Applied Categorical Structures

, Volume 22, Issue 1, pp 29–42 | Cite as

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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References

  1. 1.
    Auslander, M., Goldman, O.: The Brauer group of a commutative ring. Trans. Am. Math. Soc. 97, 367–409 (1960)CrossRefMathSciNetGoogle Scholar
  2. 2.
    Azumaya, G.: On maximally central algebras. Nagoya Math. J. 2, 119–150 (1951)MATHMathSciNetGoogle Scholar
  3. 3.
    Böhm, G.: Hopf algebroids. Handb. Algebr. 6, 173–235 (2009)CrossRefGoogle Scholar
  4. 4.
    Caenepeel, S.: Brauer groups, Hopf algebras and Galois theory. In: K-Monographs in Mathematics, vol. 4. Kluwer Academic, Dordrecht (1998)Google Scholar
  5. 5.
    Caenepeel, S., Militaru, G., Zhu, S.: Frobenius and separable functors for generalized module categories and nonlinear equations. In: Lecture Notes in Math., vol. 1787. Springer, Berlin (2002)Google Scholar
  6. 6.
    Donin, J., Mudrov, A.: Quantum groupoids and dynamical categories. J. Algebra 296, 348–384 (2006)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Kassel, C.: Quantum groups. In: Graduate Texts Math., vol. 155. Springer, Berlin (1995)Google Scholar
  8. 8.
    Knus, M.A., Ojanguren, M.: Théorie de la descente et algèbres d’Azumaya. In: Lecture Notes in Math., vol. 389. Springer, Berlin (1974)Google Scholar
  9. 9.
    Montgomery, S.: Hopf algebras and their actions on rings. American Mathematical Society, Providence (1993)MATHGoogle Scholar
  10. 10.
    Nǎstǎsescu, C., Van den Bergh, M., Van Oystaeyen, F.: Separable functors applied to graded rings. J. Algebra 123, 397–413 (1989)CrossRefMathSciNetGoogle Scholar
  11. 11.
    Rafael, M.D.: Separable functors revisited. Commun. Algebra 18, 1445–1459 (1990)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Silver, L.: Noncommutative localization and applications. J. Algebra 7, 44–76 (1967)CrossRefMATHMathSciNetGoogle Scholar

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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