Cohomology and Support Varieties for Restricted Lie Superalgebras

Abstract

Let \((\mathfrak{g}, [p]) \) be a restricted Lie superalgebra over an algebraically closed field k of characteristic p > 2. Let \(\mathfrak{u}(\mathfrak{g})\) denote the restricted enveloping algebra of \(\mathfrak{g}\). In this paper we prove that the cohomology ring \(\operatorname{H}^\bullet(\mathfrak{u}(\mathfrak{g}), k)\) is finitely generated. This allows one to define support varieties for finite dimensional \(\mathfrak{u}(\mathfrak{g})\)-supermodules. We also show that support varieties for finite dimensional \(\mathfrak{u}(\mathfrak{g})\)- supermodules satisfy the desirable properties of a support variety theory.