Skip to main content
SpringerLink
Log in

Classification of Pointed Hopf Algebras of Dimension p 2 over any Algebraically Closed Field

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

Abstract

Let p be a prime. We complete the classification of pointed Hopf algebras of dimension p 2 over an algebraically closed field k. When char k ≠ p, our result is the same as the well-known result for char k = 0. When char k = p, we obtain 14 types of pointed Hopf algebras of dimension p 2, including a unique noncommutative and noncocommutative type.

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. Andruskiewitsch, N.: About finite dimensional Hopf algebras. Contemp. Math. 294, 1–57 (2002)

    Article  MathSciNet  Google Scholar 

  2. Andruskiewitsch, N., Schneider, H.-J.: Hopf algebras of order p 2 and braided Hopf algebras of order p. J. Algebra 199, 430–454 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  3. Beattie, M.: A survey of Hopf algebras of low dimension. Acta Appl. Math. 108, 19–31 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  4. Beattie, M., Garcia, G.: Techniques for classifying Hopf algebras and applications to dimension p 3 (preprint). arXiv:1108.6037

  5. Beattie, M., Garcia, G.: Classifying Hopf algebras of a given dimension (preprint). arXiv:1206.6529

  6. Cheng, Y.-L., Ng, S.-H.: On Hopf algebras of dimension 4p. J. Algebra 328, 399–419 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  7. Jacobson, N.: Lie Algebras. Dover Publications Inc., New York (1979)

    Google Scholar 

  8. Krop, L., Radford, D.: Finite-dimensional Hopf algebras of rank one in characteristic zero. J. Algebra 302, 214–230 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  9. Masuoka, A.: Semisimple Hopf algebras of dimension 6, 8. Israel J. Math. 92, 361–373 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  10. Masuoka, A.: The p n theorem for semisimple Hopf algebras. Proc. Am. Math. Soc. 124, 735–738 (1996)

    MATH  MathSciNet  Google Scholar 

  11. Montgomery, S.: Hopf Algebras and Their Actions on Rings, CBMS Regional Conference Series in Mathematics 82. American Mathematical Soceity, Providence, Rhode Island (1993)

    Google Scholar 

  12. Ng, S.-H.: Non-semisimple Hopf algebras of dimension p 2. J. Algebra 255, 182–197 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  13. Ng, S.-H.: Hopf algebras of dimension pq II. J. Algebra 319, 2772–2788 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  14. Nichols, W., Zoeller, M.: A Hopf algebra freeness theorem. Am. J. Math. 111, 381–385 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  15. Radford, D.: Operators on Hopf algebras. Am. J. Math 99, 139–158 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  16. Scherotzke, S.: Classification of pointed rank one Hopf algebras. J. Algebra 319, 2889–2912 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  17. Ştefan, D.: Hopf subalgebras of pointed Hopf algebras and applications. Proc. Am. Math. Soc. 125, 3191–3194 (1997)

    Article  MATH  Google Scholar 

  18. Taft, E.: The order of the antipode of finite-dimensional Hopf algebra. Proc. Natl. Acad. Sci. U.S.A. 68, 2631–2633 (1971)

    Article  MATH  MathSciNet  Google Scholar 

  19. Wang, X.: Connected Hopf algebras of dimension p 2. J. Algebra 391, 93–113 (2013). doi:10.1016/j.jalgebra.2013.06.008

    Article  MATH  MathSciNet  Google Scholar 

  20. Zhu, Y.: Hopf algebras of prime dimension. Int. Math. Res. Not. 1, 53–59 (1994)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Southeastern Louisiana University, Hammond, LA, 70402, USA

    Linhong Wang

  2. Department of Mathematics, University of Washington, Seattle, WA, 98195, USA

    Xingting Wang

Authors
  1. Linhong Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xingting Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xingting Wang.

Additional information

Xingting Wang, partially supported by U.S. National Science Foundation. Research of the first author supported by the Louisiana BoR [LEQSF(2012-15)-RD-A-20].

Rights and permissions

Reprints and permissions

About this article

Cite this article

Wang, L., Wang, X. Classification of Pointed Hopf Algebras of Dimension p 2 over any Algebraically Closed Field. Algebr Represent Theor 17, 1267–1276 (2014). https://doi.org/10.1007/s10468-013-9446-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10468-013-9446-7

Keyword

Mathematics Subject Classification (2010)

Search

Navigation