Groups, Rings, Lie and Hopf Algebras pp 121-127 | Cite as

# Nilpotent Subsets of Hopf Module Algebras

## Abstract

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.

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### References

- [B]G. M. Bergman, On Jacobson radicals of graded rings, unpublished manuscript.Google Scholar
- [CM]M. Cohen and S. Montgomery,
*Group graded rings*,*smash products*,*and group actions*, Trans AMS,**282**(1984), 237–258.MathSciNetCrossRefMATHGoogle Scholar - [EG]P. Etingof and S. Gelaki,
*On finite-dimensional semisimple and cosemisimple Hopf algebras in prime characteristic*, Inter. Math. Research Notices,**16**(1998), 851–864.MathSciNetCrossRefGoogle Scholar - [F]J. R. Fisher,
*A Jacobson radical for Hopf module algebras*, J. Algebra,**34**(1975), 217–231.MathSciNetCrossRefMATHGoogle Scholar - [J]N. Jacobson, Lie Algebras, Interscience, 1962.Google Scholar
- [La]R. Larson,
*Characters of Hopf algebras*, J. Algebra,**17**(1971), 352–368.MathSciNetCrossRefMATHGoogle Scholar - [LaR]R. Larson and D. Radford,
*Semisimple cosemisimple Hopf algebras*, American J. Math.,**110**(1988), 187–195.MathSciNetCrossRefMATHGoogle Scholar - [L]V. Linchenko, On
*H*-module algebras, Ph D thesis, University of Southern California, 2001.Google Scholar - [LM]V. Linchenko and S. Montgomery,
*A**Frobenius-Schur theorem for Hopf algebras*, Alg. Rep. Theory,**3**, (2000), 347 - 355.MathSciNetCrossRefMATHGoogle Scholar - [M]S. Montgomery, Hopf Algebras and Their Actions on Rings, CBMS Lecture Notes
**82**, Amer. Math. Soc., Providence, 1993.Google Scholar - [MS]S. Montgomery and H.-J. Schneider,
*Skew derivations of finite-dimensional algebras and actions of the Taft Hopf algebra*, Tsukuba J. Math.,**25**(2002), 337–358.MathSciNetGoogle 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