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
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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.