bounded weight modules over the Lie superalgebra of Cartan W-type

TL;DR

This paper classifies simple bounded weight modules over the Lie superalgebra of Cartan W-type, showing they are quotients of tensor modules constructed from modules over Weyl superalgebras and general linear algebras.

Contribution

It provides a complete classification of simple bounded weight modules over the Witt superalgebra W_{m,n} in terms of tensor modules derived from modules over Weyl superalgebras and gl_m, gl_n modules.

Findings

All such modules are simple quotients of tensor modules F(P,L(V_1⊗V_2))

Modules are constructed from simple weight modules over Weyl superalgebras

Classification covers modules with respect to the standard Cartan algebra

Abstract

Let $A_{m,n}$ be the tensor product of the polynomial algebra in $m$ even variables and the exterior algebra in $n$ odd variables over the complex field $\C$, and the Witt superalgebra $W_{m,n}$ be the Lie superalgebra of superderivations of $A_{m,n}$. In this paper, we classify the non-trivial simple bounded weight $W_{m,n}$ modules with respect to the standard Cartan algebra of $W_{m,n}$. Any such module is a simple quotient of a tensor module $F(P,L(V_1\otimes V_2))$ for a simple weight module $P$ over the Weyl superalgebra $\mathcal K_{m,n}$, a finite-dimensional simple $\gl_m$-module $V_1$ and a simple bounded $\gl_n$-module $V_2$.