Cohomologies and derived brackets of Leibniz algebras

In this thesis, we work on the structure of Leibniz algebras and develop cohomology theories for them. The motivation comes from:

• Roytenberg, Stienon-Xu and Ginot-Grutzmann's work on standard and naive cohomology of Courant algebroids (Courant-Dorfman algebras).

• Kosmann-Schwarzbach, Roytenberg and Alekseev-Xu's constructions of derived brackets for Courant algebroids.

• The classical equivariant cohomology theory and the generalized geometry theory.

This thesis consists of three parts:

1. We introduce standard cohomology and naive cohomology for a Leibniz algebra. We discuss their properties and show that they are isomorphic. By similar methods, we prove a generalization of Ginot-Grutzmann's theorem on transitive Courant algebroids, which was conjectured by Stienon-Xu. The relation between standard complexes of a Leibniz algebra and its corresponding crossed product is also discussed.

2. We observe a canonical 3-cochain in the standard complex of a Leibniz algebra. We construct a bracket on the subspace consisting of so-called representable cochains, and prove that the subspace becomes a graded Poisson algebra. Finally we show that for a fat Leibniz algebra, the Leibniz bracket can be represented as a derived bracket.

3. In spired by the notion of a Lie algebra action and the idea of generalized geometry, we introduce the notion of a generalized action of a Lie algebra g on a smooth manifold M, to be a homomorphism of Leibniz algebras from g to the generalized tangent bundle TM+T*M. We define the interior product and Lie derivative so that the standard complex of TM+T*M becomes a g differential algebra, then we discuss its equivariant cohomology. We also study the equivariant cohomology for a subcomplex of a Leibniz complex.