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

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 ${\mathfrak{sl}}_d({\mathbb{C}}_q)\oplus H{C}_1({\mathbb{C}}_q)$ 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 $\tau ={\mathfrak{gl}}_d({\mathbb{C}}_q)\oplus H{C}_1({\mathbb{C}}_q)$. Let $Der({\mathbb{C}}_q)$ be the Lie algebra of all derivations of ${\mathbb{C}}_q$. Now we consider the Lie algebra $\tilde{\tau }=\tau ⋊Der({\mathbb{C}}_q)$, 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.

20 pages