An algebra is called finitary if it consists of finite-rank transformations of a vector space. We classify finitary simple Lie algebras over a field of characteristic 0. We also describe finitary irreducible Lie algebras.

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We prove that any simple Lie subalgebra of a locally finite associative algebra is either finitedimensional or isomorphic to the commutator algebra of the Lie algebra of skew symmetric elements of some involution simple locally finite associative algebra. The ground field is assumed to be algebraically closed of characteristic 0. This result can be viewed as a classification theorem for simple Lie algebras that can be embedded in locally finite associative algebras. We also establish a link between this class of Lie algebras and that of Lie algebras graded by finite root systems.  2004 Elsevier Inc. All rights reserved.