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An Application of U(g)-bimodules to Representation Theory of Affine Lie Algebras

Research paper by Dražen Adamović

Indexed on: 01 Oct '04Published on: 01 Oct '04Published in: Algebras and Representation Theory



Abstract

Let \(\hat {\mathfrak{g}}\) be the affine Lie algebra associated to the simple finite-dimensional Lie algebra \({\mathfrak{g}}\). We consider the tensor product of the loop \(\hat {\mathfrak{g}}\)-module \(\overline {V\left( \mu \right)} \) associated to the irreducible finite-dimensional \({\mathfrak{g}}\)-module V(μ) and the irreducible highest weight \(\hat {\mathfrak{g}}\)-module Lk,λ. Then Lk,λ can be viewed as an irreducible module for the vertex operator algebra Mk,0. Let A(Lk,λ) be the corresponding \(A\left( {M_{k,0} } \right)\left( { = U\left( {\mathfrak{g}} \right)} \right)\)-bimodule. We prove that if the \({U\left( {\mathfrak{g}} \right)}\)-module \(A\left( {L_{k,0} } \right) \otimes _{U\left( \mathfrak{g} \right)} V\left( \mu \right)\) is zero, then the \({\hat {\mathfrak{g}}}\)-module \(\left( {L_{k,0} } \right) \otimes _{U\left( {\mathfrak{g}} \right)} V\left( \mu \right)\)is irreducible. As an example, we apply this result on integrable representations for affine Lie algebras.