幂零Leibniz代数与Lie racks的同痕性

The local integration of Leibniz algebras

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 the integrating manifold is a Lie group. In his article \cite{Kinyon}, M.K. Kinyon solves the particula…

Leibniz algebras, Lie racks, and digroups

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 algebras are the corresponding tangent algebra structures. The difficulty is determining exactly what properties this generalization should have. Here we show that \emph{Lie racks}, smooth left distributive structures, have Leibniz algebra structures on their tangent spaces at certain distinguished points. One way of producing racks is by conjugation in \emph{digroups}, a generalization of group whi…

LIE n-RACKS

In this paper, we introduce the category of Lie $n$-racks and generalize several results known on racks. In particular, we show that the tangent space of a Lie $n$-Rack at the neutral element has a Leibniz $n$-algebra structure. We also define a cohomology theory of $n$-racks..

On the conjectural cohomology for groups

The goal of this paper is to present results which are consistent with conjectures about the Leibniz (co)homology for discrete groups stated by J. L. Loday. We show that rack cohomology has properties very close to the properties expected for the conjectural Leibniz cohomology. In particular, we prove the existence of a graded dendriform algebra structure on rack cohomology, and we construct a graded associative algebra morphism from group cohomology to rack cohomology which is injective in degree 1.

李分配代数、李布尼亚克架与诺特第一定理

Lie Quandles, Leibniz Racks and Noether's First Theorem

本文探讨了李分配代数与李布尼亚克架的线性与非线性对应关系，分类了其一般化对象，并描述了诺特第一定理的非线性类比结果。

两步幂零Leibniz代数

Two-step nilpotent Leibniz algebras

本文基于Kronecker模完成两步幂零Leibniz代数分类，推广经典Heisenberg李代数得到不可分解Heisenberg Leibniz代数的复实情形，证明实幂零Leibniz代数可整体积分并给出应用，还揭示了积分Leibniz代数的李quandle与李群共轭的关系。

非幂零李代数与一维导出子代数

Non-nilpotent Leibniz algebras with one-dimensional derived subalgebra

本文研究了非幂零非李李代数的一维导出子代数，证明其可分解为二维非幂零非李代数与阿贝尔代数的直和，并扩展了[11]的结果，同时研究了导出代数的李代数、自同构群及双导出代数。