Minimal W-superalgebras and modular representations of basic Lie superalgebras

TL;DR

This paper constructs explicit generators for minimal W-superalgebras associated with basic Lie superalgebras and demonstrates that the minimal dimensions of their modular representations are achievable in large characteristic fields.

Contribution

It provides explicit generators and relations for minimal W-superalgebras and applies these to establish the attainability of lower bounds in modular representation dimensions.

Findings

Explicit generators of minimal W-superalgebras are given.

Lower bounds of modular representation dimensions are shown to be attainable.

Results extend super Kac-Weisfeiler property to these structures.

Abstract

Let $\mathfrak{g}=\mathfrak{g}_{\bar 0}+\mathfrak{g}_{\bar 1}$ be a basic Lie superalgebra over $\mathbb{C}$, and $e$ a minimal nilpotent element in $\mathfrak{g}_{\bar 0}$. Set $W_\chi'$ to be the refined $W$-superalgebra associated with the pair $(\mathfrak{g},e)$, which is called a minimal $W$-superalgebra. In this paper we present a set of explicit generators of minimal $W$-superalgebras and the commutators between them. In virtue of this, we show that over an algebraically closed field $\mathds{k}$ of characteristic $p\gg0$, the lower bounds of dimensions in the modular representations of basic Lie superalgebras with minimal nilpotent $p$-characters are attainable. Such lower bounds are indicated in \cite{WZ} as the super Kac-Weisfeiler property.