Elisabeth Remm
We study a special class of weakly associative algebras: the symmetric Leibniz algebras. We describe the structure of the commutative and skew symmetric algebras associated with the polarization-depolarization principle. We also give a structure theorem for the symmetric Leibniz algebras and we study formal deformations in the context of deformation quantization.
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Adjoint functors between the categories of crossed modules of dialgebras and Leibniz algebras are constructed. The well-known relations between the categories of Lie, Leibniz, associative algebras and dialgebras are extended to the respective categories of crossed modules.
In this paper we define the basic concepts for left or right Leibniz algebras and prove some of the main results. Our proofs are often variations of the known proofs and several results seem to be new.
This paper is devoted to the complete algebraic and geometric classification of complex $4$-dimensional nilpotent weakly associative, complex $4$-dimensional symmetric Leibniz algebras, and complex $5$-dimensional nilpotent symmetric Leibniz algebras. In particular, we proved that the variety of complex $4$-dimensional symmetric Leibniz algebras has no Vergne--Grunewald--O'Halloran Property (there is an irreducible component formed by only nilpotent algebras), but on the other hand, it has Vergne Property (there are no rigid nilpotent algebras).
The aim of this paper is to introduce $n$-ary BiHom-algebras, generalizing BiHom-algebras. We introduce an alternative concept of BiHom-Lie algebra called BiHom-Lie-Leibniz algebra and study various type of $n$-ary BiHom-Lie algebras and BiHom-associative algebras. We show that $n$-ary BiHom-Lie-Leibniz algebra can be represented by BiHom-Lie-Leibniz algebra through fundamental objets. Moreover, we provide some key constructions and study $n$-ary BiHom-Lie algebras induced by $(n-1)$-ary BiHom-Lie algebras.
The geometric classification of non-associative algebras: a survey
本文综述了幂零、Lie、Jordan等多种非结合代数簇（含$n$元代数、超代数、Poisson型代数）的几何分类研究，并涵盖了退化水平分类，系统梳理该领域成果。
This paper concerns the algebraic structure of finite-dimensional complex Leibniz algebras. In particular, we introduce left central and symmetric Leibniz algebras, and study the poset of Lie subalgebras using an associative bilinear pairing taking values in the Leibniz kernel.
A pseudo-Euclidean non-associative algebra $(\mathfrak{g}, \bullet)$ is a real algebra of finite dimension that has a metric, i.e., a bilinear, symmetric, and non-degenerate form $\langle\;\rangle$. The metric is considered $\mathrm{L}$-invariant (resp. $\mathrm{R}$-invariant) if all left multiplications (resp. right multiplications) are skew-symmetric. The metric is called associative if $\langle u\bullet v,w\rangle= \langle u,v\bullet w\rangle$ for all $u, v, w \in \mathfrak{g}$. These three notions coincide when $\mathfrak{g}$ is a Lie algebra and in this case $\mathfrak{g}$ endowed with th…
The Leibniz algebras appeared as a generalization of the Lie algebras. In this work we deal with the classification of nilpotent complex Leibniz algebras of low dimensions. Namely, the classification of nilpotent complex Leibniz algebras dimensions less than 3 is extended to the dimension four. {\it AMS Subject Classifications}: 16D70, 17A30, 17A60, 17B30 {\it Key words:} Leibniz algebra, associative algebra, nilpotence, nulfiliform Leibniz algebra, filiform Leibniz algebra.
We initiate the study of $ξ$-groups and Hu-Liu Leibniz algebras, claim that alomost all simple Leibniz algebras and simple Hu-Liu Leibniz algebras are linear, and establish two passages. One is the passage from a special $\mathcal{Z}_2$-graded associative algebra to a Hu-Liu Leibniz algebra. The other one is the passage from a linear $ξ$-group to its tangent space which is a Hu-Liu Leibniz algebra.
This paper is a contribution to the development of the non associative algebras theory. More precisely, this work deals with the classification of the complex 4-dimensional Leibniz algebras. Note that the classification of 4-dimensional nilpotent complex Leibniz algebras was obtained in [1]. Therefore we will only consider non nilpotent case in this work.
We study some properties on $\mathsf{Lie}$-centroids related to central $\mathsf{Lie}$-derivations, generalized $\mathsf{Lie}$-derivations and almost inner $\mathsf{Lie}$-derivations. We also determine the $\mathsf{Lie}$-centroid of the tensor product of a commutative associative algebra and a Leibniz algebra.
The notion of non-abelian Hom-Leibniz tensor product is introduced and some properties are established. This tensor product is used in the description of the universal ($α$-)central extensions of Hom-Leibniz algebras. We also give its application to the Hochschild homology of Hom-associative algebras.
The purpose of this paper is to generalize some results on ternary Leibniz algebras to the case of ternary Leibniz color algebras. In particular, we study color Lie triple systems. In order to produce examples of ternary Leibniz color algebras from Leibniz color algebras, several results on Leibniz color algebras are given. Then we introduce and give some constructions of ternary Leibniz-Nambu-Poisson color algebras. The relationship between associative trialgebras and -1- tridendriform algebras are investigated. Moreover, we give some methods of constructing modules over ternary Leibniz-Nambu…
A Frattini theory for non-associative algebras was developed by Towers and results for particular classes of algebras have appeared in various articles. Especially plentiful are results on Lie algebras. It is the purpose of this paper to extend some of the Lie algebra results to Leibniz algebras.
In this paper we study symmetric Leibniz and related algebras, namely symmetric dialgebras and symmetric Perm-algebras. We also calculate their Koszul duals, if not known. This will give us Lie-admissible algebras and new types of algebras, which we call commutative and associative admissible algebras. Finally we prove that all operads describing the considered algebras are Koszul.
We study deformation quantization of nonassociative algebras whose associator satisfies some symmetric relations. This study is expanded to a larger class of nonassociative algebras includind Leibniz algebras. We apply also to this class the rule of polarization-depolarization.
In this paper, we study the category of trigroups as a generalization of the notion of digroup [4] and analyze their relationship with 3-racks [1] and Leibniz 3-algebras [6]. Trigroups are essentially associative trioids in which there are bar-units and bar-inverses. We prove that 3-racks can be constructed by conjugating trigroups. We also prove that trigroups equipped with a smooth manifold structure produce Leibniz 3-algebras via their associated Lie 3-racks.
Anti-Leibniz algebras: A non-commutative version of mock-Lie algebra
本文提出反莱布尼茨代数作为模拟李代数的非交换推广，完成低维分类，并利用平均算子与嵌入张量构造出反结合二代数、反结合三代数及反莱布尼茨三代数等新结构。
The aim of this paper is to introduce and study BiHom-associative dialgebras. We give various constructions and study its connections with BiHom-Poisson dialgebras and BiHom-Leibniz algebras. Next we discuss the central extensions of BiHom-diassociative and we describe the classification of n-dimensional BiHom-diassociative algebras for n less or equal to 4. Finally, we discuss derivations of BiHom-associative dialgebras.
In this paper we introduce color Hom-Akivis algebras and prove that the commutator of any color non-associative Hom-algebra structure map leads to a color Hom-akivis algebra. We give various constructions of color Hom-Akivis algebras. Next we study flexible and alternative color Hom-Akivis algebras. Likewise color Hom-Akivis algebras, we introduce non-commutative color Hom-Leibniz-Poisson algebras and presente several constructions. Moreover we give the relationship between Hom-dialgebras and Hom-Leibniz-Poisson algebras; i.e. a Hom-dialgebras give rise to a Hom-Leibniz-Poisson algebra. Finall…
In this paper we prove that in prime characteristic there is a functor $-_{p-Leib}$ from the category of diassociative algebras to the category of restricted Leibniz algebras, generalizing the functor from associative algebras to restricted Lie algebras. Moreover we define the notion of restricted universal enveloping diassociative algebra $Ud_{p}(g)$ of a restricted Leibniz algebra $g$ and we show that $Ud_{p}$ is left adjoint to the functor $-_{p-Leib}$. We also construct the restricted enveloping algebra, which classifies the restricted Leibniz modules. In the last section we put a restrict…
We study the automorphism groups of finite-dimensional cyclic Leibniz algebras. In this connection, we consider the relationships between groups, modules over associative rings and Leibniz algebras.
$δ$-Leibniz algebras and related $δ$-type algebras
本文提出$δ$-Leibniz代数及其相关$δ$-型代数的概念，研究其结构和性质，并建立与nil代数的联系，为非结合代数提供统一框架。
In this paper, the definitions of algebras of quotients and Martandale-like qoutients of Leibniz algebras are introduced and the interactions between the two quotients are determined. Firstly, some important properties which not only hold for a Leibniz algebras but also can been lifted to its algebras of quotients are investigated. Secondly, for any semiprime Leibniz algebra, its maximal algebra of quotients is constucted and a Passman-like characterization of the maximal algebra is described. Thirdly, the relationship between a Leibniz algebra and the associative algebra which is generated by…
The universal enveloping algebra functor between Leibniz and associative algebras defined by Loday and Pirashvili is extended to crossed modules. We prove that the universal enveloping crossed module of algebras of a crossed module of Leibniz algebras is its natural generalization. Then we construct an isomorphism between the category of representations of a Leibniz crossed module and the category of left modules over its universal enveloping crossed module of algebras. Our approach is particularly interesting since the actor in the category of Leibniz crossed modules does not exist in general…
The paper concerns extra special associative algebras, an analogue of the Heisenberg Lie algebra. In particular, we say that an associative algebra is extra special if its center is equal to its derived ideal and the center is 1-dimensional. In this paper, we classify extra special associative algebras by proving that their structure is equivalent to that of extra special Leibniz algebras. We then characterize their (Schur) multipliers via dimension and completely determine their capability. We connect this with the related notion of unicentral algebras and discuss the problem of classifying e…
Recently, relative Rota-Baxter (Lie/associative) algebras are extensively studied in the literature from cohomological points of view. In this paper, we consider relative Rota-Baxter Leibniz algebras (rRB Leibniz algebras) as the object of our study. We construct an $L_\infty$-algebra that characterizes rRB Leibniz algebras as its Maurer-Cartan elements. Then we define representations of an rRB Leibniz algebra and introduce cohomology with coefficients in a representation. As applications of cohomology, we study deformations and abelian extensions of rRB Leibniz algebras.
Poisson superalgebras are known as a $\mathbb{Z}_2$-graded vector space with two operations, an associative supercommutative multiplication and a super bracket tied up by the super Leibniz relation. We show that we can consider a single nonassociative multiplication containing all these datas and then consider Poisson superalgebras as non associative algebras.
Hom-unitality and hom-associative structures
本文研究了非结合代数上的同态结合结构，探讨了一侧和双侧单位代数的特性，提出新的特征化方法，并应用于多种非结合代数类。
We introduce the notion of weakly associative algebra and its relations with the notion of nonassociative Poisson algebras.
Tensor products of Leibniz bimodules and Grothendieck rings
本文定义了三种李代数双模张量积概念，引入弱双模修正自然张量积问题，并证明其诱导非结合乘法，揭示有限维李代数双模范畴的格罗滕迪克环性质。
Construction of some non-associative algebras from Associative Algebras with an endomorphism operator, a differential operator or a left averaging operator
本文引入自同态、左平均、微分和Rota-Baxter算子等概念，在特定结合代数上构造这些线性映射，以此诱导出Lie、Pre-Lie等非结合代数结构，并给出了方阵子空间M3(R)上的实例。
A Hom-algebra structure is a multiplication on a vector space where the structure is twisted by a homomorphism. The structure of Hom-Lie algebra was introduced by Hartwig, Larsson and Silvestrov and extended by Larsson and Silvestrov to quasi-hom Lie and quasi-Lie algebras. In this paper we introduce and study Hom-associative, Hom-Leibniz, and Hom-Lie admissible algebraic structures which generalize the well known associative, Leibniz and Lie admissible algebras. Also, we characterize the flexible Hom-algebras in this case. We also explain some connections between Hom-Lie algebras and Santilli…
On Dextral Symmetric Algebra
本文受对称环启发引入右对称代数概念，并给出Leavitt路径代数及低维右Leibniz代数的完整分类，同时证明有限维右对称右Leibniz代数可解与较弱幂零性等价。
Poisson algebra is usually defined to be a commutative algebra together with a Lie bracket, and these operations are required to satisfy the Leibniz rule. We describe Poisson structures in terms of a single bilinear operation. This enables us to explore Poisson algebras in the realm of non-associative algebras. We study their algebraic and cohomological properties, their deformations as non-associative algebras, and give a classification in low dimensions
We classify finite dimensional division real associative $\mathcal{Z}_2$-algebras, introduce composition $\mathcal{Z}_2$-algebras, and extend the Campbell-Baker-Hausdorff series and Lie correspondence in the context of linear Hu-Liu Leibniz algebras.
Construction of diassociative bialgebras from antisymmetric infinitesimal bialgebras and related algebra structures
本文证明反对称无穷小双代数与二次 perm 代数的张量积具有双结合双代数结构，且其结构性质与原反对称无穷小双代数对应，还明确了几类双代数及辛代数间的密切关系。
Hom-associative algebras, Admissibility and Relative averaging operators
本文在Hom-结合代数上引入相对平均算子，通过图和Nijenhuis算子刻画，并利用可容许性构造Hom-Jordan及Hom-Leibniz代数。
In hom-associative structures, the associativity condition $(xy)z=x(yz)$ is twisted to $α(x)(yz) = (xy)α(z)$, with $α$ a map in the appropriate category. In the present paper, we consider two different unitality conditions for hom-associative algebras. The first one, existence of a unit in the classical sense, is stronger than the second one, which we call weak unitality. We show associativity conditions connected to the size of the image of the twisting map for unital hom-associative algebras. Also the problem of embedding arbitrary hom-associative algebras into unital or weakly unital ones i…
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。
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.
Skew-symmetric Frobenius dialgebras, diassociative bialgebras and Yang-Baxter equations
本文系统建立了双结合双代数理论，引入双结合双代数概念，证明其等价于Frobenius双代数的双重构造和双代数的匹配对，并给出斜对称Rota-Baxter Frobenius双代数与三角/可分解双代数的联系。
A subalgebra $B$ of a Leibniz algebra $L$ is called a weak c-ideal of $L$ if there is a subideal $C$ of $L$ such that $L=B+C$ and $B\cap C\subseteq B_{L}$ where $B_{L}$ is the largest ideal of $L$ contained in $B.$ This is analogous to the concept of a weakly c-normal subgroup, which has been studied by a number of authors. We obtain some properties of weak c-ideals and use them to give some characterisations of solvable and supersolvable Leibniz algebras generalising previous results for Lie algebras. We note that one-dimensional weak c-ideals are c-ideals, and show that a result of Turner cl…
We introduce hom-associative Ore extensions as non-unital, non-associative Ore extensions with a hom-associative multiplication, and give some necessary and sufficient conditions when such exist. Within this framework, we construct families of hom-associative quantum planes, universal enveloping algebras of a Lie algebra, and Weyl algebras, all being hom-associative generalizations of their classical counterparts, as well as prove that the latter are simple. We also provide a way of embedding any multiplicative hom-associative algebra into a multiplicative, weakly unital hom-associative algebr…
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。
In this review article, first we give the concrete formulas of representations and cohomologies of associative algebras, Lie algebras, pre-Lie algebras, Leibniz algebras and 3-Lie algebras and some of their strong homotopy analogues. Then we recall the graded Lie algebras and graded associative algebras that characterize these algebraic structures as Maurer-Cartan elements. The corresponding Maurer-Cartan element equips the graded Lie or associative algebra with a differential. Then the deformations of the given algebraic structures are characterized as the Maurer-Cartan elements of the result…
A central extension is a regular epimorphism in a Barr exact category $\mathscr{C}$ satisfying suitable conditions involving a given Birkhoff subcategory of $\mathscr{C}$ (joint work with G. M. Kelly, 1994). In this paper we take $\mathscr{C}$ to be the category of (not-necessarily-unital) algebras over a (unital) commutative ring and consider central extensions with respect to the category of commutative algebras. We propose a new approach that avoids the intermediate notion of central extension due to A. Fröhlich in showing that $α:A\to B$ is a central extension if and only if $aa'=a'a$ for …
We investigate the class of finite dimensional not necessary associative algebras that have slowly growing length, that is, for any algebra in this class its length is less than or equal to its dimension. We show that this class is considerably big, in particular, finite dimensional Lie algebras as well as many other important classical finite dimensional algebras belong to this class, for example, Leibniz algebras, Novikov algebras, and Zinbiel algebras. An exact upper bounds for the length of these algebras is proved. To do this we transfer the method of characteristic sequences to non-unita…
In this note we define a notion of Courant pair as a Courant algebra over the Lie algebra of linear derivations on an associative algebra. We study formal deformations of Courant pairs by constructing a cohomology bicomplex with coefficients in a module from the cochain complexes defining Hochschild cohomology and Leibniz cohomology.
Action accessible and weakly action representable varieties of algebras
本文研究非结合代数变种中作用可及性与弱作用可表示性关系，证明k- nilpotent李代数和n-可解李代数变种不构成弱作用可表示范畴，首次展示非结合代数变种中作用可及但非弱作用可表示的例子。
Geometry of Deformations via Incidence Varieties
本文通过构造 GL-等变双线性 incidence 变换，统一研究了有限维向量空间上四种代数结构的几何变形问题，并证明了其与上同调理论的同构关系。
Kupershmidt operators on $3$-BiHom-Poisson color algebras and $3$-BiHom-pre-Poisson color algebras structures
该研究引入非交换3-BiHom-Poisson色代数并研究其表示与Kupershmidt算子，同时提出非交换3-BiHom-预Poisson色代数概念，借助Kupershmidt算子揭示两类代数的关联。
Weak representability of actions of non-associative algebras
该文研究域上非结合代数簇中内部作用的弱可表示性，刻画了满足二次恒等式的相关代数簇，构造了外部弱作用子及对应函子单态射，并阐明其与泛严格一般作用子的关系。
A hom-associative structure is a set $A$ together with a binary operation $\star$ and a selfmap $α$ such that an $α$-twisted version of associativity is fulfilled. In this paper, we assume that $α$ is surjective. We show that in this case, under surprisingly weak additional conditions on the multiplication, the binary operation is a twisted version of an associative operation. As an application, an earlier result by Yael Fregier and the author on weakly unital hom-algebras is recovered with a different proof. In the second section, consequences for the deformation theory of hom-algebras with s…
In this paper we construct "structural" pre-braidings characterizing different algebraic structures: a rack, an associative algebra, a Leibniz algebra and their representations. Some of these pre-braidings seem original. On the other hand, we propose a general homology theory for pre-braided vector spaces and braided modules, based on the quantum co-shuffle comultiplication. Applied to the structural pre-braidings above, it gives a generalization and a unification of many known homology theories. All the constructions are categorified, resulting in particular in their super- and co-versions. L…
Quantum deformations of the structure constants for a class of associative noncommutative algebras are studied. It is shown that these deformations are governed by the quantum central systems which has a geometrical meaning of vanishing Riemann curvature tensor for Christoffel symbols identified with the structure constants. A subclass of isoassociative quantum deformations is described by the oriented associativity equation and, in particular, by the WDVV equation. It is demonstrated that a wider class of weakly (non)associative quantum deformations is connected with the integrable soliton eq…
Bialgebras, Manin triples of Malcev-Poisson algebras and post-Malcev-Poisson algebras
本文引入 Malcev-Poisson 双代数与 post-Malcev-Poisson 代数，建立匹配对、Manin 三元组与双代数的等价，并揭示其作为加权相对 Rota-Baxter 算子底层结构的作用。
Jean-Louis Loday introduced a class of symmetric operads generated by one bilinear operation subject to one relation making each left-normed product of three elements equal to a linear combination of right-normed products: \[ (a_1a_2)a_3=\sum_{σ\in S_3}x_σ\, a_{σ(1)}(a_{σ(2)}a_{σ(3)})\ ; \] such an operad is called a parametrized one-relation operad. For a particular choice of parameters $\{x_σ\}$, this operad is said to be regular if each of its components is the regular representation of the symmetric group; equivalently, the corresponding free algebra on a vector space $V$ is, as a graded v…
$D$-bialgebras, dendrification and embeddings into AWB of almost Poisson algebras
本文引入了几乎Poisson Drinfel'd bialgebras（$D$-bialgebras）并建立了匹配对、Manin三元组与该结构的等价性，同时定义了新的代数结构，并展示了如何通过平均算子将几乎Poisson代数嵌入到AWB中。
Nonassociative algebras of anti-biderivation-type
本文研究一类乘积为反双导子的Jacobi-Jordan-容许代数（$\mathcal A{\rm BD}$-代数），刻画其性质，证明包含若干已知代数类，引入post-Jacobi-Jordan结构，并给出复数3维情形的代数分类。
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