Braidings on the Category of Bimodules, Azumaya Algebras and Epimorphisms of Rings

Let A be an algebra over a commutative ring k. We prove that braidings on the category of A-bimodules are in bijective correspondence to canonical R-matrices, these are elements in A⊗A⊗A satisfying certain axioms. We show that all braidings are symmetries. If A is commutative, then there exists a braiding on AMA if and only if k → A is an epimorphism in the category of rings, and then the corresponding R-matrix is trivial. If the invariants functor G = (−) : AMA → Mk is separable, then A admits a canonical R-matrix; in particular, any Azumaya algebra admits a canonical R-matrix. Working over a field, we find a remarkable new characterization of central simple algebras: these are precisely the finite dimensional algebras that admit a canonical R-matrix. Canonical R-matrices give rise to a new class of examples of simultaneous solutions for the quantum Yang-Baxter equation and the braid equation.