A categorical approach to Weyl modules

TL;DR

This paper provides a homological framework for defining Weyl modules associated with affine Lie algebras and explores their properties and relationships to tensor products, extending previous algebraic constructions.

Contribution

It introduces a homological characterization of local and global Weyl modules, enabling a functorial approach and analysis of their exactness properties.

Findings

Homological definitions of Weyl modules are natural and effective.

The Weyl functor relates modules over commutative algebras to Lie algebra modules.

The functors are generally not left exact, even for coordinate rings of affine varieties.

Abstract

Global and local Weyl Modules were introduced via generators and relations in the context of affine Lie algebras in a work by the first author and Pressley and were motivated by representations of quantum affine algebras. A more general case was considered by Feigin and Loktev by replacing the polynomial ring with the coordinate ring of an algebraic variety. We show that there is a natural definition of the local and global modules via homological properties. This characterization allows us to define the Weyl functor from the category of left modules of a commutative algebra to the category of modules for a simple Lie algebra. As an application we are able to understand the relationships of these functors to tensor products, generalizing previous results. Finally an analysis of the fundamental Weyl modules proves that the functors are not left exact in general, even for coordinate rings of affine varieties.