I. S. Rakhimov, I. M. Rikhsiboev, W. Basri
In this paper we present a complete classification (isomorphism classes with some isomorphism invariants) of complex associative algebras up to dimension five (including both cases: unitary and non-unitary). In some symbolic computations we used Maple software.
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On three-dimensional associative algebras
本文研究了任意基域上三维结合代数的分类问题，列举了具有标准代表元的同构类，并与复数域上的最新分类及任意基域上三维 nilpotent 情况进行比较。
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。
We compare two generalizations of Tate homology: stable homology and the J-completion of Tor, also known as complete homology. For finitely generated modules, we show that the two theories agree over Artin algebras and over commutative noetherian rings that are Gorenstein, or local and complete.
We describe the groups of automorphisms of two generated free braided associative algebras with involutive diagonal braidings over a field of characteristic $\neq 2$. Depending on the form of the diagonal involutive braiding, five different automorphism groups arise as automorphism groups of two generated free braided associative algebras. This list covers all three automorphism groups that arise in the case of quantum planes \cite{AC}.
Associative Rota--Baxter operators on the Sweedler algebra $H_4$
本文研究了Sweedler代数$H_4$上的Rota-Baxter算子，通过分类和核的分析，给出了所有算子的完整描述。
We consider associative algebras over a field. An algebra variety is said to be {\em Lie nilpotent} if it satisfies a polynomial identity of the kind $[x_1, x_2, ..., x_n] = 0$ where $[x_1,x_2] = x_1x_2 - x_2x_1$ and $[x_1, x_2, ..., x_n]$ is defined inductively by $[x_1, x_2, ..., x_n]=[[x_1, x_2, ..., x_{n-1}],x_n]$. By Zorn's Lemma every non-Lie nilpotent variety contains a minimal such variety, called {\em almost Lie nilpotent}, as a subvariety. A description of almost Lie nilpotent varieties for algebras over a field of characteristic 0 was made up by Yu.Mal'cev. We find a list of non-pri…
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.
We give a complete description of degenerations of complex $5$-dimensional nilpotent associative commutative algebras.
In this paper, we study moduli spaces of 2-dimensional complex associative algebras. We give a complete calculation of the cohomology of every element in the moduli space, as well as compute their versal deformations.
In this paper, we study the moduli space of $1|1$-dimensional complex associative algebras. We give a complete calculation of the cohomology of every element in the moduli space, as well as compute their versal deformations.
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…
Derivations, Cohomology and Deformation of BiHom-Associative Dialgebra
本文定义并分类了2维和3维BiHom-结合双代数的导子、广义导子与拟导子，将BiHom-结合代数的上同调推广到双代数情形，并利用上同调研究了其单参数形式形变。
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…
Countable separation property for associative algebras
本文研究结合代数的可数分离性质，证明了带平凡自同态和零化子单模的结合代数满足该性质，并推广到任意域上有限或可数生成的自由结合代数，突破了此前仅在诺特代数下研究的局限。
Varieties of associative algebras with an identity of third degree
本文研究了特征为零的域上满足第三度多项式恒等式的结合代数的变种结构。
Gradings on associative triple systems of the second kind
本文研究第二类结合三元系，通过建立单三元系与 Loos 带对合 $3$-分次结合代数的自同构群概形同构，实现可结构化代数分次的完全同构分类。
Let $\mathfrak{a},\mathfrak{b},\mathfrak{e}$ be algebras over a field $k$. Then $\mathfrak{e}$ is an extension of $\mathfrak{a}$ by $\mathfrak{b}$ if $\mathfrak{a}$ is an ideal of $\mathfrak{e}$ and $\mathfrak{b}$ is isomorphic to the quotient algebra $\mathfrak{e}/\mathfrak{a}$. In this paper, by using Gröbner-Shirshov bases theory for associative (resp. Lie) algebras, we give complete characterizations of associative (resp. Lie) algebra extensions of $\mathfrak{a}$ by $\mathfrak{b}$, where $\mathfrak{b}$ is presented by generators and relations.
Naturally graded nilpotent associative algebras of nilindex $n-3$
研究分类了特征序列为 $(n-3,2,1)$ 的 $n$ 维自然分次幂零结合代数，从而补全 $n>6$ 时幂零指数 $n-3$ 情形的分类。
In this paper, we study the moduli space of $1|2$-dimensional complex associative algebras, which is also the moduli space of codifferentials on the tensor coalgebra of a $2|1$-dimensional complex space. We construct the moduli space by considering extensions of lower dimensional algebras. We also construct miniversal deformations of these algebras. This gives a complete description of how the moduli space is glued together via jump deformations.
In this paper, we study the moduli space of $2|1$-dimensional complex associative algebras, which is also the moduli space of codifferentials on the tensor coalgebra of a $1|2$-dimensional complex space. We construct the moduli space by considering extensions of lower dimensional algebras. We also construct miniversal deformations of these algebras. This gives a complete description of how the moduli space is glued together via jump deformations.
In this paper we examine how the notion of algebra of quotients for Lie algebras ties up with the corresponding well-known concept in the associative case. Specifically, we completely characterize when a Lie algebra $Q$ is an algebra of quotients of a Lie algebra $L$ in terms of the associative algebras generated by the adjoint operators of $L$ and $Q$ respectively. In a converse direction, we also provide with new examples of algebras of quotients of Lie algebras and these come from associative algebras of quotients. In the course of our analysis, we make use of the notions of density and mul…
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。
"Fusion rules" are laws of multiplication among eigenspaces of an idempotent. This terminology is relatively new and is closely related to axial algebras, introduced recently by Hall, Rehren and Shpectorov. Axial algebras, in turn, are closely related to $3$-transposition groups and Vertex operator algebras. In this paper we consider fusion rules for semisimple idempotents, following Albert in the power-associative case. We examine the notion of an axis in the non-commutative setting and show that the dimension $d$ of any algebra $A$ generated by a pair $a,b$ of (not necessarily Jordan) axes o…
Associative Pentagon Algebras
本文研究$(S,\ast)$也为半群的结合五角代数，引入两类由半群$(S,\ast)$性质强决定的结合五角代数，用半群方程给出其完整刻画，还提供构造方法与若干类例子。
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).
Lie conformal algebras appear in the theory of vertex algebras. Their relation is similar to that of Lie algebras and their universal enveloping algebras. Associative conformal algebras play a role in conformal representation theory. We introduce the notions of conformal identity and unital associative conformal algebras and classify finitely generated simple unital associative conformal algebras of linear growth. These are precisely the complete algebras of conformal endomorphisms of finite modules.
We provide a complete classification of three-dimensional associative algebras over the real and complex number fields based on a complete elementary proof. We list up all the multiplication tables of the algebras up to isomorphism. We compare our results with those given by mathematicians in the 19th century.
Degenerations of 3-dimensional nilpotent associative algebras over an algebraically closed field
本研究确定了特征非2的代数闭域上三维幂零结合代数簇的完整退化图景，开发了适用于任意代数闭域的替代论证技术，发现结果随域特征是否为2呈现二分性。
We describe a method for classifying the Novikov algebras with a given associated Lie algebra. Subsequently we give the classification of the Novikov algebras of dimension 3 over R and C, as well as the classification of the 4-dimensional Novikov algebras over C whose associated Lie algebra is nilpotent. In particular this includes a list of all 4-dimensional commutative associative algebras over C.
Transposed $δ$-Poisson algebra structures on null-filiform associative algebras
本文研究了零丝结合代数上的转置 $δ$-Poisson 代数结构，给出了由多项式 $δ^3 - 3δ^2 + 2δ$ 的根刻画的完整分类，并构造了所有 $δ$-Poisson 代数结构，证明其为平凡结构。
Let $\boldsymbolΛ_3(\mathbb C)\,(=\mathbb C^{27})$ be the space of structure vectors of $3$-dimensional algebras over $\mathbb C$ considered as a $G$-module via the action of $G={\rm GL}(3,\mathbb C)$ on $\boldsymbolΛ_3(\mathbb C)$ `by change of basis'. We determine the complete degeneration picture inside the algebraic subset $\mathcal A^s_3$ of $\boldsymbolΛ_3(\mathbb C)$ consisting of associative algebra structures via the corresponding information on the algebraic subsets $\mathcal L_3$ and $\mathcal J_3$ of $\boldsymbolΛ_3(\mathbb C)$ of Lie and Jordan algebra structures respectively. Thi…
This is an English translation of the author's Ph.D. thesis, accumulating his results on a construction of Cohen-Macaulay modules over a polynomial ring that appeared in the study of Cauchy-Fueter equations. This construction is generalized from quaternions to arbitrary finite-dimensional associative algebras. We show that for maximally central algebras (as introduced by Azumaya) this construction produces Cohen-Macaulay modules and is an exact functor (tensoring with a bimodule, actually) and this class of algebras cannot be enlarged. For this class several invariants of the resulting modules…
The paper is devoted to classify nilpotent Jordan algebras of dimension up to five over an algebraically closed field of characteristic not 2. We obtained a list of 35 isolated non-isomorphic 5-dimensional nilpotent non-associative Jordan algebras and 6 families of non-isomorphic 5-dimensional nilpotent non-associative Jordan algebras depending either on one or two parameters over an algebraically closed field of characteristic not 2 or 3. In addition to these algebras we obtained two non-isomorphic 5-dimensional nilpotent non-associative Jordan algebras over an algebraically closed field of c…
Deformations of the structure constants for a class of associative noncommutative algebras generated by Deformation Driving Algebras (DDA's) are defined and studied. These deformations are governed by the Central System (CS). Such a CS is studied for the case of DDA being the algebra of shifts. Concrete examples of deformations for the three-dimensional algebra governed by discrete and mixed continuous-discrete Boussinesq (BSQ) and WDVV equations are presented. It is shown that the theory of the Darboux transformations, at least for the BSQ case, is completely incorporated into the proposed sc…
In this paper, by using the Composition-Diamond lemma for non-associative algebras invented by A. I. Shirshov in 1962, we give Gröbner-Shirshov bases for free Pre-Lie algebras and the universal enveloping non-associative algebra of an Akivis algebra, respectively. As applications, we show I.P. Shestakov's result that any Akivis algebra is linear and D. Segal's result that the set of all good words in $X^{**}$ forms a linear basis of the free Pre-Lie algebra $PLie(X)$ generated by the set $X$. For completeness, we give the details of the proof of Shirshov's Composition-Diamond lemma for non-ass…
A complete set of primitive orthogonal idempotents plays an important role in the representation theory of an associative algebra. In this paper, we construct a complete set of primitive orthogonal idempotents for any finite Brandt semigroup algebra.
Lie, associative and commutative quasi-isomorphism
本文证明特征零下交换 dg 代数的交换与结合拟同构等价，并给出同伦完备 dg 李代数的 Koszul 对偶结果，从而确定有理同伦型并推出幂零李代数由泛包络代数识别。
We study the quotient Q_i(A) of a free algebra A by the ideal M_i(A) generated by relation that the i-th commutator of any elements is zero. In particular, we completely describe such quotient for i=4 (for i<=3 this was done previously by Feigin and Shoikhet). We also study properties of the ideals M_i(A), e.g. when M_i(A)M_j(A) is contained in M_{i+j-1}(A) (by a result of Gupta and Levin, it is always contained in M_{i+j-2}(A)).
Axial Algebras: Questions and Conjectures
本文综述轴向代数领域，提出一系列开放问题与猜想，探讨其与群论的深刻联系。
A Classification of 2-dimensional endo-commutative straight algebras of type I
本文对任意域上二维 I 型自同态交换直代数进行了完整分类，并列出同构意义下的全部乘法表。
In a series of papers \cite{BRV1}, \cite{BRV2}, \cite{BRV3} we introduced full quivers and pseudo-quivers of representations of algebras, and used them as tools in describing PI-varieties of algebras. In this paper we apply them to obtain a complete proof of Belov's solution of Specht's problem for affine algebras over an arbitrary Noetherian ring. The inductive step relies on a theorem that enables one to find a "$\bar q$-characteristic coefficient-absorbing polynomial in each T-ideal $Γ$," i.e., a non-identity of the representable algebra $A$ arising from $Γ$, whose ideal of evaluations in $…
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…
Derived representation schemes with arbitrary coefficients and associative smoothness
研究通过任意有限维代数系数的表示同调探讨结合光滑性，证明有限生成形式光滑代数高阶同调消失，任意系数检验更强，对有限维代数而言其表示同调完全刻画形式光滑性，为相关理念提供证据。
Inner isotopes associated with automorphisms of commutative associative algebras
本文利用由代数自同态定义的内部同痕，从交换结合多项式代数构造新的交换非结合代数，建立其同构类与维数整数分拆的双射，并完整刻画幂等元及其谱。
Classification of AS-regular algebras is one of the main interests in noncommutative algebraic geometry. We say that a $3$-dimensional quadratic AS-regular algebra is of Type EC if its point scheme is an elliptic curve in $\mathbb{P}^{2}$. In this paper, we give a complete list of geometric pairs and a complete list of twisted superpotentials corresponding to such algebras. As an application, we show that there are only two exceptions up to isomorphism among all $3$-dimensional quadratic AS-regular algebras which cannot be written as a twist of a Calabi-Yau AS-regular algebra by a graded algeb…
We introduce a class of noncommutatative algebras called representation complete intersections (RCI). A graded associative algebra A is said to be RCI provided there exist arbitrarily large positive integers n such that the scheme Rep_n(A), of n-dimensional representations of A, is a complete intersection. We discuss examples of RCI algebras, including those arising from quivers. There is another interesting class of associative algebras called noncommutative complete intersections (NCCI). We prove that any graded RCI algebra is NCCI. We also obtain explicit formulas for the Hilbert series of …
Let X be a smooth affine algebraic variety over a field K of characteristic 0, and let R be a complete parameter K-algebra (e.g. R = K[[h]]). We consider associative (resp. Poisson) R-deformations of the structure sheaf O_X. The set of R-deformations has a crossed groupoid (i.e. strict 2-groupoid) structure. Our main result is that there is a canonical equivalence of crossed groupoids from the Deligne crossed groupoid of normalized polydifferential operators (resp. polyderivations) of X to the crossed groupoid of associative (resp. Poisson) R-deformations of O_X. The proof relies on a careful …
Structure and Invariants of Weyl-Type Algebras with Hyperbolic Sine Generators
本文研究了具有双曲正弦生成元的Weyl型代数的结构与不变量，证明其为Azumaya代数并探讨了其Brauer群中的表示，同时计算了Gelfand-Kirillov维度并提出了两个开放问题。
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。
Ternary Idempotent $Γ$-Semirings, Non-Reducibility, and Higher-Order Path Algebras
本文提出三元幂等Γ半环，证明其严格扩展经典半环路径代数，并展示高阶路径问题的不可约性与复杂性类差异。
Associative algebras and the representation theory of grading-restricted vertex algebras
本文通过构造与分级受限顶点代数关联的结合代数及其有限维子代数，建立了下界广义顶点代数模与相应分级结合代数模之间的等价对应，并将不可约性与完全可约性转化为结合代数模的相应性质。
Over a field of characteristic zero, we show that the forgetful functor from the homotopy category of commutative dg algebras to the homotopy category of dg associative algebras is faithful. In fact, the induced map of derived mapping spaces gives an injection on all homotopy groups at any basepoint. We prove similar results both for unital and non-unital algebras, and also Koszul dually for the universal enveloping algebra functor from dg Lie algebras to dg associative algebras. An important ingredient is a natural model for these derived mapping spaces as Maurer-Cartan spaces of complete fil…
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