Integrable Representations for Toroidal Lie Algebras Co-ordinated by Rational Quantum Torus

Abstract

We classify irreducible integrable modules with finite-dimensional weight spaces for toroidal Lie algebras coordinated by rational quantum torus with trivial central action. Let $\mathbb{C}_q$ denote the rational quantum torus associated with a rational quantum matrix $q$, and let $\hat{\tau}(d,q)$ be the toroidal Lie algebra coordinated by rational quantum torus obtained by adjoining the derivation space $D$ to the universal central extension $\tilde{\tau}(d,q)=\mathfrak{sl}_d(\mathbb{C}_q)\oplus HC_1(\mathbb{C}_q)$ of $\mathfrak{sl}_d(\mathbb{C}_q)$. The case of nontrivial central action was previously classified by S. Eswara Rao and K. Zhao. The present work completes the classification by describing all irreducible integrable $\hat{\tau}(d,q)$-modules with finite-dimensional weight spaces in the case where the $n$-dimensional center $C$ acts trivially on the modules.