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Irreducible actions and faithful actions of hopf algebras

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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.

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Author information

Authors and Affiliations

  1. DePaul University, 60614, Chicago, IL, USA

    Jeffrey Bergen

  2. Ben Gurion University of the Negev, Beer Sheva, Israel

    Miriam Cohen

  3. Weizmann Institute of Science, Rehovot, Israel

    Davida Fischman

Authors
  1. Jeffrey Bergen
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  2. Miriam Cohen
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  3. Davida Fischman
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Additional information

The first author was supported by the Faculty Research and Development Fund of the College of Liberal Arts and Sciences and the University Research Council at DePaul University and NSF Grant No. 8521704. He also wishes to thank Ben Gurion University for its hospitality, where much of this work was done.

The second and third authors were supported by the Fund for Basic Research administered by the Israel Academy of Sciences and Humanities.

Part of this author’s contribution is included in her Ph.D. thesis at Ben Gurion University under the supervision of the second author.

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Bergen, J., Cohen, M. & Fischman, D. Irreducible actions and faithful actions of hopf algebras. Israel J. Math. 72, 5–18 (1990). https://doi.org/10.1007/BF02764609

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  • DOI: https://doi.org/10.1007/BF02764609

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