Advertisement

Nichols algebras over weak Hopf algebras

  • Zhi-xiang Wu
Article
  • 25 Downloads

Abstract

In this paper, we study a Yetter-Drinfeld module V over a weak Hopf algebra ℍ. Although the category of all left ℍ-modules is not a braided tensor category, we can define a Yetter-Drinfeld module. Using this Yetter-Drinfeld modules V, we construct Nichols algebra B(V) over the weak Hopf algebra ℍ, and a series of weak Hopf algebras. Some results of [8] are generalized.

Keywords

quantum enveloping algebra Nichols algebra weak Hopf algebra 

MR Subject Classification

17B37 81R50 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgements

This article was written in 2013 while I was visiting Department of Mathematics, Hamburg University. I thank Peter Schwergert for valuable discussions and reference. The author would like to thank the referee for careful reading and useful comments on this paper.

References

  1. [1]
    N Aizawa, PS Isaac. Weak Hopf algebras corresponding to Uq[sln], J Math Phys, 2003, 44(11): 5250–5267.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    N Andruskiewitsch, H-J Schneider. Pointed Hopf algebras, In: New Directions in Hopf algebras, MSRI Publications, Vol 43, Cambridge University Press, 2002.Google Scholar
  3. [3]
    N Andruskiewitsch, H-J Schneider. Finite quantum group and Cartan matrices, Adv Math, 2000, 154: 1–45.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    N Bergeron, Y Gao, N Hu. Drinfeld doubles and Luszting’s symetries of two-parameter quantum groups, J Algebra, 2006, 301(1): 378–405.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    G Benkart, S-J Kang, D Melville. Quantized envloping algebras for Borcherds superalgebras, Trans Amer Math Soc, 1998, 350(8): 3279–3319.CrossRefzbMATHGoogle Scholar
  6. [6]
    G Benkart, S Witherspoon. Two-parameter quantum groups and Drinfeld doubles, Algebr Represent Theory, 2004, 7: 261–286.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    G Böhm, F Nill, S zlachányi. Weak Hopf algebras: I. integral theory and C-structure, J Algebra, 1999, 221(2): 385–438.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    I Heckenberger. Lustig isomorphisms for Drinfeld double of bosonizations of Nichols algebras of diagonal type, J Algebra, 2010, 323(8): 2130–2182.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    JM Howie. Fundamentals of Semigroup Theory, Clarendon Press, Oxford, 1995.zbMATHGoogle Scholar
  10. [10]
    A Joseph. Quantum Groups and Their Primitive Ideals, Ergeb Math Grenzgeb (3), Vol 29, Springer-Verlag, Berlin, 1995.Google Scholar
  11. [11]
    M Kashiwara. On crystal bases of the q-analogue of universal enveloping algebras, DukeMath J, 1991, 63: 465–516.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    SM Khoroshkin, VN Tolstoy. Universal R-matrix for quantized (super)algebras, Comm Math Phys, 1991, 141: 599–617.MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    F Li, S Duplij. Weak Hopf algebras and singular solutions of quantum Yang-Baxter equation, Comm Math Phys, 2002, 225(1): 191–217.MathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    S Montgomery. Hopf algebras and their actions on rings, CBMS Regional Conference Series in Mathematics, Vol 82, 1993.Google Scholar
  15. [15]
    WD Nichols. Bialgebras of tpye one, Comm Algebra, 1978, 6: 1521–1552.MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    SL Woronowicz. Differential calculus on compactmatrix pseudogroups (quantum groups), Comm Math Phys, 1989, 122: 125–170.MathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    Z Wu. Construct weak Hopf algebras by using Borcherds matrix, Acta Math Sin (Engl Ser), 2009, 8: 1337–1352.MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    Z Wu. Weak quantum enveloping algebras of Borcherds superalgebras, Acta Appl Math, 2009, 106(2): 185–198.MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    Z Wu. A class of weak Hopf algebras related to a Borcherds-Cartan matrix, J Phys A, 2006, 39(47): 14611–14626.MathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    S Yang. Weak Hopf algebras corresponding to Cartan matrices, J Math Phys, 2005, 46(7): 1–18.MathSciNetzbMATHGoogle Scholar

Copyright information

© Editorial Committee of Applied Mathematics-A Journal of Chinese Universities and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mathematical ScienceZhejiang UniversityHangzhouChina

Personalised recommendations