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

TL;DR

This work addresses the classification of irreducible integrable modules with finite-dimensional weight spaces for toroidal Lie algebras $\hat{\tau}(d,q)$ coordinated by the rational quantum torus $\mathbb{C}_q$ under trivial center action. It develops a framework that uses central-extension analysis, highest-weight theory, and central-operator techniques to reduce to finite-dimensional quotients over $\mathfrak{gl}_d(\mathbb{C}_q)$ and then reconstructs graded modules from these quotients via tensoring with the Laurent ring $A_n$. The main contributions are: (i) showing that trivial action of the center $C$ implies trivial action of the central extension $HC_1(\mathbb{C}_q)$, (ii) establishing the finite-dimensional quotient structure and a corresponding $\mathfrak{gl}_d(\mathbb{C}_q)$-module description, (iii) deriving a precise two-type final classification that combines with the Rao-Zhao result for nontrivial center action. Together, these results complete the classification of irreducible integrable $\hat{\tau}(d,q)$-modules with finite-dimensional weight spaces in the trivial center case, linking graded and non-graded theories and extending toroidal representation theory.

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 $\hatτ(d,q)$ be the toroidal Lie algebra coordinated by rational quantum torus obtained by adjoining the derivation space $D$ to the universal central extension $\tildeτ(d,q)=\mathfrak{sl}_d(\mathbb{C}_q)\oplus HC_1(\mathbb{C}_q)$ 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 $\hatτ(d,q)$-modules with finite-dimensional weight spaces in the case where the $n$-dimensional center $C$ acts trivially on the modules.

Theorems & Definitions (45)

• Proposition 2.1
• Lemma 2.2
• Definition 3.1
• Proposition 3.2
• Definition 3.3
• Lemma 3.4
• Lemma 3.5
• proof
• Proposition 4.1
• proof