Support varieties and representations of tame basic classical Lie superalgebras

TL;DR

This paper establishes the finite generation of cohomology algebras for basic classical Lie superalgebras over fields of positive characteristic, enabling support variety theory and classifying representations for certain cases.

Contribution

It proves the cohomology algebra is finitely generated and classifies indecomposable representations of u(osp(1|2)), advancing the understanding of Lie superalgebra representations in positive characteristic.

Findings

Cohomology algebra is finitely generated for basic classical Lie superalgebras.

Restricted enveloping algebra is wild except for specific cases.

All indecomposable restricted representations of u(osp(1|2)) are classified.

Abstract

Let k be an algebraically closed field of characteristic p> 0. For a basic classical Lie superalgebra, we show its restricted cohomology algebra is a finitely generated algebra. Thus the cohomological support theory can be established. As a consequence, we show that the restricted enveloping algebra of a basic classical Lie superalgebra g is always wild except g=sl(2) or g=osp(1|2) or g=C(2). Moreover, all finite dimensional indecomposable restricted representations of u(osp(1|2)), the restricted enveloping algebra of Lie superalgebra osp(1|2), are determined.