The Verlinde ring of a (let us say) simply connected simple compact Lie group has as underlying additive group the Grothendieck group of representations of the central extension $\widehat{LG}$ of the loop of $G$ with the 'positive energy' condition. I'm trying to find a concise mathematical definition of the product on this ring, the so-called *fusion product*. In Freed-Hopkins-Teleman's *Loop groups and twisted K-theory III* they define an $R(G)$-module structure on this additive group using induction of representations (hence a functorial description).

All references I'm reading just mention 'fusion rules', and cite Verlinde, whose 1988 paper is in the journal Nuclear Physics B. Surely we have a more recent discussion along the lines of the FHT construction mentioned above, and not in terms of linear combinations of coefficients of some irreps considered as generators...?

Confusion and Connes Fusions, golem.ph.utexas.edu/string/archives/000718.html) $\endgroup$ – David Roberts Feb 25 '15 at 9:54