Constants of Lie algebra actions

Abstract

Let R be an associative algebra over a field k and let L be a Lie algebra over k acting on R as derivations. We will be interested in studying the relationship between R and the ring of constants RL. Our situation is a special case of the study of rings of invariants of Hopf algebra actions and is also analogous to another special case of Hopf algebra actions, namely, the study of fixed rings of finite group actions. Many of the results in this paper are analogs of results on group actions and give rise to more general questions regarding Hopf algebra actions. In Section 1, we study actions on non-nilpotent algebras. We first show that any algebraic derivation has non-zero constants and then prove that any finite dimensional nilpotent Lie algebra of algebraic derivations must act with non-zero constants. It then follows, as a corollary, that any finite dimensional nilpotent restricted Lie algebra must act with non-zero constants. These are analogs of results on groups proved in [ 12, 5). In Section 2, we prove a necessary condition for the existence of nonzero invariants of certain Hopf algebra actions. We then use this result to prove the converse of our theorem on nilpotent restricted Lie algebras. We conclude the section with several questions on general Hopf algebra actions.