Schur-Weyl Duality for Toroidal Algebras of Type $A$

TL;DR

This paper establishes a Schur-Weyl duality for a quotient of the 2-toroidal Lie algebra of type A and proposes a method to extend this duality to higher toroidal cases, broadening the understanding of algebraic dualities.

Contribution

It introduces a Schur-Weyl duality for 2-toroidal Lie algebras of type A and extends the framework to m-toroidal algebras, advancing the theory of algebraic dualities.

Findings

Proved Schur-Weyl duality for a quotient of the 2-toroidal Lie algebra of type A.

Developed a method to extend duality to m-toroidal Lie algebras.

Established foundational results for future exploration of toroidal algebra representations.

Abstract

We state and prove an analog of the Schur-Weyl duality for a quotient of the classical $2$-toroidal Lie algebra of type $A$. We then provide a method to extend this duality to the $m$-toroidal case, $m > 2$.