For $G$ a suitable Lie group, the Verlinde ring is the collection of isomorphism classes of positive energy representations of the corresponding loop group, equipped with the “fusion” tensor product.
The Verlinde ring is also understood as being the ring of equivariant twisted K-theory classes on $G$ (FHT) and, essentially equivalently, of Chan-Paton gauge fields over D-branes in the WZW model (see there for further references).
Due to
See also
Domenico Fiorenza, Alessandro Valentino, $(3,2,1)$-TQFTs and Verlinde algebras (MO question, MO answer)
Dan Freed, Mike Hopkins, Constantin Teleman, Loop Groups and Twisted K-Theory
Wikipedia, Verlinde algebra