A conjectural presentation of fusion algebras

TL;DR

This paper investigates the structure of the kernel of the homomorphism from the representation ring of a semisimple Lie algebra to its fusion algebra at a fixed level, proposing conjectures for classical and some exceptional groups.

Contribution

It formulates conjectures for the presentation of the kernel of the homomorphism for classical groups and G2, extending previous results and providing partial results for F4 and E series.

Findings

Generators for the kernel are known for types A and C.

Conjectures are proposed for other classical groups and G2.

Partial results are obtained for F4 and E series.

Abstract

Let g be a semisimple Lie algebra over the complex numbers. Fix a positive integer l (called the level). Let R(l,g) be the fusion algebra at level l. Then, there is an algebra homomorphism from the representation ring R(g) of g to R(l,g). We study a presentation of its kernel. The generators for the kernel were given by Gepner, Gepner-Schwimmer, Bourdeau-Mlawer-Riggs-Schnitzer for g of type A and C series. We make a conjecture for other classical groups and also for g of type G2. We also have some partial results for F4 and E series.