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.
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