Tensor categories of affine Lie algebras beyond admissible level

Thomas Creutzig, Jinwei Yang

Published 2020-02-13Version 1

We show that if $V$ is a vertex operator algebra of positive energy such that all the irreducible ordinary $V$-modules are $C_1$-cofinite and all the grading-restricted generalized Verma modules for $V$ are of finite length, then the category of finite length generalized $V$-modules has a braided tensor category structure. By applying the general theorem to the simple affine vertex operator algebra associated to $\mathfrak{g}$ at level $k$ and the category $KL_k(\mathfrak{g})$ of its ordinary modules, we discover several families of $KL_k(\mathfrak{g})$ at non-admissible levels $k$, having braided tensor category structures. In particular, $KL_k(\mathfrak{g})$ has a braided tensor category structure if it is semisimple or more generally if each object in $KL_k(\mathfrak{g})$ is of finite length. We also prove the rigidity and determine the fusion rules of some categories $KL_k(\mathfrak{g})$, including the category $KL_{-1}(\mathfrak{sl}_n)$. Using these results, we construct a rigid tensor category structure on $KL_1(\mathfrak{sl}(n|m))$.