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 $W_{\chi }^{\prime }$ 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.