Leibniz algebras and dialgebras


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.

1 - Leibniz algebras
2 - Leibniz homology and cohomology
3 - Dialgebras
4 - Dialgebra and symmetric (co)homology

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