Skip to main content
SpringerLink
Log in

Twisted partial coactions of Hopf algebras

  • Research Article
  • Published:
Frontiers of Mathematics in China Aims and scope Submit manuscript

    We’re sorry, something doesn't seem to be working properly.

    Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

Abstract

In this paper, the notion of a twisted partial Hopf coaction is introduced. The conditions on partial cocycles are established in order to construct partial crossed coproducts. Then the classification of partial crossed coproducts is discussed. Finally, some necessary and sufficient conditions for a class of partial crossed coproducts to be quasitriangular bialgebras are given.

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. Agore A L. Coquasitriangular structures for extensions of Hopf algebras. Applications. Glasg Math J, 2013, 55: 201–215

    Article  MathSciNet  MATH  Google Scholar 

  2. Alves M M S, Batista E. Partial Hopf actions, partial invariants and a Morita context. Algebra Discrete Math, 2009, 3: 1–19

    MathSciNet  MATH  Google Scholar 

  3. Alves M M S, Batista E. Enveloping actions for partial Hopf actions. Comm Algebra, 2010, 38: 2872–2902

    Article  MathSciNet  MATH  Google Scholar 

  4. Alves M M S, Batista E. Globalization theorems for partial Hopf (co)actions and some of their applications. Contemp Math, 2011, 537: 13–30

    Article  MathSciNet  MATH  Google Scholar 

  5. Alves M M S, Batista E, Dokuchaev M, Paques A. Twisted partial actions of Hopf algebras. Israel J Math, 2013, 197: 263–308

    Article  MathSciNet  MATH  Google Scholar 

  6. Bagio D, Lazzarin J, Paques A. Crossed products by twisted partial actions: separability, semisimplicity and Frobenius properties. Comm Algebra, 2010, 38: 496–508

    Article  MathSciNet  MATH  Google Scholar 

  7. Beggs E, Majid S. Quasitriangular and differential structures on bicrossproduct Hopf algebras. J Algebra, 1999, 219: 682–727

    Article  MathSciNet  MATH  Google Scholar 

  8. Blattner R J, Cohen M, Montgomery S. Crossed products and inner actions of Hopf algebras. Trans Amer Math Soc, 1986, 298: 671–711

    Article  MathSciNet  MATH  Google Scholar 

  9. Caenepeel S, Janssen K. Partial (co)actions of Hopf algebras and partial Hopf-Galois theory. Comm Algebra, 2008, 36: 2923–2946

    Article  MathSciNet  MATH  Google Scholar 

  10. Chen H X. Quasitriangular structures of bicrossed coproducts. J Algebra, 1998, 204: 504–531

    Article  MathSciNet  MATH  Google Scholar 

  11. Chen Q G, Wang D G. Constructing quasitriangular Hopf algebras. Comm Algebra, 2015, 43(4): 1698–1722

    Article  MathSciNet  MATH  Google Scholar 

  12. Chen Q G, Wang D G. A class of coquasitriangular Hopf group algebras. Comm Algebra, 2016, 44(1): 310–335

    Article  MathSciNet  MATH  Google Scholar 

  13. Dascalescu S, Militaru G, Raianu S. Crossed coproducts and cleft coextensions. Comm Algebra, 1996, 24: 1229–1243

    Article  MathSciNet  MATH  Google Scholar 

  14. Doi Y. Equivalent crossed products for a Hopf algebra. Comm Algebra, 1989, 17(12): 3053–3085

    Article  MathSciNet  MATH  Google Scholar 

  15. Dokuchaev M, Exel R, Piccione P. Partial representations and partial group algebras. J Algebra, 2000, 226: 505–532

    Article  MathSciNet  MATH  Google Scholar 

  16. Dokuchaev M, Exel R. Associativity of crossed products by partial actions, enveloping actions and partial representations. Trans Amer Math Soc, 2005, 357: 1931–1952

    Article  MathSciNet  MATH  Google Scholar 

  17. Dokuchaev M, Exel R, Simón J J. Crossed products by twisted partial actions and graded algebras. J Algebra, 2008, 320: 3278–3310

    Article  MathSciNet  MATH  Google Scholar 

  18. Dokuchaev M, Exel R, Simón J J. Globalization of twisted partial actions. Trans Amer Math Soc, 2010, 362(8): 4137–4160

    Article  MathSciNet  MATH  Google Scholar 

  19. Dokuchaev M, Ferrero M, Paques A. Partial actions and Galois theory. J Pure Appl Algebra, 2007, 208: 77–87

    Article  MathSciNet  MATH  Google Scholar 

  20. Dokuchaev M, Novikov B. Partial projective representations and partial actions. J Pure Appl Algebra, 2010, 214: 251–268

    Article  MathSciNet  MATH  Google Scholar 

  21. Dokuchaev M, Novikov B. Partial projective representations and partial actions II. J Pure Appl Algebra, 2012, 216: 438–455

    Article  MathSciNet  MATH  Google Scholar 

  22. Drinfeld V. Quantum Groups. Proc Int Congr Math Berkeley, Vol 1. 1987, 798–820

    Google Scholar 

  23. Exel R. Circle actions on C* -algebras, partial automorphisms and generalized Pimsner-Voiculescu exact sequences. J Funct Anal, 1994, 122: 361–401

    Article  MathSciNet  MATH  Google Scholar 

  24. Exel R. Twisted partial actions: a classification of regular C* -algebraic bundles. Proc Lond Math Soc, 1997, 74: 417–443

    Article  MathSciNet  MATH  Google Scholar 

  25. Exel R. Partial actions of groups and actions of inverse semigroups. Proc Amer Math Soc, 1998, 126: 3481–3494

    Article  MathSciNet  MATH  Google Scholar 

  26. Jiao Z M. The quasitriangular structures for a class of T-smash product Hopf algebras. Israel J Math, 2005, 146: 125–148

    Article  MathSciNet  MATH  Google Scholar 

  27. Lin L P. Crossed coproducts of Hopf algebras. Comm Algebra, 1982, 10: 1–17

    Article  MathSciNet  MATH  Google Scholar 

  28. McClanahan K. K-theory for partial crossed products by discrete groups. J Funct Anal, 1995, 130: 77–117

    Article  MathSciNet  MATH  Google Scholar 

  29. Montgomery S. Hopf Algebras and their Actions on Rings. CBMS Reg Conf Ser Math, No 82. Providence: Amer Math Soc, 1993

Download references

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Qufu Normal University, Qufu, 273165, China

    Quanguo Chen, Dingguo Wang & Xiaodan Kang

Authors
  1. Quanguo Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Dingguo Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Xiaodan Kang
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chen, Q., Wang, D. & Kang, X. Twisted partial coactions of Hopf algebras. Front. Math. China 12, 63–86 (2017). https://doi.org/10.1007/s11464-016-0597-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11464-016-0597-9

Keywords

MSC

Search

Navigation