Fusion rules for admissible representations of affine algebras: the case of $A_2^{(1)}$

TL;DR

This paper derives fusion rules for admissible representations of the affine algebra _2^{(1)} at fractional levels, using an affine Weyl group interpretation, and discusses related algebraic structures.

Contribution

It introduces a new formula for fusion rules of admissible representations of _2^{(1)} at fractional levels based on an affine Weyl group framework.

Findings

Fusion rules formulae expressed via affine Weyl group

Interpretation of fusion rules in terms of affine Weyl group

Discussion of a hidden finite dimensional graded algebra

Abstract

We derive the fusion rules for a basic series of admissible representations of $\hat{sl}(3)$ at fractional level $3/p-3$. The formulae admit an interpretation in terms of the affine Weyl group introduced by Kac and Wakimoto. It replaces the ordinary affine Weyl group in the analogous formula for the fusion rules multiplicities of integrable representations. Elements of the representation theory of a hidden finite dimensional graded algebra behind the admissible representations are briefly discussed.