Cohomology and Support Varieties for Lie Superalgebras of Type W(n)

TL;DR

This paper computes the cohomology ring of the Lie superalgebra W(n), establishes support varieties for its modules, and shows these varieties align with known notions of atypicality, providing new insights into the geometric and combinatorial structure of superalgebras.

Contribution

It calculates the cohomology ring of W(n), defines support varieties for its modules, and demonstrates their equivalence with Serganova's atypicality, extending support variety theory to Cartan type superalgebras.

Findings

Cohomology ring of W(n) is a finitely generated polynomial ring.

Support varieties for W(n)-supermodules are fully characterized.

Support varieties coincide with Serganova's notion of atypicality.

Abstract

Boe, Kujawa and Nakano recently investigated relative cohomology for classical Lie superalgebras and developed a theory of support varieties. The dimensions of these support varieties give a geometric interpretation of the combinatorial notions of defect and atypicality due to Kac, Wakimoto, and Serganova. In this paper we calculate the cohomology ring of the Cartan type Lie superalgebra W(n) relative to the degree zero component W(n)_{0} and show that this ring is a finitely generated polynomial ring. This allows one to define support varieties for finite dimensional W(n)-supermodules which are completely reducible over W(n)_{0}. We calculate the support varieties of all simple supermodules in this category. Remarkably our computations coincide with the prior notion of atypicality for Cartan type superalgebras due to Serganova. We also present new results on the realizability of support varieties which hold for both classical and Cartan type superalgebras.