Association schemes and fusion algebras. (An introduction)
Fusion algebras are finite dimensional commutative and associative algebras (over complex number field) which appear in conformal field theory in mathematical physics. The main purpose of the reviewed paper is to start with the study of fusion algebras from a purely algebraic viewpoint. The author considers an algebraic object, which he calls fusion algebra at algebraic level, and finds its connection with another purely algebraic object, called character algebra, which is closely connected with Bose-Mesner algebras (Hecke algebras) of an association scheme. There is the matrix \(S\) in fusion algebra which plays an important role in conformal field theory. The author shows, contrary to the theory of fusion algebras in conformal field theory, that the matrix \(S\) of a fusion algebra at algebraic level is not necessarily unitary nor symmetric and gives exact characterizations of when the matrix \(S\) is unitary or symmetric in terms of the corresponding character algebra. Moreover he discusses the modular invariance of a fusion algebra since this property is very important in conformal field theory; for example the fusion algebras of well-known conformal theories have this property.
scientific article; zbMATH DE number 482245