Tensor modules over Witt superalgebras

TL;DR

This paper classifies simple tensor modules over Witt superalgebras by analyzing modules constructed from Weyl superalgebras and general linear Lie superalgebras, completing classification problems in their weight representation theory.

Contribution

It provides necessary and sufficient conditions for the simplicity of tensor modules over Witt superalgebras and classifies their simple subquotients, advancing the understanding of their representation theory.

Findings

Characterized when tensor modules are simple.

Determined all simple subquotients when not simple.

Contributed to the classification of weight modules over Witt superalgebras.

Abstract

In this paper, we study the tensor module $P\otimes M$ over the Witt superalgebra $W_{m,n}^+$ (resp. $W_{m,n}$), where $P$ is a simple module over the Weyl superalgebra $K_{m,n}^+$ (resp. $K_{m,n}$) and $M$ is simple weight module over the general linear Lie superalgebra $\mathfrak{gl}(m,n)$. We obtain the necessary and sufficient conditions for $P\otimes M$ to be simple, and determine all simple subquotient of $P\otimes M$ when it is not simple. All the work leads to completion of some classification problems on the weight representation theory of $W_{m,n}^+$ and $W_{m,n}$.