Delta methods in enveloping algebras of Lie superalgebras

Abstract:

Let $L$ be a Lie superalgebra over a field $K$ of characteristic $\ne 2$ . We define \[ \Delta (L) = \{ l \in L|{\dim _K}[L,l] < \infty \}. \] Then $\Delta (L)$ is a Lie ideal of $L$ and is restricted if $L$ is restricted. $\Delta (L)$ is the Lie superalgebra analog of the Lie delta ideal, used by the authors in the study of enveloping rings, and also of the finite conjugate center of a group, used in the study of group algebras and crossed products. In this paper we examine $U(L)$, where depending upon $\operatorname {char}K$, $U(L)$ denotes either the enveloping algebra or the restricted enveloping algebra of $L$. We show that $\Delta (L)$ controls certain properties of $U(L)$. Specifically, we consider semiprimeness, primeness, almost constants, almost centralizers, central closures, and the Artinian condition.