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