Vladimir Dergachev
We study pairs of associative algebras and linear functionals. New results together with corrected proofs for previously published material are presented. In particular, we prove the identity ind Mat(n) (#) A=n * ind A$ for finite-dimensional unital associative algebra A with index 1. [ Symbol (#) denotes tensor product ]
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It is well-known that the Jacobson radical of a unital ring is its largest superfluous right ideal. It is recorded here that the result carries over to non-unital rings provided the notion of "superfluous" is taken relative to all regular ideals.
Associative/Lie radical (co)invariance under tracial (co)actions
该研究针对霍普夫代数上的有限维结合或李（共）模代数，在特征合适且（共）作用保迹的条件下，证明了多种根基的（共）不变性结果，推广了已有准则并提供统一框架。
We prove the existence of the Hopf PI-exponent for finite dimensional associative algebras $A$ with a generalized Hopf action of an associative algebra $H$ with $1$ over an algebraically closed field of characteristic $0$ assuming only the invariance of the Jacobson radical $J(A)$ under the $H$-action and the existence of the decomposition of $A/J(A)$ into the sum of $H$-simple algebras. As a consequence, we show that the analog of Amitsur's conjecture holds for $G$-codimensions of finite dimensional associative algebras over a field of characteristic $0$ with an action of an arbitrary group $…
We prove that if A is a finite dimensional associative H-comodule algebra over a field F for some involutory Hopf algebra H not necessarily finite dimensional, where either char F = 0 or char F > dim A, then the Jacobson radical J(A) is an H-subcomodule of A. In particular, if A is a finite dimensional associative algebra over such a field F, graded by any group, then the Jacobson radical J(A) is a graded ideal of A. Analogous results hold for nilpotent and solvable radicals of finite dimensional Lie algebras over a field of characteristic 0. We use the results obtained to prove the analog of …
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.
Associative-Yamaguti algebras
本文引入结合-Yamaguti代数，研究其基本性质、上同调与形变，推广得到非对称operad上的Yamaguti乘法与树状-Yamaguti代数，并通过相对Rota-Baxter算子建立二者的关联。
In 1992, following earlier conjectures of Lichtman and Makar-Limanov, Klein conjectured that a noncommutative domain must contain a free, multiplicative, noncyclic subsemigroup. He verified the conjecture when the center is uncountable. In this note we consider the existence (or not) of free subsemigroups in associative $k$-algebras $R$, where $k$ is a field not algebraic over a finite subfield. We show that $R$ contains a free noncyclic subsemigroup in the following cases: (1) $R$ satisfies a polynomial identity and is noncommutative modulo its prime radical. (2) $R$ has at least one nonartin…
For an arbitrary countable field, we construct an associative algebra that is graded, generated by finitely many degree-1 elements, is Jacobson radical, is not nil, is prime, is not PI, and has Gelfand-Kirillov dimension two. This refutes a conjecture attributed to Goodearl.
Bialgebra theory for nearly associative algebras and $LR$-algebras: equivalence, characterization, and $LR$-Yang-Baxter Equation
该研究构建近结合代数与$LR$-代数的双代数理论，证明二者双代数等价，引入近结合$L$-代数并刻画相关双代数，还提出余边界近结合双代数作为研究YBE的框架。
Deformations and extensions of homotopy associative algebras
本文发展了结合2-代数的表示与上同调理论，并详细研究了其形变与阿贝尔扩张。
These are the lecture notes from my short course of the same title at the CIMPA Research School on Associative and Nonassociative Algebras and Dialgebras: Theory and Algorithms - In Honour of Jean-Louis Loday (1946-2012), held at CIMAT, Guanajuato, Mexico, February 17 to March 2, 2013. The underlying motivation is to apply the theory of noncommutative Grobner bases in free associative algebras to the construction of universal associative envelopes for nonassociative structures defined by multilinear operations. Trilinear operations were classified by the author and Peresi in 2007. In her Ph.D.…
In this paper, we study local systems of locally finite associative algebras over fields of characteristic p\ge0. We describe the perfect local systems and study the relation between them and their corresponding locally finite associative algebras. 1-perfect and conical local systems are also be considered and described briefly.
In this article we describe indecomposable objects of the derived categories of a branch class of associative algebras. To this class belong such known classes of algebras as gentle algebras, skew-gentle algebras and certain degenerations of tubular algebras.
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。
Let A be a finite dimensional symmetric cllular algebras. We construct a nilpotent ideal in A. The ideal connects the radicals of cell modules with the radical of the algebra. It also reveals some information on the dimensions of simple modules of A.
On radicals of Novikov algebras
本文研究Novikov代数的根理论，证明素非结合Novikov代数的非零理想均非结合，并证明有限维情形下Baer根与左拟正则根重合，同时证明右拟正则根不存在。
Radicals in flip subalgebras
本文研究 Matsuo 代数中 flip 子代数的结构性质，提出判定其单性及根维数的方法。
Algebraic derivations and radicals of automorphism groups of algebras
本文研究半素结合代数的代数导子与相关自同构群的根性质，证明其素根与弱可解根重合，并联系Martindale商环内导子描述。
Motivated by the Mathieu conjecture [Ma], the image conjecture [Z3] and the well-known Jacobian conjecture [K] (see also [BCW] and [E]), the notion of Mathieu subspaces as a natural generalization of the notion of ideals has been introduced recently in [Z4] for associative algebras. In this paper, we first study algebraic elements in the radicals of Mathieu subspaces of associative algebras over fields and prove some properties and characterizations of Mathieu subspaces with algebraic radicals. We then give some characterizations or classifications for strongly simple algebras (the algebras wi…
In [1], finite associative rings wih identity and such that the set of all zero-divisors form and ideal M, called the Jacobson Radical, of cube zero and square non-zero, were constructed for all the characteristics. These rings are completely primary and we call them rings with property(T). In this paper, we associate with each ring with property(T) and characteristic p, invariants (integers) and determine (in certain cases) the number of isomorphism classes of these rings with given invariants.
Near-Trinions: Complete Classification of Unital Three-Dimensional Real Associative Algebras
本文完全分类了三维实结合代数（称为近三元数），得到六个同构类，通过根维数分层和皮尔斯分解给出规范表示与不变量。
In this paper, we define and study (co)homology theories of a compatible associative algebra $A$. At first, we construct a new graded Lie algebra whose Maurer-Cartan elements are given by compatible associative structures. Then we define the cohomology of a compatible associative algebra $A$ and as applications, we study extensions, deformations and extensibility of finite order deformations of $A$. We end this paper by considering compatible presimplicial vector spaces and the homology of compatible associative algebras.
We give the algebraic classification of alternative, left alternative, Jordan, bicommutative, left commutative, assosymmetric, Novikov and left symmetric central extensions of null-filiform associative algebras.
In this paper we characterize the maximal modular ideals of an evolution algebra $A\,\ $in order to describe its Jacobson radical, \ $Rad(A).$ We characterize semisimple evolution algebras (i.e. those such that $% Rad(A)=\{0\}$)as well as radical ones. We introduce two elemental notions of spectrum of an element $a$ in an evolution algebra $A$, namely the spectrum $% σ^{A}(a)$ and the m-spectrum $σ_{m}^{A}(a)$ (they coincide for associative algebras, but in general $σ^{A}(a)\subseteq σ_{m}^{A}(a),$ and we show examples where the inclusion is strict). We prove that they are non-empty and descri…
We study Cartan-Subalgebras of Lie-Algebras associated to associative algebras.
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。
We define a cup product on the Hochschild cohomology of an associative conformal algebra $A$, and show the cup product is graded commutative. We define a graded Lie bracket with the degree $-1$ on the Hochschild cohomology $\HH^{\ast}(A)$ of an associative conformal algebra $A$, and show that the Lie bracket together with the cup product is a Gerstenhaber algebra on the Hochschild cohomology of an associative conformal algebra. Moreover, we consider the Hochschild cohomology of split extension conformal algebra $A\hat{\oplus}M$ of $A$ with a conformal bimodule $M$, and show that there exist an…
We introduce the notion of weakly associative algebra and its relations with the notion of nonassociative Poisson algebras.
For algebras with radical square zero, we give a combinatorial description to the Tate-Hochschild cohomology. We compute the Gerstenhaber algebra structure on the Tate-Hochschild cohomology for some classes of such algebras.
In this paper, we consider compatible Hom-associative algebras as a twisted version of compatible associative algebras. Compatible Hom-associative algebras are characterized as Maurer-Cartan elements in a suitable bidifferential graded Lie algebra. We also define a cohomology theory for compatible Hom-associative algebras generalizing the classical case. As applications of cohomology, we study abelian extensions and deformations of compatible Hom-associative algebras.
We classify, up to isomorphism, the 2-dimensional algebras over a field K. We focuse also on the case of characteristic 2, identifying the matrices of GL(2,F_2) with the elements of the symmetric group S_3. The classification is then given by the study of the orbits of this group on a 3-dimensional plane, viewed as a Fano plane. As applications, we establish classifications of Jordan algebras, algebras of Lie type or Hom-Associative algebras.
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 $K$.
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。
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…
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.
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.
We provide an explicit construction for a class of commutative, non-associative algebras for each of the simple Chevalley groups of simply laced type. Moreover, we equip these algebras with an associating bilinear form, which turns them into Frobenius algebras. This class includes a 3876-dimensional algebra on which the Chevalley group of type E8 acts by automorphisms. We also prove that these algebras admit the structure of (axial) decomposition algebras.
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.
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 or Jordan algebras generated by two idempotents are described precisely.
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。
(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。
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.
We prove that the Lie commutator subalgebra of the associative algebra containing a matrix subalgebra is perfect.
In this paper we explore a new method of analysis of associative algebras.
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。
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…
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.
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…
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。
Basic definitions and properties of nearly associative algebras are described. Nearly associative algebras are proved to be Lie-admissible algebras. Two-dimensional nearly associative algebras are classified, and its main classes are derived. The bimodules, matched pairs and Manin triple of a nearly associative algebras are derived and their equivalence with nearly associative bialgebras is proved. Basic definitions and properties of nearly Hom-associative algebras are described. Related bimodules and matched pairs are given, and associated identities are established.
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.
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.
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。
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.
$σ$-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。
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…
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 …
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。
We introduce an equivariant version of Hochschild cohomology as the deformation cohomology to study equivariant deformations of associative algebras equipped with finite group actions.
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