Finite W-superalgebras for basic Lie superalgebras

TL;DR

This paper establishes the PBW theorem for finite W-superalgebras associated with basic Lie superalgebras over complex and positive characteristic fields, highlighting the role of parity in their construction.

Contribution

It provides a detailed PBW theorem for finite W-superalgebras, clarifying their construction and the impact of parity on their structure, which aids in understanding modular representation theory.

Findings

PBW theorem for finite W-superalgebras proved

Construction differs based on parity of certain subspace dimension

Foundation for future work on super Kac-Weisfeiler property

Abstract

We consider the finite $W$-superalgebra $U(\mathfrak{g_\bbf},e)$ for a basic Lie superalgebra ${\ggg}_\bbf=(\ggg_\bbf)_\bz+(\ggg_\bbf)_\bo$ associated with a nilpotent element $e\in (\ggg_\bbf)_{\bar0}$ both over the field of complex numbers $\bbf=\mathbb{C}$ and over $\bbf={\bbk}$ an algebraically closed field of positive characteristic. In this paper, we mainly present the PBW theorem for $U({\ggg}_\bbf,e)$. Then the construction of $U({\ggg}_\bbf,e)$ can be understood well, which in contrast with finite $W$-algebras, is divided into two cases in virtue of the parity of $\text{dim}\,\mathfrak{g_\bbf}(-1)_{\bar1}$. This observation will be a basis of our sequent work on the dimensional lower bounds in the super Kac-Weisfeiler property of modular representations of basic Lie superalgebras (cf. \cite[\S7-\S9]{ZS}).