Cohomologies and derived brackets of Leibniz algebras

Doctoral thesis (2016)

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 ... [more ▼]

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. [less ▲]