Nilpotent Subsets of Hopf Module Algebras
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.
KeywordsHopf Algebra Jacobson Radical Module Algebra Smash Product Left Regular Representation
Unable to display preview. Download preview PDF.
- [B]G. M. Bergman, On Jacobson radicals of graded rings, unpublished manuscript.Google Scholar
- [J]N. Jacobson, Lie Algebras, Interscience, 1962.Google Scholar
- [L]V. Linchenko, On H-module algebras, Ph D thesis, University of Southern California, 2001.Google Scholar
- [M]S. Montgomery, Hopf Algebras and Their Actions on Rings, CBMS Lecture Notes 82, Amer. Math. Soc., Providence, 1993.Google Scholar
- [S]H.-J. Schneider, Lectures on Hopf Algebras, Universidad de Cordoba Trabajos de Matematica, No. 31/95, gdoba (Argentina), 1995.Google Scholar
- [Sw]M. Sweedler, Hopf Algebras, W.A. Benjamin, NY, 1969.Google Scholar