On finite-dimensional commutative non-hermitian fusion algebras Original Research Article

We characterize the three and four dimensional commutative non-hermitian fusion algebras and construct some new examples of these objects. These algebras arise naturally in the study of graphs, specially those associated with von Neumann algebras. Characterisations of hermitian fusion algebras have been given earlier by Sunder and Wildberger. Commutative finite-dimensional non-herimitian fusion algebras are algebraically isomorphic to certain special Cartan ʹsubalgebras of matrices. Every Cartan subalgebra of Mn is a conjugate of the standard Cartan algebra by an orthogonal matrix. We characterize the orthogonal matrices that can occur here and thus characterize the four dimensional non-hermitian fusion algebras. The three dimensional ones are parametrized by the hyperbola (x, y) : y2 − x2 = 1 and x, y > 0. By restricting to a special subclass of orthogonal matrices obtained by the above characterization, we construct a family of new commutative finite-dimensional non-hermitian fusion algebras.