SEMISIMPLICITY OF RESTRICTED ENVELOPING-ALGEBRAS OF LIE-SUPERALGEBRAS

Let L = L(0) circle plus L(1) be a restricted Lie superalgebra over a held of characteristic p > 2. We let u(L) denote the restricted enveloping algebra of L 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 L finite dimensional, then L(1) = 0. Combining this with a result of Hochschild, we will obtain a complete description of those finite dimensional L 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.