On the structure of simple pseudo Lie algebras and their invariant bilinear forms

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Abstract
By definition simple pseudo Lie algebras do not contain any nontrivial ideal. We show that ’’graded simplicity’’ implies ’’simplicity’’ and discuss the uniqueness of invariant bilinear forms on a simple pseudo Lie algebra. A lot of examples of simple pseudo Lie algebras is given together with their invariant bilinear forms. Under certain general assumptions we derive that the Lie algebra g contained in a simple pseudo Lie algebra a is reductive. Assuming that g is reductive, we prove that the ’’adjoint representation of g in the odd subspace of a’’ is completely reducible with at most two irreducible components. Finally we show that the pseudo Lie algebras with nondegenerate ’’generalized Killing form’’ are direct products of simple pseudo Lie algebras.

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