Branching rules of semi-simple Lie algebras using affine extensions

TL;DR

This paper introduces a new closed formula for computing branching coefficients of semi-simple Lie algebra embeddings using affine extensions, providing an alternative proof of an existing algorithm and exploring applications in WZW models.

Contribution

It presents a novel closed formula for branching coefficients based on affine extensions, offering an alternative proof of a known algorithm and linking to NIM-reps in WZW models.

Findings

Derived a closed formula for branching coefficients using affine extensions.

Provided an alternative proof of the Racah-Speiser algorithm for branching rules.

Connected the approach to NIM-reps and Verlinde-like formulas in WZW models.

Abstract

We present a closed formula for the branching coefficients of an embedding p in g of two finite-dimensional semi-simple Lie algebras. The formula is based on the untwisted affine extension of p. It leads to an alternative proof of a simple algorithm for the computation of branching rules which is an analog of the Racah-Speiser algorithm for tensor products. We present some simple applications and describe how integral representations for branching coefficients can be obtained. In the last part we comment on the relation of our approach to the theory of NIM-reps of the fusion rings of WZW models with chiral algebra g_k. In fact, it turns out that for these models each embedding p in g induces a NIM-rep at level k to infinity. In cases where these NIM-reps can be be extended to finite level, we obtain a Verlinde-like formula for branching coefficients.