On finite-dimensional commutative nonhermitian fusion algebras

We characterize the three and four dimensional commutative non-hermitian fu sion algebras and construct some new examples of these objects. These algeb ras 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 spe cial Cartan subalgebras of matrices. Every Cartan subalgebra of M-n is a co njugate of the standard Cartan algebra by an orthogonal matrix. We characte rize 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) : y(2) - x(2) = 1 and x, y > 0}. B y restricting to a special subclass of orthogonal matrices obtained by the above characterization, we construct a family of new commutative finite-dim ensional non-hermitian fusion algebras. (C) 1999 Elsevier Science Inc. All rights reserved.