Classification of some 2-graded lie algebras

Abstract

We continue the study of the structure of 2-graded Lie algebras (see next section for the definition) which was begun by Hochschild [6] and Hochschild and the author [4]. AU semi-simple 2-graded Lie algebras over an algebraically closed field of characteristic 0 have been determined in [4]. In this paper we introduce the analog of the Killing form for 2-graded Lie algebras. AS in the case of ordinary Lie algebras, this form plays an important role in the structure theory. For semi-simple 2-graded Lie algebras this form is non-degenerate although the converse is not true. Semi-simplicity of 2-graded Lie algebras is preserved under both extension and restriction of scalars. Our main result is the classification of indecomposable 2-graded Lie algebras L over an algebraically closed field of characteristic 0 under the condition that the zero component Lo of L be simple (Theorem 8). There are seven infinite series of such algebras. This classification is accomplished in two steps. The first step is the classification of those 2-graded Lie algebras L which are simple and have Lo simple (Theorem 6). In the proof of this theorem we need the following fact: If‘ L, is a simple Lie algebra then the multiplicity 172 r) (resp. nzl ) of L, in the symmetric power S2(Lo) (resp. the exterior power A-(Lo)) is nzo = I if Lo =: rl ,t (112 2),