Minimal W-superalgebras and the modular representations of basic lie superalgebras

Let g = g0 + g1 be a basic Lie superalgebra over ℂ, and e a minimal nilpotent element in g0. 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.