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

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

2011-08-12

arxiv.org/abs/1108.2575v2

Mathematics

Quantum Algebra

12 pages, new version

Scientific paper

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.

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