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 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 .