Fusion algebras with negative structure constants

We introduce fusion algebras with not necessarily positive structure constants and without identity element. We prove that they are semisimple when tensored with C and that their characters satisfy orthogonality relations. Then we define the proper notion of subrings and factor rings for such algebras. For certain algebras R we prove the existence of a ring R^′ with nonnegative structure constants such that R is a factor ring of R^′. We give some examples of interesting factor rings of the representation ring of the quantum double of a finite group. Then, we investigate the algebras associated to Hadamard matrices. For an n × n-matrix the corresponding algebra is a factor ring of a subalgebra of Z [(Z / 2 Z)^{n - 2}].