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Journal Article

Branching rules of semi-simple Lie algebras using affine extensions


Quella,  Thomas
Quantum Gravity & Unified Theories, AEI-Golm, MPI for Gravitational Physics, Max Planck Society;

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Quella, T. (2002). Branching rules of semi-simple Lie algebras using affine extensions. Journal of Physics A, 35, 3743-3753.

Cite as: https://hdl.handle.net/11858/00-001M-0000-0013-54F5-A
We present a closed formula for the branching coefficients of an embedding [fraktur p] [hookrightarrow] [fraktur g] of two finite-dimensional semi-simple Lie algebras. The formula is based on the untwisted affine extension of [fraktur p] . It leads to an alternative proof of a simple algorithm for the computation of branching rules, which is an analogue 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 ring in WZW models with chiral algebra [{mathfrak g}}_k] . In fact, it turns out that for these models each embedding [fraktur p] [hookrightarrow] [fraktur g] induces a NIM-rep at level k [rightarrow] [infty] . In cases where these NIM-reps can be extended to finite level, we obtain a Verlinde-like formula for branching coefficients. Reviewing this question, we propose a solution to a puzzle which remained open in related work by Alekseev, Fredenhagen, Quella and Schomerus.