Nilpotent Subsets of Hopf Module Algebras

  • V. Linchenko
Part of the Mathematics and Its Applications book series (MAIA, volume 555)


Let H be a Hopf algebra over the field k and let A be an H-module algebra. In this paper we consider when various nilpotent subsets of A are stable under the action of H. We first show an Engel-type theorem about triangularizing H-stable subalgebras of nilpotent matrices. We also prove that if A is finite-dimensional, H is involutory, and the field is of characteristic zero or characteristic p relatively prime to the dimension of A, then the Jacobson radical of A is H-stable. This fact was known earlier in several special cases; when H = kG,a group algebra, it is trivially true, and when H = (k [G]) * it was proved in [CM], answering a question of Bergman [B]. The general situation of H-stable radicals was considered by Fisher [F], with some partial results. It is false without some hypotheses, as we show with some examples.


Hopf Algebra Jacobson Radical Module Algebra Smash Product Left Regular Representation 
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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Copyright information

© Kluwer Academic Publishers 2003

Authors and Affiliations

  • V. Linchenko
    • 1
  1. 1.Department of MathematicsUniversity of Southern CaliforniaLos AngelesUSA

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