Complex finitary simple Lie algebras

Abstract.

An algebra is called finitary if it consists of finite-rank transformations of a vector space. We classify finitary simple Lie algebras over an algebraically closed field of zero characteristic. It is shown that any such algebra is isomorphic to one of the following¶ (1) a special transvection algebra $\frak t(V,\mit\Pi )$;¶ (2) a finitary orthogonal algebra $\frak {fso} (V,q)$; ¶ (3) a finitary symplectic algebra $\frak {fsp} (V,s)$.¶Here V is an infinite dimensional K-space; q (respectively, s) is a symmetric (respectively, skew-symmetric) nondegenerate bilinear form on V; and $\Pi$ is a subspace of the dual V ^* whose annihilator in V is trivial: $0=\{{v}\in V\mid \Pi {v}=0\}$.