Classically semisimple locally finite Lie superalgebras

Abstract

We introduce the class of classically semisimple locally finite Lie superalgebras over an algebraically closed field K of characteristic 0 and classify all Lie superalgebras in this class. By definition, a countably dimensional locally finite Lie superalgebra g= g0 ⊕ g1 over K is classically semisimple if it is semisimple (i.e. its locally solvable radical equals zero) and in addition it admits a generalized root decomposition, such that the root spaces generate g, and g0 is a root-reductive Lie algebra in the sense of [DP] and [PS]. We prove that any classically semisimple Lie superalgebra is isomorphic to a direct sum of copies of the infinite dimensional Lie superalgebras sl (∞|n), sl (∞|∞), osp(m|∞), osp(∞|2k), osp(∞|∞), sp(∞), sq(∞) and of copies of simple finite dimensional Lie superalgebras with reductive even part. In particular, the above listed infinite dimensional Lie superalgebras are all (up to isomorphism) countably dimensional simple locally finite Lie superalgebras which admit a generalized root decomposition with root-reductive even part. Finally, we describe all generalized root decompositions of any classically semisimple locally finite Lie superalgebra.