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

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.