Integrable representations of toroidal Lie algebras co-ordinatized by rational quantum tori

Abstract

For any positive integers d, n with d>1 and n>1, we fix an n by n complex matrix q=(qij) satisfying qii=1 and qij=qji−1 with all qij roots of unity. Let τ˜(d,q) be the universal central extension of the Lie subalgebra sld(Cq) of gld(Cq) with trace in [Cq,Cq], where Cq is the rational quantum torus associated to q, and let τˆ(d,q) be the Lie algebra by adding the n degree derivations to τ˜(d,q) with respect to the n non-commuting variables in Cq. The Lie algebra τˆ(d,q), called the toroidal Lie algebra co-ordinatized by the rational quantum torus Cq, has an n-dimensional center C. In this paper, we obtain a classification of irreducible integrable modules with finite dimensional weight spaces and with non-zero center action over the toroidal Lie algebra τˆ(d,q).