The structure of symmetric Lie algebras

Abstract

In Winter [7], a certain class of Lie algebras, s~*mmrtric Lie ab and then we use the theory of algebraic groups and the theory of Lie rootsystems to pro\.p the following theorem, which expresses the structure of a symmetric Lie aigebra L in terms of a classica! Lie algebra L,S, a semisimple symmetric Lie algebra Lw whose root system is a Witt rootsystem (defined below) and solvable ideals. In the case of a ground field of characteristic 0, the theorem simply says that a symmetric Lie algebra of characteristic 0 is of the form L = L, @ Solv L with L, semisimple, which follows from Levi’s Theorem. So. we restrict ourselves in this paper to the much more difficult context of a ground field k of prime characteristic p > 3 (and sometimes 7).