Braidings on the category of bimodules, Azumaya algebras and epimorphisms of rings

TL;DR

This paper characterizes braidings on bimodule categories via canonical R-matrices, showing their relation to Azumaya algebras, epimorphisms, and central simple algebras, and introduces new solutions to quantum Yang-Baxter and braid equations.

Contribution

It establishes a bijective correspondence between braidings and canonical R-matrices, and characterizes when such braidings exist, especially for Azumaya and central simple algebras.

Findings

All braidings are symmetries.

Existence of braidings linked to ring epimorphisms and separability.

Canonical R-matrices provide solutions to quantum Yang-Baxter and braid equations.

Abstract

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 $A\ot A\ot A$ satisfying certain axioms. We show that all braidings are symmetries. If $A$ is commutative, then there exists a braiding on ${}_A\Mm_A$ if and only if $k\to A$ is an epimorphism in the category of rings, and then the corresponding $R$-matrix is trivial. If the invariants functor $G = (-)^A:\{}_A\Mm_A\to \Mm_k$ 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.