Weyl modules and Weyl functors for Lie superalgebras

TL;DR

This paper introduces Weyl functors and modules for map superalgebras derived from specific Lie superalgebras, establishing their properties and conditions for finite dimensionality and finite generation.

Contribution

It defines Weyl functors and modules for map superalgebras associated with certain Lie superalgebras, and proves their universal, tensor product, and finiteness properties.

Findings

Weyl modules satisfy universal and tensor product properties.

Conditions for finite dimensionality of local Weyl modules.

Conditions for finite generation of global Weyl modules.

Abstract

Given an algebraically closed field $\Bbbk$ of characteristic zero, a Lie superalgebra $\mathfrak{g}$ over $\Bbbk$ and an associative, commutative $\Bbbk$-algebra $A$ with unit, a Lie superalgebra of the form $\mathfrak{g} \otimes_\Bbbk A$ is known as a map superalgebra. Map superalgebras generalize important classes of Lie superalgebras, such as, loop superalgebras (where $A=\Bbbk[t, t^{-1}]$), and current superalgebras (where $A=\Bbbk[t]$). In this paper, we define Weyl functors, global and local Weyl modules for all map superalgebras where $\mathfrak{g}$ is either $\mathfrak{sl} (n,n)$ with $n \ge 2$, or a finite-dimensional simple Lie superalgebra not of type $\mathfrak{q}(n)$. Under certain conditions on the triangular decomposition of these Lie superalgebras we prove that global and local Weyl modules satisfy certain universal and tensor product decomposition properties. We also give necessary and sufficient conditions for local (resp. global) Weyl modules to be finite dimensional (resp. finitely generated).