Cohomologies and derived brackets of Leibniz algebras

2016

PhDThesisXiongweiCAI.pdf

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Keywords :

Leibniz algebra; Courant-Dorfman algebra; standard cohomology; naive cohomology; crossed product; derived bracket; equivariant cohomology; generalized action

Abstract :

[en] 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.

Disciplines :

Mathematics

Cai, Xiongwei ^{}; University of Luxembourg > Faculty of Science, Technology and Communication (FSTC) ; University of Luxembourg > Faculty of Science, Technology and Communication (FSTC) > Mathematics Research Unit

Language :

English

Title :

Cohomologies and derived brackets of Leibniz algebras

Defense date :

13 December 2016

Institution :

Unilu - University of Luxembourg, Luxembourg

Degree :

Docteur en Mathématiques

Promotor :

President :

Jury member :

Liu, Zhangju

Xu, Ping

Laurent-Gengoux, Camille

Available on ORBilu :

since 11 January 2017