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

TL;DR

This paper classifies all irreducible integrable modules with finite-dimensional weight spaces for toroidal Lie algebras coordinated by rational quantum tori, specifically when the center acts trivially, completing previous classifications.

Contribution

It provides a complete classification of modules for the specified toroidal Lie algebras with trivial central action, extending prior work on nontrivial cases.

Findings

Classified irreducible integrable modules with trivial central action.

Described modules with finite-dimensional weight spaces for rational quantum torus-coordinated Lie algebras.

Extended the classification to include cases previously not covered.

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.