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.

## Keywords

Hopf Algebra Jacobson Radical Module Algebra Smash Product Left Regular Representation## Preview

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