Semisimplicity of restricted enveloping algebras of Lie superalgebras

Jeffery Marc Bergen

Vol. 162 (1994), No. 1, 1–11

DOI: 10.2140/pjm.1994.162.1

Abstract

Let L = L₀ ⊕L₁ be a restricted Lie superalgebra over a field 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 L₁ = 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.

Mathematical Subject Classification 2000

Primary: 17B35

Secondary: 16W30, 17A70

Milestones

Received: 25 August 1991

Published: 1 January 1994