# 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…

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…

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

- Mathematics
- 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

- 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…

## References

SHOWING 1-10 OF 62 REFERENCES

Leibniz algebras, Lie racks, and digroups

- Mathematics
- 2004

The "coquecigrue" problem for Leibniz algebras is that of finding an appropriate generalization of Lie's third theorem, that is, of finding a generalization of the notion of group such that Leibniz…

Universal enveloping algebras of Leibniz algebras and (co)homology

- Mathematics
- 1993

The homology of Lie algebras is closely related to the cyclic homology of associative algebras [LQ]. In [L] the first author constructed a "noncommutative" analog of Lie algebra homology which is,…

THE TENSOR CATEGORY OF LINEAR MAPS AND LEIBNIZ ALGEBRAS

- Mathematics
- 1998

AbstractWe equip the category
$$\mathcal{L}\mathcal{M}$$
of linear maps of vector spaces with a tensor product which makes it suitable for various constructions related to Leibniz algebras. In…

Leibniz algebras, Courant algebroids, and multiplications on reductive homogeneous spaces

- Mathematics
- 2000

We show that the skew-symmetrized product on every Leibniz algebra ε can be realized on a reductive complement to a subalgebra in a Lie algebra. As a consequence, we construct a nonassociative…

Abelian extensions of infinite-dimensional Lie groups

- Mathematics
- 2004

In the present paper we study abelian extensions of connected Lie groups G modeled on locally convex spaces by smooth G-modules A. We parametrize the extension classes by a suitable cohomology group…

RACKS AND LINKS IN CODIMENSION TWO

- Mathematics
- 1992

A rack, which is the algebraic distillation of two of the Reidemeister moves, is a set with a binary operation such that right multiplication is an automorphism. Any codimension two link has a…

The Complex of a Group Relative to a Set of Generators: Part II

- Mathematics
- 1951

In order to describe the contents of the present paper we begin with some definitions. An extension of a local group Y is a pair (E, 4) where E is a local group and-? is a strong homomorphism of E…

A COURSE IN HOMOLOGICAL ALGEBRA

- Mathematics

The fascinating thing is that zeroth syzygies and first syzygies have an intrinsic significance in terms of the duality functor A 7→ A∗ = HomΛ(A, Λ). Namely, a left Λ-module A is a first syzygy if…

Trunks and classifying spaces

- MathematicsAppl. Categorical Struct.
- 1995

The theory of racks can be used to define invariants of knots and links since any invariant of the rack space of the fundamental rack of a knot or link is ipso facto an invariante of the Knot or link.