Acta Mathematica Sinica

, Volume 12, Issue 3, pp 244–248 | Cite as

Free quadratic bialgebra

  • Hao Zhifeng
  • Tong Wenting
Article
  • 20 Downloads

Abstract

In this paper, we obtain the following main theorem for a free quadratic bialgebraJ:
  1. (a)

    Forp≠0,J is a pointed cosemisimple coalgebra. Forp=0,J is a hyperalgebra.

     
  2. (b)

    Forp≠0 andq≠0,J has antipodeS iffp·q+2=0 andS(x)=x. Forp=0 orq=0,J has antipode andS(x)=×.

     
  3. (c)

    All leftJ *-modules are rational.

     

Also, we give some applications in homological theory and algebraicK-theory.

Keywords

Free quadratic bialgebra Antipode Cosemisimple Rational module 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Abe A. Hopf Algebras. Cambridge, New York; Cambridge Univ Press, 1980.MATHGoogle Scholar
  2. [2]
    Caenepeel S. Computing the Brauer-Long group of a Hopf algebra I: The cohological theory.Israel J Math, 1990,72: 167–195.CrossRefMathSciNetGoogle Scholar
  3. [3]
    Childs L N. The Brauer group of graded Azumaya algebras II: Graded Galois extensions.Trans Amer Math Soc, 1975,204: 137–160.MATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    Doi Y. Homological coalgebra.J Math Soc Japan, 1981,33(1): 31–50.MATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    Kreimer H K. Quadratic Hopf algebra and Galois extension.Contem Math, 1982,13: 353–361.MATHMathSciNetGoogle Scholar
  6. [6]
    Manin Y I. Quantum groups and non-commutative geometry. Tech Report, CRM, de Montreal, 1988.Google Scholar
  7. [7]
    Nakajima A. Bialgebra and Galois extension.Math J Okayama Univ, 1991,33: 37–46.MATHMathSciNetGoogle Scholar
  8. [8]
    Nakajima A. On isomorphism class groups of non-commutative quadratic Galois extension.Math J Okayama Univ, 1991,33: 47–64.MATHMathSciNetGoogle Scholar
  9. [9]
    Silvester J R. Introduction to AlgebraicK-theory. London and New York: Chapman and Hall, 1981.Google Scholar
  10. [10]
    Tate J T, and Oort F. Group schemes of prime order.Ann Sci Ec Norm Sup (4iėme sėrie), 1970,3: 1–21.MATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 1996

Authors and Affiliations

  • Hao Zhifeng
    • 1
  • Tong Wenting
    • 2
  1. 1.Department of Applied MathematicsSouth China University of TechnologyGuangzhouChina
  2. 2.Department of MathematicsNanjing UniversityNanjingChina

Personalised recommendations