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

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 L _k,λ. Then L _k,λ can be viewed as an irreducible module for the vertex operator algebra M _k,0. Let A(L _k,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.