Biproducts and Two-cocycle Twists of Hopf Algebras

  • David E. Radford
  • Hans Jürgen Schneider
Part of the Trends in Mathematics book series (TM)


Let H be a Hopf algebra with bijective antipode over a field k and suppose that R#H is a bi-product. Then R is a bialgebra in the Yetter-Drinfel’d category \( _H^H \mathcal{Y}D \). We describe the bialgebras (R#H) op and (R#H) o explicitly as bi-products \( R^{\underline {op} } \# H^{op} \) and \( R^{\underline o } \# H^o \) respectively where \( R^{\underline {op} } \) is a bialgebra in \( _{H^{op} }^{H^{op} } \mathcal{Y}D \) and \( R^{\underline o } \) is a bialgebra in \( _{H^o }^{H^o } \mathcal{Y}D \). We use our results to describe two-cocycle twist bialgebra structures on the tensor product of bi-products.


Linear Form Hopf Algebra Left Ideal Monoidal Category Primitive Element 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    E. Abe, “Hopf Algebras,” 74, Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, UK, 1980.Google Scholar
  2. [2]
    Nicolás Andruskiewitsch and Hans-Jürgen Schneider, Finite quantum groups over abelian groups of prime exponent, Ann. Sci. École Norm. Sup. (4) 35 (2002), 1–26.zbMATHMathSciNetGoogle Scholar
  3. [3]
    Nicolás Andruskiewitsch and Hans-Jürgen Schneider, Pointed Hopf algebras. New directions in Hopf algebras, 1–68, Math. Sci. Res. Inst. Publ. 43, Cambridge Univ. Press, Cambridge, 2002.Google Scholar
  4. [4]
    M. Beattie, Duals of pointed Hopf algebras, J. of Algebra 262 (2003), 54–76.zbMATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    Yukio Doi and Mitsuhiro Takeuchi, Multiplication alteration by two-cocycles — the quantum version, Comm. Algebra 22 (1994), 5715–5732.zbMATHCrossRefMathSciNetGoogle Scholar
  6. [6]
    Larry A. Lambe and David E. Radford, Introduction to the quantum Yang-Baxter equation and quantum groups: an algebraic approach. Mathematics and its Applications 423, Kluwer Academic Publishers, Dordrecht (1997), xx+293 pp.Google Scholar
  7. [7]
    Shahn Majid, Algebras and Hopf Algebras in Braided Categories. “Advances in Hopf Algebras”, Lecture Notes in Pure and Applied Mathematics 158, 55–105, Marcel Dekker, 1994.MathSciNetGoogle Scholar
  8. [8]
    S. Montgomery, “Hopf Algebras and their actions on rings,” 82, Regional Conference Series in Mathematics, AMS, Providence, RI, 1993.Google Scholar
  9. [9]
    David E. Radford, Coreflexive coalgebras, J. of Algebra 26 (1973), 512–535.zbMATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    D.E. Radford, The structure of Hopf algebras with a projection, J. Algebra 92(1985), 322–347.zbMATHCrossRefMathSciNetGoogle Scholar
  11. [11]
    David E. Radford and Hans-Jürgen Schneider, On the Simple Representations of Generalized Quantum Groups and Quantum Doubles, to appear in J. Algebra.Google Scholar
  12. [12]
    Moss E. Sweedler, Hopf algebras. Mathematics Lecture Note Series W. A. Benjamin, Inc., New York (1969), vii+336 pp.Google Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2008

Authors and Affiliations

  • David E. Radford
    • 1
  • Hans Jürgen Schneider
    • 2
  1. 1.Department of Mathematics, Statistics and Computer ScienceUniversity of Illinois at ChicagoChicagoUSA
  2. 2.Mathematisches InstitutLudwig-Maximilians-Universität MünchenMünchenGermany

Personalised recommendations