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

Let g = g((0) over bar) + g((1) over bar) be a basic Lie superalgebra over C, and e a minimal nilpotent element in g((0) over bar). Set W-x' to be the refined W-superalgebra associated with the pair (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. By virtue of this, we show that over an algebraically closed field K of characteristic p >> 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.