Constants of Lie algebra actions
Let R be an associative algebra (not necessarily with 1) over a field k. Section 1 of the present article considers actions of finite-dimensional nilpotent Lie algebras L by algebraic derivations on R. The main result states that if R is non-nilpotent then \(R^ L=\{r\in R|\) \(\ell \circ r=0\) for all \(\ell \in L\}\), the ring of constants of L on R, is non- zero. Section 2 deals, more generally, with actions of Hopf algebras H on R. This covers the case of Lie algebra actions by taking H to be the enveloping algebra of L (restricted when L is restricted) as well as the case of group actions, with H being the group algebra. Under a suitable technical assumption on H, the author constructs a non-nilpotent algebra R on which H acts with zero constants, \(R^ H=\{0\}\). This justifies in particular the nilpotency assumption on L in Section 1. The article concludes with a number of open questions.
actions of finite-dimensional nilpotent Lie algebras
scientific article; zbMATH DE number 4065213