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Classification of simple Harish-Chandra modules over the Ovsienko-Roger superalgebra

Munayim Dilxat, Liangyun Chen, Dong Liu

Abstract

With the $Ω$-operators for the Virasoro algebra \cite{BF} and the super Virasoro algebra in \cite{CL, CLL}, we get the $Ω$-operators for the Ovsienko-Roger superalgebras in this paper and then use it to classify all simple cuspidal modules for the $\bZ$-graded and $\frac12\bZ$-graded Ovsienko-Roger superalgebras. By this result, we can easily classify all simple Harish-Chandra modules over some related Lie superalgebras, including the $N=1$ BMS$_3$ algebra, the super $W(2,2)$, etc.

Classification of simple Harish-Chandra modules over the Ovsienko-Roger superalgebra

Abstract

With the -operators for the Virasoro algebra \cite{BF} and the super Virasoro algebra in \cite{CL, CLL}, we get the -operators for the Ovsienko-Roger superalgebras in this paper and then use it to classify all simple cuspidal modules for the -graded and -graded Ovsienko-Roger superalgebras. By this result, we can easily classify all simple Harish-Chandra modules over some related Lie superalgebras, including the BMS algebra, the super , etc.
Paper Structure (8 sections, 12 theorems, 18 equations)

This paper contains 8 sections, 12 theorems, 18 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Simple cuspidal $\mathfrak L$-module
  4. Simple Harish-Chandra module
  5. Applications
  6. The Lie superalgebra $\frak q$
  7. The $N=1$ BMS$_3$ algebra
  8. The super $W(2,2)$ algebra

Key Result

Lemma 2.1

$($BF$)$ Let $\Omega_{k, s}^{(m)}=\sum\limits_{i=0}^m(-1)^i\binom{m}{i}L_{k-i}L_{s+i}$. For every $\ell\in\mathbb{Z}_+$ there exists $m\in\mathbb{Z}_+$ such that for all $k, s\in\mathbb{Z}$, $\Omega_{k,s}^{(m)}$ annihilate every cuspidal $W$-module with a composition series of length $\ell$.

Theorems & Definitions (23)

  • Lemma 2.1
  • Theorem 2.2
  • Lemma 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Theorem 3.4
  • proof
  • Lemma 4.1
  • ...and 13 more