# The local integration of Leibniz algebras

@article{Covez2010TheLI, title={The local integration of Leibniz algebras}, author={Simon Covez}, journal={arXiv: Rings and Algebras}, year={2010} }

This article gives a local answer to the coquecigrue problem. Hereby we mean the problem, formulated by J-L. Loday in \cite{LodayEns}, is that of finding a generalization of the Lie's third theorem for Leibniz algebra. That is, we search a manifold provided with an algebraic structure which generalizes the structure of a (local) Lie group, and such that the tangent space at a distinguished point is a Leibniz algebra structure. Moreover, when the Leibniz algebra is a Lie algebra, we want that…

## 44 Citations

A New Approach to Leibniz Bialgebras

- Mathematics
- 2016

A study of Leibniz bialgebras arising naturally through the double of Leibniz algebras analogue to the classical Drinfeld’s double is presented. A key ingredient of our work is the fact that the…

Leibniz Algebras with Invariant Bilinear Forms and Related Lie Algebras

- Mathematics
- 2016

In ([11]), we have studied quadratic Leibniz algebras that are Leibniz algebras endowed with symmetric, nondegenerate, and associative (or invariant) bilinear forms. The nonanticommutativity of the…

Deformation quantization of Leibniz algebras

- Mathematics
- 2013

This paper has two parts. The first part is a review and extension of the methods of integration of Leibniz algebras into Lie racks, including as new feature a new way of integrating 2-cocycles (see…

Structure Theory of Rack-Bialgebras

- Mathematics
- 2014

In this paper we focus on a certain self-distributive multiplication on coalgebras, which leads to so-called rack bialgebra. Inspired by semi-group theory (adapting the Suschkewitsch theorem), we do…

A Comment on the Integration of Leibniz Algebras

- Mathematics
- 2010

In this note we point out that the definition of the universal enveloping dialgebra for a Leibniz algebra is consistent with the interpretation of a Leibniz algebra as a generalization not of a Lie…

On Leibniz cohomology

- Mathematics
- 2019

In this paper we prove the Leibniz analogue of Whitehead's vanishing theorem for the Chevalley-Eilenberg cohomology of Lie algebras. As a consequence, we obtain the second Whitehead lemma for Leibniz…

Q A ] 1 1 O ct 2 01 8 Structure theory of Rack-Bialgebras

- 2018

In this paper we focus on a certain self-distributive multiplication on coalgebras, which leads to so-called rack bialgebra. Inspired by semi-group theory (adapting the Suschkewitsch theorem), we do…

Kupershmidt operators and related structures on Leibniz algebras

- Mathematics
- 2020

Kupershmidt operator is a key to extend a Leibniz algebra by its representation. In this paper, we investigate several structures related to Kupershmidt operators on Leibniz algebras and introduce…

Itô’s theorem and metabelian Leibniz algebras

- Mathematics
- 2014

We prove that the celebrated Itô’s theorem for groups remains valid at the level of Leibniz algebras: if is a Leibniz algebra such that , for two abelian subalgebras and , then is metabelian, i.e. .…

Enhanced Leibniz Algebras: Structure Theorem and Induced Lie 2-Algebra

- Physics, MathematicsCommunications in Mathematical Physics
- 2019

An enhanced Leibniz algebra is an algebraic struture that arises in the context of particular higher gauge theories describing self-interacting gerbes. It consists of a Leibniz algebra $$({\mathbb…

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