Semisimplicity of restricted enveloping algebras of Lie superalgebras

Abstract

Let = Lo Θ L be a restricted Lie superalgebra over a field of characteristic p > 2. We let u(L) denote the restricted enveloping algebra of and we will be concerned with when u(L) is semisimple, semiprime, or prime. The structure of u(L) is sufficiently close to that of a Hopf algebra that we will obtain ring theoretic information about u(L) by first applying basic facts about finite dimensional Hopf algebras to Hopf algebras of the form u(L) # G. Our main result along these lines is that if u(L) is semisimple with finite dimensional, then L = 0. Combining this with a result of Hochschild, we will obtain a complete description of those finite dimensional such that u(L) is semisimple. In the infinite dimensional case, we will obtain various necessary conditions for u(L) to be prime or semiprime.