Weyl modules for Lie superalgebras

TL;DR

This paper introduces and studies global and local Weyl modules for Lie superalgebras tensorized with associative commutative algebras, establishing their fundamental properties and highlighting novel features in the superalgebra context.

Contribution

It defines Weyl modules for Lie superalgebras of the form g ⊗ A and proves their universality, finite-dimensionality, and tensor product decomposition, extending classical results to the super setting.

Findings

Proved universality and finite-dimensionality of Weyl modules for superalgebras.

Established tensor product decomposition properties.

Identified new features unique to the superalgebra case.

Abstract

We define global and local Weyl modules for Lie superalgebras of the form $\mathfrak{g} \otimes A$, where $A$ is an associative commutative unital $\mathbb{C}$-algebra and $\mathfrak{g}$ is a basic Lie superalgebra or $\mathfrak{sl}(n,n)$, $n \ge 2$. Under some mild assumptions, we prove universality, finite-dimensionality, and tensor product decomposition properties for these modules. These properties are analogues of those of Weyl modules in the non-super setting. We also point out some features that are new in the super case.