J. Q. Adashev, B. A. Omirov, A. Kh. Khudoyberdiyev
In this work nul-filiform and filiform Zinbiel algebras are described up to isomorphism. Moreover, the classification of complex Zinbiel algebras is extended from dimensions $\leq 3$ up to the dimension $4.$
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The paper deals with the classification of a subclass of finite-dimensional Zinbiel algebras: the naturally graded p-filiform Zinbiel algebras. A Zinbiel algebra is the dual to Leibniz algebra in Koszul sense. We prove that there exists, up to isomorphism, only one family of naturally graded p-filiform Zinbiel algebras under hypothesis $n-p \geq 4.$
Nilpotency and nil subvarieties of binary Zinbiel algebras
本文证明了特征零域上有限维二元Zinbiel代数必为幂零，并刻画了其幂零子簇的充要条件，还推广了Itô定理。
The notions of Zinbiel 2-algebras and 2-term $Z_{\infty}$-algebras are introduced. It is proved that the category of Zinbiel 2-algebras and the category of $2$-term $Z_{\infty}$-algebras are equivalent to each other. Crossed module extensions of Zinbiel algebras are studied. Relationship between Zinbiel 2-algebras and dendriform 2-algebras are found.
We describe all central extensions of all $3$-dimensional non-zero complex Zinbiel algebras. As a corollary, we have a full classification of $4$-dimensional non-trivial complex Zinbiel algebras and a full classification of $5$-dimensional non-trivial complex Zinbiel algebras with $2$-dimensional annihilator, which gives the principal step in the algebraic classification of $5$-dimensional Zinbiel algebras.
In this paper, we compare the abelian subalgebras and ideals of maximal dimension for finite-dimensional Zinbiel algebras. We study Zinbiel algebras containing maximal abelian subalgebras of codimension $1$ and supersolvable Zinbiel algebras in which such subalgebras have codimension $2$, and we also analyze the case of filiform Zinbiel algebras. We give examples to clarify some results, including listing the values for $α$ and $β$ for the low dimensional Zinbiel algebras over the complex field that have been classified.
The notion of symmetric Zinbiel superalgebras is introduced. We prove that the nilpotency index of a symmetric Zinbiel superalgebra is not greater than 4 and describe two-generated symmetric Zinbiel algebras and odd generated superalgebras. We discuss identities of mono and binary symmetric Zinbiel and Leibniz algebras. It is proven that each quadratic Zinbiel algebra is 2-step nilpotent. Also, we study double extensions of symmetric Zinbiel algebras.
Quasi-Centroids and Quasi-Derivations of low-dimensional Zinbiel algebras
本文引入Zinbiel代数的拟中心与拟导子概念，描述了低维代数的拟中心和拟导子，并利用相关性质对具有小拟中心的代数进行分类，进而识别出拟特征幂零的重要子类。
In this paper we describe central extensions (up to isomorphism) of all complex null-filiform and filiform Zinbiel algebras. It is proven that every non-split central extension of an $n$-dimensional null-filiform Zinbiel algebra is isomorphic to an $(n+1)$-dimensional null-filiform Zinbiel algebra. Moreover, we obtain all pairwise non isomorphic quasi-filiform Zinbiel algebras.
This work is a continuation of the description of some classes of nilpotent Zinbiel algebras. We focus on the study of Zinbiel algebras with restrictions to gradation and characteristic sequence. Namely, the classification of naturally graded Zinbiel algebras with characteristic sequence equal to $ (n-p, p)$ is obtained.
This work provides a characterization of left and right Zinbiel algebras.Basic identities are established and discussed, showing that Zinbiel algebras are center-symmetric, and therefore Lie-admissible algebras. Their bimodules are given, and used to build a Zinbiel algebra structure on the direct sum of the underlying vector space and a finite-dimensional vector space. In addition, their matched pair is built, and related to the matched pair of their sub-adjacent Lie algebras. Besides, Zinbiel coalgebras are introduced, and linked to their underlying Lie coalgebras and coassociative coalgebra…
The extending structures and unified products for Zinbiel algebras are developed. Some special cases of unified products such as crossed products and matched pair of Zinbiel algebras are studied. It is proved that the extending structures can be classified by some non-abelian cohomology theory. One dimensional flag extending structures of Zinbiel algebras are also investigated.
Zinbiel bialgebras, relative Rota-Baxter operators and the related Yang-Baxter Equation
本文引入Zinbiel双代数概念，证明其与匹配对和Manin三元组等价，并研究余边界情形引出经典Yang-Baxter方程类比，进而定义拟三角和可分解Zinbiel双代数，建立二次Rota-Baxter Zinbiel代数与可分解双代数的一一对应。
We present the classification of a subclass of $n$-dimensional naturally graded Zinbiel algebras. This subclass has the nilindex $n-3$ and the characteristic sequence $(n-3,2,1).$ In fact, this result completes the classification of naturally graded Zinbiel algebras of nilindex $n-3.$
In this paper we show that every finite-dimensional Zinbiel algebra over an arbitrary field is nilpotent, extending a previous result by other authors that they are solvable.
An Algorithmic Approach to Inner Derivations of Low-Dimensional Zinbiel Algebras
本文提出一种以矩阵形式刻画任意n维Zinbiel代数内导子的算法，并将其应用于低维复Zinbiel代数，得到内导子的显式描述。
This paper is devoted to the complete algebraic and geometric classification of complex $5$-dimensional Zinbiel algebras. In particular, we proved that the variety of complex $5$-dimensional Zinbiel algebras has dimension $24$, it is defined by $16$ irreducible components and it has $11$ rigid algebras.
We define a cup-product in Hom-Leibniz cohomology and show that the cup-product satisfies the graded Hom-Zinbiel relation.
We study finite group actions on Leibniz algebras, define equivariant cohomology groups associated to such actions. We show that there exists a cup-product operation on this graded cohomology groups which makes it a graded zinbiel algebra.
We describe degenerations of four-dimensional Zinbiel and four-dimensional nilpotent Leibniz algebras over C. In particular, we describe all irreducible components in the corresponding varieties.
Zinbiel superalgebras
本文将 Zinbiel 代数研究推广到超代数，证明其幂零性，并刻画复数域上零-filiform 单生成元超代数的唯一性及低维分类。
Free Novikov-Zinbiel algebra
本文研究带有可逆导子的交换结合代数上的运算，构造自由诺维科夫-津比埃尔代数的显式基，证明其多线性恒等式可由定义恒等式导出。
Central derivations of low-dimensional Zinbiel algebras
本文分类了维数不超过4的复Zinbiel代数的中心导子代数结构与维数，并据此判定各中心化子的可分解性。
Infinite-dimensional pre-Lie bialgebras induced from Leibniz-dendriform bialgebras and Zinbiel-dendriform bialgebras
通过张量积构造，在Leibniz-dendriform双代数与Zinbiel代数、Zinbiel-dendriform双代数与Leibniz代数上建立完备pre-Lie双代数结构，并利用YBE解构造S-方程的解。
The extending structures problem for Zinbiel 2-algebras is studied. We introduce the concept of unified products for Zinbiel 2-algebras. Some special cases of unified products such as crossed product and matched pair of Zinbiel 2-algebras are investigated. It is proved that the extending problem for Zinbiel 2-algebras can be classified by some non-abelian cohomology theory.
In this work the description up to isomorphism of complex naturally graded quasi-filiform Zinbiel algebras is obtained.
On Pre-Novikov Algebras and Derived Zinbiel Variety
本文研究代数簇及其二元 operad 的导出构造，给出导出簇代数可嵌入相应微分代数的充分条件，并证明 Zinbiel 导出簇即 pre-Novikov 代数类中存在不可嵌入带导子 Zinbiel 代数的例子。
The Rota-Baxter algebra and the shuffle product are both algebraic structures arising from integral operators and integral equations. Free commutative Rota-Baxter algebras provide an algebraic framework for integral equations with the simple Riemann integral operator. The Zinbiel algebras form a category in which the shuffle product algebra is the free object. Motivated by algebraic structures underlying integral equations involving multiple integral operators and kernels, we study commutative matching Rota-Baxter algebras and construct the free objects making use of the shuffle product with m…
We build, using the notion of zinbiel algebra, some commutative subalgebras $C_{u,v}$ inside an algebra of formal iterated integrals. There is a quotient map from this algebra of formal iterated integrals to the algebra of motivic multiple zeta values. Restricting this quotient map to the subalgebras $C_{u,v}$ of A$_{1,0}$ gives a morphism of graded commutative algebras with the same graded dimension. This is conjectured to be generically an isomorphism. When u+v = 0, the image is instead a sub-algebra of the algebra of motivic multiple zeta values.
Some constructions of antisymmetric infinitesimal bialgebras and their applications to Lie bialgebras
本文提出利用Zinbiel双代数和双结合双代数构造反对称无穷小双代数及Lie双代数的方法，并证明其保持拟三角、三角和可分解性质。
Affinization of Zinbiel bialgebras and pre-Poisson bialgebras, infinite-dimensional Poisson bialgebras
本文通过pre-Poisson代数的affinization构造无限维Poisson bialgebras，并研究了相关代数结构的对应关系。
On the pre-commutative envelopes of commutative algebras
本文证明了幂零交换代数可嵌入预交换（Zinbiel）代数，并指出在有限维情形下幂零性是存在预交换包络的必要条件。
A criterion for elements of free Zinbiel algebras to be Lie or Jordan is established. This criterion is used in studying speciality problems of Tortkara algebras. We construct a base of free special Tortkara algebras. Furthermore, we prove analogues of classical Cohn's and Shirshov's theorems for Tortkara algebras.
On tortkara triple systems
本文研究自由Zinbiel代数中的tortkara三元积，通过计算机代数构造五元运算下的关系TT(a,b,c,d,e)，证明对应三元系由二次三元operad定义，还发现七元运算下的新关系，提出更高元数是否存在新恒等式的开放问题。
Malcev classification for the variety of left-symmetric algebras
本文研究了左对称代数变种中的三个子变种，揭示其与交替、asso对称和Zinbiel代数的关系，并推导了相关运算在度数至4时的恒等式。
For a commutative algebra which comes from a Zinbiel algebra the exponential series can be written without denominators. When lifted to dendriform algebras this new series satisfies a functional equation analogous to the Baker-Campbell-Hausdorff formula. We make it explicit by showing that the obstruction series is the sum of the brace products. In the multilinear case we show that the role the Eulerian idempotent is played by the iterated pre-Lie product.
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。
We show that for fields that are of characteristic 0 or algebraically closed of characteristic greater than 5, that certain classes of Leibniz algebras are 2-recognizeable. These classes are solvable, strongly solvable and super solvable. These results hold in Lie algebras and in general for groups.
Isomorphism Classes of Idempotent Evolution Algebras
本文证明了幂等演化代数同构类与对称群和环面半直积群轨道的双射关系，推导出一般计数公式，并给出了任意有限域上2至4维同构类数量的显式公式。
Since the introduction of the concept of isotopism of algebras by Albert in 1942, a prolific literature on the subject has been developed for distinct types of algebras. Nevertheless, there barely exists any result on the problem of distributing Lie algebras into isotopism classes. The current paper is a first step to deal with such a problem. Specifically, we define a new series of isotopism invariants and we determine explicitly the distribution into isotopism classes of $n$-dimensional filiform Lie algebras, for $n\leq 7$. We also deal with the distribution of such algebras into isomorphism…
We investigate criteria for von-Neumann finiteness and reversibility in some classes of non-associative algebras. We show that all finite-dimensional alternative algebras, as well as all algebras obtained from the real numbers via the standard Cayley-Dickson doubling process, are von-Neumann finite. Precise criteria for von-Neumann finiteness and reversibility of involutive algebras are given, in terms of isomorphism types of their 3-dimensional subalgebras.
The paper is an implementation in low dimensional cases of the classification method presented before by Rakhimov and Bekbaev. We give a complete classification of a subclass of complex filiform Leibniz algebras obtained from the naturally graded non-Lie filiform Leibniz algebras. A hypothetic formula for the adapted number of isomorphism classes is given.
We extend results related to maximal subalgebras and ideals from Lie to Leibniz algebras. In particular, we classify minimal non-elementary Leibniz algebras and Leibniz algebras with a unique maximal ideal. In both cases, there are types of these algebras with no Lie algebra analogue. We also give a classification of E-Leibniz algebras which is very similiar to its Lie algebra counterpart.
The calculus of classes and closure operations has proved to be a useful tool in group theory and has led to a deep theory in the study of finite soluble groups. More recently, parallel theories have started to be developed in various varieties of algebras, such as Lie, Leibniz, and Malcev algebras. This paper seeks to investigate the extent to which these later theories can be generalised to the variety of all non-associative algebras.
In this paper we classify filiform associative algebras of degree $k$ over a field of characteristic zero. Moreover, we also classify naturally graded complex filiform and quasi-filiform nilpotent associative algebras which are described by the characteristic sequence $C(\mathcal{A})=(n-2,1,1)$ or $C(\mathcal{A})=(n-2,2)$.
Schunck classes, saturated formations and projectors were originally defined for finite soluble groups for which they provided families of intravariant subgroups. In this paper, I set out the analogous theory for finite-dimensional soluble restricted Lie algebras and investigate its links to the corresponding theory for ordinary Lie algebras over the same field.
I set out the theory of Schunck classes and projectors for soluble Leibniz algebras, parallel to that for Lie algebras. Primitive Leibniz algebras come in pairs, one (Lie) symmetric, the other antisymmetric. A Schunck formation containing one member of a pair also contains the other. Projectors for a Schunck formation are intravariant.
We classify all finite-dimensional connected Hopf algebras with large abelian primitive spaces. We show that they are Hopf algebra extensions of restricted enveloping algebras of certain restricted Lie algebras. For any abelian matched pair associated with these extensions, we construct a cohomology group, which classifies all the extensions up to equivalence. Moreover, we present a $1$-$1$ correspondence between the isomorphism classes and a group quotient of the cohomology group deleting some exceptional points, where the group respects the automorphisms of the abelian matched pair and the e…
In this paper we focus on algebraic aspects of contractions of Lie and Leibniz algebras. The rigidity of algebras plays an important role in the study of their varieties. The rigid algebras generate the irreducible components of this variety. We deal with Leibniz algebras which are generalizations of Lie algebras. In Lie algebras case, there are different kind of rigidities (rigidity, absolutely rigidity, geometric rigidity and e.c.t.). We explore the relations of these rigidities with Leibniz algebra rigidity. Necessary conditions for a Lie algebra to be Leibniz rigid are discussed.
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…
Let A be a finite dimensional algebra over an algebraically closed field with the radical nilpotent of index 2. It is shown that A has finitely many conjugacy classes of left ideals if and only if A is of finite representation type provided that all simple A-modules have dimension at least 6. This is a revised version.
In this paper we classify the isomorphism classes of four dimensional nilpotent associative algebras over a field F, studying regular subgroups of the affine group AGL_4(F). In particular we provide explicit representatives for such classes when F is a finite field, the real field R or an algebraically closed field.
Local isomorphism classes of fractional ideals of orders in étale algebras
本文研究 étale 代数中序的分式理想局部同构类，借助剩余域上的线性代数对象完成分类，并提出计算类代表元的高效递归算法。
The paper concerns the classification problem of a subclass of complex filiform Leibniz algebras in dimensions 7 and 8. This subclass arises from naturally graded filiform Lie algebras. We give a complete list of isomorphism classes of algebras including Lie case. In parametric families cases, the corresponding orbit functions (invariants) are given. In discrete orbits case, we show representatives of the orbits.
We introduce characteristic classes for the spectral sequence associated to a split short exact sequence of Hopf algebras. We show that these characteristic classes can be seen as obstructions for the vanishing of differentials in the spectral sequence and prove a decomposition theorem. We also interpret our results in the settings of group and Lie algebra extensions and prove some interesting corollaries concerning the collapse of the (Lyndon-)Hochschild-Serre spectral sequence and the order of characteristic classes.
Ideal classes of orders in quaternion algebras
提出一种算法，计算全局域上四元数代数中任意序的所有右理想类代表元，包括不可逆情形，并推广至更一般代数。
In this paper we classify graded reflexive ideals, up to isomorphism and shift, in certain three dimensional Artin-Schelter regular algebras. This classification is similar to the classification of right ideals in the first Weyl algebra, a problem that was completely settled recently. The situation we consider is substantially more complicated however.
This paper is primarily concerned with studying finite-dimensional anti-commutative nonassociative algebras in which every centralizer is an ideal. These are shown to be anti-associative and are classified over a general field $F$; in particular, they are nilpotent of class at most $3$ and metabelian. These results are then applied to show that a Leibniz algebra over a field of charactersitic zero in which all centralizers are ideals is solvable.
A homotope, or a mutation, of a $k$-algebra is a new algebra with the same underlying space, but with the multiplication law dependent on the multiplication law of the original algebra. In this paper, we show that a generic finite-dimensional algebra of dimension greater than 3 has infinitely many non-isotopic homotopes, and that, more generally, a similar result is true for generic trilinear forms. We also study a particular class of homotopes called $Δ$-homotopes, where $Δ$ is an element of the algebra, and show that there are algebras with infinitely many non-isomorphic homotopes even under…
Let $J$ be a finite-dimensional semi-simple Jordan algebra over an algebraically closed field of characteristic $0$. In this article, the diagonal action of the automorphism group of $J$ on the $n$-fold product $J\times\ldots \times J$ is studied. In particular, it is shown that the orbit through an $n$-tuple $x=(x_1,,\ldots,x_n)$ is closed if and only if the Jordan subalgebra generated by the elements $x_1,\ldots, x_n$ is semi-simple.
Coherent Conditions: Algebraic Geometry for Arbitrary Classes of Algebras
该研究将通用代数几何推广到任意代数类的谱理论，建立无量词命题与Zariski闭集的对应，并探讨方程Noether类与不可约谱。
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