Applied Categorical Structures

, Volume 22, Issue 1, pp 29–42

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

Article

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

Keywords

Braided category Epimorphism of rings Azumaya algebra Separable functor Quantum Yang–Baxter equation 

Mathematics Subject Classifications (2010)

16T10 16T05 16S40 

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Copyright information

© Springer Science+Business Media Dordrecht 2012

Authors and Affiliations

  1. 1.Faculty of EngineeringVrije Universiteit BrusselBrusselsBelgium
  2. 2.Faculty of Mathematics and Computer ScienceUniversity of BucharestBucharest 1Romania

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