Braided tensor categories of admissible modules for affine Lie algebras

TL;DR

This paper constructs a braided tensor category structure for modules of affine Lie algebras at admissible levels, proving key properties in the case of ap;sl2, and discusses conjectures on rigidity, modularity, and quantum group equivalence.

Contribution

It develops a braided tensor category framework for admissible affine Lie algebra modules, including proofs of rigidity and modularity for ap;sl2.

Findings

Established a braided tensor category structure for admissible modules.

Proved rigidity and modularity for ap;sl2 case.

Formulated conjectures on the category's modularity and quantum group equivalence.

Abstract

Using the tensor category theory developed by Lepowsky, Zhang and the second author, we construct a braided tensor category structure with a twist on a semisimple category of modules for an affine Lie algebra at an admissible level. We conjecture that this braided tensor category is rigid and thus is a ribbon category. We also give conjectures on the modularity of this category and on the equivalence with a suitable quantum group tensor category. In the special case that the affine Lie algebra is $\widehat{\mathfrak{sl}}_2$, we prove the rigidity and modularity conjectures.