# Representations of the Fermion-Virasoro algebras

Yaohui Xue, Kaiming Zhao

###### Abstract.

Let $\delta=0$ or $\frac{1}{2}$. In this paper, we introduce the Fermion
algebra $F(\delta)$ and the Fermion-Virasoro algebra $\mathcal{S}(\delta)$.
They are infinite-dimensional Lie superalgebras. All simple smooth
$F(\delta)$-modules, all simple weight $F(\delta)$-modules, all simple smooth
$\mathcal{S}(\delta)$-modules of nonzero level, and all simple Harish-Chandra
$\mathcal{S}(\delta)$-modules are determined. Surprisingly, we have found four
different $\mathcal{S}(\delta)$-module structures on free
${\mathbb{C}}[L_{0}]$-modules of rank $2$.

MSC2020: 17B10, 17B65, 17B66, 17B68, 17B81

Keywords: Fermion algebra, Fermion-Virasoro algebra, smooth module, Harish-
Chandra module

##  1\. Introduction

Throughout the paper we denote by
${\mathbb{Z}},{\mathbb{Z}}^{*},{\mathbb{Z}}_{+},{\mathbb{N}},{\mathbb{R}},{\mathbb{C}},$
and ${\mathbb{C}}^{*}$ the sets of integers, nonzero integers, non-negative
integers, positive integers, real numbers, complex numbers, and nonzero
complex numbers, respectively. All vector spaces and algebras are assumed to
be over ${\mathbb{C}}$. For a Lie superalgebra $\mathfrak{g}$, the universal
enveloping algebra of $\mathfrak{g}$ is denoted by $U(\mathfrak{g})$.

In this paper we will introduce and study the Lie superalgebras
$\mathcal{S}(0)$ and $\mathcal{S}(\frac{1}{2})$, which we consider to be the
super version of twisted Heisenberg-Virasoro algebra $\mathfrak{D}$ and are
called Fermion-Virasoro algebras. We first recall the Lie algebra
$\mathfrak{D}$ from [1].

######  Definition 1.1.

The twisted Heisenberg-Virasoro algebra ${{\mathfrak{D}}}$ is a Lie algebra
with a basis

$\left\\{d_{m},h_{r},c_{1},c_{2},{c}_{3}:m,r\in{\mathbb{Z}}\right\\}$

and subject to the commutation relations

(1.1)
$\displaystyle[d_{m},d_{n}]=(m-n)d_{m+n}+\delta_{m+n,0}\frac{m^{3}-m}{12}c_{1},$
$\displaystyle[d_{m},h_{r}]=-rh_{m+r}+\delta_{m+r,0}(m^{2}+m)c_{2},$
$\displaystyle[h_{r},h_{s}]=r\delta_{r+s,0}{c}_{3},$
$\displaystyle[c_{1},{{\mathfrak{D}}}]=[c_{2},{{\mathfrak{D}}}]=[c_{3},{{\mathfrak{D}}}]=0,$

for $m,n,r,s\in{\mathbb{Z}}$.

It is clear that ${{\mathfrak{D}}}$ contains a copy of the Virasoro subalgebra
$\mathrm{Vir}=\text{span}\\{c_{1},d_{i}:i\in{\mathbb{Z}}\\}$ and the
Heisenberg algebra
${\mathcal{H}}=\bigoplus_{r\in{\mathbb{Z}}}{\mathbb{C}}h_{r}\oplus{\mathbb{C}}{c}_{3}$.

Similarly, for $\delta=0$ or $\frac{1}{2}$, the Fermion-Virasoro algebra
$\mathcal{S}(\delta)$ contains a copy of the Virasoro subalgebra
$\mathrm{Vir}$ and the Fermion algebra $F(\delta)$, see Definition 2.1.

The representation theory over various infinite-dimensional Lie
(super)algebras has been studied by many mathematicians and physicists
recently. For example, Simple smooth modules for the Virasoro algebra, the
Neveu-Schwarz algebra were determined in [13, 7]. Y. Billig and V. Futorny
classified the simple Harish-Chandra modules over the Witt algebra of rank $n$
in [2]; H. Tan, Y. Yao and K. Zhao classified the simple smooth modules over
the mirror and twisted Heisenberg-Virasoro algebras with nonzero level in
[14]. For more related results, see the references within these papers.

Let $\delta=0$ or $\frac{1}{2}$. In this paper, we classify all simple smooth
$F(\delta)$-modules, all simple weight $F(\delta)$-modules, all simple smooth
$\mathcal{S}(\delta)$-modules of nonzero level, all simple Harish-Chandra
$\mathcal{S}(\delta)$-modules, and all $\mathcal{S}(\delta)$-modules
structures on free ${\mathbb{C}}[L_{0}]$-modules of rank $2$ with non-trivial
odd spaces.

This paper is arranged as follows. In Section 2, the Fermion algebras and
Fermion-Virasoro algebras are introduced, and related definitions are
collected. In Section 3, we give some property for the representations of
$F(\delta)$. In particular, we classify all simple smooth $F(\delta)$-modules
in Lemma 3.4 and Lemma 3.6, and classify simple weight $F(\delta)$-modules in
Lemma 3.11. Also we show that any smooth $F(\delta)$-module of nonzero level
is completely reducible in Lemma 3.9. In Section 4, using vertex operators
(infinite sum of fermionic oscillators) we classify all simple smooth
$\mathcal{S}(\delta)$-modules of nonzero level in Theorems 4.3 and 4.4. In
Section 5, using properties of weight $F(\delta)$-modules we classify all
simple Harish-Chandra $\mathcal{S}(\delta)$-modules in Theorems 5.4 and 5.5.
In Section 6, we determine all $\mathcal{S}(\delta)$-modules structures on
free ${\mathbb{C}}[L_{0}]$-modules of rank $2$ with non-trivial odd spaces.
Surprisingly we have found four different classes of such modules in Theorems
6.12.

At the end of this paper we propose two questions which may be interesting to
readers.

##  2\. The Fermion-Virasoro algebras

Throughout this paper, $\delta=0$ or $\frac{1}{2}$. Now we define the Fermion
algebra $F(\delta)$ that was used implicitly in Section 3.4 to construct
unitary highest weight representation of the Virasoro algebra in [5].

######  Definition 2.1.

The Fermion algebra $F(\delta)=F(\delta)_{\bar{0}}\oplus F(\delta)_{\bar{1}}$
is a Lie superalgebra with a basis

$\left\\{\psi_{m},z:m\in{\mathbb{Z}}+\delta\right\\}$

and subject to the commutation relations

(2.1) $[\psi_{m},\psi_{n}]=\delta_{m,-n}z,\,\,[F(\delta),z]=0,$

for $m,n\in{\mathbb{Z}}+\delta$, where
$F(\delta)_{\bar{1}}=\sum_{i\in{\mathbb{Z}}+\delta}{\mathbb{C}}\psi_{i}$ and
$F(\delta)_{\bar{0}}={\mathbb{C}}z$.

Write $F:=F(0)$ for short. We can consider the Fermion algebra $F$ as the
super version of the Heisenberg algebra ${\mathcal{H}}$. The operators
$\psi_{m}$ are generally called fermionic oscillators.

Now we define the Fermion-Virasoro algebra $\mathcal{S}(\delta)$ that was
implicitly used in Section 3.4 in [5] although the algebra
$\mathcal{S}(\delta)$ was not defined there.

######  Definition 2.2.

The Fermion-Virasoro algebra
$\mathcal{S}(\delta)=\mathcal{S}(\delta)_{\bar{0}}\oplus\mathcal{S}(\delta)_{\bar{1}}$
is the Lie superalgebra with basis
$\\{L_{m},\psi_{n},c,z:m\in{\mathbb{Z}},n\in{\mathbb{Z}}+\delta\\}$, where
$L_{m},c,z\in\mathcal{S}(\delta)_{\bar{0}},\psi_{n}\in\mathcal{S}(\delta)_{\bar{1}}$,
and subject to the commutation relations

$\displaystyle{}[L_{m},L_{n}]$
$\displaystyle=(m-n)L_{m+n}+\delta_{m,-n}\frac{m^{3}-m}{12}c,$
$\displaystyle[L_{m},\psi_{n}]$ $\displaystyle=(-n-\frac{m}{2})\psi_{m+n},$
$\displaystyle[\psi_{m},\psi_{n}]$ $\displaystyle=\delta_{m,-n}z,$
$\displaystyle[\mathcal{S}(\delta),c]$
$\displaystyle=[\mathcal{S}(\delta),z]=0.$

Write $\mathcal{S}:=\mathcal{S}(0)$ for short. The results in this paper are
proved for $\mathcal{S}$ and $F$ while their proofs are also applicable to
those for $\mathcal{S}(\frac{1}{2})$ and $F(\frac{1}{2})$. The main results
are given in both $\delta=0$ and $\frac{1}{2}$.

The Lie superalgebra $\mathcal{S}(0)/{\mathbb{C}}z$ is isomorphic to a
quotient algebra of a special case of the Lie superalgebras studied in [3].
The Lie superalgebras $\mathcal{S}(\delta)/{\mathbb{C}}z$ were discussed in
[4], where their simple Harish-Chandra modules were classified.

It is clear that $\mathcal{S}$ contains a copy of the Virasoro subalgebra
$\mathrm{Vir}=\text{span}\\{c,L_{i}:i\in{\mathbb{Z}}\\}$ and a copy of the
Fermion algebra
$F=\bigoplus_{r\in{\mathbb{Z}}}{\mathbb{C}}\psi_{r}\oplus{\mathbb{C}}z$. We
know that $\mathcal{S}$ has a ${\mathbb{Z}}$-gradation
$\mathcal{S}=\oplus_{k\in{\mathbb{Z}}}\mathcal{S}_{k}$, where

$\mathcal{S}_{k}={\mathbb{C}}L_{k}\oplus{\mathbb{C}}\psi_{k},\forall
k\in{\mathbb{Z}}^{*};\,\,\,\,\mathcal{S}_{0}={\mathbb{C}}L_{0}\oplus{\mathbb{C}}\psi_{0}\oplus{\mathbb{C}}c\oplus{\mathbb{C}}z.$

Write
$\mathcal{S}_{-}:=\oplus_{k\in-{\mathbb{N}}}\mathcal{S}_{k},\mathcal{S}_{+}:=\oplus_{k\in{\mathbb{N}}}\mathcal{S}_{k}$.
Then $\mathcal{S}$ has a triangular decomposition

$\mathcal{S}=\mathcal{S}_{-}\oplus\mathcal{S}_{0}\oplus\mathcal{S}_{+}.$

Set
$\mathrm{Vir}_{\pm}=\mathrm{Vir}\cap\mathcal{S}_{\pm},F_{\pm}=F\cap\mathcal{S}_{\pm}$.

For any $m\in{\mathbb{Z}}_{+},n\in{\mathbb{Z}}$, define the following
subalgebras of $\mathcal{S}$:

$\displaystyle\mathcal{S}^{(m,-\infty)}=\bigoplus_{i\in{\mathbb{Z}}_{+}}{\mathbb{C}}L_{m+i}\oplus\bigoplus_{j\in{\mathbb{Z}}}{\mathbb{C}}\psi_{j}\oplus{\mathbb{C}}c\oplus{\mathbb{C}}z,$
$\displaystyle
F^{(n)}=\bigoplus_{i\in{\mathbb{Z}}_{+}}{\mathbb{C}}\psi_{n+i}\oplus{\mathbb{C}}z,$
$\displaystyle
F_{[m]}=\bigoplus_{i=-m}^{m}{\mathbb{C}}\psi_{i}\oplus{\mathbb{C}}z.$

Let $\mathfrak{g}=\mathcal{S}(\delta),F(\delta)$ or
$F_{[m]},m\in{\mathbb{Z}}_{+}$. For any $\mu\in{\mathbb{C}}^{*}$, there is an
automorphism $\sigma_{\mu}$ of $\mathfrak{g}$ given by

$\sigma_{\mu}(L_{k})=L_{k},\sigma_{\mu}(\psi_{l})=\mu\psi_{l},\sigma_{\mu}(c)=c,\sigma_{\mu}(z)=\mu^{2}z,k\in{\mathbb{Z}},l\in{\mathbb{Z}}+\delta.$

For any $\mathfrak{g}$-module $V$ and any $\mu\in{\mathbb{C}}^{*}$, define the
$\mathfrak{g}$-module $V^{\mu}$ by

$x\cdot v=\sigma_{\mu}(x)v,x\in\mathfrak{g},v\in V.$

######  Definition 2.3.

An $\mathcal{S}$-module $V$ is called to be of level $\ell$ and with central
charge $\dot{c}$ if $z,c$ acts on $V$ as scalar $\ell,\dot{c}$ respectively.

######  Definition 2.4.

An $\mathcal{S}$-module $V$ is called a weight module if $L_{0}$ acts on $V$
diagonally, i.e., $V=\oplus_{\alpha\in{\mathbb{C}}}V_{\alpha}$, where
$V_{\alpha}=\\{v\in V:L_{0}\cdot v=\alpha v\\}$. For a weight
$\mathcal{S}$-module $V$, write

$\mathrm{Supp}(V)=\\{\alpha\in{\mathbb{C}}:V_{\alpha}\neq 0\\},$

which is called the support set of $V$. A weight $\mathcal{S}$-module $V$ is
called a highest (resp. lowest) weight module if
$\mathrm{Supp}(V)\subset\alpha+{\mathbb{Z}}_{+}$ (resp.
$\alpha-{\mathbb{Z}}_{+}$) for some $\alpha\in{\mathbb{C}}$.

##  3\. Representations of $F(\delta)$

######  Definition 3.1.

Let $V$ be a module over a Lie superalgebra $\mathfrak{g}$, $\mathfrak{h}$ be
a subalgebra of $\mathfrak{g}$, and $S$ be a subset of $V$. We denote

$\mathrm{Ann}_{V}(\mathfrak{h})=\\{v\in
V:\mathfrak{h}v=0\\},\,\,\,\mathrm{Ann}_{\mathfrak{g}}(S)=\\{x\in\mathfrak{g}:xS=0\\}.$

######  Definition 3.2.

Let $\mathfrak{g}=\oplus_{i\in{\mathbb{Z}}}\mathfrak{g}_{i}$ be a
${\mathbb{Z}}$-graded Lie superalgebra. A $\mathfrak{g}$-module $V$ is called
a smooth module if for any $v\in V$ there exists $n\in{\mathbb{Z}}_{+}$ such
that $\mathfrak{g}_{i}v=0$ for $i>n$. The category of smooth modules over
$\mathfrak{g}$ will be denoted by $\mathfrak{R}_{\mathfrak{g}}$.

Smooth modules for affine Kac-Moody algebras $\mathfrak{g}$ were introduced
and studied by D. Kazhdan and G. Lusztig in [6].

Let $\mathscr{V}(\delta)$ be the exterior algebra generated by the odd
elements $\xi_{i},i\in{\mathbb{Z}}_{+}+\delta$. Then $\mathscr{V}(\delta)$ has
a basis

$1,\xi_{i_{1}}\cdots\xi_{i_{k}},\,\,\,0\leqslant
i_{1}<\dots<i_{k},k\in{\mathbb{N}}.$

Now $\mathscr{V}(\delta)$ has an irreducible $F(\delta)$-module structure
defined by

$\displaystyle\psi_{n}\cdot v=\partial/\partial\xi_{n}\cdot v,$
$\displaystyle\psi_{-n}\cdot v=\xi_{n}\cdot v,$ $\displaystyle\psi_{0}\cdot
v=\frac{1}{\sqrt{2}}(\xi_{0}+\partial/\partial{\xi_{0}})\cdot v\,\,(\text{if
}\delta=0),$ $\displaystyle z\cdot v=v,$

where $n\in({\mathbb{Z}}_{+}+\delta)\setminus\\{0\\},v\in\mathscr{V}(\delta)$.
Write $\mathscr{V}:=\mathscr{V}(0)$ for short.

######  Definition 3.3.

For $m\in{\mathbb{Z}}_{+}$, define in the same manner above the
$F_{[m]}$-submodule $\mathscr{V}_{[m]}$ of $\mathscr{V}$ by

$\mathscr{V}_{[m]}=\mathrm{span}\\{1,\xi_{i_{1}}\cdots\xi_{i_{k}}\ :\
0\leqslant i_{1}<\dots<i_{k}\leqslant m\\}.$

Then $\mathscr{V}_{[m]}$ is a finite-dimensional simple $F_{[m]}$-module.

######  Lemma 3.4.

  * (1)

Suppose $V$ is a simple $F_{[m]}$-module such that $zV=0$. Then $F_{[m]}V=0$.
Consequently $\dim V=1$.

  * (2)

Suppose $V$ is a simple $F(\delta)$-module such that $zV=0$. Then
$F(\delta)V=0$. Consequently $\dim V=1$.

###### Proof.

Since $z=0$, $\psi_{i}V$ is a submodule of $V$, which implies $\psi_{i}V=0$ or
$\psi_{i}V=V$. The latter cannot occur since $\psi_{i}^{2}V=0$. ∎

######  Lemma 3.5.

  * (1)

Suppose $V$ is a simple $F_{[m]}$-module such that $z$ acts on $V$ as a scalar
$\lambda\in{\mathbb{C}}^{*}$. If there is a nonzero vector $v\in V_{\bar{0}}$
such that $\psi_{k}\cdot v=0$ for all $k\in\\{1,\dots,m\\}$, then
$V\cong\mathscr{V}_{[m]}^{\sqrt{\lambda}}$ for some
$\sqrt{\lambda}\in{\mathbb{C}}$ with $(\sqrt{\lambda})^{2}=\lambda$.

  * (2)

Suppose $V$ is a simple $F$-module such that $z$ acts on $V$ as a scalar
$\lambda\in{\mathbb{C}}^{*}$. If there is a nonzero vector $v\in V_{\bar{0}}$
such that $\psi_{k}\cdot v=0$ for all $k\in{\mathbb{N}}$, then
$V\cong\mathscr{V}^{\sqrt{\lambda}}$ for some $\sqrt{\lambda}\in{\mathbb{C}}$
with $(\sqrt{\lambda})^{2}=\lambda$.

###### Proof.

(1) It suffices to prove that
$V^{\frac{1}{\sqrt{\lambda}}}\cong\mathscr{V}_{[m]}$. Note that $z$ acts on
$V^{\frac{1}{\sqrt{\lambda}}}$ as the identity map. Without loss of
generality, assume that $\lambda=1$.

Since $\psi_{0}^{2}\cdot v=\frac{1}{2}[\psi_{0},\psi_{0}]\cdot
v=\frac{1}{2}v$, ${\mathbb{C}}v\oplus{\mathbb{C}}\psi_{0}v$ is a simple
$F_{[0]}$-module. Also, the fact that
$\psi_{k}\cdot({\mathbb{C}}v\oplus{\mathbb{C}}\psi_{0}v)=0$ for all
$k=1,\dots,m$ implies that ${\mathbb{C}}v\oplus{\mathbb{C}}\psi_{0}v$ is a
simple $F_{[0]}\oplus F_{[m]+}$-module, where $F_{[m]+}=F_{[m]}\cap F_{+}$. So
$V$ is a simple quotient of $\rm{Ind}_{F_{[0]}\oplus
F_{[m]+}}^{F_{[m]}}({\mathbb{C}}v\oplus{\mathbb{C}}\psi_{0}v)$.

Let $f:{\mathbb{C}}v\oplus{\mathbb{C}}\psi_{0}v\rightarrow\mathscr{V}_{[0]}$
be a linear map defined by $f(v)=1,f(\psi_{0}v)=\frac{1}{\sqrt{2}}\xi_{0}$.
Then $f$ is an isomorphism of $(F_{[0]}\oplus F_{[m]+})$-modules. Thus, $f$
induces an $F_{[m]}$-module homomorphism

$\begin{split}\tilde{f}:\rm{Ind}_{F_{[0]}\oplus
F_{[m]+}}^{F_{[m]}}({\mathbb{C}}v\oplus{\mathbb{C}}\psi_{0}v)&\rightarrow\mathscr{V}_{[m]}\\\
\psi_{i_{1}}\dots\psi_{i_{k}}v&\mapsto\xi_{-i_{1}}\dots\xi_{-i_{k}},\\\
\psi_{i_{1}}\dots\psi_{i_{k}}\psi_{0}v&\mapsto\frac{1}{\sqrt{2}}\xi_{-i_{1}}\dots\xi_{-i_{k}}\xi_{0},i_{1}<\dots<i_{k}<0.\\\
\end{split}$

Note that $\tilde{f}$ maps a basis to a basis. So $\tilde{f}$ is an
$F_{[m]}$-module isomorphism. Since $\mathscr{V}_{[m]}$ is a simple
$F_{[m]}$-module, we have $\rm{Ind}_{F_{[0]}\oplus
F_{[m]+}}^{F_{[m]}}({\mathbb{C}}v\oplus{\mathbb{C}}\psi_{0}v)$ is a simple
module and $V\cong\mathscr{V}_{[m]}$.

(2) This could be proved similarly. ∎

######  Lemma 3.6.

  * (1)

Suppose $V$ is a simple $F_{[m]}$-module for some $m\in{\mathbb{Z}}_{+}$ such
that $z$ acts on $V$ as a scalar $\lambda\in{\mathbb{C}}^{*}$. Then
$V\cong\mathscr{V}_{[m]}^{\sqrt{\lambda}}$ for some
$\sqrt{\lambda}\in{\mathbb{C}}$ with $(\sqrt{\lambda})^{2}=\lambda$.

  * (2)

Suppose $V$ is a simple smooth $F$-module such that $z$ acts on $V$ as a
scalar $\lambda\in{\mathbb{C}}^{*}$. Then $V\cong\mathscr{V}^{\sqrt{\lambda}}$
for some $\sqrt{\lambda}\in{\mathbb{C}}$ with $(\sqrt{\lambda})^{2}=\lambda$.

###### Proof.

(1) Note that $\psi_{i}^{2}\cdot V=0$ and that the action of $\psi_{i}$ and
$\psi_{j}$ are anti-commutative for all $i,j\in\\{1,\dots,m\\}$. So there
exists a nonzero homogeneous vector $v\in V$ such that $\psi_{k}\cdot
v=0,k=1,\dots,m$. Replace $v$ with $\psi_{0}v$ if necessary, and assume that
$v\in V_{\bar{0}}$. Then (1) follows from Lemma 3.5 (1).

(2) Let $v$ be a nonzero homogeneous vector in $V$. Since $V$ is a smooth
$F$-module, there exists an $m\in{\mathbb{N}}$ such that $\psi_{k}\cdot v=0$
for any $k>m$. Then $\psi_{k},k>m,$ acts trivially on the finite-dimensional
$F_{[m]}$-module $U(F_{[m]})v$. Clearly, $U(F_{[m]})v$ has a nonzero even
vector $w$ such that $\psi_{k}\cdot w=0,k=1,\dots,m$. So $\psi_{k}\cdot w=0$
for all $k\in{\mathbb{N}}$. Then (2) follows from Lemma 3.5 (2). ∎

With similar arguments, we have the following result.

######  Lemma 3.7.

Suppose $V$ is a simple smooth $F(\frac{1}{2})$-module such that $z$ acts on
$V$ as a scalar $\lambda\in{\mathbb{C}}^{*}$. Then
$V\cong\mathscr{V}(\frac{1}{2})^{\sqrt{\lambda}}$ for some
$\sqrt{\lambda}\in{\mathbb{C}}$ with $(\sqrt{\lambda})^{2}=\lambda$.

Note that the Fermion algebra $F_{[m]}$ has very good properties such that
every $F_{[m]}$-module with nonzero action of the center over $F_{[m]}$ is
completely reducible.

######  Lemma 3.8.

Let $m\in{\mathbb{Z}}_{+}$, and $V$ be an $F_{[m]}$-module such that $z$ acts
on $V$ as a scalar $\lambda\in{\mathbb{C}}^{*}$. Then $V$ is completely
reducible.

###### Proof.

$\mathbf{Claim}$. $U(F_{[m]})v$ is completely reducible for any homogeneous
$v\in V$.

For any $k\in\\{1,\dots,m\\}$, let
$X_{k}=\\{\psi_{k},\psi_{-k}\psi_{k}\\},Y_{k}=\\{\psi_{-k},\psi_{k}\psi_{-k}\\}$.
Then for any $Z_{k}\in\\{X_{k},Y_{k}\\},k=1,\dots,m$, the subspace

(3.1) $\text{span}\\{x_{1}\dots x_{m}v,\psi_{0}x_{1}\dots x_{m}v:x_{j}\in
Z_{j},j=1,\dots,m\\}$

is an $F_{[m]}$-submodule of $U(F_{[m]})v$. By Lemma 3.6(1), any simple
$F_{[m]}$-module is isomorphic to $\mathscr{V}_{[m]}^{\sqrt{\lambda}}$ for
some $\sqrt{\lambda}\in{\mathbb{C}}$ with $(\sqrt{\lambda})^{2}=\lambda$.
Comparing the dimensions of $\mathscr{V}_{[m]}^{\sqrt{\lambda}}$ and (3.1), we
have (3.1) is zero or is a simple module. Note that
$\psi_{-k}\psi_{k}+\psi_{k}\psi_{-k}=z$ in the enveloping algebra for any
$k\in\\{1,\dots,m\\}$. Then

$\bigoplus_{Z_{1}\in\\{X_{1},Y_{1}\\}}\dots\bigoplus_{Z_{m}\in\\{X_{m},Y_{m}\\}}\mathrm{span}\\{x_{1}\dots
x_{m}v,\psi_{0}x_{1}\dots x_{m}v:x_{j}\in Z_{j},j=1,\dots,m\\}$

contains $v$, and therefore equals $U(F_{[m]})v$. Thus, $U(F_{[m]})v$ is
completely reducible.

By the Claim, for any homogeneous $v\in V\setminus\\{0\\}$, $U(F_{[m]})v$ is a
sum of simple modules. Then $V$ is also a sum of simple modules, which means
that $V$ is completely reducible. ∎

######  Lemma 3.9.

Let $V$ be a smooth $F$-module such that $z$ acts on $V$ as a scalar
$\lambda\in{\mathbb{C}}^{*}$. Then $V$ is completely reducible.

###### Proof.

It suffices to prove that $U(F)v$ is a sum of simple $F$-modules for any
homogeneous $v\in V$. Let $v$ be a nonzero homogeneous vector in $V$. Then
there exists $m\in{\mathbb{N}}$ such that $\psi_{m+k}v=0$ for all
$k\in{\mathbb{Z}}_{+}$. By Lemma 3.8, $U(F_{[m]})v$ is completely reducible.
From Lemma 3.6 (1), there are $x_{1},\dots,x_{r}\in U(F_{[m]})v$ satisfying
$\psi_{k}x_{i}=0$ for $k\in{\mathbb{Z}}_{+}$ such that
$U(F_{[m]})v=U(F_{[m]})x_{1}+\dots+U(F_{[m]})x_{r}$. We see that

$U(F)v=U(F)x_{1}+\dots+U(F)x_{r}.$

Note that each $U(F)x_{i}$ is isomorphic to $\mathscr{V}^{\sqrt{\lambda}}$ for
some $\sqrt{\lambda}\in{\mathbb{C}}$ with $(\sqrt{\lambda})^{2}=\lambda$. So
$U(F)v$ is a sum of simple $F$-modules. ∎

For any $I\subset{\mathbb{Z}}_{+}$, let

$S_{I}=\\{J\subset{\mathbb{Z}}_{+}:|I\setminus J|<\infty,|J\setminus
I|<\infty\\}.$

Define the $F$-module $\mathscr{V}_{I}$ as follows: $\mathscr{V}_{I}$ has a
basis $\\{\xi_{J}:J\in S_{I}\\}$, the parity of $\xi_{J}$ is defined as
$|\xi_{J}|=\overline{|I\setminus J|+|J\setminus I|}$, and the action of $F$ on
$\mathscr{V}_{I}$ is given by

$\displaystyle\psi_{k}\cdot\xi_{J}=$ $\displaystyle\left\\{\begin{aligned}
(-1)^{|\\{i\in J:i<k\\}|}\xi_{J\setminus\\{k\\}},&\ \text{if}\ k\in J,\\\ 0,&\
\text{if}\ k\notin J,\end{aligned}\right.$
$\displaystyle\psi_{-k}\cdot\xi_{J}=$ $\displaystyle\left\\{\begin{aligned}
(-1)^{|\\{i\in J:i<k\\}|}\xi_{J\cup\\{k\\}},&\ \text{if}\ k\notin J,\\\ 0,&\
\text{if}\ k\in J,\end{aligned}\right.$ $\displaystyle\psi_{0}\cdot\xi_{J}=$
$\displaystyle\left\\{\begin{aligned}
\frac{1}{\sqrt{2}}\xi_{J\setminus\\{0\\}},&\ \text{if}\ 0\in J,\\\
\frac{1}{\sqrt{2}}\xi_{J\cup\\{0\\}},&\ \text{if}\ 0\notin
J,\end{aligned}\right.$ $\displaystyle z\cdot\xi_{J}=$ $\displaystyle\xi_{J},$

$k\in{\mathbb{N}}$. Then $\mathscr{V}_{I}$ is a simple $F$-module (see the
proof of Lemma 3.10). Moreover, $\mathscr{V}_{I}$ is isomorphic to the
$F$-module $\mathscr{V}$ if $I$ is a finite set, and $\mathscr{V}_{I}$ is not
a smooth $F$-module if $I$ is not a finite set.

######  Lemma 3.10.

Suppose $V$ is a simple $F$-module such that $z$ acts on $V$ as the identity
map. If there is $v\in V_{\bar{0}}\setminus\\{0\\}$ and
$I\subset{\mathbb{Z}}_{+}$ such that

$\psi_{l}\cdot v=0,\psi_{-k}\cdot v=0,\forall l\in{\mathbb{N}}\setminus I,k\in
I\setminus\\{0\\},$

then $V\cong\mathscr{V}_{I}$.

###### Proof.

Let $f_{I}$ be the Lie superalgebra isomorphism of $F$ defined by

$f_{I}(\psi_{k})=\psi_{-k},f_{I}(\psi_{-k})=\psi_{k},f_{I}(\psi_{l})=\psi_{l},\forall
k\in I,l\in{\mathbb{Z}}\setminus(I\cup-I).$

Then $f_{I}^{2}$ is the identity map on $F$. Let $V^{f_{I}}$ be the simple
$F$-module given by $V^{f_{I}}=V$ and

$x\cdot w=f_{I}(x)w,\forall x\in F,w\in V.$

Then for $v\in V^{f_{I}}$ we have

$\psi_{k}\cdot v=0,\forall k\in{\mathbb{N}}.$

By Lemma 3.5(2), $V^{f_{I}}\cong\mathscr{V}$. So $V\cong\mathscr{V}^{f_{I}}$.

Note that $\mathscr{V}_{I}$ is a simple $F$-module with the given property.
Then $V\cong\mathscr{V}^{f_{I}}\cong\mathscr{V}_{I}$. ∎

Suppose $V$ is an $F$-module. For any
$\lambda=(\lambda_{0},\lambda_{1},\dots)\in{\mathbb{C}}^{\infty}$, let

$V_{(\lambda)}=\\{v\in V:\psi_{-k}\psi_{k}\cdot
v=\lambda_{k}v,k\in{\mathbb{Z}}_{+}\\}.$

$V$ is called a weight $F$-module if
$V=\oplus_{\lambda\in{\mathbb{Z}}^{\infty}}V_{(\lambda)}$. By definition, for
any $I\subset{\mathbb{Z}}_{+}$, $\mathscr{V}_{I}$ is a weight $F$-module.

######  Lemma 3.11.

Suppose $V$ is a weight $F$-module such that $z$ acts as the identity map on
$V$.

  * (1)

$\lambda_{k}=0$ or $1$ for any $\lambda\in{\mathbb{C}}^{\infty}$ with
$V_{(\lambda)}\neq 0$ and any $k\in{\mathbb{N}}$.

  * (2)

For any $v\in V_{(\lambda)}\setminus\\{0\\}$, $U(F)v$ is isomorphic to
$\mathscr{V}_{I}$ for some $I\subset{\mathbb{Z}}_{+}$ and $v$ is corresponding
to $\xi_{I}$.

  * (3)

If $V$ is further simple, then $V$ is isomorphic to $\mathscr{V}_{I}$ for some
$I\subset{\mathbb{Z}}_{+}$.

###### Proof.

(1) Suppose $v\in V_{(\lambda)}\setminus\\{0\\}$ for some
$\lambda\in{\mathbb{C}}^{\infty}$. For any $k\in{\mathbb{N}}$,

$\lambda_{k}^{2}v=\psi_{-k}\psi_{k}\psi_{-k}\psi_{k}v=\psi_{-k}[\psi_{k},\psi_{-k}]\psi_{k}v=\psi_{-k}\psi_{k}v=\lambda_{k}v.$

So $\lambda_{k}=0$ or $1$ and (1) is proved.

(2) Replacing $v$ with $\psi_{0}v$ if necessary, assume that $|v|=\bar{0}$. If
$\psi_{-k}\psi_{k}v=0$, then $\psi_{k}v=\psi_{k}\psi_{-k}\psi_{k}v=0$. If
$\psi_{-k}\psi_{k}v=v$, then $\psi_{-k}v=0$. Let
$I=\\{k\in{\mathbb{N}}:\lambda_{k}=1\\}$ and let $J(I)$ be the left ideal of
$U(F)$ generated by $\psi_{-i},\psi_{j},i\in I,j\in{\mathbb{N}}\setminus I$.
It is easy to see that $U(F)/(J(I)+\langle z-1\rangle)\cong\mathscr{V}_{I}$.
Then $U(F)v\cong\mathscr{V}_{I}$ and $v$ is corresponding to $\xi_{I}$. So (2)
is proved.

Part (3) follows from (2). ∎

##  4\. smooth modules over $\mathcal{S}(\delta)$

The $F(\delta)$-module $\mathscr{V}(\delta)$ constructed last section has an
$\mathcal{S}(\delta)$-module structure as follows. For $k\in{\mathbb{Z}}$, we
have the following vertex operator

(4.1)
$\bar{L}_{k}=\frac{(1-2\delta)}{16}\delta_{k,0}+\frac{1}{2}\sum_{j\in{\mathbb{Z}}+\delta}j:\psi_{-j}\psi_{j+k}:$

as an operation on $\mathscr{V}(\delta)$, where the normal ordering is defined
by

$\displaystyle:\psi_{j}\psi_{k}:=$ $\displaystyle\psi_{j}\psi_{k}$
$\displaystyle\ \text{if}\ k\geqslant j$ $\displaystyle=$
$\displaystyle-\psi_{k}\psi_{j}$ $\displaystyle\ \text{if}\ k<j.$

######  Lemma 4.1.

[5, Proposition 3.7] Define the action of $\mathrm{Vir}$ on
$\mathscr{V}(\delta)$ by

$L_{k}\cdot v=\bar{L}_{k}\cdot v,\,\,\,c\cdot
v=\frac{1}{2}v,\,\,\,v\in\mathscr{V}(\delta).$

Then $\mathscr{V}(\delta)$ is an $\mathcal{S}(\delta)$-module, i.e.,

(4.2)
$\displaystyle[\bar{L}_{m},\bar{L}_{n}]=(m-n)\bar{L}_{m+n}+\delta_{m,-n}\frac{m^{3}-m}{24},$
$\displaystyle[\bar{L}_{m},\psi_{n}]=(-n-\frac{m}{2})\psi_{m+n}.$

Note that the action of $L_{0}$ on $\mathscr{V}(\delta)$ is
$\frac{(1-2\delta)}{16}+\sum_{j\in{\mathbb{Z}}_{+}+\delta\setminus\\{0\\}}j\psi_{-j}\psi_{j}$.
Then $\xi_{i_{1}}\cdots\xi_{i_{k}}$ is an eigenvector of $L_{0}$ of eigenvalue
$\frac{(1-2\delta)}{16}+i_{1}+\dots+i_{k}$. Also, for any
$v\in\mathscr{V}(\delta)$, $L_{k}\cdot 1=0=\psi_{k}\cdot 1$ for sufficiently
large $k$.

######  Lemma 4.2.

[10, Lemma 8] Let $W$ be a module over a Lie superalgebra $L$, $\mathfrak{g}$
be a subalgebra of $L$, and $B$ be a $\mathfrak{g}$-module. Then the
$L$-module homomorphism $\mathrm{Ind}_{\mathfrak{g}}^{L}(W\otimes
B)\rightarrow W\otimes\mathrm{Ind}_{\mathfrak{g}}^{L}(B)$ induced from the
inclusion map $W\otimes B\rightarrow
W\otimes\mathrm{Ind}_{\mathfrak{g}}^{L}(B)$ is an $L$-module isomorphism.

For any $\mathrm{Vir}$-module $V$, define the $\mathcal{S}(\delta)$-module
structure on $V$ by letting $F(\delta)\cdot V=0$. Denote the resulting module
by $V^{\mathcal{S}(\delta)}$.

######  Theorem 4.3.

Let $M$ be a simple smooth $\mathcal{S}$-module of level
$\lambda\in{\mathbb{C}}^{*}$. Then $M\cong\mathscr{V}^{\sqrt{\lambda}}\otimes
V^{\mathcal{S}}$ for some simple smooth $\mathrm{Vir}$-module $V$ and some
$\sqrt{\lambda}\in{\mathbb{C}}$ with $(\sqrt{\lambda})^{2}=\lambda$, where
$\mathscr{V}^{\sqrt{\lambda}}$ is considered as an $\mathcal{S}$-module.

###### Proof.

Let $v$ be a nonzero homogeneous vector in $M$. Then there is an
$m\in{\mathbb{N}}$ such that $\psi_{j}\cdot v=0$ for all $j>m$. By Lemma 3.6,
$U(F_{[m]})v$ has a simple $F_{[m]}$-submodule that is isomorphic to
$\mathscr{V}_{[m]}^{\sqrt{\lambda}}$. Then $U(F_{[m]})v$ has a nonzero even
vector $w$ with $\psi_{j}\cdot w=0,j\in{\mathbb{N}}$. So $U(F)w$ is a simple
$F$-quotient of
$\mathrm{Ind}_{F^{(0)}}^{F}({\mathbb{C}}w\oplus{\mathbb{C}}\psi_{0}w)\cong\mathscr{V}^{\sqrt{\lambda}}$.
Thus, $U(F)w\cong\mathscr{V}^{\sqrt{\lambda}}$ as $F$-modules.

Let $n\in{\mathbb{N}}$ with $n\geqslant 2m$ such that $L_{i}\cdot w=0$ for all
$i\geqslant n$. Then $U(F)w$ is a simple module over
$\mathcal{S}^{(n,-\infty)}$. Suppose $c$ acts on $M$ as a scalar $\mu$. Define
the one-dimensional $\mathcal{S}^{(n,-\infty)}$-module ${\mathbb{C}}u$ by

$L_{i}\cdot u=F\cdot u=0,\,\,\,c\cdot u=(\mu-\frac{1}{2})u,i\geqslant n.$

Then $U(F)w\cong\mathscr{V}^{\sqrt{\lambda}}\otimes{\mathbb{C}}u$ as
$\mathcal{S}^{(n,-\infty)}$-modules. By Lemma 4.2,

$\mathrm{Ind}_{\mathcal{S}^{(n,-\infty)}}^{\mathcal{S}}U(F)w\cong\mathrm{Ind}_{\mathcal{S}^{(n,-\infty)}}^{\mathcal{S}}(\mathscr{V}^{\sqrt{\lambda}}\otimes{\mathbb{C}}u)\cong\mathscr{V}^{\sqrt{\lambda}}\otimes\mathrm{Ind}_{\mathcal{S}^{(n,-\infty)}}^{\mathcal{S}}{\mathbb{C}}u.$

So $M$ is a simple quotient of
$\mathscr{V}^{\sqrt{\lambda}}\otimes\mathrm{Ind}_{\mathcal{S}^{(n,-\infty)}}^{\mathcal{S}}{\mathbb{C}}u$.
Note that
$\mathrm{Ind}_{\mathcal{S}^{(n,-\infty)}}^{\mathcal{S}}{\mathbb{C}}u$ is a
smooth $\mathrm{Vir}$-module. It suffices to prove that any submodule of
$\mathscr{V}^{\sqrt{\lambda}}\otimes\mathrm{Ind}_{\mathcal{S}^{(n,-\infty)}}^{\mathcal{S}}{\mathbb{C}}u$
is of the form $\mathscr{V}^{\sqrt{\lambda}}\otimes W$, where $W$ is a
$\mathrm{Vir}$-submodule of
$\mathrm{Ind}_{\mathcal{S}^{(n,-\infty)}}^{\mathcal{S}}{\mathbb{C}}u$.

Let $M^{\prime}$ be a nonzero submodule of
$\mathscr{V}^{\sqrt{\lambda}}\otimes\mathrm{Ind}_{\mathcal{S}^{(n,-\infty)}}^{\mathcal{S}}{\mathbb{C}}u$.
We claim that $M^{\prime}=\mathscr{V}^{\sqrt{\lambda}}\otimes V$, where
$V=\\{x\in\mathrm{Ind}_{\mathcal{S}^{(n,-\infty)}}^{\mathcal{S}}{\mathbb{C}}u:\mathscr{V}^{\sqrt{\lambda}}\otimes{\mathbb{C}}x\subset
M^{\prime}\\}$. It follows that $V$ is a $\mathrm{Vir}$-submodule of
$\mathrm{Ind}_{\mathcal{S}^{(n,-\infty)}}^{\mathcal{S}}{\mathbb{C}}u$ and we
are done.

Let $a$ be any nonzero homogeneous vector in $M^{\prime}$. Then $a$ is of the
form $\sum_{i=1}^{k}v_{i}\otimes u_{i}$, where
$v_{i}\in\mathscr{V}^{\sqrt{\lambda}},u_{i}\in(\mathrm{Ind}_{\mathcal{S}^{(n,-\infty)}}^{\mathcal{S}}{\mathbb{C}}u)\setminus\\{0\\}$
and $v_{1},\dots,v_{k}$ are linearly independent. Since $u_{1},\dots,u_{k}$
are of the same parity, so $v_{1},\dots,v_{k}$ are of the same parity. By
repeatedly letting $\psi_{i},i\in{\mathbb{Z}}_{+}$, act on $a$, we get
$1\otimes u_{i}\in M^{\prime}$ up to a scalar for some $i\in\\{1,\dots,k\\}$.
So $\mathscr{V}^{\sqrt{\lambda}}\otimes u_{i}\subset M^{\prime}$ and $u_{i}\in
V$. Using induction on $k$, we have $u_{1},\dots,u_{k}\in V$. Therefore,
$M^{\prime}=\mathscr{V}^{\sqrt{\lambda}}\otimes V$. ∎

With similar arguments, we have the following result.

######  Theorem 4.4.

Let $M$ be a simple smooth $\mathcal{S}(\frac{1}{2})$-module of level
$\lambda\in{\mathbb{C}}^{*}$. Then
$M\cong\mathscr{V}(\frac{1}{2})^{\sqrt{\lambda}}\otimes
V^{\mathcal{S}(\frac{1}{2})}$ for some simple smooth $\mathrm{Vir}$-module $V$
and some $\sqrt{\lambda}\in{\mathbb{C}}$ with $(\sqrt{\lambda})^{2}=\lambda$,
where $\mathscr{V}(\frac{1}{2})^{\sqrt{\lambda}}$ is considered as an
$\mathcal{S}(\frac{1}{2})$-module.

######  Definition 4.5.

A module $M$ over a Lie superalgebra $A$ is called strictly simple if it is a
simple module over the algebra $A$ (forgetting the
$\mathbb{Z}_{2}$-gradation).

Let ${\mathfrak{g}}={\mathfrak{a}}\ltimes{\mathfrak{b}}$ be a Lie superalgebra
where ${\mathfrak{a}}$ is a subalgebra of ${\mathfrak{g}}$ and
${\mathfrak{b}}$ is an ideal of ${\mathfrak{g}}$. Let $M$ be a
${\mathfrak{g}}$-module with a ${\mathfrak{b}}$-submodule $H$ so that the
${\mathfrak{b}}$-submodule structure on $H$ can be extended to a
${\mathfrak{g}}$-module structure on $H$. We denote this
${\mathfrak{g}}$-module by $H^{\mathfrak{g}}$. For any ${\mathfrak{a}}$-module
$U$, we can make it into a ${\mathfrak{g}}$-module by ${\mathfrak{b}}U=0$. We
denote this ${\mathfrak{g}}$-module by $U^{\mathfrak{g}}$. Then by Lemma 3.3
in [8] we have

######  Lemma 4.6.

Let ${\mathfrak{g}}={\mathfrak{a}}\ltimes{\mathfrak{b}}$ be a countable
dimensional Lie superalgebra. Let $M$ be a simple ${\mathfrak{g}}$-module with
a strictly simple ${\mathfrak{b}}$-submodule $H$ so that an $H^{\mathfrak{g}}$
exists. Then $M\cong U^{\mathfrak{g}}\otimes H^{\mathfrak{g}}$ as
$\mathfrak{g}$-modules for some simple ${\mathfrak{a}}$-module $U$.

Note that Theorem 4.3 can not be proved directly by using Lemma 4.6 because
the $F$-module $\mathscr{V}^{\sqrt{\lambda}}$ is not strictly simple.

We know that simple smooth $\mathrm{Vir}$-modules were classified in [13,
Theorem 2].

Applying Theorem 4.3 to highest weight $\mathcal{S}$-modules we obtain the
following theorem.

######  Theorem 4.7.

Let $V$ be a simple highest weight $\mathcal{S}$-module.

  * (1)

If $z$ acts on $V$ as a scalar $\lambda\in{\mathbb{C}}^{*}$, then
$V\cong\mathscr{V}^{\sqrt{\lambda}}\otimes M^{\mathcal{S}}$ for some simple
highest weight $\mathrm{Vir}$-module $M$ and some
$\sqrt{\lambda}\in{\mathbb{C}}$ with $(\sqrt{\lambda})^{2}=\lambda$, where
$\mathscr{V}^{\sqrt{\lambda}}$ is considered as an $\mathcal{S}$-module.

  * (2)

If $zV=0$, then $FV=0$ and $V$ is a simple highest weight
$\mathrm{Vir}$-module.

###### Proof.

(1) This follows from Theorem 4.3 since highest weight $\mathrm{Vir}$-module
are smooth modules.

(2) Let $v$ be a nonzero homogeneous highest weight vector of $V$. Then
$\psi_{k}\cdot v=0$ for all $k>0$. Replacing $v$ with $\psi_{0}v$ if
necessary, assume that $\psi_{0}v=0$. By the simplicity of $V$ and the PBW
theorem, $V=U(\mathrm{Vir}_{-})U(F_{-})v$. Then

$F\cdot V=F\cdot U(\mathrm{Vir}_{-})U(F_{-})v\subset
U(\mathrm{Vir}_{-})U(F_{-})F_{-}v.$

Considering the support set of $U(\mathrm{Vir}_{-})U(F_{-})F_{-}v$ we have
$v\notin F\cdot V$. So $F\cdot V=0$ and we are done. ∎

We remark that in this paper we are not able to determine all simple smooth
$\mathcal{S}(\delta)$-modules $V$ of level $0$.

##  5\. Harish-Chandra modules

A weight $\mathcal{S}(\delta)$-module is called a Harish-Chandra module if its
weight spaces are all finite-dimensional. A weight
$\mathcal{S}(\delta)$-module is called a bounded module if the dimensions of
its weight spaces are uniformly bounded.

######  Lemma 5.1.

A simple Harish-Chandra $\mathcal{S}$-module is either a highest weight
module, a lowest weight module or a bounded module.

###### Proof.

Suppose $M$ is a simple Harish-Chandra $\mathcal{S}$-module that is not
bounded. Then $V=U(\mathrm{Vir})\cdot(\oplus_{\alpha\neq 0}M_{\alpha})$ is the
unique minimal $\mathrm{Vir}$-submodule of $M$ such that $M/V$ is a trivial
$\mathrm{Vir}$-module. Let $T$ be the maximal trivial $\mathrm{Vir}$-submodule
of $V$. Then $\bar{V}=V/T$ is an Harish-Chandra $\mathrm{Vir}$-module such
that $\bar{V}$ and $\bar{V}^{*}$ do not contain trivial
$\mathrm{Vir}$-submodules, where $\bar{V}^{*}$ is the restricted dual of
$\bar{V}$. By Proposition 3.3 in [11], $\bar{V}=V^{-}\oplus B\oplus V^{+}$,
where $V^{+}$ (resp. $V^{-}$) is the maximal $\mathrm{Vir}$-submodule of
$\bar{V}$ with lower (resp. upper) bounded weights, and $B$ is a bounded
$\mathrm{Vir}$-submodule of $\bar{V}$.

Without loss of generality, assume $V^{+}\neq 0$. Then $V$ contains a nonzero
homogeneous weight vector $v$ such that $L_{k}\cdot v=0$ for all
$k\in{\mathbb{N}}$, and $U(\mathrm{Vir})v$ is not a trivial module. Then
$U(\mathrm{Vir})v$ has a nontrivial simple $\mathrm{Vir}$-subquotient that is
a highest weight $\mathrm{Vir}$-module. So $U(\mathrm{Vir})v$ is not bounded.
Suppose $v\in V_{h}$. Then there is a $k\in{\mathbb{N}}$ such that

$\dim(U(\mathrm{Vir})v)_{h+k}>2\dim M_{h}.$

Then there exists $w\in(U(\mathrm{Vir})v)_{h+k}$ such that $L_{k}\cdot
w=\psi_{k}\cdot w=0$. Also, for any $j\in{\mathbb{N}}$, $L_{k+j}\cdot w=0$. So

$0=[L_{k+j},\psi_{k}]\cdot w=-\frac{1}{2}(3k+j)\psi_{2k+j}\cdot w.$

Thus, $\mathcal{S}_{n}\cdot w=0$ for all $n\geqslant 2k$. By Lemma 1.6 in
[12], $M$ has a highest weight. Then $M$ is a highest weight module. ∎

######  Lemma 5.2.

Suppose $V$ is a simple bounded $\mathcal{S}$-module. Then $zV=0$.

###### Proof.

Since $V$ is simple, $z$ acts on $V$ as a scalar. Suppose that $zV\neq 0$ and,
without loss of generality, assume that $z$ acts on $V$ as the identity map.

Let $\lambda\in\mathrm{Supp}(V)$. Then $\dim V_{\lambda}<\infty$ and
$\psi_{-k}\psi_{k}\cdot V_{\lambda}\subset V_{\lambda}$ for all
$k\in{\mathbb{Z}}_{+}$. So $V_{\lambda}$ contains a nonzero homogeneous common
eigenvector $v$ of $\psi_{-k}\psi_{k},k\in{\mathbb{Z}}_{+}$. By Lemma 3.11,
$U(F)v$ is isomorphic to $\mathscr{V}_{I}$ for some $I\subset{\mathbb{Z}}_{+}$
and $v$ is corresponding to $\xi_{I}$. Identify elements in $\mathscr{V}_{I}$
with that in $U(F)v$. For any $J\in S_{I}$, $\xi_{J}\in V_{\mu}$, where

$\mu=\lambda+\sum_{k\in J\setminus I}k-\sum_{l\in I\setminus J}l.$

Then $U(F)v$ is a weight ${\mathbb{C}}L_{0}$-module.

We will now prove that $U(F)v$ is not bounded, which will lead to a
contradiction and the lemma follows. Suppose $U(F)v$ is bounded with maximal
weight space dimension $d$. Let $\mu\in\mathrm{Supp}(V)$ with
$\dim(U(F)v)_{\mu}=d$. Then $(U(F)v)_{\mu}$ has a basis
$\xi_{J_{1}},\dots,\xi_{J_{d}}$ satisfying

$\mu=\lambda+\sum_{k\in J_{i}\setminus I}k-\sum_{l\in I\setminus J_{i}}l.$

Let $m\in{\mathbb{N}}$ such that $J_{i}\setminus\\{0,\dots,m\\}$ are the same
for all $i\in\\{1,\dots,d\\}$.

If $I\subset\\{0,\dots,m\\}$, then
$\xi_{J_{i}\cup\\{m+1,m+2\\}},\xi_{J_{i}\cup\\{2m+3\\}}\in(U(F)v)_{\mu+2m+3}$
for all $i\in\\{1,\dots,d\\}$. So

$\dim(U(F)v)_{\mu+2m+3}\geqslant 2d,$

which is a contradiction. Assume that $I\not\subset\\{0,\dots,m\\}$ and,
without loss of generality, that $m+1\in I$.

Case 1. There is $n\in{\mathbb{N}}\setminus\\{0,\dots,m+1\\}$ such that $n\in
I$ and $m+1+n\in I$.

Then
$\xi_{J_{i}\setminus\\{m+1,n\\}},\xi_{J_{i}\setminus\\{m+1+n\\}}\in(U(F)v)_{\mu-m-1-n}$
for all $i\in\\{1,\dots,d\\}$. So

$\dim(U(F)v)_{\mu-m-1-n}\geqslant 2d,$

which is a contradiction.

Case 2. There is $n\in{\mathbb{N}}\setminus\\{0,\dots,m+1\\}$ such that
$n\notin I$ and $m+1+n\in I$.

Then
$\xi_{J_{i}\setminus\\{m+1\\}},\xi_{(J_{i}\setminus\\{m+1+n\\})\cup\\{n\\}}\in(U(F)v)_{\mu-m-1}$
for all $i\in\\{1,\dots,d\\}$. So

$\dim(U(F)v)_{\mu-m-1}\geqslant 2d,$

which is a contradiction.

Case 3. For any $n\in{\mathbb{N}}\setminus\\{0,\dots,m+1\\}$ we have
$m+1+n\notin I$.

Then $({\mathbb{N}}\setminus\\{0,\dots,2m+2\\})\cap I=\varnothing$, i.e.,
$I\subset\\{0,\dots,2m+2\\}$. As the proof above, this also leads to a
contradiction.

Thus, $U(F)v$ is not bounded. ∎

Now we recall the following results from [3, 4].

######  Lemma 5.3.

Suppose $V$ is a simple bounded $\mathcal{S}$-module of level $0$. Then $FV=0$
and $V$ is a simple $\mathrm{Vir}$-module of the intermediate series.

We are ready to state the classification for simple Harish-Chandra
$\mathcal{S}$-modules.

######  Theorem 5.4.

Any simple Harish-Chandra $\mathcal{S}$-module is isomorphic to one of the
following $\mathcal{S}$-modules.

  * (1)

$V^{\mathcal{S}}$, where $V$ is a simple Harish-Chandra $\mathrm{Vir}$-module
and $FV=0$;

  * (2)

$\mathscr{V}^{\lambda}\otimes V^{\mathcal{S}}$, where
$\lambda\in{\mathbb{C}}^{*}$ and $V$ is a simple highest weight
$\mathrm{Vir}$-module, and $\mathscr{V}^{\lambda}$ is considered as an
$\mathcal{S}$-module;

  * (3)

the restricted dual of the module in (2).

###### Proof.

This follows from Lemma 5.1, Lemma 5.2, Lemma 5.3 and Theorem 4.7. ∎

We have similar results for the Fermion-Virasoro algebra
$\mathcal{S}(\frac{1}{2})$.

######  Theorem 5.5.

Any simple Harish-Chandra $\mathcal{S}(\frac{1}{2})$-module is isomorphic to
one of the following $\mathcal{S}(\frac{1}{2})$-modules.

  * (1)

$V^{\mathcal{S}(\frac{1}{2})}$, where $V$ is a simple Harish-Chandra
$\mathrm{Vir}$-module and $F(\frac{1}{2})V=0$;

  * (2)

$\mathscr{V}(\frac{1}{2})^{\lambda}\otimes V^{\mathcal{S}(\frac{1}{2})}$,
where $\lambda\in{\mathbb{C}}^{*}$ and $V$ is a simple highest weight
$\mathrm{Vir}$-module, and $\mathscr{V}(\frac{1}{2})^{\lambda}$ is considered
as an $\mathcal{S}(\frac{1}{2})$-module;

  * (3)

the restricted dual of the module in (2).

Note that the modules in (2) and (3) of Theorems 5.4, 5.5 do not occur in the
classifications for the algebras in [3, 4].

##  6\. free ${\mathbb{C}}[L_{0}]$-modules over Fermion-Virasoro algebras

Our purpose in this section is to construct modules over $\mathcal{S}(0)$ and
$\mathcal{S}(\frac{1}{2})$ that are free ${\mathbb{C}}[L_{0}]$-modules by
restriction.

For $\lambda\in{\mathbb{C}}^{*},a\in{\mathbb{C}}$, define the free
${\mathbb{C}}[L_{0}]$-module $\Omega(\lambda,a):={\mathbb{C}}[x]$ of rank $1$
over $\mathrm{Vir}$ by

$c\cdot f(x)=0,\ L_{m}\cdot f(x)=\lambda^{m}(x+ma)f(x+m),\
m\in{\mathbb{Z}},f(x)\in{\mathbb{C}}[x].$

We have the following lemma.

######  Lemma 6.1.

[15, Theorem 3] Any $\mathrm{Vir}$-module that is a free
${\mathbb{C}}[L_{0}]$-module of rank $1$ is isomorphic to $\Omega(\lambda,a)$
for some $\lambda\in{\mathbb{C}}^{*}$ and $a\in{\mathbb{C}}$.

For $\lambda\in{\mathbb{C}}^{*}$ and $a,b\in{\mathbb{C}}$, let
$\Omega_{\mathcal{S}(\delta)}(\lambda,a,b):={\mathbb{C}}[x_{0}]\oplus{\mathbb{C}}[x_{1}]$
with $\Omega_{\mathcal{S}(\delta)}(\lambda,a,b)_{\bar{0}}={\mathbb{C}}[x_{0}]$
and $\Omega_{\mathcal{S}(\delta)}(\lambda,a,b)_{\bar{1}}={\mathbb{C}}[x_{1}]$.
Fix some $\sqrt{\lambda}\in{\mathbb{C}}^{*}$ with
$(\sqrt{\lambda})^{2}=\lambda$, and set
$\lambda^{r}=\lambda^{r-\frac{1}{2}}\sqrt{\lambda}$ for any
$r\in\frac{1}{2}+{\mathbb{Z}}$. Define an $\mathcal{S}(\delta)$-action on
$\Omega_{\mathcal{S}(\delta)}(\lambda,a,b)$ as follows:

$\begin{split}&L_{m}\cdot f(x_{0})=\lambda^{m}(x_{0}+ma)f(x_{0}+m),\\\
&L_{m}\cdot g(x_{1})=\lambda^{m}(x_{1}-\frac{m}{2}+ma)g(x_{1}+m),\\\
&\psi_{r}\cdot f(x_{0})=b\lambda^{r}f(x_{1}+r),\\\ &\psi_{r}\cdot
g(x_{1})=0,\\\
&c\cdot\Omega_{\mathcal{S}(\delta)}(\lambda,a,b)=z\cdot\Omega_{\mathcal{S}(\delta)}(\lambda,a,b)=0,\end{split}$

for any $m\in{\mathbb{Z}},r\in\delta+{\mathbb{Z}}$ and any
$f(x_{0})\in{\mathbb{C}}[x_{0}],g(x_{1})\in{\mathbb{C}}[x_{1}]$.

For $\lambda\in{\mathbb{C}}^{*}$ and $a,b\in{\mathbb{C}}$, let
$\Omega^{\prime}_{\mathcal{S}(\delta)}(\lambda,a,b):={\mathbb{C}}[y_{0}]\oplus{\mathbb{C}}[y_{1}]$
with
$\Omega^{\prime}_{\mathcal{S}(\delta)}(\lambda,a,b)_{\bar{0}}={\mathbb{C}}[y_{0}]$
and
$\Omega^{\prime}_{\mathcal{S}(\delta)}(\lambda,a,b)_{\bar{1}}={\mathbb{C}}[y_{1}]$.
Fix some $\sqrt{\lambda}\in{\mathbb{C}}^{*}$ with
$(\sqrt{\lambda})^{2}=\lambda$, and set
$\lambda^{r}=\lambda^{r-\frac{1}{2}}\sqrt{\lambda}$ for any
$r\in\frac{1}{2}+{\mathbb{Z}}$. Define an $\mathcal{S}(\delta)$-action on
$\Omega^{\prime}_{\mathcal{S}(\delta)}(\lambda,a,b)$ as follows:

$\begin{split}&L_{m}\cdot f(y_{0})=\lambda^{m}(y_{0}+ma)f(y_{0}+m),\\\
&L_{m}\cdot g(y_{1})=\lambda^{m}(y_{1}-\frac{3m}{2}+ma)g(y_{1}+m),\\\
\end{split}$ $\begin{split}&\psi_{r}\cdot
f(y_{0})=b\lambda^{r}(y_{1}-2ra+3r)f(y_{1}+r),\\\ &\psi_{r}\cdot
g(y_{1})=0,\\\
&c\cdot\Omega^{\prime}_{\mathcal{S}(\delta)}(\lambda,a,b)=z\cdot\Omega^{\prime}_{\mathcal{S}(\delta)}(\lambda,a,b)=0\end{split}$

for any $m\in{\mathbb{Z}},r\in\delta+{\mathbb{Z}}$ and any
$f(y_{0})\in{\mathbb{C}}[y_{0}],g(y_{1})\in{\mathbb{C}}[y_{1}]$.

For $\lambda\in{\mathbb{C}}^{*}$ and $b\in{\mathbb{C}}$, let
$\Omega^{\prime\prime}_{\mathcal{S}(\delta)}(\lambda,b):={\mathbb{C}}[z_{0}]\oplus{\mathbb{C}}[z_{1}]$
with
$\Omega^{\prime\prime}_{\mathcal{S}(\delta)}(\lambda,b)_{\bar{0}}={\mathbb{C}}[z_{0}]$
and
$\Omega^{\prime\prime}_{\mathcal{S}(\delta)}(\lambda,b)_{\bar{1}}={\mathbb{C}}[z_{1}]$.
Fix some $\sqrt{\lambda}\in{\mathbb{C}}^{*}$ with
$(\sqrt{\lambda})^{2}=\lambda$, and set
$\lambda^{r}=\lambda^{r-\frac{1}{2}}\sqrt{\lambda}$ for any
$r\in\frac{1}{2}+{\mathbb{Z}}$. Define an $\mathcal{S}(\delta)$-action on
$\Omega^{\prime\prime}_{\mathcal{S}(\delta)}(\lambda,b)$ as follows:

$\begin{split}&L_{m}\cdot f(z_{0})=\lambda^{m}(z_{0}+m)f(z_{0}+m),\\\
&L_{m}\cdot g(z_{1})=\lambda^{m}(z_{1}-\frac{3}{2}m)g(z_{1}+m),\\\
&\psi_{r}\cdot f(z_{0})=b\lambda^{r}(z_{1}+3r)(z_{1}+r)f(z_{1}+r),\\\
&\psi_{r}\cdot g(z_{1})=0,\\\
&c\cdot\Omega^{\prime\prime}_{\mathcal{S}(\delta)}(\lambda,b)=z\cdot\Omega^{\prime\prime}_{\mathcal{S}(\delta)}(\lambda,b)=0\end{split}$

for any $m\in{\mathbb{Z}},r\in\delta+{\mathbb{Z}}$ and any
$f(z_{0})\in{\mathbb{C}}[z_{0}],g(z_{1})\in{\mathbb{C}}[z_{1}]$.

For $\lambda\in{\mathbb{C}}^{*}$ and $b\in{\mathbb{C}}$, let
$\tilde{\Omega}^{\prime\prime}_{\mathcal{S}(\delta)}(\lambda,b):={\mathbb{C}}[\tilde{z}_{0}]\oplus{\mathbb{C}}[\tilde{z}_{1}]$
with
$\tilde{\Omega}^{\prime\prime}_{\mathcal{S}(\delta)}(\lambda,b)_{\bar{0}}={\mathbb{C}}[\tilde{z}_{0}]$
and
$\tilde{\Omega}^{\prime\prime}_{\mathcal{S}(\delta)}(\lambda,b)_{\bar{1}}={\mathbb{C}}[\tilde{z}_{1}]$.Fix
some $\sqrt{\lambda}\in{\mathbb{C}}^{*}$ with $(\sqrt{\lambda})^{2}=\lambda$,
and set $\lambda^{r}=\lambda^{r-\frac{1}{2}}\sqrt{\lambda}$ for any
$r\in\frac{1}{2}+{\mathbb{Z}}$. Define an $\mathcal{S}(\delta)$-action on
$\tilde{\Omega}^{\prime\prime}_{\mathcal{S}(\delta)}(\lambda,b)$ as follows:

$\begin{split}&L_{m}\cdot
f(\tilde{z}_{0})=\lambda^{m}(\tilde{z}_{0}+\frac{5}{2}m)f(\tilde{z}_{0}+m),\\\
&L_{m}\cdot g(\tilde{z}_{1})=\lambda^{m}\tilde{z}_{1}g(\tilde{z}_{1}+m),\\\
&\psi_{r}\cdot
f(\tilde{z}_{0})=b\lambda^{r}\tilde{z}_{1}(\tilde{z}_{1}-2r)f(\tilde{z}_{1}+r),\\\
&\psi_{r}\cdot g(\tilde{z}_{1})=0,\\\
&c\cdot\tilde{\Omega}^{\prime\prime}_{\mathcal{S}(\delta)}(\lambda,b)=z\cdot\tilde{\Omega}^{\prime\prime}_{\mathcal{S}(\delta)}(\lambda,b)=0\end{split}$

for any $m\in{\mathbb{Z}},r\in\delta+{\mathbb{Z}}$ and any
$f(\tilde{z}_{0})\in{\mathbb{C}}[\tilde{z}_{0}],g(\tilde{z}_{1})\in{\mathbb{C}}[\tilde{z}_{1}]$.

######  Lemma 6.2.

$\Omega_{\mathcal{S}(\delta)}(\lambda,a,b),\Omega^{\prime}_{\mathcal{S}(\delta)}(\lambda,a,b),\Omega^{\prime\prime}_{\mathcal{S}(\delta)}(\lambda,b)$
and $\tilde{\Omega}^{\prime\prime}_{\mathcal{S}(\delta)}(\lambda,b)$ are
$\mathcal{S}(\delta)$-modules.

###### Proof.

Clearly, both ${\mathbb{C}}[x_{0}]$ and ${\mathbb{C}}[y_{0}]$ are isomorphic
to the $\mathrm{Vir}$-module $\Omega(\lambda,a)$ defined before Lemma 6.1. For
any $m,n\in\delta+{\mathbb{Z}},x\in\mathcal{S}(\delta)$ and any
$f(x_{0})\in{\mathbb{C}}[x_{0}],g(x_{1})\in{\mathbb{C}}[x_{1}]$ we have

$(\psi_{m}x-(-1)^{|x|-1}x\psi_{m})\cdot g(x_{1})=0=[\psi_{m},x]\cdot
g(x_{1}),$ $(\psi_{m}\psi_{n}+\psi_{n}\psi_{m})\cdot
f(x_{0})=0=\delta_{m,-n}z\cdot f(x_{0})=[\psi_{m},\psi_{n}]\cdot f(x_{0}).$

To prove that $\Omega_{\mathcal{S}(\delta)}(\lambda,a,b)$ is an
$\mathcal{S}(\delta)$-module, it suffice to verify that

$(L_{m}\psi_{n}-\psi_{n}L_{m})\cdot f(x_{0})=[L_{m},\psi_{n}]\cdot
f(x_{0}),\forall
m\in{\mathbb{Z}},n\in\delta+{\mathbb{Z}},f(x_{0})\in{\mathbb{C}}[x_{0}].$

Similarly, to prove that
$\Omega^{\prime}_{\mathcal{S}(\delta)}(\lambda,a,b),\Omega^{\prime\prime}_{\mathcal{S}(\delta)}(\lambda,b)$
and $\tilde{\Omega}^{\prime\prime}_{\mathcal{S}(\delta)}(\lambda,b)$ are
$\mathcal{S}(\delta)$-modules, it suffices to verify respectively that

$(L_{m}\psi_{n}-\psi_{n}L_{m})\cdot g(y_{0})=[L_{m},\psi_{n}]\cdot
g(y_{0}),\forall
m\in{\mathbb{Z}},n\in\delta+{\mathbb{Z}},g(y_{0})\in{\mathbb{C}}[y_{0}],$
$(L_{m}\psi_{n}-\psi_{n}L_{m})\cdot h(z_{0})=[L_{m},\psi_{n}]\cdot
h(z_{0}),\forall
m\in{\mathbb{Z}},n\in\delta+{\mathbb{Z}},h(z_{0})\in{\mathbb{C}}[z_{0}],$
$(L_{m}\psi_{n}-\psi_{n}L_{m})\cdot\tilde{h}(\tilde{z}_{0})=[L_{m},\psi_{n}]\cdot\tilde{h}(\tilde{z}_{0}),\forall
m\in{\mathbb{Z}},n\in\delta+{\mathbb{Z}},\tilde{h}(\tilde{z}_{0})\in{\mathbb{C}}[\tilde{z}_{0}].$

Let $m\in{\mathbb{Z}},n\in\delta+{\mathbb{Z}}$ and
$f(x_{0})\in{\mathbb{C}}[x_{0}],g(y_{0})\in{\mathbb{C}}[y_{0}],h(z_{0})\in{\mathbb{C}}[z_{0}],\tilde{h}(\tilde{z}_{0})\in{\mathbb{C}}[\tilde{z}_{0}]$.
Then we have

$\begin{split}(L_{m}\psi_{n}&-\psi_{n}L_{m})\cdot f(x_{0})\\\ =&L_{m}\cdot
b\lambda^{n}f(x_{1}+n)-\psi_{n}\cdot\lambda^{m}(x_{0}+ma)f(x_{0}+m)\\\
=&b\lambda^{mn}(x_{1}-\frac{m}{2}+ma)f(x_{1}+m+n)-b\lambda^{m+n}(x_{1}+n+ma)f(x_{1}+n+m)\\\
=&b\lambda^{m+n}(-n-\frac{m}{2})f(x_{1}+m+n)\\\ =&[L_{m},\psi_{n}]\cdot
f(x_{0}),\\\ (L_{m}\psi_{n}&-\psi_{n}L_{m})\cdot g(y_{0})\\\ =&L_{m}\cdot
b\lambda^{n}(y_{1}-2na+3n)g(y_{1}+n)-\psi_{n}\cdot\lambda^{m}(y_{0}+ma)g(y_{0}+m)\\\
=&b\lambda^{m+n}(y_{1}-\frac{3m}{2}+ma)(y_{1}+m-2na+3n)g(y_{1}+m+n)\\\
&-b\lambda^{m+n}(y_{1}-2na+3n)(y_{1}+n+ma)g(y_{1}+n+m)\\\
=&(-n-\frac{m}{2})b\lambda^{m+n}(y_{1}-(m+n)(2a-3))g(y_{1}+m+n)\\\
=&[L_{m},\psi_{n}]\cdot g(y_{0}),\\\ (L_{m}\psi_{n}&-\psi_{n}L_{m})\cdot
h(z_{0})\\\ =&L_{m}\cdot
b\lambda^{n}(z_{1}+3n)(z_{1}+n)h(z_{1}+n)-\psi_{n}\cdot\lambda^{m}(z_{0}+m)h(z_{0}+m)\\\
=&b\lambda^{m+n}(z_{1}-\frac{3}{2}m)(z_{1}+m+3n)(z_{1}+m+n)h(z_{1}+m+n)\\\
&-b\lambda^{m+n}(z_{1}+3n)(z_{1}+n)(z_{1}+m+n)h(z_{1}+m+n)\\\
=&(-n-\frac{m}{2})b\lambda^{m+n}(z_{1}+3m+3n)(z_{1}+m+n)h(z_{1}+m+n)\\\
=&[L_{m},\psi_{n}]\cdot h(z_{0}),\\\
(L_{m}\psi_{n}&-\psi_{n}L_{m})\cdot\tilde{h}(\tilde{z}_{0})\\\ =&L_{m}\cdot
b\lambda^{n}\tilde{z}_{1}(\tilde{z}_{1}-2n)\tilde{h}(\tilde{z}_{1}+n)-\psi_{n}\cdot\lambda^{m}(\tilde{z}_{0}+\frac{5}{2}m)\tilde{h}(\tilde{z}_{0}+m)\\\
=&b\lambda^{m+n}\tilde{z}_{1}(\tilde{z}_{1}+m)(\tilde{z}_{1}-2n+m)\tilde{h}(\tilde{z}_{1}+m+n)\\\
&-b\lambda^{m+n}\tilde{z}_{1}(\tilde{z}_{1}-2n)(\tilde{z}_{1}+n+\frac{5}{2}m)\tilde{h}(\tilde{z}_{1}+m+n)\\\
=&(-n-\frac{m}{2})b\lambda^{m+n}\tilde{z}_{1}(\tilde{z}_{1}-2m-2n)\tilde{h}(\tilde{z}_{1}+m+n)\\\
=&[L_{m},\psi_{n}]\cdot\tilde{h}(\tilde{z}_{0}).\end{split}$

So
$\Omega_{\mathcal{S}(\delta)}(\lambda,a,b),\Omega^{\prime}_{\mathcal{S}(\delta)}(\lambda,a,b),\Omega^{\prime\prime}_{\mathcal{S}(\delta)}(\lambda,b)$
and $\tilde{\Omega}^{\prime\prime}_{\mathcal{S}(\delta)}(\lambda,b)$ are
$\mathcal{S}(\delta)$-modules. ∎

Let $V$ be an $\mathcal{S}(\delta)$-module such that $V_{\bar{0}}$ and
$V_{\bar{1}}$ are free ${\mathbb{C}}[L_{0}]$-modules of rank $1$ by
restriction, and that $\mathcal{S}(\delta)_{\bar{1}}V\neq 0$. Next, we will
show that $V$ is isomorphic to
$\Omega_{\mathcal{S}(\delta)}(\lambda,a,b),\Omega^{\prime}_{\mathcal{S}(\delta)}(\lambda,a,b),\Omega^{\prime\prime}_{\mathcal{S}(\delta)}(\lambda,b)$
or $\tilde{\Omega}^{\prime\prime}_{\mathcal{S}(\delta)}(\lambda,b)$ up to a
parity-change for some $\lambda,b\in{\mathbb{C}}^{*}$ and some
$a\in{\mathbb{C}}$.

Let $\mathbf{1}_{0}$ be the ${\mathbb{C}}[L_{0}]$-basis of $V_{\bar{0}}$, and
let $\mathbf{1}_{1}$ be the ${\mathbb{C}}[L_{0}]$-basis of $V_{\bar{1}}$. Then
$V={\mathbb{C}}[L_{0}]\mathbf{1}_{0}\oplus{\mathbb{C}}[L_{0}]\mathbf{1}_{1}$.

From Lemma 6.1 we know that $c\cdot V_{\bar{0}}=c\cdot V_{\bar{1}}=0$, and
there are $\lambda,\lambda^{\prime}\in{\mathbb{C}}^{*}$ and
$a,a^{\prime}\in{\mathbb{C}}$ such that

$\displaystyle L_{m}\cdot
f(L_{0})\mathbf{1}_{0}=\lambda^{m}(L_{0}+ma)f(L_{0}+m)\mathbf{1}_{0},$
$\displaystyle L_{m}\cdot f(L_{0})\mathbf{1}_{1}=\lambda^{\prime
m}(L_{0}+ma^{\prime})f(L_{0}+m)\mathbf{1}_{1},$

for all $m\in{\mathbb{Z}},f(L_{0})\in{\mathbb{C}}[L_{0}].$

For any $m\in\delta+{\mathbb{Z}}$, let $f_{m},g_{m}\in{\mathbb{C}}[L_{0}]$ be
such that

$\psi_{m}\cdot\mathbf{1}_{0}=f_{m}(L_{0})\mathbf{1}_{1},\psi_{m}\cdot\mathbf{1}_{1}=g_{m}(L_{0})\mathbf{1}_{0}.$

Using the fact that $\psi_{m}f(L_{0})=f(L_{0}+m)\psi_{m}$ in
$U(\mathcal{S}(\delta))$, we have the following formulas.

######  Lemma 6.3.

For any $m\in\delta+{\mathbb{Z}},f(L_{0})\in{\mathbb{C}}[L_{0}]$, we have

$\displaystyle\psi_{m}\cdot
f(L_{0})\mathbf{1}_{0}=f_{m}(L_{0})f(L_{0}+m)\mathbf{1}_{1},$
$\displaystyle\psi_{m}\cdot
f(L_{0})\mathbf{1}_{1}=g_{m}(L_{0})f(L_{0}+m)\mathbf{1}_{0}.$

######  Lemma 6.4.

If $g_{r}(L_{0})=0$ (resp. $f_{r}(L_{0})=0$) for some
$r\in\delta+{\mathbb{Z}}$, then $g_{m}(L_{0})=0$ (resp. $f_{m}(L_{0})=0$) for
all $m\in\delta+{\mathbb{Z}}$.

###### Proof.

Suppose $g_{r}(L_{0})=0$. For any $n\in{\mathbb{Z}}$ we have

$(-r-\frac{n}{2})g_{r+n}(L_{0})\mathbf{1}_{0}=(-r-\frac{n}{2})\psi_{r+n}\cdot\mathbf{1}_{1}=[L_{n},\psi_{r}]\cdot\mathbf{1}_{1}=L_{n}\psi_{r}\mathbf{1}_{1}-\psi_{r}L_{n}\mathbf{1}_{1}=0.$

So $g_{r+n}(L_{0})=0$ for all $n\in{\mathbb{Z}}\setminus\\{-2r\\}$. Then

$-\frac{1}{2}g_{-r}(L_{0})\mathbf{1}_{0}=-\frac{1}{2}\psi_{-r}\cdot\mathbf{1}_{1}=[L_{-2r-1},\psi_{r+1}]\cdot\mathbf{1}_{1}=0.$

Therefore, $g_{-r}(L_{0})=0$. So $g_{m}(L_{0})=0$ for all
$m\in\delta+{\mathbb{Z}}$.

Similarly, if $f_{r}(L_{0})=0$, then $f_{m}(L_{0})=0$ for all
$m\in\delta+{\mathbb{Z}}$. ∎

By Lemma 6.3, for any $m\in\delta+{\mathbb{Z}}$ we have

$0=\psi_{m}^{2}\cdot\mathbf{1}_{0}=g_{m}(L_{0})f_{m}(L_{0}+m)\mathbf{1}_{0}.$

So $g_{m}(L_{0})f_{m}(L_{0}+m)=0$. Since the polynomial algebra
${\mathbb{C}}[L_{0}]$ has no nonzero zero divisors, we have $g_{m}(L_{0})=0$
or $f_{m}(L_{0}+m)=0$. Without loss of generality, assume that
$g_{m}(L_{0})=0$. Then $f_{m}(L_{0})\neq 0$. In fact, if $f_{m}(L_{0})=0$,
then $f_{r}(L_{0})=g_{r}(L_{0})=0$ for all $r\in\delta+{\mathbb{Z}}$ by Lemma
6.4. This contradicts with $\mathcal{S}(\delta)_{\bar{1}}V\neq 0$.

For any $m\in\delta+{\mathbb{Z}}$, let $d_{m}=\deg(f_{m})$.

######  Lemma 6.5.

(1). $a^{\prime}-a=-d_{m}-\frac{1}{2}$ for any $m\in\delta+{\mathbb{Z}}$; (2).
$\lambda^{\prime}=\lambda.$

###### Proof.

(1). For any $m\in\delta+{\mathbb{Z}}$ we have

$\begin{split}0=&[L_{-2m},\psi_{m}]\cdot\mathbf{1}_{0}\\\
=&L_{-2m}\psi_{m}\cdot\mathbf{1}_{0}-\psi_{m}L_{-2m}\cdot\mathbf{1}_{0}\\\
=&L_{-2m}\cdot
f_{m}(L_{0})\mathbf{1}_{1}-\psi_{m}\cdot\lambda^{-2m}(L_{0}-2ma)\mathbf{1}_{0}\\\
=&\lambda^{\prime-2m}(L_{0}-2ma^{\prime})f_{m}(L_{0}-2m)\mathbf{1}_{1}-\lambda^{-2m}(L_{0}+m-2ma)f_{m}(L_{0})\mathbf{1}_{1}.\end{split}$

So

(6.1)
$\lambda^{\prime-2m}(L_{0}-2ma^{\prime})f_{m}(L_{0}-2m)=\lambda^{-2m}(L_{0}+m-2ma)f_{m}(L_{0})\in{\mathbb{C}}[L_{0}].$

Comparing the coefficients of $L_{0}^{d_{m}+1}$ in (6.1), we get
$\lambda^{\prime-2m}=\lambda^{-2m}$ for any $m\in\delta+{\mathbb{Z}}$. So

$\lambda^{\prime 2}=\lambda^{2}\text{ if }\delta=0,\text{ and
}\lambda^{\prime}=\lambda\text{ if }\delta=\frac{1}{2}.$

Let $f_{m}(L_{0})=\mu(L_{0}-x_{1})\dots(L_{0}-x_{d_{m}})$ for some
$\mu\in{\mathbb{C}}^{*}$ and some $x_{1},\dots,x_{d_{m}}\in{\mathbb{C}}$. From
(6.1) we have

$(L_{0}-2ma^{\prime})(L_{0}-2m-x_{1})\dots(L_{0}-2m-x_{d_{m}})=(L_{0}+m-2ma)(L_{0}-x_{1})\dots(L_{0}-x_{d_{m}}).$

Then
$2ma^{\prime}+(2m+x_{1})+\dots+(2m+x_{d_{m}})=(2ma-m)+x_{1}+\dots+x_{d_{m}},$
i. e.,

$a^{\prime}-a=-d_{m}-\frac{1}{2},\forall m\in\delta+{\mathbb{Z}}.$

(2). If $\delta=\frac{1}{2}$ we already have $\lambda^{\prime}=\lambda$. Now
suppose $\delta=0$. By (1), $d_{1}=d_{0}$. Note that

$\begin{split}-\frac{1}{2}f_{1}(L_{0})\mathbf{1}_{1}=&[L_{1},\psi_{0}]\cdot\mathbf{1}_{0}\\\
=&L_{1}\cdot
f_{0}(L_{0})\mathbf{1}_{1}-\psi_{0}\cdot\lambda(L_{0}+a)\mathbf{1}_{0}\\\
=&\lambda^{\prime}(L_{0}+a^{\prime})f_{0}(L_{0}+1)\mathbf{1}_{1}-\lambda(L_{0}+a)f_{0}(L_{0})\mathbf{1}_{1}.\end{split}$

Comparing the coefficients of $L_{0}^{d_{0}+1}$ we get
$\lambda^{\prime}=\lambda$. ∎

######  Lemma 6.6.

Suppose $m\in\delta+{\mathbb{Z}}$ and $n\in{\mathbb{Z}}$. Then

$(-m-\frac{n}{2})f_{m+n}(L_{0})=\lambda^{n}(L_{0}+na^{\prime})f_{m}(L_{0}+n)-\lambda^{n}(L_{0}+m+na)f_{m}(L_{0}).$

###### Proof.

This follows from the following computation:

$\begin{split}(-m-\frac{n}{2})&f_{m+n}(L_{0})\mathbf{1}_{1}=(-m-\frac{n}{2})\psi_{m+n}\cdot\mathbf{1}_{0}\\\
=&L_{n}\psi_{m}\cdot\mathbf{1}_{0}-\psi_{m}L_{n}\cdot\mathbf{1}_{0}\\\
=&L_{n}\cdot
f_{m}(L_{0})\mathbf{1}_{1}-\psi_{m}\cdot\lambda^{n}(L_{0}+na)\mathbf{1}_{0}\\\
=&\lambda^{n}(L_{0}+na^{\prime})f_{m}(L_{0}+n)\mathbf{1}_{1}-\lambda^{n}(L_{0}+m+na)f_{m}(L_{0})\mathbf{1}_{1}.\end{split}$

So

$(-m-\frac{n}{2})f_{m+n}(L_{0})=\lambda^{n}(L_{0}+na^{\prime})f_{m}(L_{0}+n)-\lambda^{n}(L_{0}+m+na)f_{m}(L_{0}).$

∎

Write $d=d_{m}$ for any $m\in\delta+{\mathbb{Z}}$.

######  Lemma 6.7.

Suppose $d>0$. For any $m\in\delta+{\mathbb{Z}}$ with $m>0$, the polynomial
$f_{m}(x)$ has roots $x_{1},\dots,x_{d}\in{\mathbb{C}}$ such that
$x_{i+1}-x_{i}=2m,i=1,\dots,d-1$, and that $x_{1}=2ma^{\prime},x_{d}=2ma-3m$.

###### Proof.

Let $x_{1},\dots,x_{d}$ be the roots of $f_{m}(x)$ with the positive part of
$x_{i}$ being smaller or equal to $x_{i+1}$, $i=1,\dots,d-1$. From (6.1) we
have

$(L_{0}-2ma^{\prime})f_{m}(L_{0}-2m)=(L_{0}+m-2ma)f_{m}(L_{0}).$

Then we have

$\\{2ma^{\prime},x_{1}+2m,\dots,x_{d}+2m\\}=\\{2ma-m,x_{1},\dots,x_{d}\\}.$

It is easy to see that $x_{1}\notin\\{x_{1}+2m,\dots,x_{d}+2m\\}$ and that
$x_{d}+2m\notin\\{x_{1},\dots,x_{d}\\}$. So $x_{1}=2ma^{\prime}$ and
$x_{d}+2m=2ma-m$. By comparing the real parts of elements in these two sets,
we have

$x_{1}+2m=x_{2},\dots,x_{d-1}+2m=x_{d}.$

∎

######  Lemma 6.8.

$d\leqslant 2$.

###### Proof.

Let $r\in\delta+{\mathbb{Z}}$ with $r>0$.

Suppose $d\geqslant 3$. By Lemma 6.7, $f_{r}(x)$ has roots $x_{1},\dots,x_{d}$
such that $x_{i+1}-x_{i}=2r,i=1,\dots,d-1$. Then $f_{r}(x+1)$ has roots
$x_{1}-2r,x_{1},\dots,x_{d-1}$.

Let $m=r$ and $n=1$ in Lemma 6.6 and we get

(6.2)
$(-r-\frac{1}{2})f_{r+1}(L_{0})=\lambda(L_{0}+a^{\prime})f_{r}(L_{0}+1)-\lambda(L_{0}+r+a)f_{r}(L_{0}).$

Then $x_{1},x_{2}$ are roots of $f_{r+1}(x)$. By Lemma 6.7,

$2r=x_{2}-x_{1}\in(2r+2){\mathbb{Z}},$

which is a contradiction. So $d\leqslant 2$. ∎

Fix some $\sqrt{\lambda}\in{\mathbb{C}}^{*}$ with
$(\sqrt{\lambda})^{2}=\lambda$, and set
$\lambda^{r}=\lambda^{r-\frac{1}{2}}\sqrt{\lambda}$ for any
$r\in\frac{1}{2}+{\mathbb{Z}}$.

######  Lemma 6.9.

Suppose $d=2$. Then $a=1$ or $\frac{5}{2}$, and $V$ is isomorphic to
$\Omega^{\prime\prime}_{\mathcal{S}(\delta)}(\lambda,b)$ or
$\tilde{\Omega}^{\prime\prime}_{\mathcal{S}(\delta)}(\lambda,b)$ for some
$b\in{\mathbb{C}}^{*}$.

###### Proof.

Let $r\in\delta+{\mathbb{Z}}$ with $r>0$. Suppose $d=2$. By Lemma 6.7, the
roots of $f_{r}(x)$ are $2ra^{\prime}$ and $2ra-3r$. Then

$f_{r}(x)=\mu(x-2ra^{\prime})(x-2ra+3r)$

for some $\mu\in{\mathbb{C}}^{*}$. From (6.2) we have

(6.3)
$\begin{split}(-r-\frac{1}{2})f_{r+1}(L_{0})=&\mu\lambda(L_{0}+a^{\prime})(L_{0}+1-2ra^{\prime})(L_{0}+1-2ra+3r)\\\
&-\mu\lambda(L_{0}+r+a)(L_{0}-2ra^{\prime})(L_{0}-2ra+3r).\end{split}$

By Lemma 6.7, the roots of $f_{r+1}(x)$ are $(r+1)(2a-5)$ and $(r+1)(2a-3)$.
Then

$0=(-r-\frac{1}{2})f_{r+1}((r+1)(2a-5))=\mu\lambda(a-1)(2a-5)(2r+1).$

So $a=1$ or $\frac{5}{2}$.

Let $b=\mu\lambda^{-r}$. Then

$f_{r}(x)=b\lambda^{r}(x-2ra^{\prime})(x-2ra+3r).$

From Lemma 6.6 we have

$f_{m}(L_{0})=b\lambda^{m}(L_{0}+3m)(L_{0}+m),\forall m\in\delta+{\mathbb{Z}}$

if $a=1$, and

$f_{m}(L_{0})=b\lambda^{m}L_{0}(L_{0}-2m),\forall m\in\delta+{\mathbb{Z}}$

if $a=\frac{5}{2}$. By the definition of
$\Omega^{\prime\prime}_{\mathcal{S}(\delta)}(\lambda,b)$ and
$\tilde{\Omega}^{\prime\prime}_{\mathcal{S}(\delta)}(\lambda,b)$, $V$ is
isomorphic to $\Omega^{\prime\prime}_{\mathcal{S}(\delta)}(\lambda,b)$ or
$\tilde{\Omega}^{\prime\prime}_{\mathcal{S}(\delta)}(\lambda,b)$ as
$\mathcal{S}(\delta)$-modules. ∎

######  Lemma 6.10.

Suppose $d=1$. Then $V$ is isomorphic to
$\Omega^{\prime}_{\mathcal{S}(\delta)}(\lambda,a,b)$ for some
$b\in{\mathbb{C}}^{*}$.

###### Proof.

Let $r\in\delta+{\mathbb{Z}}$ with $r>0$. By Lemma 6.7, the root of $f_{r}(x)$
is $2ra-3r$. Then

$f_{r}(L_{0})=\mu(L_{0}-2ra+3r)$

for some $\mu\in{\mathbb{C}}^{*}$.

By Lemma 6.6, for any $n\in{\mathbb{Z}}$ we have

$\begin{split}(-r-\frac{n}{2})f_{n+r}(L_{0})=&\lambda^{n}(L_{0}+na^{\prime})f_{r}(L_{0}+n)-\lambda^{n}(L_{0}+na+r)f_{r}(L_{0})\\\
=&\lambda^{n}\mu(L_{0}+na^{\prime})(L_{0}-2ra+3r+n)\\\
&-\lambda^{n}\mu(L_{0}+na+r)(L_{0}-2ra+3r)\\\
=&\lambda^{n}\mu(-r-\frac{n}{2})(L_{0}-(2a-3)(n+r)).\end{split}$

So

$f_{n+r}(L_{0})=\lambda^{n}\mu(L_{0}-(2a-3)(n+r))$

if $n\neq-2r$. In particular,

$f_{3r}(L_{0})=\lambda^{2r}\mu(L_{0}-3(2a-3)r).$

Let $m=3r,n=-4r$ in Lemma 6.6 and we get

$\begin{split}-rf_{-r}(L_{0})=&\lambda^{-4r}(L_{0}-4ra^{\prime})f_{3r}(L_{0}-4r)-\lambda^{-4r}(L_{0}+3r-4ra)f_{3r}(L_{0})\\\
=&-r\mu\lambda^{-2r}(L_{0}+2ra-3r).\end{split}$

Hence,

$f_{n+r}(L_{0})=\lambda^{n}\mu(L_{0}-2(n+r)a+3(n+r)),\forall
n\in{\mathbb{Z}}.$

Then

$f_{m}(L_{0})=b\lambda^{m}(L_{0}-2ma+3m),\forall m\in\delta+{\mathbb{Z}},$

where $b=\mu\lambda^{-r}$. By the definition of
$\Omega^{\prime}_{\mathcal{S}(\delta)}(\lambda,a,b)$, $V$ is isomorphic to
$\Omega^{\prime}_{\mathcal{S}(\delta)}(\lambda,a,b)$ as
$\mathcal{S}(\delta)$-modules. ∎

######  Lemma 6.11.

Suppose $d=0$. Then $V$ is isomorphic to
$\Omega_{\mathcal{S}(\delta)}(\lambda,a,b)$ for some $b\in{\mathbb{C}}^{*}$.

###### Proof.

Let $r\in\delta+{\mathbb{Z}}\setminus\\{0\\}$. Then
$f_{r}(L_{0})=b\lambda^{r}$ for some $b\in{\mathbb{C}}^{*}$. By Lemma 6.6, for
any $n\in{\mathbb{Z}}$ we have

$(-r-\frac{n}{2})f_{r+n}(L_{0})=b\lambda^{n+r}(L_{0}+na^{\prime})-b\lambda^{n+r}(L_{0}+r+na)=b\lambda^{n+r}(-r-\frac{n}{2}).$

So $f_{r+n}(L_{0})=b\lambda^{n+r}$ for any $n\neq-2r$. In particular,
$f_{3r}(L_{0})=b\lambda^{3r}$. Let $m=3r,n=-4r$ in Lemma 6.6 and we get

$-rf_{-r}(L_{0})=\lambda^{-4r}(L_{0}-4ra^{\prime})f_{3r}(L_{0}-4r)-\lambda^{-4r}(L_{0}+3r-4ra)f_{3r}(L_{0})=-rb\lambda^{-r}.$

Hence

$f_{m}(L_{0})=b\lambda^{m},\forall m\in\delta+{\mathbb{Z}}.$

By the definition of $\Omega_{\mathcal{S}(\delta)}(\lambda,a,b)$, $V$ is
isomorphic to $\Omega_{\mathcal{S}(\delta)}(\lambda,a,b)$ as
$\mathcal{S}(\delta)$-modules. ∎

From the discussion above, we have

######  Theorem 6.12.

Suppose $V$ is an $\mathcal{S}(\delta)$-module such that $V_{\bar{0}}$ and
$V_{\bar{1}}$ are free ${\mathbb{C}}[L_{0}]$-modules of rank $1$ by
restriction. If $\mathcal{S}(\delta)_{\bar{1}}V\neq 0$, then $V$ is isomorphic
to $\Omega_{\mathcal{S}(\delta)}(\lambda,a,b)$,
$\Omega^{\prime}_{\mathcal{S}(\delta)}(\lambda,a,b),\Omega^{\prime\prime}_{\mathcal{S}(\delta)}(\lambda,b)$
or $\tilde{\Omega}^{\prime\prime}_{\mathcal{S}(\delta)}(\lambda,b)$ up to a
parity-change for some $\lambda,b\in{\mathbb{C}}^{*}$ and some
$a\in{\mathbb{C}}$.

It is easy to see that all these $\mathcal{S}(\delta)$-modules are not simple
since $V_{\bar{1}}$ is a $\mathcal{S}(\delta)$-submodule of $V$.

We conclude this paper with the following questions.

  * •

Is there any way to construct new simple smooth $\mathcal{S}(\delta)$-modules
of nonzero level? Or can we classify such $\mathcal{S}(\delta)$-modules?

  * •

Is there any way to construct new simple $F(\delta)$-modules of nonzero level?
Or can we classify such $\mathcal{S}(\delta)$-modules?

Acknowledgement. The authors would like to thank Rencai Lü and Yao Ma for
helpful discussions. Part of the research in this paper was carried out during
the visit of the first author at Wilfrid Laurier University in 2022. The first
author is partially supported by CSC of China (No. 202006920055), and National
Natural Science Foundation of China (Grants No. 11971440). The second author
is partially supported by National Natural Science Foundation of China (Grant
No. 11871190) and NSERC (311907-2020).

## References

  * [1] E. Arbarello, C. De Concini, V. G. Kac, C. Procesi, Moduli spaces of curves and representation theory, Comm. Math. Phys. 117 (1988), no. 1, 1-36.
  * [2] Y. Billig, V. Futorny, Classification of irreducible representations of Lie algebra of vector fields on a torus, J. Reine Angew. Math. 720 (2016), 199-216.
  * [3] Y. Cai, R. Lü, Y. Wang, Classification of simple Harish-Chandra modules for map (super)algebras related to the Virasoro algebra, J. Algebra 570 (2021), 397-415.
  * [4] M. Dilxat, L. Chen, D. Liu, Classification of simple Harish-Chandra modules over the Ovsienko-Roger superalgebra, Proceedings of the Royal Society of Edinburgh Section A: Mathematics, 1-11, doi:10.1017/prm.2023.20.
  * [5] V. Kac, A. Raina, N. Rozhkovskaya, Bombay lectures on highest weight representations of infinite dimensional Lie algebras. Second edition. Advanced Series in Mathematical Physics, 29. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2013. xii+237 pp.
  * [6] D. Kazhdan, G. Lusztig, Tensor structures arising from affine Lie algebras, I. II, J. Amer. Math. Soc. 6 (1993), no. 4, 905-947, 949-1011.
  * [7] D. Liu, Y. Pei, L. Xia, Simple restricted modules for Neveu-Schwarz algebra, J. Algebra, 546 (2020), 341-356.
  * [8] D. Liu, Y. Pei, L. Xia, K. Zhao, Simple smooth modules over the superconformal current algebra, arXiv:2305.16662.
  * [9] R. Lü, K. Zhao, Irreducible Virasoro modules from irreducible Weyl modules, J. Algebra 414 (2014), 271–287.
  * [10] R. Lü, K. Zhao, Generalized oscillator representations of the twisted Heisenberg-Virasoro algebra, Algebr. Represent. Theory 23 (2020), no. 4, 1417-1442.
  * [11] C. Martin, A. Piard, Nonbounded indecomposable admissible modules over the Virasoro algebra, Lett. Math. Phys. 23 (1991), no. 4, 319-324.
  * [12] O. Mathieu, Classification of Harish-Chandra modules over the Virasoro Lie algebra, Invent. Math. 107 (1992), no. 2, 225-234.
  * [13] V. Mazorchuk, K. Zhao, Simple Virasoro modules which are locally finite over a positive part, Selecta Math.(N.S.) 20 (2014), 839-854.
  * [14] H. Tan, Y. Yao, K. Zhao, Simple restricted modules over the Heisenberg-Virasoro algebra as VOA modules, arXiv:2110.05714.
  * [15] H. Tan, K. Zhao, $\mathcal{W}_{n}^{+}$\- and $\mathcal{W}_{n}$-module structures on $U(\mathfrak{h}_{n})$, J. Algebra 424(2015), 357–375.

Y. Xue: School of Sciences, Nantong University, Nantong, Jiangsu, 226019, P.
R. China.

Email address<EMAIL_ADDRESS>
K. Zhao: Department of Mathematics, Wilfrid Laurier University, Waterloo, ON,
Canada N2L 3C5, and School of Mathematical Science, Hebei Normal (Teachers)
University, Shijiazhuang, Hebei, 050024 P. R. China.

Email address<EMAIL_ADDRESS>


# Attention Augmented Convolutional Transformer for Tabular Time-series

Sharath M Shankaranarayana1 and Davor Runje1,2 1 Airt Research, Zagreb,
Croatia2 Algebra University College, Zagreb, Croatia

###### Abstract

Time-series classification is one of the most frequently performed tasks in
industrial data science, and one of the most widely used data representation
in the industrial setting is tabular representation. In this work, we propose
a novel scalable architecture for learning representations from tabular time-
series data and subsequently performing downstream tasks such as time-series
classification. The representation learning framework is end-to-end, akin to
bidirectional encoder representations from transformers (BERT) in language
modeling, however, we introduce novel masking technique suitable for
pretraining of time-series data. Additionally, we also use one-dimensional
convolutions augmented with transformers and explore their effectiveness,
since the time-series datasets lend themselves naturally for one-dimensional
convolutions. We also propose a novel timestamp embedding technique, which
helps in handling both periodic cycles at different time granularity levels,
and aperiodic trends present in the time-series data. Our proposed model is
end-to-end and can handle both categorical and continuous valued inputs, and
does not require any quantization or encoding of continuous features.

##  I Introduction

Industrial entities in domains such as finance, telecommunication, and
healthcare usually log a large amount of data of their customers or patients.
The data is typically in the form of events data, capturing interactions their
users have with different entities. The events could be specific to the
respective domains, for example, financial institutions log all financial
transactions made by their customers, telecommunication companies log all the
interactions of the individual customers, national healthcare systems keep
logs of all visits and diagnoses patients had in different healthcare
institutions, etc. These kinds of data mostly employ tabular representation. A
multivariate time-series data represented in tabular form is often referred to
as dynamic tabular data [1] or tabular time-series. These datasets are rich
for performing knowledge discovery, doing analytics, and also building
predictive models using machine learning (ML) techniques. Even though there
exists a large amount of data, the data often remains unexplored due to
various inherent difficulties (such as unstructuredness, noise, sparse and
missing values). Because of the complexities, usually only a subset of the
data is employed for building ML models. Thus, a large amount of potentially
rich and relevant data is unused for ML modeling. Moreover, machine learning
using such data is frequently performed by first extracting hand-crafted or
engineered features and later building task-specific machine learning models.
This engineering of the features is known to be one of the biggest challenges
in industrial machine learning, since it requires domain knowledge and time-
consuming experimentation for every different use case. Additionally, it has
limited scalability because models cannot be automatically created and
maintained. Another important issue is that most machine learning models
require data to be in the form of fixed-length vectors or sequences of
vectors. This is not straightforward and is difficult to obtain for such large
time-series datasets. To this end, we propose a framework for learning vector
representations from large time-series data. Our framework can later employ
these learned representations on downstream tasks such as time-series
classification.

Learning numerical vector representations (embeddings), or representation
learning, is one the major areas of research, specifically in the natural
language processing (NLP) domain. In the seminal work called word2vec [2],
vector representations of words are learnt from huge quantities of text. In
word2vec, each word is mapped to a $d$-dimensional vector such that
semantically similar words have geometrically closer vectors. This is achieved
by predicting either the context words appearing in a window around a given
target word (skip-gram model), or the target word given the context (called
continuous bag of words or CBOW model). In this model, it is assumed that the
words appearing frequently in similar contexts share statistical properties
and thus this model leverages word co-occurrence statistics.

NLP has since then seen massive improvements in terms of results by building
upon embeddings-related works. For example, the work [3] proposed a transfer
learning scheme for text classification and thus heralded a new era by making
the transfer learning in NLP on par with the transfer learning tasks in
computer vision. Another recent work [4] proposed contextualized word
representations, as opposed to static word representations.

The current state-of-the-art techniques in NLP learn vector representations of
words using transformers and attention mechanism [5] on large datasets with
the task of reconstructing an input text with some of the words in it randomly
masked [6].

Figure 1: Attention-augmented convolution: For each temporal location, $N_{h}$
attention maps over the image are computed from queries and keys. These
attention maps are used to compute $N_{h}$ weighted averages of the values
$V$. The results are then concatenated, reshaped to match the original input’s
dimensions and mixed with a pointwise convolution. Multi-head attention is
applied in parallel to a standard convolution operation and the outputs are
concatenated.

Upon seeing the immense gains of employing models with the attention mechanism
and transformers in NLP, there have been a few works employing them for
tabular data. In the work [7], the authors propose a network called TabNet,
which is a transformer based model for self-supervised learning for tabular
data. They report significant gains in prediction accuracy when the labelled
data is sparse, however, the work does not learn any sort of embeddings
inherently and translates the use of transformer architecture from NLP domain
to tabular domain. Another similar very recent work [8] proposes similar use
of transformers for the tabular data. In this work, although the authors do
intend to learn embeddings, it is only for encoding categorical features
present in the tabular data. Whereas in our work, we learn the embeddings for
specific agents by contextualizing the interactions between the various
agents. Thus, the embeddings we obtain are very informative, unlike broad
encoding of categorical values. Most recently, the authors in [9] proposed the
use of BERT architecture modified for time-series anomaly detection task.

To the best of our knowledge, the closest work to ours is TabBERT proposed in
the work [1], in which the authors employ hierarchical BERT [10, 11] to first
perform pretraining on time-series data and then later employ the embeddings
on downstream tasks. Although similar, our proposed work differs in several
aspects:

  * •

We do not employ hierarchical BERT but single encoder layer consisting of
transformers, and make use of embeddings to transform raw inputs in order to
be fed to BERT, making its computational cost significantly smaller.

  * •

We propose a masking technique specifically suited for time-series data
resulting in higher performance of the model.

  * •

TabBERT requires input data to be already encoded as categorical values,
whereas our proposed framework can handle both discrete and continuous inputs.

Another relevant work to ours is [12], where the authors propose transformers
for time-series representation learning, but the pretraining loss function
employed is mean squared error loss for both continuous and discrete features,
whereas in our method we apply classification loss for discrete type features
and regression loss for floating features. To the best of our knowledge, ours
is the first work that employs attention augmented convolutions [13] (as a
part of BERT layer) for time-series data and also the first work that proposes
timestamp embedding block.

In summary, the main contributions of our paper are as follows:

  * •

We propose a novel BERT framework employing attention augmented convolutions
for time-series tabular data TabAConvBERT.

  * •

We propose a masking scheme that is better suitable for time-series tabular
data.

  * •

We propose a novel timestamp embedding block along with positional encoding.
This timestamp embedding block helps in handling of both periodic cycles at
different time granularity levels and aperiodic trends.

  * •

The proposed framework can handle both discrete and continuous input types.

Figure 2: Our proposed architecture of TabAConvBERT

##  II Methodology

###  II-A Attention Augmented Convolution

One-dimensional (1-D) convolutions have long been employed in time-series
tasks and have shown reasonably good results [14]. In computer vision,
employing attention mechanisms along with convolutional neural networks have
given significant improvements in various vision tasks [15, 16], [17, 18].
Recently, the authors of the work [13] proposed attention augmented
convolutional networks that can jointly attend to both spatial and feature
subspaces, in contrast to previous convolutional attention mechanisms, which
either perform only channel-wise reweighing [15, 16] or perform reweighing of
both channels and spatial positions independently[17, 18]. In this work, we
propose a 1-D convolution version of attention augmented convolutions [13].
Similar to [13], the proposed attention augmented convolution

  1. 1.

is equivariant to translation, and

  2. 2.

can readily operate on inputs of different temporal dimensions.

As shown in [13], if we consider an original convolution operator with kernel
size $k$, $F_{in}$ input filters, and $F_{out}$ output filters, the
corresponding attention augmented convolution can simply be written as:

$AAConv(X)=Concat[Conv(X),MHA(X)]$ (1)

where $Conv$ denotes convolution and $MHA$ denotes multi-head attention.

###  II-B Architecture

The proposed architecture is shown in Fig. 2. The architecture consists of
embedding network, which can encode both categorical inputs (by using simple
embedding neural network) and continuous inputs (by using simple shallow
neural network). The architecture also consists of a special timestamp
embedding block, which is described in more detail in Fig. 3. We first break
the original raw timestamp into multiple components such as year, month, day,
weekday, week, hour, minute and seconds. These broken up components are
discrete and have a finite set of values. For example, month feature has $12$
values, week feature has $52$ values, weekday feature has $7$ values, and so
on. Each of these broken up discrete features are passed through an embedding
layer. The outputs of each of the embedding layers are then summed-up.
Additionally, we also create normalized timestamp features based on dates and
time. These float values are then passed through a shallow neural network with
“activity regularization” to obtain the vector representations having the same
output dimensions of embedding representation. The resulting outputs from
these shallow neural networks are also added to the summed-up embedding vector
to obtain the final time embedding. The obtained time embedding is then added
with input features’ embedding and also positional encoding. The resulting
summed value is then passed to attention augmented convolution layer and feed
forward neural network layer. This combination of attention augmented
convolution layer and feed forward neural network layer can be stacked $N$
times to obtain the encoder layer of BERT. Finally, the dense layers in the
output part of the network are added to aid pretraining or downstream tasks.

Figure 3: Our proposed Timestamp Embedding Block

###  II-C Masking

For learning representations from the time-series tabular data, we propose a
procedure called masked data modeling (MDM), akin to masked language modeling
(MLM) employed in NLP. One of the foremost steps in pretraining, as the name
suggests, is masking. In MLM, masking is straightforward, since the languages
contain only sequence of words. However, in multivariate time-series data, we
propose two kinds of masking. For the first kind of masking, we mask out
certain percentage of the features at random from the tabular time-series
data. This masking is done independently for all the features and is similar
to the one performed in [7]. For the second kind of masking, we randomly mask
out certain percentage of entire rows of features for the time-series inputs.
Almost all the previous works perform only first kind of masking for the
tabular data. But only masking out individual features may not be very
effective since often the features are slightly correlated and hence it
becomes easier for MDM to predict missing features. The masking of entire row
is similar to masking a word in MLM since, in tabular data, the analogue of a
”word” is an entire row of features. The figure Fig. 4 gives a clear picture
of two types of masking employed in our work.

Figure 4: Our proposed Masking methodology

Additionally, different from other previous works, our framework has the
ability to handle continuous type of inputs as is, without resorting to
binning. Although for masking the categorical data we can simply have a
specific integer token similar to MASK token employed in MLM, for continuous
type of inputs, we mask the data by replacing the original values with the
mean value of the particular continuous feature.

##  III Experiments and Results

For our experiments, we employ the dataset provided in the work [1]. The
dataset has 24 million transactions from 20,000 users. Each transaction (row)
has 12 fields (columns) consisting of both continuous and discrete nominal
attributes, such as merchant name, merchant address, transaction amount, etc.
For easier comparison, we employ the similar procedures for sampling as
performed in [1] by creating samples as sliding windows of 10 transactions,
with a stride of 5. The strategies employed in masking are different since we
propose custom masking methodology, however, we perform an experiment by
quantizing and creating a vocabulary same as [1]. For masking, we perform
$30\%$ masking of sample’s fields and $15\%$ masking of sample’s entire rows.
While performing pretraining, we omit the label column to prevent biasing. In
our proposed TabAConvBERT architecture, we only employ a single AAConvBERT
layer to keep the number of parameters low.

For the downstream task of fraud classification, we again perform similar
procedures by creating samples by combining 10 contiguous rows (with a stride
of 10) in a time-dependent manner for each user, and thus obtain 2.4M samples,
with 29,342 labeled as fraudulent. For evaluation, we use F1 binary score, on
a test set consisting of 480K samples for better comparison. For the
downstream task, we remove the final dense layers employed during the
pretraining stage and a single dense layer for binary classification. For
downstream classification task, after pretraining, we freeze all embedding
layers and finetune the layers from attention augmented convolution layer. We
perform three different kinds of experiments. First, we employ the same
quantization and same masking employed in [1], with the main difference being
the BERT architecture having attention augmented convolutional layers. Next,
we employ our proposed masking scheme but keep the same quantized data. This
helps in gauging the improvements offered by our masking scheme. Lastly, we do
not perform any encoding and keep all the continuous valued features as is,
and also employ timestamp encoding as described in the previous section. Since
we employ the same data partitions as described in [1], we directly compare
our results from their paper.

As seen from Table I, the proposed methods give significantly improved results
on the downstream fraud classification task. The improvements can be seen by
directly employing our TabAConvBERT architecture on the quantized or encoded
data. The masking procedure also helps in providing a slight improvement in
the F1 score. The best results can be seen from our TabAConvBERT with no data
preprocessing, i.e., keeping the continuous and categorical values as is and
employing timestamp embedding.

##  IV Conclusion

Method | Fraud F1 Score
---|---
Baseline MLP [1] | 0.74
Baseline LSTM [1] | 0.73
TabBERT MLP [1] | 0.83
TabBERT LSTM [1] | 0.86
Proposed TabAConvBERT (Only Encoded Data) | 0.888
Proposed TabAConvBERT (With custom masking) | 0.892
Proposed TabAConvBERT | 0.896
TABLE I: Comparison of various methods

In this work, we proposed a novel end-to-end BERT based architecture for time-
series tasks. For the first time, we proposed the use of attention augmented
convolutions for tabular time-series data and also proposed major
modifications to the masking methodology for tabular time-series data. From
our experiments, we showed that each of the individual modifications lead to
improved results. Our method has a major advantage that it can be directly
used with raw data, without resorting to techniques such as feature
quantization or encoding. In future, we would like to rigorously evaluate on
even larger scale industrial datasets.

## References

  * [1] I. Padhi, Y. Schiff, I. Melnyk, M. Rigotti, Y. Mroueh, P. Dognin, J. Ross, R. Nair, and E. Altman, “Tabular transformers for modeling multivariate time series,” in _ICASSP 2021-2021 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)_. IEEE, 2021, pp. 3565–3569.
  * [2] T. Mikolov, K. Chen, G. Corrado, and J. Dean, “Efficient estimation of word representations in vector space,” _arXiv preprint arXiv:1301.3781_ , 2013\.
  * [3] J. Howard and S. Ruder, “Universal language model fine-tuning for text classification,” _arXiv preprint arXiv:1801.06146_ , 2018.
  * [4] M. E. Peters, M. Neumann, M. Iyyer, M. Gardner, C. Clark, K. Lee, and L. Zettlemoyer, “Deep contextualized word representations,” _arXiv preprint arXiv:1802.05365_ , 2018.
  * [5] A. Vaswani, N. Shazeer, N. Parmar, J. Uszkoreit, L. Jones, A. N. Gomez, L. Kaiser, and I. Polosukhin, “Attention is all you need,” _arXiv preprint arXiv:1706.03762_ , 2017.
  * [6] J. Devlin, M.-W. Chang, K. Lee, and K. Toutanova, “Bert: Pre-training of deep bidirectional transformers for language understanding,” _arXiv preprint arXiv:1810.04805_ , 2018.
  * [7] S. O. Arik and T. Pfister, “Tabnet: Attentive interpretable tabular learning,” _arXiv preprint arXiv:1908.07442_ , 2019.
  * [8] X. Huang, A. Khetan, M. Cvitkovic, and Z. Karnin, “Tabtransformer: Tabular data modeling using contextual embeddings,” _arXiv preprint arXiv:2012.06678_ , 2020.
  * [9] W. Dang, B. Zhou, L. Wei, W. Zhang, Z. Yang, and S. Hu, “Ts-bert: Time series anomaly detection via pre-training model bert,” in _International Conference on Computational Science_. Springer, 2021, pp. 209–223.
  * [10] X. Zhang, F. Wei, and M. Zhou, “Hibert: Document level pre-training of hierarchical bidirectional transformers for document summarization,” _arXiv preprint arXiv:1905.06566_ , 2019.
  * [11] R. Pappagari, P. Zelasko, J. Villalba, Y. Carmiel, and N. Dehak, “Hierarchical transformers for long document classification,” in _2019 IEEE Automatic Speech Recognition and Understanding Workshop (ASRU)_. IEEE, 2019, pp. 838–844.
  * [12] G. Zerveas, S. Jayaraman, D. Patel, A. Bhamidipaty, and C. Eickhoff, “A transformer-based framework for multivariate time series representation learning,” in _Proceedings of the 27th ACM SIGKDD Conference on Knowledge Discovery & Data Mining_, 2021, pp. 2114–2124.
  * [13] I. Bello, B. Zoph, A. Vaswani, J. Shlens, and Q. V. Le, “Attention augmented convolutional networks,” in _Proceedings of the IEEE/CVF international conference on computer vision_ , 2019, pp. 3286–3295.
  * [14] S. Bai, J. Z. Kolter, and V. Koltun, “An empirical evaluation of generic convolutional and recurrent networks for sequence modeling,” _arXiv preprint arXiv:1803.01271_ , 2018.
  * [15] J. Hu, L. Shen, and G. Sun, “Squeeze-and-excitation networks,” in _Proceedings of the IEEE conference on computer vision and pattern recognition_ , 2018, pp. 7132–7141.
  * [16] J. Hu, L. Shen, S. Albanie, G. Sun, and A. Vedaldi, “Gather-excite: Exploiting feature context in convolutional neural networks,” _arXiv preprint arXiv:1810.12348_ , 2018.
  * [17] J. Park, S. Woo, J.-Y. Lee, and I. S. Kweon, “Bam: Bottleneck attention module,” _arXiv preprint arXiv:1807.06514_ , 2018.
  * [18] S. Woo, J. Park, J.-Y. Lee, and I. S. Kweon, “Cbam: Convolutional block attention module,” in _Proceedings of the European conference on computer vision (ECCV)_ , 2018, pp. 3–19.