Simple Restricted Modules for the Deformed 𝔟𝔪𝔰₃ Algebra

Abstract

In this paper, we construct a large class of simple restricted modules over the deformed 𝔟𝔪𝔰₃ algebra $ℬ$, which contain the highest weight modules and Whittaker modules. Moreover, we obtain several equivalent statements for simple restricted modules over $ℬ$.

1. Introduction

Throughout the paper, we denote by $\mathbb{C},\mathbb{Z},\mathbb{N},{\mathbb{Z}}_{+}$ the sets of complex numbers, integers, positive integers and non-negative integers, respectively. All vector spaces are assumed to be over $\mathbb{C}$. For a Lie algebra $\mathcal{L}$, denote by $\mathcal{U}\left(\mathcal{L}\right)$ the universal enveloping algebra of $\mathcal{L}$. For a subset X of $\mathcal{L}$, denote by $\mathcal{U}\left(X\right)$ the universal enveloping algebra of the subalgebra of $\mathcal{L}$ generated by X.

Infinite-dimensional Lie algebras play a major role in a variety of physical systems and has developed in numerous specialized directions over recent decades and many infinite-dimensional Lie algebras can be obtained by means of the Lie algebra expansion method. For example, the ${\mathfrak{bms}}_3$ algebra can be gained through an S-expansion of the Virasoro algebra for an appropriate semigroup S. The deformed ${\mathfrak{bms}}_3$ $\mathcal{B}$, which was first introduced in [1], is a Lie algebra with basis $\{L_m,J_m,I_m,C,C_1,C_2\mid m\in \mathbb{Z}\}$ satisfying the following relations

$$\begin{array}{l}[L_m,L_n]=(n-m)L_{m+n}+{\delta }_{m+n,0}\frac{m^3-m}{12}C,\\ [L_m,J_n]=(n-m)J_{m+n}+{\delta }_{m+n,0}\frac{m^3-m}{12}{C}_1,\\ [L_m,I_n]=(n-m)I_{m+n}+{\delta }_{m+n,0}\frac{m^3-m}{12}{C}_2,\\ [J_m,J_n]=(m-n)I_{m+n}+{\delta }_{m+n,0}\frac{m^3-m}{12}{C}_2,\\ [I_m,I_n]=[I_m,J_n]=0,\forall m,n\in \mathbb{Z}.\end{array}$$

Actually, $\mathcal{B}$ is equivalent to an infinite-dimensional lift of the $(2+1)$-dimensional Maxwell algebra. One can see from the definition that the subalgebra ${\mathrm{span}}_{\mathbb{C}}\{L_m,I_m,C,C_2\mid m\in \mathbb{Z}\}$ is isomorphic to $W(2,2)$ [2]. Moreover, $\mathcal{B}$ is also a truncated loop algebra of the Virasoro algebra. Obviously, $\mathcal{B}$ has a $\mathbb{Z}$-gradation according to the adjoint action of $L_0$:

$$\mathcal{B}={\oplus }_{m\in \mathbb{Z}}{\mathcal{B}}_m,{\mathcal{B}}_m={\mathrm{span}}_{\mathbb{C}}\{L_m,J_m,I_m\mid m\in \mathbb{Z}\}\oplus {\delta }_{m,0}(\mathbb{C}C\oplus \mathbb{C}{C}_1\oplus \mathbb{C}{C}_2).$$

The local derivations and left-symmetric algebra structures of $\mathcal{B}$ were investigated in [3] and [4], respectively.

It is universally acknowledged that the highest weight modules and Whittaker modules are two kinds of the most significant modules for many infinite-dimensional Lie algebras. The most accessible modules for a Lie algebra are the highest weight modules, which arise first as an auxiliary tool for the construction of simple modules (cf. [5,6,7,8,9]). In 1978, Whittaker modules were first introduced by Kostant [10] for finite dimensional simple Lie algebras. However, these modules appeared earlier for the algebra $s{l}_2$ by [11]. Since then, there have been many works on studying similar modules over kinds of Lie algebras (cf. [12,13,14,15,16,17,18,19,20,21,22,23,24,25]).

Observe that the highest weight modules and Whittaker modules have a common characteristic: the actions of elements in the positive part of the algebra are locally finite, thus they could be considered by unified approach, which leads to the definition of restricted modules. Understanding restricted modules for an infinite-dimensional Lie algebra is one of the leading topics in Lie theory, as this class of modules are closely connected with the modules for the corresponding vertex operator algebras. The first stage of studying restricted modules is to classify all restricted simple modules for a Lie algebra, but this is a difficult challenge. As far as I know, a classification of all simple restricted modules for the Virasoro algebra was given in [9], and a classification of all simple restricted modules over the mirror Heisenberg–Virasoro algebra was obtained recently in [26,27]. Moreover, there are some partial results of simple restricted modules for other Lie algebras, such as Lie algebras of Cartan type $W,S,H$ [28,29], the Heisenberg–Virasoro algebra [5,27,30], the Schrödinger–Virasoro algebra [31], the Neveu–Schwarz algebra [32], and the planar Galilean conformal algebra [33,34]. The aim of the present paper is to construct a large family of simple restricted $\mathcal{B}$-modules. Some ideas we use come from [5,9].

The paper is organized as follows. In Section 2, we recall some fundamental definitions and introduce some total orders for later use. In Section 3, we construct some simple restricted $\mathcal{B}$-modules by exploiting the technique of induction. The idea, which arises in similar ways in other types of modules, is to start with an easily constructed family of modules over the finite-dimensional quotient algebras of some subalgebras and then induce to $\mathcal{B}$. In Section 4, we first give equivalent characterizations for simple restricted $\mathcal{B}$-modules. Then we discuss the problem about the characterization of the simple restricted $\mathcal{B}$-modules under some conditions. In Section 5, we present some examples using our results established in Section 4. In the final section, we give the conclusions.

2. Preliminaries

Definition 1.

For any $\mathbb{Z}$-graded Lie algebra $\mathcal{L}={\displaystyle \underset{i\in \mathbb{Z}}{⨁}}{\mathcal{L}}_i$, we say that an $\mathcal{L}$-module M is restricted if for any $v\in M$, there exists $n\in \mathbb{N}$ such that ${\mathcal{L}}_{i}v=0$ for any $i\ge n$. Let ${\mathcal{R}}_{\mathcal{L}}$ denote the subcategory of $\mathcal{L}$-module category consisting of restricted $\mathcal{L}$-modules.

Definition 2.

Let V be a module over a Lie algebra $\mathcal{L}$ and $x\in \mathcal{L}$.

(1) The action of x on V is called locally nilpotent if for any $v\in V$ there exists $n\in {\mathbb{Z}}_{+}$, such that $x^{n}v=0$. Similarly, the action of $\mathcal{L}$ on V is called locally nilpotent if for any $v\in V$ there exists $n\in {\mathbb{Z}}_{+}$, such that ${\mathcal{L}}^{n}v=0$.

(2) The action of x on V is locally finite if for any $v\in V$, we have $\dim \left({\sum }_{n\in {\mathbb{Z}}_{+}}\mathbb{C}{x}^{n}v\right)<+\infty$. Similarly, the action of $\mathcal{L}$ on V is called locally finite if for any $v\in V$ we have $\dim \left({\sum }_{n\in {\mathbb{Z}}_{+}}{\mathcal{L}}^{n}v\right)<+\infty$.

For any $d_1,d_2\in {\mathbb{Z}}_{+}$ with $d_1\ge 2d_2$, let

$${\mathcal{B}}_{d_1,d_2}=\sum _{i\in {\mathbb{Z}}_{+}}(\mathbb{C}{I}_{i-d_1}\oplus \mathbb{C}{J}_{i-d_2}\oplus \mathbb{C}{L}_i)\oplus \mathbb{C}{C}_1\oplus \mathbb{C}{C}_2\oplus \mathbb{C}{C}_3$$

It is easy to see that ${\mathcal{B}}_{d_1,d_2}$ is a subalgebra of $\mathcal{B}$.

Letting V be a simple ${\mathcal{B}}_{d_1,d_2}$-module, we have the induced $\mathcal{B}$-module

$$\mathrm{Ind}\left(V\right)=\mathcal{U}\left(\mathcal{B}\right){\otimes }_{\mathcal{U}\left({\mathcal{B}}_{d_1,d_2}\right)}V.$$

Since we usually consider simple modules for the algebra $\mathcal{B}$ or one of its subalgebras containing the central elements $C,C_1,C_2$, we always assume that the actions of $C,C_1$ and $C_2$ are scalars $c,c_1$, and $c_2$, respectively.

Denote $\mathbb{T}=\{\mathbf{i}:=(\dots ,i_s,\dots ,i_2,i_1)\mid i_s\in {\mathbb{Z}}_{+}\}$, where the number of non-zero entries of $\mathbf{i}$ is finite. In particular, denote

$$\begin{array}{cc}\hfill \mathbf{0}& =(\dots ,0,0)\in \mathbb{T},\hfill \\ \hfill {ϵ}_i& =(\dots ,0,\stackrel{i}{\overbrace{1,0,\dots ,0)}}\in \mathbb{T}.\hfill \end{array}$$

For any $\mathbf{i}\in \mathbb{T}$, we define

$$\begin{array}{cc}\hfill \mathbf{wt}\left(\mathbf{i}\right)& =\sum _{s\in \mathbb{N}}s·i_s.\hfill \end{array}$$

For any non-zero $\mathbf{i}\in \mathbb{T}$, we define

$$\begin{array}{cc}\hfill p& =\max \{s\mid i_s\ne 0\},\hfill \\ \hfill q& =\min \{s\mid i_s\ne 0\},\hfill \\ \hfill {\mathbf{i}}^{\prime }& =\mathbf{i}-{ϵ}_p,\hfill \\ \hfill {\mathbf{i}}^{″}& =\mathbf{i}-{ϵ}_q.\hfill \end{array}$$

Given any $\mathbf{i},\mathbf{j}\in \mathbb{T}$, define

$$\mathbf{j}>\mathbf{i}\iff \mathrm{there}\mathrm{exists}r\in \mathbb{N}\mathrm{such}\mathrm{that}{j}_r>i_r\mathrm{and}{j}_s=i_s,\forall s>r$$

and

$$\mathbf{j}\succ \mathbf{i}\iff \mathrm{there}\mathrm{exists}r\in \mathbb{N}\mathrm{such}\mathrm{that}{j}_r>i_r\mathrm{and}{j}_s=i_s,\forall 1\le s<r.$$

Then, ${\mathbb{T}}^3$ carries a natural total order, still denoted by ≻:

$$\begin{array}{ccc}& & ({\mathbf{i}}_1,{\mathbf{j}}_1,{\mathbf{l}}_1)\succ ({\mathbf{i}}_2,{\mathbf{j}}_2,{\mathbf{l}}_2)⟺({\mathbf{j}}_1,\mathbf{wt}\left({\mathbf{j}}_1\right),{\mathbf{l}}_1,\mathbf{wt}\left({\mathbf{l}}_1\right))\succ ({\mathbf{j}}_2,\mathbf{wt}\left({\mathbf{j}}_2\right),{\mathbf{l}}_2,\mathbf{wt}\left({\mathbf{l}}_2\right))\mathrm{or}\hfill \\ & & ({\mathbf{j}}_1,\mathbf{wt}\left({\mathbf{j}}_1\right),{\mathbf{l}}_1,\mathbf{wt}\left({\mathbf{l}}_1\right))=({\mathbf{j}}_2,\mathbf{wt}\left({\mathbf{j}}_2\right),{\mathbf{l}}_2,\mathbf{wt}\left({\mathbf{l}}_2\right))\mathrm{and}{\mathbf{i}}_1>{\mathbf{i}}_2\hfill \end{array}$$

for any $({\mathbf{i}}_1,{\mathbf{j}}_1,{\mathbf{l}}_1),({\mathbf{i}}_2,{\mathbf{j}}_2,{\mathbf{l}}_2)\in {\mathbb{T}}^3$.

Fix $d_1,d_2\in {\mathbb{Z}}_{+}$ and let V be a simple ${\mathcal{B}}_{d_1,d_2}$-module. For $\mathbf{i},\mathbf{j},\mathbf{l}\in \mathbb{T}$, we denote

$$I^{\mathbf{i}}{J}^{\mathbf{j}}{L}^{\mathbf{l}}=\cdots I_{-d_1-2}^{i_2}{I}_{-d_1-1}^{i_1}\cdots J_{-d_2-2}^{j_2}{J}_{-d_2-1}^{j_1}\cdots L_{-2}^{l_2}{L}_{-1}^{l_1}\in \mathcal{U}\left(\mathcal{B}\right).$$

Using $\mathrm{PBW}$ Theorem, $\mathrm{Ind}\left(V\right)$ consists of all vectors of the form

$$\sum _{(\mathbf{i},\mathbf{j},\mathbf{l})\in {\mathbb{T}}^3}{I}^{\mathbf{i}}{J}^{\mathbf{j}}{L}^{\mathbf{l}}{v}_{\mathbf{i},\mathbf{j},\mathbf{l}},v_{\mathbf{i},\mathbf{j},\mathbf{l}}\in V.$$

(1)

For any $v\in \mathrm{Ind}\left(V\right)$ as above, set

$$\mathrm{supp}\left(v\right)=\{(\mathbf{i},\mathbf{j},\mathbf{l})\in {\mathbb{T}}^3\mid v_{\mathbf{i},\mathbf{j},\mathbf{l}}\ne 0\}.$$

For a non-zero $v\in \mathrm{Ind}\left(V\right)$, we define the degree of v $\mathrm{deg}\left(v\right)$ as the maximal element in $\mathrm{supp}\left(v\right)$ with respect to the total order on ${\mathbb{T}}^3$.

3. Construction of Simple Restricted Modules

We shall construct some simple restricted $\mathcal{B}$-modules by exploiting the technique of induction. The idea is to start with an easily constructed family of modules for a subalgebra and then induce to $\mathcal{B}$.

Theorem 1.

Let $d_1,d_2\in {\mathbb{Z}}_{+}$ and V is a simple ${\mathcal{B}}_{d_1,d_2}$-module. If there exists $k\in {\mathbb{Z}}_{+}$ such that:

(a) 

$$\left\{\begin{array}{cc}\mathit{the}\mathit{action}\mathit{of}{I}_k\mathit{on}\mathit{V}\mathit{is}\mathit{injective},\hfill & \mathit{if}k\ne 0;\hfill \\ I_{0}v\ne -\frac{n^2-1}{24}{c}_{2}v\mathit{for}\mathit{all}v\in V\mathit{and}n\in \mathbb{Z}\backslash \left\{0\right\},\hfill & \mathit{if}k=0.\hfill \end{array}\right.$$

(b) 

$I_{i}V=J_{j}V=L_{l}V=0$for all$i>k,j>k+d_2$ and $l>k+d_1$.

Then the following statements hold.

(1) 

$\mathrm{Ind}\left(V\right)$ is a simple $\mathcal{B}$-module;

(2) 

$I_i,J_j,L_l$ act locally nilpotent on $\mathrm{Ind}\left(V\right)$ for all $i>k,j>k+d_2$ and $l>k+d_1$.

Proof. 

At first, we need the following results.

Claim 1.

Assume that

$$\begin{array}{cc}\hfill \mathrm{deg}\left(v\right)& =(\mathbf{i},\mathbf{j},\mathbf{l}),\mathrm{for}\mathrm{any}v\in \mathrm{Ind}\left(V\right)\backslash V,\hfill \\ \hfill \widehat{i}& =\max \{s:i_s\ne 0\},\mathrm{if}\mathbf{i}\ne \mathbf{0},\hfill \\ \hfill \widehat{j}& =\min \{s:j_s\ne 0\},\mathrm{if}\mathbf{j}\ne \mathbf{0},\hfill \\ \hfill \widehat{l}& =\min \{s:l_s\ne 0\},\mathrm{if}\mathbf{l}\ne \mathbf{0}.\hfill \end{array}$$

(i) 

If $\mathbf{l}\ne \mathbf{0}$, then $\widehat{l}>0$ and $\mathrm{deg}\left(I_{\widehat{l}+k}v\right)=(\mathbf{i},\mathbf{j},{\mathbf{l}}^{″})$.

(ii) 

If $\mathbf{l}=\mathbf{0}, \mathbf{j}\ne \mathbf{0}$, then $\widehat{j}>0$ and $\mathrm{deg}\left(J_{\widehat{j}+d_2+k}v\right)=(\mathbf{i},{\mathbf{j}}^{″},\mathbf{0})$.

(iii) 

If $\mathbf{l}=\mathbf{j}=\mathbf{0}, \mathbf{i}\ne \mathbf{0}$, then $\widehat{i}>0$ and $\mathrm{deg}\left(L_{\widehat{i}+k+d_1}v\right)=({\mathbf{i}}^{\prime },\mathbf{0},\mathbf{0})$.

(i) According to expression (1), one need only deal with those $v_{\mathbf{a},\mathbf{b},\mathbf{c}}$ with

$$I_{\widehat{l}+k}{I}^{\mathbf{a}}{J}^{\mathbf{b}}{L}^{\mathbf{c}}{v}_{\mathbf{a},\mathbf{b},\mathbf{c}}\ne 0.$$

In the following discussion, we will frequently use the fact $I_{i}V=0$ for $i>k$. Clearly,

$$I_{\widehat{l}+k}{I}^{\mathbf{a}}{J}^{\mathbf{b}}{L}^{\mathbf{c}}{v}_{\mathbf{a},\mathbf{b},\mathbf{c}}=I^{\mathbf{a}}{J}^{\mathbf{b}}[I_{\widehat{l}+k},L^{\mathbf{c}}]v_{\mathbf{a},\mathbf{b},\mathbf{c}}.$$

If $c_{\widehat{l}}\ne 0$, then

$$\mathrm{deg}\left(I_{\widehat{l}+k}{I}^{\mathbf{a}}{J}^{\mathbf{b}}{L}^{\mathbf{c}}{v}_{\mathbf{a},\mathbf{b},\mathbf{c}}\right)=(\mathbf{a},\mathbf{b},\mathbf{c}-{ϵ}_{\widehat{l}})⪯(\mathbf{i},\mathbf{j},{\mathbf{l}}^{″}),$$

where the equality holds if, and only if, $(\mathbf{a},\mathbf{b},\mathbf{c})=(\mathbf{i},\mathbf{j},\mathbf{l})$. If $c_{\widehat{l}}=0$, let

$$\mathrm{deg}\left(I_{\widehat{l}+k}{I}^{\mathbf{a}}{J}^{\mathbf{b}}{L}^{\mathbf{c}}{v}_{\mathbf{a},\mathbf{b},\mathbf{c}}\right)=({\mathbf{a}}^{*},{\mathbf{b}}^{*},{\mathbf{c}}^{*}).$$

We can get that

$$\mathbf{wt}\left({\mathbf{c}}^{*}\right)\le \mathbf{wt}\left(\mathbf{c}\right)-\widehat{l}.$$

If $\mathbf{wt}\left(\mathbf{c}\right)<\mathbf{wt}\left(\mathbf{l}\right)$, then $\mathbf{wt}\left({\mathbf{c}}^{*}\right)<\mathbf{wt}\left({\mathbf{l}}^{″}\right)$, which means that $({\mathbf{a}}^{*},{\mathbf{b}}^{*},{\mathbf{c}}^{*})\prec (\mathbf{i},\mathbf{j},{\mathbf{l}}^{″})$. If $\mathbf{wt}\left(\mathbf{c}\right)=\mathbf{wt}\left(\mathbf{l}\right)$, then $\mathbf{c}\prec \mathbf{l}$. Let $\widehat{c}=\min \{s:c_s\ne 0\}$, which is strictly larger than $\widehat{l}$. Thus, $\mathbf{wt}\left({\mathbf{c}}^{*}\right)<\mathbf{wt}\left(\mathbf{l}\right)-\widehat{l}=\mathbf{wt}\left({\mathbf{l}}^{″}\right)$.

Consequently, we conclude that $\mathrm{deg}\left(I_{\widehat{l}+k}v\right)=(\mathbf{i},\mathbf{j},{\mathbf{l}}^{″})$.

(ii) Similarly, it suffices to deal with those $v_{\mathbf{a},\mathbf{b},\mathbf{0}}$ with

$$J_{\widehat{j}+d_2+k}{I}^{\mathbf{a}}{J}^{\mathbf{b}}{v}_{\mathbf{a},\mathbf{b},\mathbf{0}}\ne 0.$$

As $J_{\widehat{j}+d_2+k}{v}_{\mathbf{a},\mathbf{b},\mathbf{0}}=0$ for any $(\mathbf{a},\mathbf{b},\mathbf{0})\in \mathrm{supp}\left(v\right)$, we have

$$J_{\widehat{j}+d_2+k}{I}^{\mathbf{a}}{J}^{\mathbf{b}}{v}_{\mathbf{a},\mathbf{b},\mathbf{0}}=I^{\mathbf{a}}[J_{\widehat{j}+d_2+k},J^{\mathbf{b}}]v_{\mathbf{a},\mathbf{b},\mathbf{0}}.$$

First we consider $b_{\widehat{j}}\ne 0$. One can show that

$$\mathrm{deg}\left(J_{\widehat{j}+d_2+k}{I}^{\mathbf{a}}{J}^{\mathbf{b}}{v}_{\mathbf{a},\mathbf{b},\mathbf{0}}\right)=(\mathbf{a},\mathbf{b}-{ϵ}_{\widehat{j}},\mathbf{0})⪯(\mathbf{i},{\mathbf{j}}^{″},\mathbf{0}),$$

where the equality holds if, and only if, $(\mathbf{a},\mathbf{b})=(\mathbf{i},\mathbf{j})$. Now we consider $b_{\widehat{j}}=0$ and one can get

$$\mathrm{deg}\left(J_{\widehat{j}+d_2+k}{I}^{\mathbf{a}}{J}^{\mathbf{b}}{v}_{\mathbf{a},\mathbf{b},\mathbf{0}}\right)=({\mathbf{a}}^{*},{\mathbf{b}}^{*},\mathbf{0}),$$

where

$$\mathbf{wt}\left({\mathbf{b}}^{*}\right)\le \mathbf{wt}\left(\mathbf{b}\right)-\widehat{j}.$$

If $\mathbf{wt}\left(\mathbf{b}\right)<\mathbf{wt}\left(\mathbf{j}\right)$, then $\mathbf{wt}\left({\mathbf{b}}^{*}\right)<\mathbf{wt}\left({\mathbf{j}}^{″}\right)$, which means that $({\mathbf{a}}^{*},{\mathbf{b}}^{*},\mathbf{0})\prec (\mathbf{i},{\mathbf{j}}^{″},\mathbf{0})$. If $\mathbf{wt}\left(\mathbf{b}\right)=\mathbf{wt}\left(\mathbf{j}\right)$, then $\mathbf{b}\prec \mathbf{j}$. Let $\widehat{b}=\min \{s:b_s\ne 0\}$ and one can observe that $\widehat{b}>\widehat{j}$. Clearly, $\mathbf{wt}\left({\mathbf{b}}^{*}\right)<\mathbf{wt}\left(\mathbf{j}\right)-\widehat{j}=\mathbf{wt}\left({\mathbf{j}}^{″}\right)$. Thus, $\mathrm{deg}\left(J_{\widehat{j}+d_2+k}v\right)=(\mathbf{i},{\mathbf{j}}^{″},\mathbf{0})$.

(iii) Noticing that $L_{\widehat{i}+k+d_1}V=0$ and $[L_{\widehat{i}+k+d_1},I_{-d_1-\widehat{i}}]V\ne 0$, we have

$$L_{\widehat{i}+k+d_1}{I}^{\mathbf{a}}{v}_{\mathbf{a},\mathbf{0},\mathbf{0}}=a_{\widehat{i}}{I}^{{\mathbf{a}}^{\prime }}[L_{\widehat{i}+k+d_1},I_{-d_1-\widehat{i}}]v_{\mathbf{a},\mathbf{0},\mathbf{0}}\mathrm{for}{a}_{\widehat{i}}\in \mathbb{C}\backslash \left\{0\right\},(\mathbf{a},\mathbf{0},\mathbf{0})\in \mathrm{supp}\left(v\right).$$

This means the result holds and the proof is completed.

(1) follows from repeated application of Claim 1, while (2) just follows from a straightforward computation. □

Remark 1.

An important fact that needs to be pointed out here is as follows: if we do not assume that the simplicity of V as a ${\mathcal{B}}_{d_1,d_2}$-module, Claim 1 also holds.

4. Characterization of Simple Restricted Modules

For any $p,q,r\in {\mathbb{Z}}_{+}$, denote

$${\mathcal{B}}^{(p,q,r)}=\sum _{i\in {\mathbb{Z}}_{+}}(\mathbb{C}{I}_{p+i}\oplus \mathbb{C}{J}_{q+i}\oplus \mathbb{C}{L}_{r+i}),$$

which is a subalgebra of $\mathcal{B}$. We first give equivalent characterizations for simple restricted $\mathcal{B}$-modules.

Theorem 2.

Let S be a simple $\mathcal{B}$-module. Then the following conditions are equivalent:

(1)

$S\in {\mathcal{R}}_{\mathcal{B}}$.

(2)

There exists $k\in {\mathbb{Z}}_{+}$ such that the actions of $I_i,J_i,L_i,i\ge k$ on S are locally nilpotent.

(3)

There exists $k\in {\mathbb{Z}}_{+}$ such that the actions of $I_i,J_i,L_i,i\ge k$ on S are locally finite.

(4)

There exist $p,q,r\in {\mathbb{Z}}_{+}$ such that S is a locally nilpotent ${\mathcal{B}}^{(p,q,r)}$-module.

(5)

There exist $p,q,r\in {\mathbb{Z}}_{+}$ such that S is a locally finite ${\mathcal{B}}^{(p,q,r)}$-module.

Proof. 

Evidently (4) ⇒ (5) ⇒ (3), while (4) ⇒ (2) ⇒ (3). So it suffices to prove that (1) ⇒ (4) and (3) ⇒ (1).

We start with (1) implies (4). For any non-zero $v\in S$, there exists $s\in {\mathbb{Z}}_{+}$ such that $L_{i}v=J_{i}v=I_{i}v=0$ for all $i>s$. Since S is a simple $\mathcal{B}$-module, $S=\mathcal{U}\left(\mathcal{B}\right)v$. This along with $\mathrm{PBW}$ Theorem forces that S is a locally nilpotent ${\mathcal{B}}^{(p,q,r)}$-module for $p,q,r>s$.

Now we prove (3) implies (1). According to the assumption that the actions of $L_i,i\ge k$ on S are locally finite, we know that for any fixed $t\in {\mathbb{Z}}_{+}$ with $t\ge k$ there exists non-zero $v\in S$, such that $L_{t}v=\lambda v$ for some $\lambda \in \mathbb{C}$.

Choose any $j\in \mathbb{Z}$ with $j>t$ and set

$$\begin{array}{cc}\hfill N_I:& =\sum _{n\in {\mathbb{Z}}_{+}}\mathbb{C}{L}_t^n{I}_{j}v=\mathcal{U}\left(\mathbb{C}{L}_t\right)I_{j}v,\hfill \\ \hfill N_J:& =\sum _{n\in {\mathbb{Z}}_{+}}\mathbb{C}{L}_t^n{J}_{j}v=\mathcal{U}\left(\mathbb{C}{L}_t\right)J_{j}v,\hfill \\ \hfill N_L:& =\sum _{n\in {\mathbb{Z}}_{+}}\mathbb{C}{L}_t^n{L}_{j}v=\mathcal{U}\left(\mathbb{C}{L}_t\right)L_{j}v,\hfill \end{array}$$

which are all finite-dimensional. Clearly,

$$\begin{array}{cc}\hfill (j+(n-1)t)I_{j+(n+1)t}v& =[L_t,I_{j+nt}]v=L_t{I}_{j+nt}v-I_{j+nt}{L}_{t}v=(L_t-\lambda )I_{j+nt}v,\hfill \\ \hfill (j+(n-1)t)J_{j+(n+1)t}v& =[L_t,J_{j+nt}]v=L_t{J}_{j+nt}v-J_{j+nt}{L}_{t}v=(L_t-\lambda )J_{j+nt}v,\hfill \\ \hfill (j+(n-1)t)L_{j+(n+1)t}v& =[L_t,L_{j+nt}]v=L_t{L}_{j+nt}v-L_{j+nt}{L}_{t}v=(L_t-\lambda )L_{j+nt}v,\forall n\in {\mathbb{Z}}_{+},\hfill \end{array}$$

which imply that

$$X_{j+nt}v\in N_X\Rightarrow X_{j+(n+1)t}v\in N_X,\mathrm{where}X\in \{I,J,L\},$$

for $n\in {\mathbb{Z}}_{+}$ and $j>t$. Inductively, we can prove that

$$X_{j+nt}v\in N_X,\mathrm{where}X\in \{I,J,L\},\forall n\in {\mathbb{Z}}_{+}.$$

In particular,

$$\sum _{n\in {\mathbb{Z}}_{+}}\mathbb{C}{X}_{j+nt}v,\mathrm{where}X\in \{I,J,L\},$$

are all finite-dimensional for all $j>t$ and, hence,

$$\sum _{i\in {\mathbb{Z}}_{+}}\mathbb{C}{X}_{t+i}v=\mathbb{C}{X}_{t}v+\sum _{j=t+1}^{2t}\left(\sum _{n\in {\mathbb{Z}}_{+}}\mathbb{C}{X}_{j+nt}v\right),\mathrm{where}X\in \{I,J,L\},$$

are all finite-dimensional. So, we can find $h\in \mathbb{N}$ such that

$$\sum _{i\in {\mathbb{Z}}_{+}}\mathbb{C}{X}_{t+i}v=\sum _{i=0}^h\mathbb{C}{X}_{t+i}v,\mathrm{where}X\in \{I,J,L\}.$$

Denote

$$V^{\prime }=\sum _{i_0,\dots ,i_h,j_0,\dots ,j_h,l_0,\dots ,l_h\in {\mathbb{Z}}_{+}}\mathbb{C}{I}_t^{i_0}\cdots I_{t+h}^{i_h}{J}_t^{j_0}\cdots J_{t+h}^{j_h}{L}_t^{l_0}\cdots L_{t+h}^{l_h}v,$$

which is finite-dimensional by (3). Moreover, it is not hard to check that $V^{\prime }$ is a finite-dimensional ${\mathcal{B}}^{(t,t,t)}$-module. From linear algebra, we know that there exists a minimal $n\in {\mathbb{Z}}_{+}$, such that

$$(L_m+b_1{L}_{m+1}+\cdots +b_n{L}_{m+n})V^{\prime }=0$$

for some $m\ge t$ and $b_i\in \mathbb{C}$. Using $L_m$, we get

$$(b_1[L_m,L_{m+1}]+\cdots +b_n[L_m,L_{m+n}])V^{\prime }=0.$$

This forces $n=0$, i.e., $L_m{V}^{\prime }=0$. Therefore, we have

$$0=L_i{L}_m{V}^{\prime }=[L_i,L_m]V+L_m{L}_i{V}^{\prime }=(m-i)L_{m+i}{V}^{\prime },\forall i\ge t.$$

Thus, $L_{m+i}{V}^{\prime }=0$ for all $i>m$. Similarly, we have $J_{m+i}{V}^{\prime }=I_{m+i}{V}^{\prime }=0$ for all $i>m$, respectively. So there exists non-zero $w\in S$, such that $I_{i}w=J_{i}w=L_{i}w=0$ for $i>2m$. Since S is simple, it follows that $S=U\left(\mathcal{G}\right)w$. By PBW Theorem, each element of S is the linear combinations of

$$\cdots I_{2m-1}^{i_{2m-1}}{I}_{2m}^{i_{2m}}\cdots J_{2m-1}^{j_{2m-1}}{J}_{2m}^{j_{2m}}\cdots L_{2m-1}^{l_{2m-1}}{L}_{2m}^{l_{2m}}w.$$

So, for any $u\in S$, there exists sufficiently large $n\in \mathbb{Z}$, such that $I_{i}u=J_{i}u=L_{i}u=0$ for $i>n$, which means that S is a restricted $\mathcal{B}$-module, i.e., $S\in {\mathcal{R}}_{\mathcal{B}}$. This completes the proof. □

Now we give a precise characterization of simple restricted modules over $\mathcal{B}$ under certain conditions.

Theorem 3.

Assume that S is a simple restricted $\mathcal{B}$-module with $I_{0}v\ne -\frac{n^2-1}{24}{c}_{2}v$ for all $v\in S$ and $n\in \mathbb{Z}\backslash \left\{0\right\}$. Then there exist $d_1,d_2\in {\mathbb{Z}}_{+}$ and a simple ${\mathcal{B}}_{d_1,d_2}$-module V such that both conditions (a) and (b) of Theorem 1 are satisfied and $S\cong \mathrm{Ind}\left(V\right)$.

Proof. 

It follows from S is a simple restricted $\mathcal{B}$-module that

$$N_{\tilde{i},\tilde{j},\tilde{l}}:=\{v\in S\mid I_{i}v=J_{j}v=L_{l}v=0,\forall i>\tilde{i}, j>\tilde{j}, l>\tilde{l}\}\ne 0$$

for sufficiently large $\tilde{i},\tilde{j}$ and $\tilde{l}$. In addition, $N_{\tilde{i},\tilde{j},\tilde{l}}=0$ for all $\tilde{i}<0$ since we have $I_{0}v\ne 0$ for any non-zero vector $v\in S$ by the assumption that $I_{0}v\ne -\frac{n^2-1}{24}{c}_{2}v$ for all $v\in S$ and $n\in \mathbb{Z}\backslash \left\{0\right\}$. Therefore, one can find a smallest $k\in {\mathbb{Z}}_{+}$, and choose some $k^{\prime },k^{″}\ge k$ with $k^{″}-k\ge 2(k^{\prime }-k)$, such that $N_{k,k^{\prime },k^{″}}\ne 0$. Denote $d_1=k^{″}-k, d_2=k^{\prime }-k$ and $V=N_{k,k^{\prime },k^{″}}$. For any $i>k, j>k^{\prime }, l>k^{″}$ and $n\in {\mathbb{Z}}_{+}$, since $n+l-d_1>k^{″}-d_1=k, n+j-d_2>k^{\prime }-d_2=k, n+l-d_2>k^{″}-d_2\ge k^{\prime }$, we have

$$\begin{array}{cc}& I_i\left(I_{n-d_1}v\right)=J_j\left(I_{n-d_1}v\right)=0,\\ & L_l\left(I_{n-d_1}v\right)=(n-d_1-l)I_{n+l-d_1}v=0,\end{array}$$

$$\begin{array}{cc}& I_i\left(J_{n-d_2}v\right)=0,\\ & J_j\left(J_{n-d_2}v\right)=(j-n+d_2)I_{n+j-d_2}v=0,\\ & L_l\left(J_{n-d_2}v\right)=(n-d_2-l)J_{n+l-d_2}v=0,\end{array}$$

and

$$\begin{array}{cc}& I_i\left(L_{n}v\right)=(n-i)I_{n+i}v=0,\\ & J_j\left(L_{n}v\right)=(n-j)J_{n+j}v=0,\\ & L_l\left(L_{n}v\right)=(n-l)L_{n+l}v=0\end{array}$$

for any $v\in V$. Therefore, V is a ${\mathcal{B}}_{d_1,d_2}$-module.

First we assume that $k\in \mathbb{N}$. In this case, we have $I_k$ acts injectively on V. Since S is simple, it is generated by V and there exists a canonical surjective map

$$\pi :\mathrm{Ind}\left(V\right)\to S,\pi (1\otimes v)=v,\forall v\in V.$$

Hence, S is a simple quotient module of $\mathrm{Ind}\left(V\right)$. It suffices to check that $\pi$ is also injective. Assume conversely that $K:=\ker \left(\pi \right)\ne 0$. Obviously, $K\cap V=0$. Then we can choose a non-zero vector $v\in K$, such that $\mathrm{deg}\left(v\right)=(\mathbf{i},\mathbf{j},\mathbf{l})$ is minimal possible. Note that K is a $\mathcal{B}$-submodule of $\mathrm{Ind}\left(V\right)$ and, hence, is stable under the actions of $L_i,J_i,I_i$ for all $i\in \mathbb{Z}$. By Claim 1 in the proof of Theorem 1 and Remark 1, we can obtain a new vector $u\in K$ with $\mathrm{deg}\left(u\right)\prec (\mathbf{i},\mathbf{j},\mathbf{l})$, which leads a contradiction. This means $K=0$, which, in turn, forces $S\cong \mathrm{Ind}\left(V\right)$. In particular, V is simple as a ${\mathcal{B}}_{d_1,d_2}$-module.

In the remaining case $k=0$, we can deduce the same result using the assumption $I_{0}v\ne -\frac{n^2-1}{24}{c}_{2}v$ for all $v\in S$ and $n\in \mathbb{Z}\backslash \left\{0\right\}$ and similar discussion as above. □

5. Examples

In this section, we present some examples of simple restricted ${\mathcal{B}}_{d_1,d_2}$-modules. From Theorem 1, we obtain the corresponding simple restricted $\mathcal{B}$-modules.

Example 1.

Let $\mathfrak{h}={\mathrm{span}}_{\mathbb{C}}\{L_0,I_0,J_0\}$ be the Cartan subalgebra of $\mathcal{B}$. For

$$\xi =(\xi \left(L_0\right),\xi \left(I_0\right),\xi \left(J_0\right),c,c_1,c_2)\in {\mathfrak{h}}^{*}$$

with $\xi \left(I_0\right)\ne -\frac{n^2-1}{24}{c}_2$ for all $n\in \mathbb{Z}\backslash \left\{0\right\}$, we have the Verma module $M\left(\xi \right)=\mathcal{U}\left(\mathcal{B}\right){\otimes }_{\mathcal{U}\left({\mathcal{B}}_{0,0}\right)}{\mathbb{C}}_{\xi }$, where $L_i{\mathbb{C}}_{\xi }=I_i{\mathbb{C}}_{\xi }=J_i{\mathbb{C}}_{\xi }=0$ for $i>0$, and $L_0{\mathbb{C}}_{\xi }=\xi \left(L_0\right){\mathbb{C}}_{\xi },I_0{\mathbb{C}}_{\xi }=\xi \left(I_0\right){\mathbb{C}}_{\xi },J_0{\mathbb{C}}_{\xi }=\xi \left(J_0\right){\mathbb{C}}_{\xi },C{\mathbb{C}}_{\xi }-c{\mathbb{C}}_{\xi },C_1{\mathbb{C}}_{\xi }-c_1{\mathbb{C}}_{\xi },C_2{\mathbb{C}}_{\xi }-c_2{\mathbb{C}}_{\xi }$, respectively. Directly by Theorem 1 (with $d_1=d_2=k=0$), the induced module $M\left(\xi \right)$ is simple.

Example 2.

Fix ${\lambda }_1,{\lambda }_2,\mu ,\nu \in \mathbb{C}$ with $\mu \ne 0$. We denote by Q the ${\mathcal{B}}_{1,1}$-module $\mathcal{U}\left({\mathcal{B}}_{1,1}\right)/I$, where I is the left ideal generated by

$$L_1-{\lambda }_1,L_2-{\lambda }_2,L_3,\dots ,I_1-\mu ,I_2,\dots ,J_1-\nu ,J_2,\dots ,C-c,C_1-c_1,C_2-c_2.$$

Next, we want to show that Q is a simple ${\mathcal{B}}_{1,1}$-module. Take $0\ne v\in Q$ and write

$$v=\sum _{l,i_1,i_2,j_1,j_2\in {\mathbb{Z}}_{+}}{a}_{l,i_1,i_2,j_1,j_2}{L}_0^l{I}_{-1}^{i_1}{I}_0^{i_2}{J}_{-1}^{j_1}{J}_0^{j_2},$$

with only finitely many $a_{l,i_1,i_2,j_1,j_2}$ non-zero. Set $\mathrm{supp}\left(v\right)=\{(l,i_1,i_2,j_1,j_2)\mid a_{l,i_1,i_2,j_1,j_2}\ne 0\}$ and denote by $\mathrm{deg}\left(v\right)$ the maximal element in $\mathrm{supp}\left(v\right)$ under the following total order

$$\begin{array}{ccc}& & (l,i_1,i_2,j_1,j_2)\succ (p,m_1,m_2,n_1,n_2)\hfill \\ & ⟺& (l,i_1+j_1,i_1+i_2+j_1+j_2,i_1,i_2,j_1,j_2)>(p,m_1+n_1,m_1+m_2+n_1+n_2,\hfill \\ & & m_1,m_2,n_1,n_2)\hfill \end{array}$$

for any $(l,i_1,i_2,j_1,j_2),(p,m_1,m_2,n_1,n_2)\in {\mathbb{Z}}_{+}^5$. Suppose that $\mathrm{deg}\left(v\right)=(l,i_1,i_2,j_1,j_2)$.

Case 1: $l>0$.

In this case, it is not hard to see $\mathrm{deg}\left((I_1-{\mu }_1)v\right)=(l-1,i_1,i_2,j_1,j_2)$.

Case 2: $l=0,j_1+j_2\ne 0$.

If $j_1\ne 0$, then $\mathrm{deg}\left(J_{2}v\right)=(0,i_1,i_2,j_1-1,j_2)$. If $j_1=0,j_2\ne 0$, then $\mathrm{deg}\left((J_1-\nu )v\right)=(0,i_1,i_2,0,j_2-1)$.

Case 3: $l=0,j_1=j_2=0,i_1+i_2\ne 0$.

First assume that $i_1\ne 0$. If $\nu =0$ or $\nu \ne 0$ and $(0,i_1-1,i_2,1,0)\notin \mathrm{supp}\left(v\right)$, then in both cases we have $\mathrm{deg}\left((L_2-{\lambda }_2)v\right)=(0,i_1-1,i_2,0,0)$. For the case $\nu \ne 0$ and $(0,i_1-1,i_2,1,0)\in \mathrm{supp}\left(v\right)$, we get $\mathrm{deg}\left(J_{2}v\right)=(0,i_1-1,i_2,0,0)$. Assume now that $i_1=0,i_2\ne 0$. If $\nu =0$ or $\nu \ne 0$ and $(0,0,i_2-1,0,1)\notin \mathrm{supp}\left(v\right)$, then $\mathrm{deg}\left((L_1-{\lambda }_1)v\right)=(0,0,i_2-1,0,0)$ in each case. In the remaining case $\nu \ne 0$ and $(0,0,i_2-1,0,1)\in \mathrm{supp}\left(v\right)$, we get $\mathrm{deg}\left((J_1-\nu )v\right)=(0,0,i_2-1,0,0)$.

In conclusion, we see that Q is a simple ${\mathcal{B}}_{1,1}$-module. Thanks to Theorem 1 (with $d_1=d_2=k=1$), the induced module $\mathrm{Ind}\left(Q\right)$ is simple. It is obvious that $\mathrm{Ind}\left(Q\right)$ is a universal Whittaker $\mathcal{B}$-module.

Example 3.

Fix $\lambda ,{\mu }_1,{\mu }_2,{\nu }_1,{\nu }_2\in \mathbb{C}$ with ${\mu }_2\ne 0$. We denote by Q the ${\mathcal{B}}_{1,1}$-module $\mathcal{U}\left({\mathcal{B}}_{1,1}\right)/I$, where I is the left ideal generated by

$$L_2-\lambda ,L_4,\dots ,I_{-1}-{\mu }_1,I_1,I_2-{\mu }_2,I_3,\dots ,J_1-{\nu }_1,J_2-{\nu }_2,J_3,\dots ,C-c,C_1-c_1,C_2-c_2.$$

It can be verified that Q is a simple ${\mathcal{B}}_{1,1}$-module with a basis

$$\{L_0^{l_1}{L}_1^{l_2}{L}_3^{l_3}{I}_0^i{J}_{-1}^{j_1}{J}_0^{j_2}\mid l_1,l_2,l_3,i,j_1,j_2\in {\mathbb{Z}}_{+}\}.$$

From Theorem 1 (with $d_1=d_2=1$ and $k=2$), the induced module $\mathrm{Ind}\left(Q\right)$ is simple. As the actions of $L_1$ and $L_3$ are free, $\mathrm{Ind}\left(Q\right)$ is a new simple $\mathcal{B}$-module.

6. Conclusions

In this paper, we construct some simple restricted $\mathcal{B}$-modules using the technique of induction. One can see that the total order equipped with ${\mathbb{T}}^3$ plays a key role in the process. The conditions in Theorem 1 means that V can be viewed as a simple module over some finite-dimensional solvable quotient algebra of ${\mathcal{B}}_{d_1,d_2}$. We also characterize simple restricted $\mathcal{B}$-modules as simple modules with locally finite (nilpotent) actions of elements in certain positive part. Any simple restricted $\mathcal{B}$-module satisfying conditions in Theorem 3 is determined by some simple module over a certain subalgebra ${\mathcal{B}}_{d_1,d_2}$, which turns out to be some simple module over the corresponding finite-dimensional algebra. Finally, we enumerate three non-trivial examples to which Theorem 1 apply.