engleski

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

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

affine Lie algebras; vertex operator algebras; U(g)-bimodules; Frenkel-Zhu bimodule; fusion rules; irreducible representations; loop modules; tensor products

nije evidentirano