Irreducible modules over the divergence zero algebras and their $q$-analogues

TL;DR

This paper investigates irreducible modules over divergence zero algebras and their q-analogues, constructing new infinite-dimensional weight modules using Larsson's functor and establishing their irreducibility.

Contribution

It introduces a unified method to determine irreducibility of modules derived from $ ext{sl}_d$-modules, expanding the class of known irreducible weight modules.

Findings

Identified conditions for irreducibility of modules over divergence zero algebras.

Constructed new irreducible weight modules with infinite-dimensional weight spaces.

Extended results to quantum tori and q-analogues.

Abstract

In this paper, we study a class of $\Z_d$-graded modules, which are constructed using Larsson's functor from $\sl_d$-modules $V$, for the Lie algebras of divergence zero vector fields on tori and quantum tori. We determine the irreducibility of these modules for finite-dimensional or infinite-dimensional $V$ using a unified method. In particular, these modules provide new irreducible weight modules with infinite-dimensional weight spaces for the corresponding algebras.