\leq i\leq
p-1}(t-\frac{i-1}{2}+jl).$ (4.8)

###### Proof.

Since $(L_{\widehat{\mathfrak{g}}}(\mathcal{l},0),\omega_{\xi})$ is a quotient
vertex operator superalgebra of $(V(\mathcal{l},\mathbb{C}),\omega_{\xi})$,
then by Lemma 4.4, $A(L_{\widehat{\mathfrak{g}}}(\mathcal{l},0))$ is quotient
superalgebra of $A(V(\mathcal{l},\mathbb{C}))\cong\mathbb{C}[t]$. Set
$A(L_{\widehat{\mathfrak{g}}}(\mathcal{l},0))=A(V(\mathcal{l},\mathbb{C}))/I$.
By the proof of Proposition 4.6, any irreducible
$A(V(\mathcal{l},\mathbb{C}))$-module $U$ is also an irreducible
$\mathbb{C}h$-module. Then for any irreducible
$A(V(\mathcal{l},\mathbb{C}))$-module $U$, by the definition of
$L(\mathcal{l},U)$, $L(\mathcal{l},U)$ is the corresponding irreducible
$\mathbb{Q}_{+}$-graded weak $V(\mathcal{l},\mathbb{C})$-module in Theorem
4.1. From Theorem 3.3, $L(\mathcal{l},U)$ is an irreducible
$\mathbb{Q}_{+}$-graded weak
$L_{\widehat{\mathfrak{g}}}(\mathcal{l},0)$-module if and only if $f(h)U=0$
and $U$ is an irreducible $\mathbb{C}h$-module. $L(\mathcal{l},U)$ is an
irreducible $\mathbb{Q}_{+}$-graded weak
$L_{\widehat{\mathfrak{g}}}(\mathcal{l},0)$-module is equivalent to $U$ is an
irreducible $A(L_{\widehat{\mathfrak{g}}}(\mathcal{l},0))$-module, i.e.,
$I.U=0$. Then we have $I=\langle f(t)\rangle$. Since
$(L_{\widehat{\mathfrak{g}}}(\mathcal{l},0),\omega_{\xi})$ is rational, by
Proposition 4.2, $A(L_{\widehat{\mathfrak{g}}}(\mathcal{l},0))$ is semisimple.
∎

Let $B_{0}=\mathbb{C}(t^{-1}+1)\otimes f+\mathbb{C}(t^{-1}+1)\otimes
y+(t^{-2}+t^{-1})\mathbb{C}[x^{-1}]\otimes\mathfrak{g}$. Then $B_{0}$ is an
ideal of $N_{-}$, we denote
$L_{0}=N_{-}/B_{0}=\mathbb{C}T_{e}+\mathbb{C}T_{f}+\mathbb{C}T_{h}+\mathbb{C}T_{x}+\mathbb{C}T_{y},$
where
$T_{e}=e(-1)+B_{0},T_{f}=f(0)+B_{0},T_{h}=h(-1)+B_{0},T_{x}=x(-1)+B_{0},T_{y}=y(0)+B_{0}$
satisfies the following anticommutation relations:

$\displaystyle[T_{e},T_{f}]=T_{h},~{}[T_{h},T_{e}]=-2T_{e},~{}[T_{h},T_{f}]=2T_{f},$
$\displaystyle[T_{h},T_{x}]=-T_{x},~{}[T_{e},T_{x}]=0,~{}[T_{f},T_{x}]=T_{y},$
$\displaystyle[T_{h},T_{y}]=T_{y},~{}[T_{e},T_{y}]=-T_{x},~{}[T_{f},T_{y}]=0,$
$\displaystyle\\{T_{x},T_{x}\\}=-2T_{e},\\{T_{x},T_{y}\\}=T_{h},\\{T_{y},T_{y}\\}=-2T_{f}.$

Let $\pi_{1}$ be the natural quotient map from $U(N_{-})$ onto $U(L_{0})$. Set
$P_{1}(\alpha)=T_{x}T_{y}-\alpha$ and $Q_{1}(\alpha)=T_{y}T_{x}+\alpha$ for
$\alpha\in\mathbb{C}$. We have the following commutation relations in
$U(L_{0})$.

######  Proposition 4.9.

For any $\alpha,\beta,\gamma\in\mathbb{C}$, we have

$\displaystyle[P_{1}(\alpha),P_{1}(\beta)]=0,[P_{1}(\alpha),Q_{1}(\beta)]=0,[Q_{1}(\alpha),Q_{1}(\beta)]=0,$
$\displaystyle{T_{e}}^{\gamma}P_{1}(\alpha)=P_{1}(\alpha-\gamma){T_{e}}^{\gamma},{T_{e}}^{\gamma}Q_{1}(\alpha)=Q_{1}(\alpha+\gamma){T_{e}}^{\gamma},$
$\displaystyle{T_{f}}^{\gamma}P_{1}(\alpha)=P_{1}(\alpha+\gamma){T_{f}}^{\gamma},{T_{f}}^{\gamma}Q_{1}(\alpha)=Q_{1}(\alpha-\gamma){T_{f}}^{\gamma},$
$\displaystyle{T_{x}}P_{1}(\alpha)=Q_{1}(1-\alpha){T_{x}},{T_{x}}Q_{1}(\alpha)=P_{1}(-\alpha){T_{x}},$
$\displaystyle
T_{y}P_{1}(\alpha)=Q_{1}(-\alpha)T_{y},T_{y}Q_{1}(\alpha)=P_{1}(1-\alpha)T_{y},$
$\displaystyle{T_{e}}^{\gamma}T_{y}=-Q_{1}(\gamma)T_{x}{T_{e}}^{\gamma-1},T_{x}{T_{e}}^{\gamma}T_{y}=P_{1}(-\gamma){T_{e}}^{\gamma},$
$\displaystyle{T_{f}}^{\gamma}T_{x}=-P_{1}(\gamma)T_{y}{T_{f}}^{\gamma-1},T_{y}{T_{f}}^{\gamma}T_{x}=Q_{1}(-\gamma){T_{f}}^{\gamma}.$

Similar to the Lemma 4.2 in [14], we have the following proposition by
induction.

######  Proposition 4.10.

The following projection formulas hold:

$\displaystyle\pi_{1}(F_{1}(m,s))=\theta_{(m,s)}\prod_{j=1}^{s}\prod_{i=1}^{m}\\{\prod_{i+j\in
2\mathbb{Z}}P_{1}(\frac{i}{2}+jl)\prod_{i+j\in
2\mathbb{Z}+1}Q_{1}(-\frac{i-1}{2}-jl)\\}{T_{y}}^{m},$
$\displaystyle\pi_{2}(F_{2}(m,s))=\theta_{(p-m,q-s)}\prod_{j=1}^{q-s-1}\prod_{i=1}^{p-m}\\{\prod_{i+j\in
2\mathbb{Z}}Q_{1}(\frac{i}{2}+jl)\prod_{i+j\in
2\mathbb{Z}+1}P_{1}(-\frac{i-1}{2}-jl)\\}{T_{x}}^{p-m},$

where $\theta_{(m,s)}=(-1)^{\frac{m(1+(-1)^{m})+s(1+(-1)^{s})}{4}}$.

######  Theorem 4.11.

Let $\mathcal{j}=\frac{m-1}{2}-ls$ be an admissible weight. Then the
$A(L_{\widehat{\mathfrak{g}}}(\mathcal{l},0))$-bimodule
$A(L(\mathcal{l},\mathcal{j}))$ is isomorphic to the quotient space of
$\mathbb{C}[t_{1},t_{2}]$ modulo the subspace

$\mathbb{C}[t_{1},t_{2}]t_{2}^{m}+\mathbb{C}[t_{1}]f_{\mathcal{j},0}(t_{1},t_{2})+\mathbb{C}[t_{1}]f_{\mathcal{j},1}(t_{1},t_{2})+\cdots+\mathbb{C}[t_{1}]f_{\mathcal{j},m-1}(t_{1},t_{2})$
(4.9)

where

$f_{\mathcal{j},n}(t_{1},t_{2})=(\prod_{i+n+j\in 2\mathbb{Z},0\leq j\leq
q-s-1,0\leq i\leq p-m-1}(t_{1}-\frac{i+n}{2}+jl))t_{2}^{n}.$

The left and right actions of $A(L_{\widehat{\mathfrak{g}}}(\mathcal{l},0))$
on $A(L(\mathcal{l},\mathcal{j}))$ are given by Proposition 4.7.

###### Proof.

Note that $C=B_{0}\oplus\mathbb{C}e(-1)\oplus\mathbb{C}x(-1)$. Then from
Proposition 4.5, we have

$O(M(\mathcal{l},\mathcal{j}))=CM(\mathcal{l},\mathcal{j})\cong
B_{0}U(N_{-})+e(-1)U(N_{-})+x(-1)U(N_{-}).$

Since $B_{0}$ is an ideal of $N_{-}$, $U(N_{-})B_{0}=B_{0}U(N_{-})$ is an
ideal of $U(N_{-})$. From Proposition 2.7, the maximal proper submodule of
$M(\mathcal{l},\mathcal{j})$ is generated by $v_{\mathcal{j},1}$ and
$v_{\mathcal{j},2}$. Then by Lemma 4.4, we have

$\displaystyle A(L(\mathcal{l},\mathcal{j}))$ $\displaystyle\cong
M(\mathcal{l},\mathcal{j})/(CM(\mathcal{l},\mathcal{j})+U(N_{-})v_{\mathcal{j},1}+U(N_{-})v_{\mathcal{j},2})$
$\displaystyle\begin{aligned}
\cong&U(N_{-})/(B_{0}U(N_{-})+e(-1)U(N_{-})+x(-1)U(N_{-})+\\\
&U(N_{-})F_{1}(m,s)+U(N_{-})F_{2}(m,s))\end{aligned}$

as $A(L(\mathcal{l},0))$-bimodules. Note that $U(N_{-})/B_{0}U(N_{-})\cong
U(L_{0})$. Thus

$A(L(\mathcal{l},\mathcal{j}))\cong
U(L_{0})/(U(L_{0})\pi_{1}(F_{1}(m,s))+U(L_{0})\pi_{1}(F_{2}(m,s))+T_{x}U(L_{0})).$

For any $a,b,d\in\mathbb{Z}_{+}$, by Proposition 4.9 and Proposition 4.10, we
have

$\displaystyle T_{x}^{a}T_{h}^{b}T_{y}^{d}\pi_{1}(F_{1}(m,s))$
$\displaystyle=\theta_{(m,s)}T_{x}^{a}T_{h}^{b}T_{y}^{d}\prod_{j=1}^{s}\prod_{i=1}^{m}\\{\prod_{i+j\in
2\mathbb{Z}}P_{1}(\frac{i}{2}+jl)\prod_{i+j\in
2\mathbb{Z}+1}Q_{1}(-\frac{i-1}{2}-jl)\\}{T_{y}}^{m}$
$\displaystyle=\begin{cases}\theta_{(m,s)}T_{x}^{a}\prod_{j=1}^{s}\prod_{i=1}^{m}\\{\prod_{i+j\in
2\mathbb{Z}}Q_{1}(\frac{1-i-d}{2}-jl)\prod_{i+j\in
2\mathbb{Z}+1}P_{1}(\frac{i+d}{2}+jl)\\}T_{h}^{b}{T_{y}}^{d+m},~{}d\notin
2\mathbb{Z}\\\
\theta_{(m,s)}T_{x}^{a}\prod_{j=1}^{s}\prod_{i=1}^{m}\\{\prod_{i+j\in
2\mathbb{Z}}P_{1}(\frac{i+d}{2}+jl)\prod_{i+j\in
2\mathbb{Z}+1}Q_{1}(\frac{1-i-d}{2}-jl)\\}T_{h}^{b}{T_{y}}^{d+m},~{}d\in
2\mathbb{Z}\end{cases}$
$\displaystyle=\theta_{(m,s)}T_{x}^{a}\prod_{j=1}^{s}\prod_{i=d+1}^{m+d}\\{\prod_{i+j\in
2\mathbb{Z}+1}Q_{1}(-\frac{i-1}{2}-jl)\prod_{i+j\in
2\mathbb{Z}}P_{1}(\frac{i}{2}+jl)\\}T_{h}^{b}{T_{y}}^{d+m}$
$\displaystyle\equiv\theta_{(m,s)}T_{x}^{a}\prod_{j=1}^{s}\prod_{i=d+1}^{m+d}\\{\prod_{i+j\in
2\mathbb{Z}+1}(T_{h}-\frac{i-1}{2}-jl)\prod_{i+j\in
2\mathbb{Z}}(-\frac{i}{2}-jl)\\}T_{h}^{b}{T_{y}}^{d+m}~{}~{}(\mbox{mod}~{}T_{x}U(L_{0})).$

Note that $-\frac{i}{2}-jl\neq 0$ for $i+j\in 2\mathbb{Z},1\leq j\leq
s,d+1\leq i\leq m+d$, we have

$\displaystyle U(L_{0})\pi_{1}(F_{1}(m,s))+T_{x}U(L_{0})$ $\displaystyle=$
$\displaystyle
T_{x}U(L_{0})+\sum_{d=0}^{\infty}\mathbb{C}[T_{h}](\prod_{i+d+j\in
2\mathbb{Z}+1,1\leq j\leq s,1\leq i\leq
m}(T_{h}-\frac{i+d-1}{2}-jl)){T_{y}}^{d+m}.$

Similarly, let $a,b,d\in\mathbb{Z}_{+}$, if $d<p-m$, we have

$\displaystyle T_{x}^{a}T_{h}^{b}T_{y}^{d}\pi_{1}(F_{2}(m,s))$
$\displaystyle=\theta_{(p-m,q-s)}T_{x}^{a}T_{h}^{b}T_{y}^{d}\prod_{j=1}^{q-s-1}\prod_{i=1}^{p-m}\\{\prod_{i+j\in
2\mathbb{Z}}Q_{1}(\frac{i}{2}+jl)\prod_{i+j\in
2\mathbb{Z}+1}P_{1}(-\frac{i-1}{2}-jl)\\}{T_{x}}^{p-m}$
$\displaystyle=\theta_{(p-m,q-s)}T_{x}^{a}T_{h}^{b}T_{y}^{d}{T_{x}}^{p-m}\prod_{j=1}^{q-s-1}\prod_{i=0}^{p-m-1}\\{\prod_{i+j\in
2\mathbb{Z}}Q_{1}(-\frac{i}{2}+jl)\prod_{i+j\in
2\mathbb{Z}+1}P_{1}(\frac{i+1}{2}-jl)\\}$ $\displaystyle\begin{aligned}
=&\theta_{(p-m,q-s)}T_{x}^{a}T_{h}^{b}\prod_{i=1}^{d}\\{\prod_{i\in
2\mathbb{Z}}P_{1}(\frac{i}{2})\prod_{i\in
2\mathbb{Z}+1}Q_{1}(-\frac{i-1}{2})\\}{T_{x}}^{p-m-d}\prod_{j=1}^{q-s-1}\prod_{i=0}^{p-m-1}\\\
&\\{\prod_{i+j\in 2\mathbb{Z}}Q_{1}(-\frac{i}{2}+jl)\prod_{i+j\in
2\mathbb{Z}+1}P_{1}(\frac{i+1}{2}-jl)\\}\end{aligned}$
$\displaystyle\begin{aligned}
=&\theta_{(p-m,q-s)}T_{x}^{a}T_{h}^{b}{T_{x}}^{p-m-d}\prod_{i=1+p-m-d}^{p-m}\\{\prod_{i\in
2\mathbb{Z}}P_{1}(\frac{i}{2})\prod_{i\in
2\mathbb{Z}+1}Q_{1}(-\frac{i-1}{2})\\}\\\
&\prod_{j=1}^{q-s-1}\prod_{i=0}^{p-m-1}\\{\prod_{i+j\in
2\mathbb{Z}}Q_{1}(-\frac{i}{2}+jl)\prod_{i+j\in
2\mathbb{Z}+1}P_{1}(\frac{i+1}{2}-jl)\\}\end{aligned}$
$\displaystyle\begin{aligned}
=&\theta_{(p-m,q-s)}T_{x}^{a+p-m-d}(T_{h}-p+m+d)^{b}\prod_{i=1+p-m-d}^{p-m}\\{\prod_{i\in
2\mathbb{Z}}P_{1}(\frac{i}{2})\prod_{i\in
2\mathbb{Z}+1}Q_{1}(-\frac{i-1}{2})\\}\\\
&\prod_{j=1}^{q-s-1}\prod_{i=0}^{p-m-1}\\{\prod_{i+j\in
2\mathbb{Z}}Q_{1}(-\frac{i}{2}+jl)\prod_{i+j\in
2\mathbb{Z}+1}P_{1}(\frac{i+1}{2}-jl)\\}\end{aligned}$ $\displaystyle\equiv
0~{}~{}(\mbox{mod}~{}T_{x}U(L_{0})).$

If $d=n+p-m$ for some $n\in\mathbb{Z}_{+}$, we have

$\displaystyle T_{x}^{a}T_{h}^{b}T_{y}^{d}\pi_{1}(F_{2}(m,s))$
$\displaystyle\begin{aligned}
=&\theta_{(p-m,q-s)}T_{x}^{a}T_{h}^{b}{T_{y}}^{n}\prod_{i=1}^{p-m}\\{\prod_{i\in
2\mathbb{Z}}P_{1}(\frac{i}{2})\prod_{i\in
2\mathbb{Z}+1}Q_{1}(-\frac{i-1}{2})\\}\prod_{j=1}^{q-s-1}\prod_{i=0}^{p-m-1}\\\
&\\{\prod_{i+j\in 2\mathbb{Z}}Q_{1}(-\frac{i}{2}+jl)\prod_{i+j\in
2\mathbb{Z}+1}P_{1}(\frac{i+1}{2}-jl)\\}\end{aligned}$
$\displaystyle=\theta_{(p-m,q-s)}T_{x}^{a}T_{h}^{b}{T_{y}}^{n}\prod_{j=0}^{q-s-1}\prod_{i=0}^{p-m-1}\\{\prod_{i+j\in
2\mathbb{Z}}Q_{1}(-\frac{i}{2}+jl)\prod_{i+j\in
2\mathbb{Z}+1}P_{1}(\frac{i+1}{2}-jl)\\}$
$\displaystyle=\theta_{(p-m,q-s)}T_{x}^{a}\prod_{j=0}^{q-s-1}\prod_{i=n}^{d-1}\\{\prod_{i+j\in
2\mathbb{Z}}Q_{1}(-\frac{i}{2}+jl)\prod_{i+j\in
2\mathbb{Z}+1}P_{1}(\frac{i+1}{2}-jl)\\}T_{h}^{b}{T_{y}}^{n}$
$\displaystyle\equiv\theta_{(p-m,q-s)}T_{x}^{a}\prod_{j=0}^{q-s-1}\prod_{i=n}^{d-1}\\{\prod_{i+j\in
2\mathbb{Z}}(T_{h}-\frac{i}{2}+jl)\prod_{i+j\in
2\mathbb{Z}+1}(-\frac{i+1}{2}+jl)\\}T_{h}^{b}{T_{y}}^{n}~{}~{}(\mbox{mod}~{}T_{x}U(L_{0})).$

Note that $-\frac{i+1}{2}+jl\neq 0$ for $i+j\in 2\mathbb{Z}+1,0\leq j\leq
q-s-1,n\leq i\leq d-1$, we have

$\displaystyle U(L_{0})\pi_{1}(F_{2}(m,s))+T_{x}U(L_{0})$ $\displaystyle=$
$\displaystyle
T_{x}U(L_{0})+\sum_{n=0}^{\infty}\mathbb{C}[T_{h}](\prod_{i+n+j\in
2\mathbb{Z},0\leq j\leq q-s-1,0\leq i\leq
p-m-1}(T_{h}-\frac{i+n}{2}+jl)){T_{y}}^{n}.$

Thus

$\displaystyle
U(L_{0})\pi_{1}(F_{1}(m,s))+U(L_{0})\pi_{1}(F_{2}(m,s))+T_{x}U(L_{0})$
$\displaystyle\subset$ $\displaystyle
T_{x}U(L_{0})+U(L_{0})T_{y}^{m}+\sum_{n=0}^{m-1}\mathbb{C}[T_{h}](\prod_{i+n+j\in
2\mathbb{Z},0\leq j\leq q-s-1,0\leq i\leq
p-m-1}(T_{h}-\frac{i+n}{2}+jl)){T_{y}}^{n}.$

On the other hand, since
$\frac{i+d-1}{2}+jl\neq\frac{i^{\prime}+n}{2}-j^{\prime}l$ for any $i+j\in
2\mathbb{Z}+1,1\leq j\leq s,1\leq i\leq m,d\in\mathbb{Z}_{+}$ and
$i^{\prime}+j^{\prime}\in 2\mathbb{Z},0\leq j^{\prime}\leq q-s-1,0\leq
i^{\prime}\leq p-m-1,n\in\mathbb{Z}_{+}$, $\prod_{i+d+j\in 2\mathbb{Z}+1,1\leq
j\leq s,1\leq i\leq m}(t-\frac{i+d-1}{2}-jl)$ and $\prod_{i+n+j\in
2\mathbb{Z},0\leq j\leq q-s-1,0\leq i\leq p-m-1}(t-\frac{i+n}{2}+jl)$ are
relatively prime. Then we have

$\mathbb{C}[T_{h}]T_{y}^{m+i}\subseteq
U(L_{0})\pi_{1}(F_{1}(m,s))+U(L_{0})\pi_{1}(F_{2}(m,s))+T_{x}U(L_{0})$

for any $i\in\mathbb{Z}_{+}$. Thus

$\displaystyle
U(L_{0})\pi_{1}(F_{1}(m,s))+U(L_{0})\pi_{1}(F_{2}(m,s))+T_{x}U(L_{0})$
$\displaystyle\supset$ $\displaystyle
T_{x}U(L_{0})+U(L_{0})T_{y}^{m}+\sum_{n=0}^{m-1}\mathbb{C}[T_{h}](\prod_{i+n+j\in
2\mathbb{Z},0\leq j\leq q-s-1,0\leq i\leq
p-m-1}(T_{h}-\frac{i+n}{2}+jl)){T_{y}}^{n}.$

Set $t_{1}=T_{h},t_{2}=T_{y}$. Then by Proposition 4.7 and Lemma 4.4, we
complete the proof. ∎

Now we apply the Zhu’s algebras $A(L_{\widehat{\mathfrak{g}}}(\mathcal{l},0))$
and their bimodules $A(L(\mathcal{l},\mathcal{j}))$ which we obtain in Theorem
4.8 and Theorem 4.11 to calculate the fusion rules among the irreducible
ordinary modules of $(L_{\widehat{\mathfrak{g}}}(\mathcal{l},0),\omega_{\xi})$
by Theorem 4.3.

######  Proposition 4.12.

For admissible weights $\mathcal{j}_{i}=\frac{m_{i}-1}{2}-ls_{i}~{}(i=1,2)$,
the fusion rules are given as follows:

$L(\mathcal{l},\mathcal{j}_{1})\boxtimes
L(\mathcal{l},\mathcal{j}_{2})=\sum_{n={\rm max}\\{0,m_{1}+m_{2}-p\\}}^{{\rm
min}\\{m_{1}-1,m_{2}-1\\}}L(\mathcal{l},\mathcal{j}_{1}+\mathcal{j}_{2}-n)$
(4.10)

if $0\leq s_{2}\leq q-s_{1}-1$, and $L(\mathcal{l},\mathcal{j}_{1})\boxtimes
L(\mathcal{l},\mathcal{j}_{2})=0$ otherwise.

###### Proof.

For any admissible weight $\mathcal{j}$, let $\mathbb{C}v_{\mathcal{j}}$ be
the one-dimensional module for the Lie algebra $\mathbb{C}h$ such that
$hv_{\mathcal{j}}=\mathcal{j}v_{\mathcal{j}}$. From Theorem 4.3, we need to
calculate the $A(L_{\widehat{\mathfrak{g}}}(\mathcal{l},0))$-module
$A(L(\mathcal{l},\mathcal{j}_{1}))\otimes_{A(L_{\widehat{\mathfrak{g}}}(\mathcal{l},0))}\mathbb{C}v_{\mathcal{j}_{2}}$.
From Theorem 4.11, we have

$A(L(\mathcal{l},\mathcal{j}_{1}))\otimes_{A(L_{\widehat{\mathfrak{g}}}(\mathcal{l},0))}\mathbb{C}v_{\mathcal{j}_{2}}\cong\mathbb{C}[t_{1},t_{2}]/J,$

where $J$ is the subspace of $\mathbb{C}[t_{1},t_{2}]$ spanned by

$\\{\mathbb{C}[t_{1},t_{2}](t_{1}-\mathcal{j}_{2}),\mathbb{C}[t_{1},t_{2}]t_{2}^{m_{1}}+f_{\mathcal{j}_{1},n}(\mathcal{j}_{2},1)\mathbb{C}[t_{1}]t_{2}^{n},n=0,1,\cdots,m_{1}-1\\}.$

If $\mathcal{j}_{2}$ does not satisfy the relation $0\leq s_{2}\leq
q-s_{1}-1$, then

$f_{\mathcal{j}_{1},n}(\mathcal{j}_{2},1)=(\prod_{i+n+j\in 2\mathbb{Z},0\leq
j\leq q-s_{1}-1,0\leq i\leq p-m_{1}-1}(\mathcal{j}_{2}-\frac{i+n}{2}+jl))\neq
0$

for $0\leq n\leq m_{1}-1$. Thus
$A(L(\mathcal{l},\mathcal{j}_{1}))\otimes_{A(L_{\widehat{\mathfrak{g}}}(\mathcal{l},0))}\mathbb{C}v_{\mathcal{j}_{2}}=0$,
so that all the corresponding fusion rules are zero.

Suppose $0\leq s_{2}\leq q-s_{1}-1$. Note that $\mathbb{C}[t_{1}]t_{2}^{n}=0$
in $\mathbb{C}[t_{1},t_{2}]/J$ if and only if
$f_{\mathcal{j}_{1},n}(\mathcal{j}_{2},1)\neq 0$. Since
$f_{\mathcal{j}_{1},n}(\mathcal{j}_{2},1)=(\prod_{i+n+j\in 2\mathbb{Z},0\leq
j\leq q-s_{1}-1,0\leq i\leq p-m_{1}-1}(\mathcal{j}_{2}-\frac{i+n}{2}+jl))=0$
if and only if $\mathcal{j}_{2}-\frac{i+n}{2}+jl=0$ for some $i+n+j\in
2\mathbb{Z},0\leq j\leq q-s_{1}-1,0\leq i\leq p-m_{1}-1$. This implies that
$1\leq i+n+1\leq p-1$, by Remark 2.6 we have $i+n+1=m_{2},j=s_{2}$, that is
$m_{1}+m_{2}-p\leq n\leq m_{2}-1$. Therefore

$\mbox{max}\\{0,m_{1}+m_{2}-p\\}\leq n\leq\mbox{min}\\{m_{1}-1,m_{2}-1\\}.$
(4.11)

Then $\mathbb{C}[t_{1}]t_{2}^{n}\neq 0$ in $\mathbb{C}[t_{1},t_{2}]/J$ if and
only if $n$ satisfies (4.11). Thus

$\mathbb{C}[t_{1},t_{2}]/J\cong\bigoplus_{n={\rm
max}\\{0,m_{1}+m_{2}-p\\}}^{{\rm
min}\\{m_{1}-1,m_{2}-1\\}}\mathbb{C}t_{2}^{n}.$

From Proposition 4.7, we have $t\ast
t_{2}^{n}=(\mathcal{j}_{1}+\mathcal{j}_{2}-n)t_{2}^{n}$, then we complete the
proof. ∎

If $\mathcal{l}\in\mathbb{Z}_{+}$, from Proposition 2.5,
$q=1,p=2\mathcal{l}+3$, then for any admissible weights
$\mathcal{j}_{i}=\frac{m_{i}-1}{2}-ls_{i}~{}(i=1,2)$, we have
$s_{i}=0,\mathcal{j}_{i}=\frac{m_{i}-1}{2}$, hence $0\leq s_{2}\leq q-s_{1}-1$
holds automatically. Since
$m_{1}+m_{2}-p=2\mathcal{j}_{1}+2\mathcal{j}_{2}-2\mathcal{l}-1$, then by
Proposition 4.12 we have the following corollary.

######  Corollary 4.13.

Suppose $\mathcal{l}\in\mathbb{Z}_{+}$. For any admissible weight
$\mathcal{j}_{1}$ and $\mathcal{j}_{2}$, we have

$L(\mathcal{l},\mathcal{j}_{1})\boxtimes
L(\mathcal{l},\mathcal{j}_{2})=\sum_{\mathcal{j}=|\mathcal{j}_{1}-\mathcal{j}_{2}|,\mathcal{j}+\mathcal{j}_{1}+\mathcal{j}_{2}\leq
2\mathcal{l}+1}^{\mathcal{j}_{1}+\mathcal{j}_{2}}L(\mathcal{l},\mathcal{j}).$
(4.12)

######  Remark 4.14.

(i) Since $L^{\prime}(-1)=L(-1)$ the fusion rules among the weak modules with
respect two different vertex operator superalgebra structure of
$L_{\widehat{\mathfrak{g}}}(\mathcal{l},0)$ are same. Thus the fusion rules
obtained in the Proposition 4.12 and Corollary 4.13 are also those with
respect to the old vertex operator superalgebra structure.
(ii) The fusion rules among the irreducible relaxed highest-weight modules
over $L_{\widehat{\mathfrak{g}}}(\mathcal{l},0)$ with admissible level have
been calculated in [5] by viewing $L_{\widehat{\mathfrak{g}}}(\mathcal{l},0)$
as extensions of the tensor product of certain simple Virasoro vertex operator
algebra and simple affine vertex operator algebra associated to $sl_{2}$.

## References

  * [1] T. Arakawa, Rationality of admissible affine vertex algebras in the category $\mathcal{O}$, Duke Math. J. 165 (2016), no. 1, 67-93.
  * [2] T. Arakawa, J. Van Ekeren, Modularity of relatively rational vertex algebras and fusion rules of principal affine $W$-algebras, Comm. Math. Phys. 370 (2019), no. 1, 205-247.
  * [3] D. Adamović, A. Milas, Vertex operator algebras associated to modular invariant representations for $A_{1}^{(1)}$, Math. Res. Lett. 2 (1995), no. 5, 563-575.
  * [4] T. Creutzig, N. Genra, A. Linshaw, Category $\mathcal{O}$ for vertex algebras of $\mathfrak{osp}_{1|2n}$, 2022, arXiv:math.RT/2203.08188.
  * [5] T. Creutzig, S. Kanade, T. Liu, D. Ridout, Cosets, characters and fusion for admissible-level $\mathfrak{osp}(1|2)$ minimal models, Nuclear Phys. B 938 (2019), 22-55.
  * [6] C. Dong, H. Li, G. Mason, Regularity of rational vertex operator algebras, Adv. Math. 132 (1997), no. 1, 148-166.
  * [7] C. Dong, H. Li, G. Mason, Vertex operator algebras associated to admissible representations of $\hat{sl}_{2}$, Comm. Math. Phys. 184 (1997), no. 1, 65-93.
  * [8] C. Dong, Z. Zhao, Modularity of trace functions in orbifold theory for $\mathbb{Z}$-graded vertex operator superalgebras, Moonshine: the first quarter century and beyond, 128-143, London Math. Soc. Lecture Note Ser., 372, Cambridge Univ. Press, Cambridge, 2010.
  * [9] A. De Sole, V. Kac, Finite vs affine $W$-algebras, Jpn. J. Math. 1 (2006), no. 1, 137-261.
  * [10] J. Van Ekeren, Modular invariance for twisted modules over a vertex operator superalgebra, Comm. Math. Phys. 322 (2013), no. 2, 333-371.
  * [11] I. Frenkel, Y. Huang, J. Lepowsky, On axiomatic approaches to vertex operator algebras and modules, Mem. Am. Math. Soc. 104 (1993).
  * [12] I. Frenkel, Y. Zhu, Vertex operator algebras associated to representations of affine and Virasoro algebras, Duke Math. J. 66 (1992), no. 1, 123-168.
  * [13] M. Gorelik, V. Serganova, Snowflake modules and Enright functor for Kac-Moody superalgebras, Algebra Number Theory 16 (2022), no. 4, 839-879.
  * [14] K. Iohara, Y. Koga, Fusion algebras for $N=1$ superconformal field theories through coinvariants 2: $\widehat{\mathfrak{osp}}(1|2)$-symmetry, J. reine angew. Math. 531(2001), 1-34.
  * [15] V. Kac, Lie superalgebras, Adv. Math. 26 (1977), 8-96.
  * [16] V. Kac, D. Kazhdan, Highest weight representations for affine Lie algebras, Adv. Math. 34 (1979), 97-108.
  * [17] V. Kac, M. Wakimoto, Modular invariant representations of infinite-dimensional Lie algebras and superalgebras, Proc. Nat. Acad. Sci. U.S.A. 85 (1988), no. 14, 4956-4960.
  * [18] V. Kac, M. Wakimoto, Classification of modular invariant representations of affine algebras, Infinite-dimensional Lie algebras and groups (Luminy-Marseille, 1988), 138-177, Adv. Ser. Math. Phys., 7, World Sci. Publ., Teaneck, NJ, 1989.
  * [19] V. Kac, W. Wang, Vertex operator superalgebras and their representations, Mathematical aspects of conformal and topological field theories and quantum groups (South Hadley, MA, 1992), 161-191, Contemp. Math., 175, Amer. Math. Soc., Providence, RI, 1994.
  * [20] H. Li, Local systems of vertex operators, vertex superalgebras and modules, J. Pure Appl. Algebra 109 (1996), no.2, 143-195.
  * [21] H. Li, Determining fusion rules by $A(V)$-modules and bimodules, J. Algebra 212 (1999), no. 2, 515-556.
  * [22] J. Lepowsky, H. Li, Introduction to vertex operator algebras and their representations, Progress in Math., Vol. 227, Birkhäuser, Boston, 2004.
  * [23] X. Lin, Rationality of affine vertex operator superalgebras with rational conformal weights, Comm. Math. Phys. 402 (2023), no. 3, 2765-2790.
  * [24] F. Malikov, B. Feigin, D. Fuks, Singular vectors in Verma modules over Kac-Moody algebras, J. Funct. Anal. Appl. 20 (1986), no. 2, 25-37.
  * [25] S. Wood, Admissible level $\mathfrak{osp}(1|2)$ minimal models and their relaxed highest weight modules, Transform. Groups 25 (2020), no. 3, 887-943.