In mathematics, a Verlinde algebra is a finite-dimensional associative algebra introduced by . It is defined to have basis of elements ÃÂ<sub>û</sub> corresponding to primary fields of a rational two-dimensional conformal field theory, whose structure constants N describe fusion of primary fields.
In the context of modular tensor categories, there is also a Verlinde algebra. It is defined to have a basis of elements corresponding to isomorphism classes of simple obejcts and whose structure constants describe the fusion of simple objects.
In terms of the modular S-matrix for modular tensor categories, the Verlinde formula is stated as follows. Given any simple objects in a modular tensor category, the Verlinde formula relates the fusion coefficient in terms of a sum of products of -matrix entries and entries of the inverse of the -matrix, normalized by quantum dimensions.
In terms of the modular S-matrix for conformal field theory, Verlinde formula expresses the fusion coefficients as
where is the component-wise complex conjugate of .
These two formulas are equivalent because under appropriate normalization the S-matrix of every modular tensor category can be made unitary, and the S-matrix entry is equal to the quantum dimension of .
If G is a compact Lie group, there is a rational conformal field theory whose primary fields correspond to the representations û of some fixed level of loop group of G. For this special case showed that the Verlinde algebra can be identified with twisted equivariant K-theory of G.