Leibniz homology, relationship with Poisson manifolds, Gelfand-Fuchs cohomology and dialgebras

This note is a brief survey on the main known results on Leibniz algebras, introduced by J.-L.~Loday as a generalization of Lie algebras, and their relationship with two branches of classical geometry. First, we present two examples of a Leibniz algebra on multivector fields and differential forms on Poisson manifolds, found by Y.~Kosmann-Schwarzbach. Secondly we introduce a Leibniz cohomology theory for differentiable manifolds and show its relationship, recently developped by J.~Lodder, with the Gelfand-Fuchs cohomology of vector fields and some related invariants of foliations. Finally, we present the author's contribution to the computation of Leibniz homology, based on a generalization of associative algebras called dialgebras.

Contents

Introduction
1 - Leibniz algebras and (co)homology
2 - Example: Leibniz brackets extending a Poisson bracket
3 - Leibniz version of Gelfand-Fuchs cohomology
4 - Computations and dialgebra homology
References