Confluence and quantum Yang-Baxter equation Original Research Article

The X-confluence of a quadratic algebra A defined by generators and relations can be interpreted as an equality between two idempotent endomorphisms acting on tensors of degree three (X is the ordered set of generators). Those endomorphisms are simply defined from the X-reduction operator of A which is an idempotent endomorphism acting on tensors of degree two. In general, the X-confluence of A is not locally generic, even if the continuous change of relations of A is regular, i.e., the image of the X-reduction operator is not changed. For the special class of quantum algebras, we introduce a condition which implies X-confluence and which is locally generic in the regular case. A global result is derived when the change is analytic. More precise results are obtained if the quantum algebras are of Hecke type.