Local Weyl modules for equivariant map algebras with free abelian group actions

TL;DR

This paper extends the concept of local Weyl modules to equivariant map algebras with free abelian group actions, establishing their properties and isomorphisms with untwisted cases.

Contribution

It introduces twisting and untwisting functors for equivariant map algebras, generalizing local Weyl modules beyond previous specific algebra types.

Findings

Defined twisting and untwisting functors as isomorphisms.

Extended properties of local Weyl modules to new algebraic settings.

Showed tensor product and homological characterizations hold in the generalized context.

Abstract

Suppose a finite group acts on a scheme X and a finite-dimensional Lie algebra g. The associated equivariant map algebra is the Lie algebra of equivariant regular maps from X to g. Examples include generalized current algebras and (twisted) multiloop algebras. Local Weyl modules play an important role in the theory of finite-dimensional representations of loop algebras and quantum affine algebras. In the current paper, we extend the definition of local Weyl modules (previously defined only for generalized current algebras and twisted loop algebras) to the setting of equivariant map algebras where g is semisimple, X is affine of finite type, and the group is abelian and acts freely on X. We do so by defining twisting and untwisting functors, which are isomorphisms between certain categories of representations of equivariant map algebras and their untwisted analogues. We also show that other properties of local Weyl modules (e.g. their characterization by homological properties and a tensor product property) extend to the more general setting considered in the current paper.