Minimal $W$-Superalgebras and the Modular Representations of Basic Lie Superalgebras

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 $\mathbb k$ of characteristic $p \gg 0$, 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 \[33] as the super Kac–Weisfeiler property.