Actor of a crossed module of Leibniz algebras

Actor of a crossed module of dialgebras via tetramultipliers

We study the representability of actions in the category of crossed modules of dialgebras via tetramultipliers. We deduce a pair of dialgebras in order to construct an object which, under certain circumstances, is the actor (also known as the split extension classifier). Moreover, we give give a full description of actions in terms of equations. Finally, we check that under the aforementioned circumstances, the center coincides with the kernel of the canonical map from a crossed module to its actor.

H-standard cohomology for Courant-Dorfman algebras and Leibniz algebras

We introduce the notion of H-standard cohomology for Courant-Dorfman algebras and Leibniz algebras, by generalizing Roytenberg's construction. Then we generalize a theorem of Ginot-Grutzmann on transitive Courant algebroids, which was conjectured by Stienon-Xu. The relation between H-standard complexes of a Leibniz algebra and the associated crossed product is also discussed.

A non-abelian exterior product and homology of Leibniz algebras

We introduce a non-abelian exterior product of two crossed modules of Leibniz algebra and investigate its relation to the low dimensional Leibniz homology. Later this non-abelian exterior product is applied to the construction of eight term exact sequence in Leibniz homology. Also its relationship to the universal quadratic functor is established, which is applied to the comparison of the second Lie and Leibniz homologies of a Lie algebra.

关于Leibniz n-代数的笔记

Notes on Leibniz $n$-algebras

本文研究了Leibniz n-代数中遗忘函子和Daletskii-Takhtajan函子的行为，并探讨了其在同调和中心扩展中的应用。

Leibniz 2-代数的自同构与导子的Wells正合序列

Wells exact sequence for automorphisms and derivations of Leibniz 2-algebras

本文研究Leibniz 2-代数中自同构对与导子对的可诱导性，建立Wells正合序列的类比，并讨论Leibniz代数上交叉模的特殊情形。

Actions of internal groupoids in the category of Leibniz algebras

The aim of this paper is to characterize the notion of internal category (groupoid) in the category of Leibniz algebras and investigate the properties of well-known notions such as covering groupoid and groupoid operations (actions) in this category. Further, for a fixed internal groupoid $G$, we prove that the category of covering groupoids of $G$ and the category of internal groupoid actions of $G$ on Leibniz algebras are equivalent. Finally we interpret the corresponding notion of covering groupoids in the category of crossed modules of Leibniz algebras.

Unified products of Leibniz conformal algebras

The aim of this paper is to provide an answer to the $\mathbb{C}[\partial]$-split extending structures problem for Leibniz conformal algebras, which asks that how to describe all Leibniz conformal algebra structures on $E=R\oplus Q$ up to an isomorphism such that $R$ is a Leibniz conformal subalgebra. For this purpose, an unified product of Leibniz conformal algebras is introduced. Using this tool, two cohomological type objects are constructed to classify all such extending structures up to an isomorphism. Then this general theory is applied to the special case when $R$ is a free $\mathbb{C}[…

Isoclinic Lie Crossed Modules

We introduce the notion of isoclinism among crossed modules of Lie algebras, which will be called "Lie crossed modules" hereafter, and investigate some basic properties. Additionally, we introduce the notion of class preserving actor of a Lie crossed module and the relation with isoclinism.

关于 Leibniz 代数和 Poisson 代数作用可表示性

On the representability of actions of Leibniz algebras and Poisson algebras

本文证明了 Leibniz 代数范畴是弱作用可表示的，刻画了作用态射，并通过构造万有严格一般作用者研究了 Poisson 代数作用的可表示性。

幺半范畴的作用与Cartan型李代数的表示

Actions of monoidal categories and representations of Cartan type Lie algebras

本文通过交叉同态建立幺半范畴作用，统一构造Lie-Rinehart代数及Leibniz对的弱表示与容许表示，并引入交叉同态上同调理论以研究其形变，同时为广义Witt代数等提供新表示。

Left-right Noncommutative Poisson algebras

The notions of left-right noncommutative Poisson algebra ($\NP^{lr}$-algebra) and left-right algebra with bracket $\AWB^{lr}$ are introduced. These algebras are special cases of $\NLP$-algebras and algebras with bracket $\AWB$, respectively, studied earlier. An $\NP^{lr}$-algebra is a noncommutative analogue of the classical Poisson algebra. Properties of the new algebras are studied. The constructions of free objects in the corresponding categories are given. The relations between the properties of $\NP^{lr}$-algebras, the underlying $\AWB^{lr}$, associative and Leibniz algebras are investiga…

Unified products for Leibniz algebras. Applications

Let $\mathfrak{g}$ be a Leibniz algebra and $E$ a vector space containing $\mathfrak{g}$ as a subspace. All Leibniz algebra structures on $E$ containing $\mathfrak{g}$ as a subalgebra are explicitly described and classified by two non-abelian cohomological type objects: ${\mathcal H}{\mathcal L}^{2}_{\mathfrak{g}} \, (V, \, \mathfrak{g})$ provides the classification up to an isomorphism that stabilizes $\mathfrak{g}$ and ${\mathcal H}{\mathcal L}^{2} \, (V, \, \mathfrak{g})$ will classify all such structures from the view point of the extension problem - here $V$ is a complement of $\mathfrak{…

原斜纹莱布尼茨代数中的形变映射

Deformation maps in proto-twilled Leibniz algebras

该研究在原斜纹莱布尼茨代数中引入形变映射以统一研究莱布尼茨代数上各类算子的上同调，构造了对应弯曲$L_\infty$-代数并给出相关算子的Maurer-Cartan刻画，还构建了控制算子及代数与算子同时形变的两个主控$L_\infty$-代数。

n-李代数的扩张结构

Extending Structures for $n$-Lie Algebras

本文从三个角度重新研究n-李代数的上同调理论，重点探讨其扩张结构，得到统一乘积判据及相关特例，还研究了分解、形变映射等问题。

非结合代数簇中的作用：综述

Actions in varieties of non-associative algebras: a survey

本文综述域上非结合代数簇内部作用可表示性的主要结果，讨论作用可表示性等概念，构造通用严格广义作用子与外部弱作用子，适配偏代数场景，结合多种代数举例并总结结果与开放问题。