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Israel Journal of Mathematics

, Volume 72, Issue 1–2, pp 5–18 | Cite as

Irreducible actions and faithful actions of hopf algebras

  • Jeffrey Bergen
  • Miriam Cohen
  • Davida Fischman
Article

Abstract

LetH be a Hopf algebra acting on an algebraA. We will examine the relationship betweenA, the ring of invariantsA H, and the smash productA # H. We begin by studying the situation whereA is an irreducibleA # H module and, as an application of our first main theorem, show that ifD is a division ring then [D : D H]≦dimH. We next show that prime rings with central rings of invariants satisfy a polynomial identity under the action of certain Hopf algebras. Finally, we show that the primeness ofA # H is strongly related to the faithfulness of the left and right actions ofA # H onA.

Keywords

Hopf Algebra Left Ideal Prime Ring Division Ring Polynomial Identity 
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.

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References

  1. [A 80]
    E. Abe,Hopf Algebras, Cambridge Univ. Press, Cambridge, 1980.MATHGoogle Scholar
  2. [B 88]
    J. Bergen,Constants of Lie algebra actions, J. Algebra114 (1988), 452–465.MATHCrossRefMathSciNetGoogle Scholar
  3. [B 89]
    J. Bergen,Automorphic-differential identities in rings, Proc. Am. Math. Soc.106 (2) (1989), 297–305.MATHCrossRefMathSciNetGoogle Scholar
  4. [BI 73]
    G. M. Bergman and I. M. Isaacs,Rings with fixed-point-free group actions, Proc. London Math. Soc27 (1973), 69–87.MATHCrossRefMathSciNetGoogle Scholar
  5. [BM 86]
    J. Bergen and S. Montgomery,Smash products and outer derivations, Isr. J. Math.53 (1986), 321–345.MATHMathSciNetGoogle Scholar
  6. [Br]
    A. Braun, unpublished.Google Scholar
  7. [C 87]
    M. Cohen,H-Simple H-module algebras, Canadian Math. Bull.30(3) (1987), 363–366.MATHGoogle Scholar
  8. [CF 86]
    M. Cohen and D. Fischman,Hopf algebra actions, J. Algebra100 (1986), 363–379.MATHCrossRefMathSciNetGoogle Scholar
  9. [CFM 90]
    M. Cohen, D. Fischman and S. Montgomery,Hopf Galois extensions, smash products, and Morita equivalence, J. Algebra133 (1990), 351–372.MATHCrossRefMathSciNetGoogle Scholar
  10. [HN 75]
    I. N. Herstein and L. Neumann,Centralizers in rings, Ann. Mat. Pura Appl.102(4) (1975), 37–44.MathSciNetGoogle Scholar
  11. [J 56]
    N. Jacobson,Structure of Rings, Am. Math. Soc. Colloquium Publ., 1956.Google Scholar
  12. [J 62]
    N. Jacobson,Lie Algebras, Wiley-Interscience, New York, 1962.MATHGoogle Scholar
  13. [K 74]
    V. K. Kharchenko,Galois extensions and rings of quotients, Algebra i Logika13(4) (1974), 460–484, 488 (English transl. (1975), 265–281).MATHMathSciNetGoogle Scholar
  14. [L 66]
    J. Lambek,Rings and Modules, Blaisdell, Waltham, Massachusetts, 1966.MATHGoogle Scholar
  15. [M 69]
    W. Martindale,Prime rings satisfying a generalized polynomial identity, J. Algebra12 (1969), 576–584.MATHCrossRefMathSciNetGoogle Scholar
  16. [McS 71]
    J. C. McConnell and M. E. Sweedler,Simplicity of smash products, Proc. London Math. Soc.23(3) (1971), 251–266.MATHCrossRefMathSciNetGoogle Scholar
  17. [O 81]
    J. Osterburg,Central fixed rings, J. London Math. Soc.23(2) (1981), 246–248.MATHCrossRefMathSciNetGoogle Scholar
  18. [P 83]
    A. Z. Popov,Derivations of prime rings, Algebra i Logika22 (1983), 79–92.MathSciNetGoogle Scholar
  19. [S 68]
    M. E. Sweedler,Cohomology of algebras over Hopf algebras, Trans. Am. Math. Soc.133 (1968), 203–239.CrossRefMathSciNetGoogle Scholar
  20. [S 69a]
    M. E. Sweedler,Hopf Algebras, Benjamin, New York, 1969.Google Scholar
  21. [S 69b]
    M. E. Sweedler,Integrals for Hopf algebras, Ann. of Math.89(2) (1969), 323–335.CrossRefMathSciNetGoogle Scholar

Copyright information

© The Weizmann Science Press of Israel 1990

Authors and Affiliations

  • Jeffrey Bergen
    • 1
  • Miriam Cohen
    • 2
  • Davida Fischman
    • 3
  1. 1.DePaul UniversityChicagoUSA
  2. 2.Ben Gurion University of the NegevBeer ShevaIsrael
  3. 3.Weizmann Institute of ScienceRehovotIsrael

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