Fusion algebra at a rational level and cohomology of nilpotent subalgebras of sl_2

TL;DR

This paper uses cohomological methods to define and compute the fusion algebra of the WZW model at rational levels, providing a new perspective on admissible representations of affine sl_2.

Contribution

It introduces a cohomological approach to characterize the fusion algebra and admissible representations at rational levels, which is a novel method in the field.

Findings

Fusion algebra explicitly calculated at rational levels

Cohomological characterization of admissible representations

New insights into the structure of affine sl_2 representations

Abstract

We define and calculate the fusion algebra of WZW model at a rational level by cohomological methods. As a byproduct we obtain a cohomological characterization of admissible representations of $\widehat{\gtsl}_{2}$.