Lie superalgebra structures in H*(g; g)

Abstract

On a given manifold M, the Nijenhuis bracket makes the superspace of vector-valued differential forms 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 the superspaces of cochains and cohomology with coefficients in the adjoint module for any Lie superalgebra. We use a Mathematica--based package SuperLie (already proven useful in various problems) to explicitly describe these Lie superalgebras for some simple finite dimensional Lie superalgebras and their ``relatives'' (the nontrivial central extensions or derivation algebras of the considered simple ones).