Simple decompositions of simple Lie superalgebras

Abstract

In this paper we consider Lie superalgebras decomposable as the sum of two proper subalgebras. Any of these algebras has the form of the vector space sum $L=A+B$ where $A$ and $B$ are proper simple subalgebras which need not be ideals of $L$, and the sum need not be direct. The main result of this paper is the following: Let $S = {osp}(m,2n)$ be a Lie superalgebra such that $S=K+L$ where $K$, $L$ are two proper basic simple subalgebras. Then $m$ is even, $m=2k$ and $K \cong osp(2k-1,2n)$, $L \cong sl(k,n)$.