The Virasoro minimal model $mathcal{M}(2,5)$ (or in some conventions also $mathcal{M}(5,2)$ is the conformal field theory which describes the critical behaviour of the Lee-Yang edge singularity. It is described, for example, in Conformal Field Theory, by di Francesco, Mathieu and Sénéchal; albeit the description of the Lee-Yang singularity itself is perhaps a little too physicsy. Still their treatment of minimal models should be amenable to mathematicians without prior exposure to physics.
At any rate, googling Lee-Yang edge singularity might reveal other sources easier to digest. In general it is the Verlinde formula which relates the fusion ring and the Virasoro characters, and at least for the case of the Lee-Yang singularity, these can be related in turn to Temperley-Lieb algebras and Ocneanu path algebras on a suitable graph. Some details appear in this paper.
The relation between the Virasoro minimal models and the representations of $widehat{sl}_2$ goes by the name of the coset construction in the Physics conformal field theory literature or also Drinfeld-Sokolov reduction. This procedure gives a cohomology theory (a version of semi-infinite cohomology for a nilpotent subalgebra) which produces Virasoro modules from $widehat{sl}_2$ modules. Relevant words to google are W- algebras, Casimir algebras,... Of course here we are dealing with the simplest case of $widehat{sl}_2$ and Virasoro, which is the tip of a very large iceberg. The case of the Lee-Yang edge singularity is simple enough that it appears in many papers as an example from which to understand more general constructions.
I know less about the quantum group story, but this paper of Gaberdiel might be a good starting point.
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