Lie Superalgebra Structures in H? (g;g)

Abstract

Let g=vect(M) be the Lie (super)algebra of vector fields on any connected (super)manifold M; let “-” be the change of parity functor, C i and H i the space of i-chains and i-cohomology. The Nijenhuis bracket makes into a Lie superalgebra that can be interpreted as the centralizer of the exterior differential considered as a vector field on the supermanifold associated with the de Rham bundle on M. A similar bracket introduces structures of DG Lie superalgebra in L * and for any Lie superalgebra g. We use a Mathematica-based package SuperLie (already proven useful in various problems) to explicitly describe the algebras l * for some simple finite dimensional Lie superalgebras g and their “relatives” - the nontrivial central extensions or derivation algebras of the considered simple ones.