2. Preliminaries

In this section, we collect some basic definitions and results for our study.

By definition, as a vector space, the Virasoro algebra ${\rm Vir}$ has a basis $\{L_m, C \mid m\in \mathbb {Z}\}$, subject to the following relations:

(2.1)\begin{align} [L_m, L_n]=(n-m)L_{m+n}+\delta_{m+n, 0}{1\over 12}(n^3-n)C, \quad \forall m, n\in\mathbb{Z}. \end{align}

For any ${\rm Vir}$-module $V$ and $\lambda \in \mathbb {C}$, set $V^{\lambda }:=\bigl \{v\in V\bigm |L_0v=\lambda v\bigr \}$, which is generally called the weight space of $V$ corresponding to the weight $\lambda$. A ${\rm Vir}$-module $V$ is called a weight module if $V$ is the sum of all its weight spaces.

For a weight module $V$, we define

(2.2)\begin{equation} \hbox{Supp}(V):=\bigl\{\lambda\in \mathbb{C} \bigm|V^\lambda \neq 0\bigr\}. \end{equation}

Obviously, if $V$ is a weight ${\rm Vir}$-module, then there exists $\lambda \in \mathbb {C}$ such that ${\rm Supp}(V)\subset \lambda +\mathbb {Z}$. So $V=\sum _{i\in \mathbb {Z}}V_i$ is $\mathbb {Z}$-graded, where $V_i:=V^{\lambda +i}$. A weight ${\rm Vir}$-module $V=\sum V_i$ is called Harish–Chandra if all $V_i$ are finite-dimensional. If, in addition, there exists a positive integer $p$ such that

(2.3)\begin{equation} \dim V_i\le p,\quad \forall i\in\mathbb{Z},\end{equation}

the module $V$ is called cuspidal. A cuspidal module $V$ with $p\le 1$ is called a module of the intermediate series.

A ${\rm Vir}$-module $V$ is called a highest (resp. lowest) weight module, if there exists a nonzero $v \in V_{\lambda }$ such that

1. (1) $V$ is generated by $v$ as a ${\rm Vir}$-module with $L_0v=hv$ and $Cv=cv$ for some $h,c\in \mathbb {C}$;

2. (2) ${{\rm Vir}}_+v=0$ (resp. ${{\rm Vir}}_- v=0$), where ${{\rm Vir}}_+=\sum _{i>0}{{\rm Vir}}_i$, ${{\rm Vir}}_-=\sum _{i<0}{{\rm Vir}}_i$.

Clearly highest or lowest weight modules are Harish–Chandra modules. All simple Harish–Chandra modules over the Virasoro algebra were classified in [Reference Mathieu23].

It is well known that the intermediate series ${\rm Vir}$-module is

\[{\mathcal{A}}_{a,\; b}:=\displaystyle\sum_{i\in\mathbb{Z}}\mathbb{C} v_i \ {\rm with} \ L_mv_i=(a+i+bm)v_{m+i}, Cv_i=0, \forall m, i\in\mathbb{Z},\]

where $a, b\in \mathbb {C}$. ${\mathcal {A}}_{a,\; b}$ is simple if and only if $a\not \in \mathbb {Z}$ or $b\ne 0, 1$. As usual, we use $\mathcal {A}_{a,\; b}'$ to denote the irreducible sub-quotient module of ${\mathcal {A}}_{a,\; b}$ (see [Reference Kaplansky and Santharoubane17]). If $a\in \mathbb {Z}$, then ${\mathcal {A}}_{a,\; b}\cong {\mathcal {A}}_{0,\; b}$. So we always suppose that $a\notin \mathbb {Z}$ or $a=0$ in ${\mathcal {A}}_{a,\; b}$. As in [Reference Fuks12, Reference Ovsienko and Roger24], we set ${\mathcal {F}}_\lambda ={\mathcal {A}}_{0,\; \lambda }, \lambda \in \mathbb {C}$, which is called density module of the Virasoro algebra.

Motivated by [Reference Ovsienko and Roger24], for $\epsilon =0, 1/2$, we can define the Ovsienko–Roger superalgebra ${\mathfrak L}(\epsilon ):={\rm Vir}\ltimes \mathcal {F}_{-1/2}$. More precisely, ${\mathfrak L}(\epsilon )$ has a basis $\{L_n, G_r,C| n\in \mathbb {Z}, r\in \mathbb {Z}+\epsilon \}$ with the brackets

\begin{align*} [L_m,L_n]& =(n-m)L_{m+n}+\delta_{m+n,0}\frac{1}{12}(n^3-n)C,\\ [L_m, G_r]& =\left(r-\frac12m\right)G_{r+m},\\ [G_r,G_s]& =0, \quad \forall m, n\in\mathbb{Z}, r, s\in\mathbb{Z}+\epsilon. \end{align*}

Here we shall note that the odd part of ${\mathfrak L}(\epsilon )$ is spanned by $\{G_{n}\mid n\in \mathbb {Z}+\epsilon \}$.

In the case of $\epsilon =0$, ${\mathfrak L}(0)$ can be realized as the affine-Virasoro superalgebra $\mathbb {C} x\otimes \mathbb {C}[t, t^{-1}]\rtimes {\rm Vir}$, where $\mathbb {C} x$ is the one-dimensional abelian Lie superalgebra.

For the Ovsienko–Roger superalgebra, all simple Harish–Chandra modules over ${\mathfrak L}(0)$ were classified in [Reference Cai, Lü and Wang10]. For the case of $\epsilon =1/2$, it is more complicated to classify such modules as in [Reference Cai and Lü9]. This paper gives a uniform new method to consider both cases of $\epsilon =0$ and $\epsilon =1/2$. For convenience, we just write our research for the case of $\epsilon =1/2$. So from now on we denote by ${\mathfrak L}={\mathfrak L}(1/2)$ for short. Clearly, $\mathfrak L$ is a $\frac {1}{2}\mathbb {Z}$-graded Lie superalgebra with ${\mathfrak L}_i=\mathbb {C} L_i\oplus \delta _{0, i}\mathbb {C} C, \forall i\in \mathbb {Z}$ and ${\mathfrak L}_{i+1/2}=\mathbb {C} G_{i+\frac 12}, \forall i\in \mathbb {Z}$. The subalgebra of $\mathfrak L$ spanned by $\{L_i, C| i\in \mathbb {Z}\}$ is isomorphic to the Virasoro algebra ${\rm Vir}$. The following annihilation operator $\Omega _{k, s}^{(m)}$ plays an important role in the classification of simple cuspidal modules over the Virasoro algebra.

3. Simple cuspidal $\mathfrak L$-module

In this section, we shall consider cuspidal $\mathfrak L$-modules. Note that $C$ acts trivially on any cuspidal module by [Reference Kaplansky and Santharoubane17]. The following annihilation operator $\overline {\Omega }_{r, s}^{(m)}$ was given in [Reference Cai and Lü9] (also see [Reference Cai, Liu and Lü8]).

By lemma 3.1, we can easily get the following annihilation operator $\underline {\Omega }_{r, s}^{(m)}$ on cuspidal modules over $\mathfrak L$.

Lemma 3.2 Let $V$ be a cuspidal ${\mathfrak L}$-module. Then there exists $m\in \mathbb {Z}_+$ such that

(3.1)\begin{equation} \underline{\Omega}_{r, s}^{(m)}V=0, \end{equation}

where $\underline {\Omega }_{r, s}^{(m)}:=\sum _{i=0}^m(-1)^i\binom {m}{i}G_{r-i}G_{s+i}$ for any $r, s\in \mathbb {Z}+\frac 12$.

Proof. For the cuspidal module $V$, by lemma 3.1, there exists $m\in \mathbb {Z}_+$ such that for all $r\in \mathbb {Z}+\frac 12, s\in \mathbb {Z}$, $\overline {\Omega }_{r, s}^{(m)}V=0$, that is

(3.2)\begin{equation} \sum\limits_{i=0}^m({-}1)^i\binom{m}{i}G_{r-i}L_{s+i}V=0, \quad \forall r\in\mathbb{Z}+\frac12, s\in\mathbb{Z}. \end{equation}

By the action of $G_t$ on (3.2), we get

(3.3)\begin{equation} \frac12\sum\limits_{i=0}^m({-}1)^i\binom{m}{i}(s+i-2t)G_{r-i}G_{s+t+i}V=0, \quad \forall r\in\mathbb{Z}+\frac12, s\in\mathbb{Z}. \end{equation}

Choosing $t=t_1, t_2, t_1\ne t_2$ in (3.3), we get the lemma.

Now we use lemma 3.2 to classify all simple cuspidal modules over $\mathfrak L$ without complicated calculations.

Proof. By lemma 3.2, we can get (3.1). Multiply both sides of (3.1) by $G_{s+1} G_{s+2}\cdots G_{s+m}$, $G_{r-j+1}\cdots G_{r-1}G_rG_{s+j+1}\cdots G_{s+m}, 1\le j\le m$ to get

(3.4)\begin{align} & G_{r}G_{s}G_{s+1}\cdots G_{s+m}V=0, \end{align}

(3.5)\begin{align} & G_{r-j}\cdots G_{r-1}G_rG_{s+j}G_{s+j+1}\cdots G_{s+m}V=0,\quad \forall 1\le j\le m. \end{align}

Fix some $s\in \mathbb {Z}+1/2$ and set ${\mathcal {O}}_n=\{s, s+1, s+2, \cdots, s+n\}$. By (3.4) the following identity

(3.6)\begin{equation} G_{r_0}G_{r_1}\cdots G_{r_{m+1}}V=0 \end{equation}

holds for all $r_0, r_1, \cdots, r_{m+1}\in {\mathcal {O}}_{m+1}$.

By (3.4) and (3.5) we see that (3.6) holds for all $r_0, r_1, \cdots, r_{m+1}\in {\mathcal {O}}_{m+2}$.

We shall use the induction on $k$ to prove that

\[G_{r_0}G_{r_1}\cdots G_{r_{m+1}}V=0\]

for all $r_0, r_1,\cdots, r_{m+1}\in {\mathcal {O}}_{m+k}$ for all $k\ge 1$. Then, according to the arbitrariness of $s$, we get the lemma by choosing $N=m+2$.

Suppose that (3.6) holds for all $r_0, r_1, \cdots, r_{m+1}\in {\mathcal {O}}_{n}$ and some $n>m+1$. Now we shall prove that

(3.7)\begin{equation} G_{r_0}G_{r_1}\cdots G_{r_m}G_{s+n+1}V=0 \end{equation}

holds for all $r_0< r_1<\cdots < r_{m}\in {\mathcal {O}}_{n}$.

Case 1. $r_0=s+n-m$.

In this case $r_i=s+n-m+i$ for any $i=1, 2, \ldots, m$. So (3.7) follows from (3.4) directly.

Case 2. $r_0=s+n-m-k$ for some $1\le k\le n-m$.

Replacing $s, r$ by $s+n-m+1, s+n-k$ in (3.1), respectively, we get

\begin{align*} & \big(G_{s+n-k}G_{s+n-m+1}-\binom{m}{1}G_{s+n-k-1}G_{s+n-m+2}\\ & +\cdots+({-}1)^{m-1}\binom{m}{m-1}G_{s+n-k-m+1}G_{s+n}+({-}1)^mG_{r_0}G_{s+n+1}\big)V=0. \end{align*}

So we get

(3.8)\begin{equation} G_{r_0}G_{s+n+1}V\subset \left(\sum_{r_i, r_j\in\mathcal{O}_n} G_{r_i}G_{r_j}\right)V. \end{equation}

In this case (3.7) follows by inductive hypothesis.

Proof. Clearly $\dim V_i\le p$ for some positive integer $p$ holds for almost $i\in \mathbb {Z}$ and $C$ acts on $V$ as zero (see [Reference Kaplansky and Santharoubane17]). Now ${\mathfrak L}_{\bar 1}^iV$ is ${\mathfrak L}$-submodule since ${\mathfrak L}_{\bar 1}^{i+1}V\subset {\mathfrak L}_{\bar 1}^iV$ for all $i\in \mathbb {N}$. So ${\mathfrak L}_{\bar 1}V=V$ or ${\mathfrak L}_{\bar 1}V=0$.

By lemma 3.3, we get

(3.9)\begin{equation} {\mathfrak L}_{\bar1}^NV=0. \end{equation}

If ${\mathfrak L}_{\bar 1}V=V$ then ${\mathfrak L}_{\bar 1}^NV=V=0$, which is a contradiction. So ${\mathfrak L}_{\bar 1}V=0$ and the proposition follows from theorem 2.1.

4. Simple Harish–Chandra module

Now we can classify all simple Harish–Chandra modules over ${\mathfrak L}$. The following result is well-known.

Combining lemma 4.2 and theorem 3.4, we can get the following result.

5. Applications

Some Lie superalgebras were constructed in [Reference Wang, Chen and Bai28] as an application of the classification of Balinsky–Novikov super-algebras with dimension $2|2$. As applications of the above results, we can classify all Harish–Chandra modules over many Lie superalgebras listed in table 7 in [Reference Wang, Chen and Bai28].

5.1 The Lie superalgebra $\frak q$

By definition the Lie superalgebra $\frak q=\frak q_{\bar 0}+\frak q_{\bar 1}$, where $\frak q_{\bar 0}:=\mathbb {C}\{L_m, H_m, C\mid m\in \mathbb {Z}\}$ and $\frak q_{\bar 1}=\mathbb {C}\{G_p\mid p\in \mathbb {Z}+\frac 12\}$, is a subalgebra of the $N=2$ Neveu–Schwarz superconformal algebra, with the following relations:

(5.1)\begin{align} & [L_m,L_n]=(n-m)L_{n+m}+{1\over12}(n^3-n)C,\nonumber\\ & [H_m,H_n]={1\over3}m\delta_{m+n,0}C,\quad [L_m, H_n]=nH_{m+n},\nonumber\\ & [L_m,G_p]=\left(p-\frac{m}{2}\right)G_{p+m},\quad [H_m,G_p]=G_{m+p},\\ & [G_p, G_q]=0, \nonumber \end{align}

for $m,n\in \mathbb {Z},p, q\in \mathbb {Z}+\frac 12$.

Clearly $\mathcal {A}_{a, b, c}=\sum _{i\in \mathbb {Z}}\mathbb {C} v_i$ is a $\frak q$-module with

\[ L_mv_i=(a+bm+i)v_{m+i}, \quad H_mv_i=cv_{m+i}, \ G_rv_i=0, \forall m, i\in\mathbb{Z}, r\in\mathbb{Z}+\frac12. \]

Moreover ${\mathcal {A}}_{a, b, c}$ is simple if and only if $a\not \in \mathbb {Z}$, or $b\ne 0, 1$ or $c\ne 0$. We also use $\mathcal {A}_{a, b, c}'$ to denote the simple sub-quotient of $\mathcal {A}_{a, b, c}$.

Proposition 5.1 Any simple cuspidal $\frak q$-module $V$ is isomorphic to the module $\mathcal {A}_{a, b, c}'$ of the intermediate series for some $a, b, c\in \mathbb {C}$.

Proof. Clearly, the subalgebra $\frak q':={\rm span}\{L_m, G_r, C\mid m\in \mathbb {Z}, r\in \mathbb {Z}+\frac 12\}$ is isomorphic to ${\mathfrak L}$. By theorem 3.4, we can choose an irreducible $\frak q'$-module $V'$ with $G_rV'=0$ for all $r\in \mathbb {Z}+\frac 12$. In this case we have $V={\rm Ind}_{\frak q'}^{\frak q}V'$. Moreover we have $G_rV=0$ for all $r\in \mathbb {Z}+\frac 12$ by (5.1). In this case the $\frak q$-module $V$ is simple if and only if $V$ is a simple $\frak q_{\bar 0}$-module. So the proposition follows from the main theorem in [Reference Lü and Zhao22].

Remark 5.2 Proposition 5.1 palys a key role in the classification of all simple cuspidal weight modules for the $N=2$ Neveu–Schwarz superconformal algebra, see [Reference Liu, Pei and Xia21].

5.2 The $N=1$ BMS$_3$ superalgebra

The Bondi–Metzner–Sachs (BMS$_3$) algebra is the symmetry algebra of asymptotically flat three-dimensional spacetimes [Reference Bondi, Van der Burg and Metzner7]. It is the semi-direct product of the Virasoro algebra with its adjoint module. The $N=1$ super-BMS$_3$ is a minimal supersymmetric extension of the BMS$_3$ algebra, which has been introduced to describe the asymptotic structure of the $N=1$ supergravity in [Reference Barnich, Donnay, Matulich and Troncoso3, Reference Dilxat, Gao and Liu11].

Definition 5.3 The $N$=1 BMS$_3$ superalgebra $\mathcal {B}$ is a Lie superalgebra with a basis $\{L_m, I_m, Q_r, C_1, C_2\mid m\in \mathbb {Z}, r\in \mathbb {Z}+\frac 12\}$, with the following commutation relations:

\begin{align*} {[L_m, L_n]}& =(n-m)L_{m+n}+{1\over12}\delta_{m+n, 0}(n^3-n)C_1,\\ {[L_m, I_n]}& =(n-m)I_{m+n}+{1\over12}\delta_{m+n, 0}(n^3-n)C_2,\\ {[Q_r, Q_s]}& =2I_{r+s}+{1\over3}\delta_{r+s, 0}\left(r^2-\frac14\right)C_2,\\ {[L_m, Q_r]}& =-\left(\frac{m}{2}-r\right)Q_{m+r},\\ {[I_m,I_n]}& =[M_n,Q_r]=0, \quad [C_1,\frak g]=[C_2, \frak g]=0, \end{align*}

for any $m, n\in \mathbb {Z}, r, s\in \mathbb {Z}+\frac 12$.

Note that $\mathcal {B}=\mathcal {B}_{\bar 0}+\mathcal {B}_{\bar 1}$, where $\mathcal {B}_{\bar 0}:=\mathbb {C}\{L_m, I_m, C_1, C_2\mid m\in \mathbb {Z}\}$ and $\mathcal {B}_{\bar 1}=\mathbb {C}\{Q_p\mid p\in \mathbb {Z}+\frac 12\}$. The quotient algebra $\mathcal {B}/J$ is isomorphic to ${\mathfrak L}$, where $J=\mathbb {C}\{I_m, C_2\mid m\in \mathbb {Z}\}$.

Clearly the Vir-module $\mathcal {A}_{a, b}$ can become a $\mathcal {B}$-module with the trivial actions of $I_m, Q_r$ for any $m\in \mathbb {Z}, r\in \mathbb {Z}+\frac 12$.

Proposition 5.4 Any simple cuspidal $\mathcal {B}$-module $V$ is isomorphic to the module $\mathcal {A}_{a, b}'$ of the intermediate series for some $a, b\in \mathbb {C}$.

Proof. Clearly, the subalgebra $\mathcal {B}_{\bar 0}$ is isomorphic to $W(2, 2)$. By theorem 4.6 in [Reference Guo, Lü and Zhao14], we can choose an irreducible $\mathcal {B}_{\bar 0}$-module $V'$ with $I_mV'=C_1V'=C_2V'=0$ for all $m\in \mathbb {Z}$. In this case we have $V={\rm Ind}_{\mathcal {B}_{\bar 0}}^{\mathcal {B}}V'$. Moreover we have $I_mV=0$ and $[G_r, G_s]V=0$ for all $m\in \mathbb {Z}, r, s\in \mathbb {Z}+\frac 12$ by definition 5.3. In this case the $\mathcal {B}$-module $V$ is simple if and only if $V$ is a simple $\mathcal {B}/J$-module. So the proposition follows from theorem 3.4.

5.3 The super $W(2,2)$ algebra

By definition, the super $W(2,2)$ algebra is the Lie superalgebra $SW(2,2):= \mathbb {C}\{L_m, I_m, G_r, Q_r, C_1, C_2\mid m\in \mathbb {Z}, r\in \mathbb {Z}+\frac 12\}$ with the following relations:

(5.2)\begin{align} & [L_m, L_n]=(n-m)L_{m+n}+{1\over12}\delta_{m+n, 0}(n^3-n)C_1,\nonumber\\ & [L_m, I_n]=(n-m)I_{m+n}+{1\over12}\delta_{m+n, 0}(n^3-n)C_2,\nonumber\\ & [G_r, G_s]=-2L_{r+s}+{1\over3}\delta_{r+s, 0}\left(r^2-\frac14\right)C_1,\nonumber\\ & [G_r, Q_s]=2I_{r+s}+{1\over3}\delta_{r+s, 0}\left(r^2-\frac14\right)C_2,\nonumber\\ & [L_m, G_r]=-\left(\frac{m}{2}-r\right)G_{m+r}, \quad [L_m, Q_r]=-\left(\frac{m}{2}-r\right)Q_{m+r},\nonumber \\ & [I_m,G_r]=\left(\frac{m}{2}-r\right)Q_{m+r},\end{align}

for any $m, n\in \mathbb {Z}, r, s\in \mathbb {Z}+\frac 12$.

Note that $SW(2,2)=SW(2,2)_{\bar 0}+SW(2,2)_{\bar 1}$, where $SW(2,2)_{\bar 0}:=\mathbb {C}\{L_m, I_m, C_1, C_2\mid m\in \mathbb {Z}\}$ and $SW(2,2)_{\bar 1}=\mathbb {C}\{G_p, Q_p\mid p\in \mathbb {Z}+\frac 12\}$.

Clearly the subalgebra generated by $\{L_m, G_r, C~|~ m\in \mathbb {Z}, r\in \mathbb {Z}+\frac 12\}$ is isomorphic to the $N=1$ Neveu–Schwarz algebra $\frak S$ (see [Reference Liu, Pei and Xia19]). From [Reference Su26] we see that $S_{a, b}$ or $\Pi S_{a, b}$ is the Harich–Chandra module of intermediate series over $\frak S$ for some $a, b\in \mathbb {C}$, where $S_{a, b}$ is defined as follows:

\begin{align*} S_{a, b}& :=\sum_{i\in\mathbb{Z}}\mathbb{C} x_i+\sum_{k\in\mathbb{Z}+\frac12}\mathbb{C} y_k\ \hbox{with}\\ L_nx_i& =(a+bn+i)x_{i+n}, \quad L_ny_k=\left(a+\left(b+\frac12\right)n+k\right)y_{k+n},\\ G_rx_i& =(a+i+2rb)y_{r+i}, \quad G_ry_k={-}x_{r+k}, \end{align*}

for all $n, i\in \mathbb {Z}, r, k\in \mathbb {Z}+\frac 12$.

Moreover $S_{a, b}$ is simple if and only if $a\not \in \mathbb {Z}$ or $a\in \mathbb {Z}$ and $b\ne 0, \frac 12$. We also use $S_{a, b}'$ to denote the simple sub-quotient of $S_{a, b}$.

Clearly the $\frak S$-modules $S_{a, b}$ and $\Pi S_{a, b}$ become $SW(2,2)$-modules with trivial actions of $I_m, Q_{m+\frac 12}$ for any $m\in \mathbb {Z}$.

Proposition 5.5 Any simple cuspidal $SW(2, 2)$-module $V$ is isomorphic to $S_{a, b}'$ or $\Pi S_{a, b}'$ for some $a, b\in \mathbb {C}$.

Proof. Set $\frak p={\rm span}\{L_m, I_m, Q_r, C_1, C_2\mid m\in \mathbb {Z}, r\in \mathbb {Z}+\frac 12\}$. By proposition 5.4 we can choose a simple $\frak p$-module $V'$ with $I_mV'=Q_rV'=C_2V'=C_2V'=0$ for all $m\in \mathbb {Z}, r\in \mathbb {Z}+\frac 12$. In this case we have $V={\rm Ind}_{\frak p}^{SW(2,2)}V'$. Moreover we have $I_mV=Q_rV=0$ for all $m\in \mathbb {Z}, r\in \mathbb {Z}+\frac 12$ by definition 5.2. In this case the $SW(2, 2)$-module $V$ is simple if and only if $V$ is a simple $\frak S$-module. So the proposition follows from the main theorem in [Reference Su26] (also see theorem 4.5 in [Reference Cai and Lü9]).

Remark 5.6 We can easily prove that any simple Harish–Chandra module over the above Lie superalgebras is a cuspidal module, or a highest (or lowest) weight module as lemma 4.2. So all simple Harish–Chandra modules over the above Lie superalgebras are also classified.

Remark 5.7 All indecomposable modules of the intermediate series and some other representations were studied in [Reference Wang, Geng and Chen29] and [27].