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Leibniz algebras and dialgebras

**Abstract**

This note is a brief survey on the main known results on Leibniz algebras and
dialgebras, introduced by J.-L.~Loday as a generalization of Lie algebras and
associative algebras respectively, and on their natural homology theories.
Two examples of a Leibniz algebra are given: the non-commutative Steinberg algebra
introduced by J.-L.~Loday and T.~Pirashvili, and the extension of the Poisson bracket
of functions to the algebras of multivector fields and differential forms on a Poisson
manifolds, found by Y.~Kosmann-Schwarzbach.

Leibniz homology produces new invariants for Lie algebras. We present some examples
of these new invariants for some finite dimensional Lie algebras, for the Lie algebra
of vector fields on the real space, and the relationship with the rational homotopy
theory of topological spaces.
Finally, we present the author's contribution to the computation of Leibniz
homology, based on a homology theory with coefficients for dialgebras.

**Contents**

Introduction

1 - Leibniz algebras

2 - Leibniz homology and cohomology

3 - Dialgebras

4 - Dialgebra and symmetric (co)homology

References