Irreducible Integrable Modules for the full Toroidal Lie Algebras co-ordinated by Rational Quantum Torus

TL;DR

This paper classifies irreducible integrable modules with finite-dimensional weight spaces for the full toroidal Lie algebra associated with rational quantum tori, extending understanding of their representation theory.

Contribution

It provides the first classification of irreducible integrable modules with finite-dimensional weight spaces for the full toroidal Lie algebra co-ordinated by rational quantum tori.

Findings

Classification of irreducible integrable modules achieved

Modules with nonzero central action characterized

Advances understanding of representations of quantum torus-related Lie algebras

Abstract

Let $\mathbb{C}_q$ be a non-commutative Laurent polynomial ring associated with a $(n+1)\times (n+1)$ rational quantum matrix $q$. Let $\mathfrak{sl}_d(\mathbb{C}_q)\oplus HC_1(\mathbb{C}_q)$ be the universal central extension of Lie subalgebra $\mathfrak{sl}_d(\mathbb{C}_q)$ of $\mathfrak{gl}_d(\mathbb{C}_q)$. Now let us take the Lie algebra $\tau=\mathfrak{gl}_d(\mathbb{C}_q)\oplus HC_1(\mathbb{C}_q)$. Let $Der(\mathbb{C}_q)$ be the Lie algebra of all derivations of $\mathbb{C}_q$. Now we consider the Lie algebra $\tilde{\tau}=\tau\rtimes Der(\mathbb{C}_q)$, called as full toroidal Lie algebra co-ordinated by rational quantum tori. In this paper we get a classification of irreducible integrable modules with finite dimensional weight spaces for $\tilde{\tau}$ with nonzero central action on the modules.