Ling Zhang, Tao Zhang
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.
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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…
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 $4.$
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.
In this work the description up to isomorphism of complex naturally graded quasi-filiform Zinbiel algebras is obtained.
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。
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…
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。
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.
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。
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 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。
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.
Twisted Lie algebras by invertible derivations
本文提出InvDer algebra代数结构，通过可逆导子扭曲代数，定义了多种相关代数，并利用Rota-Baxter算子和自同态算子研究其结构间的关系。
Bialgebraic approach to rack cohomology
本文通过微分分次辫代数解释辫上同调复形，给出其结构结果的初等代数证明，包括两个控制结构缺陷的显式同伦，并证明杯积与Zinbiel积在拟辫上同调上的限制与下降，最后用相容余积完备化杯积。
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.
$δ$-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。
We find a natural compatible condition between the Rota-Baxter operator and Turaev's (Hopf) group-(co)algebras, which leads to the concept of Rota-Baxter Turaev's (Hopf) group-(co)algebra. Two characterizations of Rota-Baxter Turaev's group-algebras (abbr. T-algebras) are obtained: one by Atkinson factorization and the other by T-quasi-idempotent elements. The relations among some related Turaev's group algebraic structures (such as (tri)dendriform T-algebras, Zinbiel T-algebras, pre-Lie T-algebras, Lie T-algebras) are discussed, and some concrete examples from the algebras of dimensions 2,3 a…
Differential perm bialgebras, differential perm Yang-Baxter equation, and their applications in related bialgebra structures
本文引入微分 perm 双代数，通过 Manin 三元组和匹配对刻画，并利用对称解和 O-算子建立三角结构，进而构造微分无穷小双代数和 Novikov 双代数，揭示其与相关双代数结构的联系。
Maurer-Cartan characterization and cohomology of compatible LieDer and AssDer pairs
本文研究相容AssDer对、相容LieDer对及其上同调，还探讨带导子的相容dendriform代数等其他相容结构，借助Rota-Baxter算子等工具刻画了这些相容结构间的关联。
Nijenhuis deformations of Poisson algebras and $F$-manifold algebras
本文引入NS-泊松代数与NS-F-流形代数，统一推广泊松代数、预泊松代数、F-流形代数及预F-流形代数，并证明Nijenhuis算子、扭曲Rota-Baxter算子及半经典极限可自然产生这些新代数结构。
Skew category algebras, twisted tensor product algebras and induction of precosheaves
本文证明斜范畴代数可嵌入扭曲张量积代数，并将Puig和Turull的概念从群代数推广到范畴代数，研究其在斜范畴代数下的性质。
In this article we use semigroupoids to describe a notion of algebraic bundles, mostly motivated by Fell ($C^*$-algebraic) bundles, and the sectional algebras associated to them. As the main motivational example, Steinberg algebras may be regarded as the sectional algebras of trivial (direct product) bundles. Several theorems which relate geometric and algebraic constructions - via the construction of a sectional algebra - are widely generalized: Direct products bundles by semigroupoids correspond to tensor products of algebras; Semidirect products of bundles correspond to "naïve" crossed prod…
A structure theorem for streamed information
本文将自由半洗牌代数与路径泛函代数等同，发现元数为3的新恒等式以恢复Zinbiel和Tortkara恒等式，并利用Hall积分与自由李代数的对偶性证明了流信息的结构定理，即任何元素均可唯一表示为Hall积分上的多项式。
Wreath products of cocommutative Hopf algebras
本文定义余交换Hopf代数的圈积，证明其具有类似Kaloujnine-Krasner定理的泛性质，并用于分类群扩张和李代数扩张。
In this paper we study twisted algebras of multiplier Hopf ($^*$-)algebras which generalize all kinds of smash products such as generalized smash products, twisted smash products, diagonal crossed products, L-R-smash products, two-sided crossed products and two-sided smash products for the ordinary Hopf algebras appeared in [P-O].
We reformulate the definition of a zero product determined algebra in terms of tensor products and obtain necessary and sufficient conditions for an algebra to be zero product determined. These conditions allow us to prove that the direct sum \bigoplus_{i \in I} A_i of algebras for any index set I is zero product determined if and only if each of the component algebras A_i is zero product determined. As an application, every parabolic subalgebra of a finite-dimensional reductive Lie algebra, over an algebraically-closed field of characteristic zero, is zero product determined. In particular, e…
We prove that the Yoneda Ext-algebra of a Takeuchi smash product is the graded Takeuchi smash product of the Yoneda Ext-algebras of the two algebras or modules involved. As an application, we prove that graded Takeuchi smash products preserve Artin-Schelter regularity, and describe the Nakayama automorphism of the product.
We prove that assosymmetric algebras under Jordan product are Lie triple. A Lie triple algebra is called special if it is isomorphic to a subalgebra of some plus-assosymmetric algebra. We establish that Glennie identitiy is valid for special Lie triple algebras, but not for all Lie triple algebras.
In this paper we generalize the well-known construction of shuffle product algebras by using mixable shuffles, and prove that any free Baxter algebra is isomorphic to a mixable shuffle product algebra. This gives an explicit construction of the free Baxter algebra, extending the work of Rota and Cartier.
Let P be a quadratic operad. We determine an associated operad ~P such that for any P-algebra A and any ~P-algebra B then the tensor product $A \otimes B$ is a P-algebra.
The extending structures and unified products for Malcev algebras are developed. Some special cases of unified products such as crossed products and matched pair of Malcev 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 Malcev algebras are also investigated.
We discuss the Kummer subspaces of tensor products of cyclic algebras, focusing mainly on the case of cyclic algebras of degree 3. We present a family of maximal spaces in the general case, classify all the monomial spaces in the case of tensor products of cyclic algebras of degree 3 using graph theory, and provide an upper bound for the dimension in the generic tensor product of cyclic algebras of degree 3.
A five term sequence for the low degree cohomology of a smash product of (cocommutative) Hopf algebras is obtained, generalizing that of Tahara for a semi-direct product of groups.
We study BiHom-Novikov-Poisson algebras, which are twisted generalizations of Novikov-Poisson algebras and Hom-Novikov-Poisson algebras, and find that BiHom-Novikov-Poisson algebras are closed under tensor products and several kinds of perturbations. Necessary and sufficient conditions are given under which BiHom-Novikov-Poisson algebras give rise to BiHom-Poisson algebras.
The question of whether a split tensor product of quaternion algebras with involution over a field of characteristic two can be expressed as a tensor product of split quaternion algebras with involution, is shown to have an affirmative answer.
Recently G. Bhattacharyya, S.Y. Song and R. Tanaka began to study Terwilliger algebras of wreath products of one-class association schemes. K. Kim determined the structure of Terwilliger algebras of wreath products by one-class association schemes or quasi-thin schemes. In this paper, we study Terwilliger algebras of wreath products by $3$-equivalenced schemes.
The Cyclicity of Tensor Products of Cyclic $p$-Algebras
本文重新考察 Albert 关于循环 $p$-代数张量积循环性的定理，并针对素数次情形，以符号代数形式显式计算了张量积所得的循环代数。
The class of matrix algebras on a ladder $\mathcal{L}$ generalizes the class of block upper triangular matrix algebras. It was previously shown that the matrix algebra on a ladder $\mathcal{L}$ is zero product determined under matrix multiplication. In this article, we show that the matrix algebra on a one-step ladder is zero product determined under the Lie bracket.
We characterize prelie algebras in words of left ideals of the enveloping algebras and in words of modules, and use this result to prove that a simple complex finite-dimensional Lie algebra is not prelie, with the possible exception of f4.
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