Whittaker modules of central extensions of Takiff superalgebras and finite supersymmetric $W$-algebras

TL;DR

This paper explores the structure and representations of Whittaker modules for central extensions of Takiff superalgebras, establishing equivalences with modules over supersymmetric finite W-algebras, and classifying irreducible representations.

Contribution

It introduces a novel equivalence between categories of Whittaker modules for certain superalgebras and finite W-algebras, advancing the understanding of their representation theory.

Findings

Category of $rak g$-Whittaker modules is equivalent to that of $rak s$-Whittaker modules.

Established an equivalence between modules over supersymmetric finite W-algebras and principal finite W-superalgebras.

Classified and constructed irreducible representations of principal finite supersymmetric W-algebras.

Abstract

For a basic classical Lie superalgebra $\mathfrak s$, let $\mathfrak g$ be the central extension of the Takiff superalgebra $\mathfrak s\otimes\Lambda(\theta)$, where $\theta$ is an odd indeterminate. We study the category of $\mathfrak g$-Whittaker modules associated with a nilcharacter $\chi$ of $\mathfrak g$ and show that it is equivalent to the category of $\mathfrak s$-Whittaker modules associated with a nilcharacter of $\mathfrak s$ determined by $\chi$. In the case when $\chi$ is regular, we obtain, as an application, an equivalence between the categories of modules over the supersymmetric finite $W$-algebras associated to the odd principal nilpotent element at non-critical levels and the category of the modules over the principal finite $W$-superalgebra associated to $\mathfrak s$. Here, a supersymmetric finite $W$-algebra is conjecturally the Zhu algebra of a supersymmetric affine $W$-algebra. This allows us to classify and construct irreducible representations of a principal finite supersymmetric $W$-algebra.