Infinite dimensional irreducible Lie algebras containing transformations of finite rank

Let be a field of characteristic zero and let V be an infinite dimensional vector space over . A linear transformation x of V is called finitary if . The aim of this paper is to describe irreducible Lie subalgebras of containing nonzero finitary transformations. It turns out that any such algebra is a semidirect product of a finite dimensional Lie algebra and a “dense” Lie subalgebra of for some vector space W. Received January 4, 2000 / Published online March 12, 2001