Weyl modules for toroidal Lie algebras

TL;DR

This paper investigates Weyl modules for toroidal Lie algebras with multiple variables, establishing isomorphisms with Fock space representations and computing graded characters, thus extending previous work in the field.

Contribution

It demonstrates that level one global Weyl modules are isomorphic to submodules of Fock space representations and calculates their graded characters.

Findings

Level one global Weyl modules are isomorphic to Fock space submodules.

Graded characters of level one local Weyl modules are explicitly computed.

Generalizes previous results to toroidal Lie algebras with arbitrary variables.

Abstract

In this paper we study Weyl modules for a toroidal Lie algebra $\CT$ with arbitrary $n$ variables. Using the work of Rao \cite{1995}, we prove that the level one global Weyl modules of $\CT$ are isomorphic to suitable submodules of a Fock space representation of $\CT$ upto a twist. As an application, we compute the graded character of the level one local Weyl module of $\CT$, thereby generalising the work of Kodera \cite{ko}.