On a new approach to classification of associative algebras

Composition-Diamond Lemma for Non-associative Algebras over a Commutative Algebra

We establish the Composition-Diamond lemma for non-associative algebras over a free commutative algebra. As an application, we prove that every countably generated non-associative algebra over an arbitrary commutative algebra $K$ can be embedded into a two-generated non-associative algebra over $K$.

对二结合代数和三结合代数的操作进行分解

Splitting of operations for di-associative algebras and tri-associative algebras

本文研究了二结合代数和三结合代数的操作分解，引入了四 dendriform 代数和六 dendriform 代数的概念，并探讨了它们与相对平均算子的关系。

任意基域上二维结合代数与双结合代数的完全分类

Complete classification of two-dimensional associative and diassociative algebras over any basic field

该研究完成了任意基域上二维结合、双结合代数的同构意义下完全分类，推进了代数结构分类领域的工作。

Generalized associative algebras

We study diverse parametrized versions of the operad of associative algebra, where the parameter are taken in an associative semigroup $Ω$ (generalization of matching or family associative algebras) or in its cartesian square (two-parameters associative algebras). We give a description of the free algebras on these operads, study their formal series and prove that they are Koszul when the set of parameters is finite. We also study operadic morphisms between the operad of classical associative algebras and these objects, and links with other types of algebras (diassociative, dendriform, post-Li…

Structure and classification of Hom-associative algebras

The purpose of this paper is to study the structure and the algebraic varieties of Hom-associative algebras. We give characterize multiplicative simple Hom-associative algebras and show some examples deforming the $2\times 2$-matrix algebra to simple Hom-associative algebras. We provide a classification of $n$-dimensional Hom-associative algebras for $n\leq3$. Then study their derivations and compute small Hom-Type Hochschild cohomology groups. Furthermore, we discuss their irreducible components.

Hom-power associative algebras

A generalization of power associative algebra, called Hom-power associative algebra, is studied. The main result says that a multiplicative Hom-algebra is Hom-power associative if and only if it satisfies two identities of degrees three and four. It generalizes Albert's result that power associativity is equivalent to third and fourth power associativity. In particular, multiplicative right Hom-alternative algebras and non-commutative Hom-Jordan algebras are Hom-power associative.

Classification of nilpotent associative algebras of small dimension

We classify nilpotent associative algebras of dimensions up to 4 over any field. This is done by constructing the nilpotent associative algebras as central extensions of algebras of smaller dimension, analogous to methods known for nilpotent Lie algebras.

Power-associative evolution algebras

The paper is devoted to the study of evolution algebras that are power-associative algebras. We give the Wedderburn decomposition of evolution algebras that are power-associative algebras and we prove that these algebras are Jordan algebras. Finally, we use this decomposition to classify these algebras up to dimension six.

Associative and Jordan algebras generated by two idempotents

Associative or Jordan algebras generated by two idempotents are described precisely.

$A_3$-结合代数的Manin三元组、双代数和Yang-Baxter方程

Manin triples, bialgebras and Yang-Baxter equation of $A_3$-associative algebras

本文发展了$A_3$-结合代数的双代数理论，通过引入Manin三元组和双代数并证明其等价性，定义了$A_3$-结合Yang-Baxter方程，并利用相对Rota-Baxter算子构造了该方程的斜对称解。

零幂零结合代数上的（转置）Poisson代数结构

(Transposed) Poisson algebra structures on null-filiform associative algebras

本文对结合零幂零代数上的所有Poisson代数及转置Poisson代数结构进行了完整分类。

Gerstenhaber algebra structure on the cohomology of a hom-associative algebra

A hom-associative algebra is an algebra whose associativity is twisted by an algebra homomorphism. In this paper, we define a cup product on the cohomology of a hom-associative algebra. We show that the cup product together with the degree $-1$ graded Lie bracket (which controls the deformation of the hom-associative algebra structure) on the cohomology forms a Gerstenhaber algebra. This generalizes a classical fact that the Hochschild cohomology of an associative algebra carries a Gerstenhaber algebra structure.

A classification of barycentrically associative polynomial functions

We describe the class of polynomial functions which are barycentrically associative over an infinite commutative integral domain.

Lie structure of associative algebras containing matrix subalgebras

We prove that the Lie commutator subalgebra of the associative algebra containing a matrix subalgebra is perfect.

反结合型树形代数

Anti-associative dendriform algebras

本文研究反结合型树形代数，通过定义两个运算之和为反结合性，提出O-算子作为解释工具，并探讨非退化Connes余循环的反结合代数的兼容结构。

Gröbner-Shirshov bases and embeddings of algebras

In this paper, by using Gröbner-Shirshov bases, we show that in the following classes, each (resp. countably generated) algebra can be embedded into a simple (resp. two-generated) algebra: associative differential algebras, associative $Ω$-algebras, associative $λ$-differential algebras. We show that in the following classes, each countably generated algebra over a countable field $k$ can be embedded into a simple two-generated algebra: associative algebras, semigroups, Lie algebras, associative differential algebras, associative $Ω$-algebras, associative $λ$-differential algebras. Also we pro…

Derivations and Centroids of Four Dimensional Associative Algebras

In this paper, we focus on derivations and centroids of four dimensional associative algebras. Using an existing classification result of low dimensional associative algebras, we describe the derivations and centroids of four dimensional associative algebras. We also identify algebra(s) that belong to the characteristically nilpotent class among the algebras of four dimensional associative algebras.

Hom-associative algebras up to homotopy

A hom-associative algebra is an algebra whose associativity is twisted by an algebra homomorphism. In this paper, we introduce a strongly homotopy version of hom-associative algebras ($HA_\infty$-algebras in short) on a graded vector space. We describe $2$-term $HA_\infty$-algebras in details. In particular, we study 'skeletal' and 'strict' $2$-term $HA_\infty$-algebras. We also introduce hom-associative $2$-algebras as categorification of hom-associative algebras. The category of $2$-term $HA_\infty$-algebras and the category of hom-associative $2$-algebras are shown to be equivalent. An appr…

移位结合代数

Shift associative algebras

本文对满足$(xy)z=y(zx)$的移位结合代数展开全面研究，关联了相关代数类，构造了自由代数基，建立了类似Wedderburn-Artin的定理，并完成了低维复代数的分类。

Hochshild cohomology of the associative conformal algebra Cend_{1,x}

It is established in this work that second Hochshild cohomology group of the associative conformal algebra Cend_{1,x} is zero. As a corollary, this algebra split off in each extension with a nilpotent kernel. Key words: associative conformal algebra, splitting off radical, Hochshild cohomology.

4维复Hom-结合对代数的分类、$α^k$-导子与形心

Classification, $α^k$-Derivations and Centroids of $4$-dimensional Complex Hom-associative Dialgebras

该研究针对Hom-结合对代数的结构与代数簇问题，完成了n≤4维Hom-结合对代数的分类，以此刻画了其低维情形的α-导子与形心，还计算了低维结合对代数的形心。

Non-semigroup gradings of associative algebras

It is known that there are Lie algebras with non-semigroup gradings, i.e. such that the binary operation on the grading set is not associative. We provide a similar example in the class of associative algebras.

Lie algebras with associative structures. Applications to the study of 2-step nilpotent Lie algebras

We investigate Lie algebras whose Lie bracket is also an associative or cubic associative multiplication to characterize the class of nilpotent Lie algebras with a nilindex equal to 2 or 3. In particular we study the class of 2-step nilpotent Lie algebras, their deformations and we compute the cohomology which parametrize the deformations in this class.

高阶同源结合Weyl代数

The higher-order hom-associative Weyl algebras

本文研究高阶Weyl代数作为同源结合代数的变形特性，证明其自然出现于同源结合迭代微分多项式环中，并探讨其结构特征与分类。

(Co)associative $3$-ary (co)algebras and infinitesimal bialgebras: construction and main properties

The (co)associative, partially (co)associative and totally (co)associative $3$-ary (co) algebras and infinitesimal bialgebras are constructed and discussed. Their trimodules and matched pairs are defined and completely characterized. The main structural properties and relations are also deduced and analyzed.

某些结合代数上的 $σ$-匹配与可交换结构

$σ$-matching and interchangeable structures on certain associative algebras

本文刻画了含单位元代数、矩形带半群代数及自由非单位结合/交换代数等多类结合代数上的 $σ$-匹配、可互换积与完全相容积。

Anti-Frobenius Algebras and Associative Yang-Baxter Equation

Associative Yang-Baxter equation arises in different areas of algebra, e.g., when studying double quadratic Poisson brackets, non-abelian quadratic Poisson brackets, or associative algebras with cyclic 2-cocycle (anti-Frobenius algebras). Precisely, faithful representations of anti-Frobenius algebras (up to isomorphism) are in one-to-one correspondence with skew-symmetric solutions of associative Yang-Baxter equation (up to equivalence). Following the work of Odesskii, Rubtsov and Sokolov and using computer algebra system Sage, we found some constant skew-symmetric solutions of associative Yan…

Decompositions of algebras and post-associative algebra structures

We introduce post-associative algebra structures and study their relationship to post-Lie algebra structures, Rota--Baxter operators and decompositions of associative algebras and Lie algebras. We show several results on the existence of such structures. In particular we prove that there exists no post-Lie algebra structure on a pair $(\mathfrak{g},\mathfrak{n})$, where $\mathfrak{n}$ is a simple Lie algebra and $\mathfrak{g}$ is a reductive Lie algebra, which is not isomorphic to $\mathfrak{n}$. We also show that there is no post-associative algebra structure on a pair $(A,B)$ arising from a …

hom-结合Weyl代数中的理想

Ideals in hom-associative Weyl algebras

本研究提出高阶Weyl代数的hom-结合版本，推广了已有hom-结合Weyl代数的构造，并证明这类代数是单的且所有单侧理想均为主理想，丰富了hom-结合代数领域的研究成果。

Equivariant one-parameter deformations of associative algebras

We introduce an equivariant version of Hochschild cohomology as the deformation cohomology to study equivariant deformations of associative algebras equipped with finite group actions.

Simple associative conformal algebras of linear growth

We describe simple finitely generated associative conformal algebras of Gel'fand--Kirillov dimension one.

Composition-Diamond lemma for associative $n$-conformal algebras

In this paper, we study the concept of associative $n$-conformal algebra over a field of characteristic 0 and establish Composition-Diamond lemma for a free associative $n$-conformal algebra. As an application, we construct Gröbner-Shirshov bases for Lie $n$-conformal algebras presented by generators and defining relations.

Characterizing Nilpotent Associative Algebras by Their Multiplier

The paper concerns an analogue of the famous Schur multiplier in the context of associative algebras and a measure of how far its dimension is from being maximal. Applying a methodology from Lie theory, we characterize all finite-dimensional nilpotent associative algebras for which this measure is ten or less.

Noncommutative differential calculus on (co)homology of hom-associative algebras

A hom-associative algebra is an algebra whose associativity is twisted by an algebra homomorphism. It was previously shown by the author that the Hochschild cohomology of a hom-associative algebra $A$ carries a Gerstenhaber structure. In this short paper, we show that this Gerstenhaber structure together with certain operations on the Hochschild homology of $A$ makes a noncommutative differential calculus. As an application, we obtain a Batalin-Vilkovisky algebra structure on the Hochschild cohomology of a regular unital symmetric hom-associative algebra.

Central extensions of filiform associative algebras

In this paper we describe central extensions (up to isomorphism) of all complex null-filiform and filiform associative algebras.

On filtered multiplicative bases of some associative algebras

We deal with the existing problem of filtered multiplicative bases of finite-dimensional associative algebras. For an associative algebra A over a field, we investigate when the property of having a filtered multiplicative basis is hereditated by homomorphic images or by the associated graded algebra of $A$. These results are then applied to some classes of group algebras and restricted enveloping algebras.

非结合的Solomon下降代数

Non-associative Solomon's descent algebras

受相对自由Sabinin代数的泛包络代数具有自然结合内积的启发，从非结合视角重新审视Solomon下降代数、非交换对称函数与分次双代数的Patras下降代数之间的经典关系，并利用余交换连通Hopf operad上的代数语言，将这种联系推广到非结合情形。

The hom-associative Weyl algebras in prime characteristic

We introduce the first hom-associative Weyl algebras over a field of prime characteristic as a generalization of the first associative Weyl algebra in prime characteristic. First, we study properties of hom-associative algebras constructed from associative algebras by a general "twisting" procedure. Then, with the help of these results, we determine the commuter, center, nuclei, and set of derivations of the first hom-associative Weyl algebras. We also classify them up to isomorphism, and show, among other things, that all nonzero endomorphisms on them are injective, but not surjective. Last, …

Nijenhuis族Ω-结合代数的上同调理论

Cohomology theory of Nijenhuis family $Ω$-associative algebras

本文引入Nijenhuis族Ω-结合代数概念，建立其上同调理论，并证明该上同调控制形变，进而用第二上同调群刻画其阿贝尔扩张。

A short proof that the free associative algebra is Hopfian

A short proof is given of the fact that various classes of algebras including the free associative algebra are Hopfian, i.e., every epimorphism is an automorphism. This further simplifies the Dicks-Lewin solution of the Jacobian conjecture for the free associative algebra.

Three deformation functors for associative algebras

We show that three deformation functors (deformations of the product, flat deformations and deformations of the relations) assigned to an associative algebra are naturally isomorphic.

Associative Algebras with Small Derived Ideal

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…

Some classes of nilpotent 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 $C(\mathcal{A})=(n-2,2)$.

On unitality conditions for Hom-associative algebras

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…

其关联结合代数具有一维中心的四维反树状代数的分类

Classification of four-dimensional anti-dendriform algebras whose associated associative algebra has the center of dimension one

本文将反树状代数刻画为两运算之和结合的代数，并完整分类了关联于一维中心四维结合代数的四维复反树状代数。

结合代数的Mal'cev型子簇中的自由代数

Free Algebras in Mal'cev-Type Subvarieties of Associative Algebras

该研究聚焦结合代数Mal'cev分类下子簇的自由代数，为各子簇构建对应自由代数的基并刻画其对称多项式空间。

结合环上三角矩阵代数的李同构与若尔当同构

Lie and Jordan Isomorphisms of Algebras of Triangular Matrices over Associative Rings

研究结合环上三角矩阵代数的李同构与若尔当同构，描述其相关性质。

Rota-type operators on null-filiform associative algebras

We give the description of homogeneous Rota-Baxter operators, Reynolds operators, Nijenhuis operators, Average operators and differential operator of weight 1 of null-filiform associative algebras of arbitrary dimension.

The group of automorphisms of the category of free associative algebras

In this paper, the problem formulated in [8] is solved. We prove, that the group of automorphisms of the category of free associative algebras is generated by semi-inner and mirror automorphisms

BiHom-$\Omega$-结合代数及相关结构

BiHom-$Ω$-associative algebras and related structures

该研究提出四类BiHom-$\Omega$代数概念，借助权$\lambda$的Rota-Baxter族构造新的BiHom-$\Omega$结合与李代数，并将经典代数间关系推广到BiHom-$\Omega$情形，丰富了同态代数结构理论。

Classification of the direct limits of involution simple associative algebras and the corresponding dimension groups

A classification of (countable) direct limits of finite dimensional involution simple associative algebras over an algebraically closed field of arbitrary characteristic is obtained. This also classifies the corresponding dimension groups. The set of invariants consists of two supernatural numbers and two real parameters.

Finite generation of Lie derived powers of associative algebras

Let $A$ be an associative algebra over a field of characteristic $\neq 2$ that is generated by a finite collection of nilpotent elements. We prove that all Lie derived powers of $A$ are finitely generated Lie algebras.

Weakly associative and symmetric Leibniz algebras

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.

n-ary associative algebras, cohomology, free algebras and coalgebras

When $n$ is odd, a cohomology of type Hochschild for $n$-ary partially associative algebras has been defined in Gnedbaye's thesis. Unfortunately, the cohomology definition is not valid when $n$ is even. This fact is found again in the computations of the $n$-ary partially associative free algebra. In this work, we define in a first time two approachs of an Hochschild cohomology for $n$-ary partially associative algebras. First by reducing the space of cochains, secondly by using a graded version. Next we compute the free $n$-ary algebra, giving a basis of this algebra. At last we extend the no…