Classification of irreducible integrable modules for toroidal Lie algebras with finite dimensional weight spaces

Abstract

The purpose of this paper is to classify irreducible integrable modules for multi-loop algebras. Let V be a vector space over the complex numbers . Let A = t± 1 t± n be the Laurent polynomials ring in n variables. Let VA = V ⊗A and let v m = v ⊗ t for m = m1 mn ∈ n and t = t1 1 t2 2 · · · tn n . Let be a simple finite-dimensional Lie algebra over the complex numbers . Then A can be given a Lie algebra structure (see (1.1)). Let d1 dn be derivations defined by di X m = miX m and let D be the linear span of d1 dn. Then A = A ⊕ is a Lie algebra which we call a multi-loop algebra. The universal central externion of A is called a toroidal Lie algebra and is studied in [BB, BC, E3, E4, EM]. So our multiloop algebras are quotients of toroidal Lie algebras by centers. Fix a Cartan subalgebra h of . Let ψ be a -graded homomorphism of U hA → A (see Section 1 for details). Then we can define the universal highest weight module M ψ for A. Let V ψ be the unique irreducible quotient of M ψ . We then prove in Theorem 3.4 that any irreducible integrable module for A is isomorphic to V ψ by making use of a result from Chari [C]. We associate to ψ a map ψ from U hA → by evaluating at 1 and an irreducible module V ψ π for A (derivations do not act on V ψ , and therefore it cannot be extended to A). Now consider V ψ ⊗A as a