Skip to main content
SpringerLink
Log in

Crossed Products as Twisted Category Algebras

  • Published:
Algebras and Representation Theory Aims and scope Submit manuscript

Abstract

We show that certain crossed product algebras can be viewed as twisted category algebras associated with finite groupoids. The latter are always isomorphic to products of matrix algebras over twisted group algebras. This has several consequences, in particular in the case where the crossed product algebra is the bismash product associated with a group factorization of a finite group.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Benson, D.J., Witherspoon, S.: Examples of support varieties for Hopf algebras with noncommutative tensor product. Arch. Math. 102, 513–520 (2014)

  2. Brauer, R.: On groups whose order contains a prime number to the first power. Amer. J. Math. 64, 401–420 (1942)

    Article  MATH  MathSciNet  Google Scholar 

  3. Clarke, M.C.: On the algebra structure of some bismash products. J. Algebra 322, 2590–2600 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  4. Collins, M.J.: Some bismash products that are not group algebras. J. Algebra 316, 297–302 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  5. Collins, M.J.: Some bismash products that are not group algebras II. J. Algebra 375, 59–72 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  6. Conlon, S.B.: Certain representation algebras. J. Austral. Math. Soc. 5, 83–99 (1965)

    Article  MATH  MathSciNet  Google Scholar 

  7. Ganyushkin, O., Mazorchuk, V., Steinberg, B.: On the irreducible representations of a finite semigroup. Proc. Amer. Math. Soc. 137, 3585–3592 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  8. Green, J.A. In: Erdmann, K., Green, J.A., Schocker, M. (eds.): Lecture Notes in Mathematics, 830. Springer, Berlin (2007)

  9. Jedwab, A., Montgomery, S.: Modular representations indicators for bismash products. J. Algebra 377, 281–297 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  10. Kashina, Y., Mason, G., Montgomery, S.: Computing the Frobenius–Schur indicator for abelian extensions of Hopf algebras. J. Algebra 251, 888–913 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  11. Liebeck, M., Praeger, C., Saxl, J.: On factorizations of almost simple groups. J. Algebra 185(2), 409–419 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  12. Linckelmann, M.: On inverse categories and transfer in cohomology. Proc. Edinb. Math. Soc. 56, 187–210 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  13. Linckelmann, M., Stolorz, M.: On simple modules over twisted finite category algebras. Proc. Amer. Math. Soc. 140, 3725–3737 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  14. Majid, S.: Foundations of quantum group theory. Cambridge Univ. Press, Cambridge (1995)

    Book  MATH  Google Scholar 

  15. Masuoka, A.: Hopf algebra extensions and cohomology. In: New directions in Hopf algebra. vol. 43, pp. 167–209, Math. Sci. Res. Inst. Publ. Cambridge University Press, Cambridge (2002)

  16. Steinberg, B.: Möbius functions and semigroup representation theory. II. Character formulas and multiplicities. Adv. Math. 217, 1521–1557 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  17. Steinberg, B.: A groupoid approach to discrete inverse semigroup algebras. Adv. Math. 223, 689–727 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  18. Nagao, H., Tsushima, Y.: Representations of finite groups. Academic Press, New York (1989)

    MATH  Google Scholar 

  19. Witherspoon, S.J.: Products in Hochschild cohomology and Grothendieck rings of group crossed products. Adv. Math. 185, 136–158 (2004)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Kaiserslautern, P.O. Box 3049, 67653, Kaiserslautern, Germany

    Susanne Danz

  2. Mathematical Institute, University of Oxford, ROQ, Oxford, OX2 6GG, UK

    Karin Erdmann

Authors
  1. Susanne Danz
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Karin Erdmann
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Susanne Danz.

Additional information

Communicated by: Susan Montgomery

Presented by Susan Montgomery

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Danz, S., Erdmann, K. Crossed Products as Twisted Category Algebras. Algebr Represent Theor 18, 281–296 (2015). https://doi.org/10.1007/s10468-014-9493-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10468-014-9493-8

Keywords

Mathematics Subject Classification (2010)

Search

Navigation