Associative and Jordan algebras generated by two idempotents

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。

Skew idempotent functionals of ordered semirings

Skew idempotent functionals of ordered semirings are studied. Different associative and non-associative semirings are considered. Theorems about properties of skew idempotent functionals are proved. Examples are given.

Jordan-Lie inner ideals of finite dimensional associative algebras

We study Jordan-Lie inner ideals of finite dimensional associative algebras and the corresponding Lie algebras and prove that they admit Levi decompositions. Moreover, we classify Jordan-Lie inner ideals satisfying a certain minimality condition and show that they are generated by pairs of idempotents.

2-dimensional algebras. Application to Jordan, G-associative and 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.

Jordan structures and non-associative geometry

In this survey paper we give an overview over constructions of geometries associated to Jordan structures (algebras, triple systems and pairs), featuring analogs of these constructions with the Lie functor on the one hand and with the approach of non-commutative geometry on the other hand.

Jordan derivations on semirings of triangular matrices

We explore Jordan derivations of triangular matrices with entries from an additively idempotent semiring. The main result states that for any matrix A over additively idempotent semiring, if we put all the elements of the family of dense submatrices of A to be zeroes, we find a derivative of A. The set of derivations of this type is established.

Construction of Nilpotent Jordan Algebras Over any Arbitrary Fields

We give a computional method to construct and classify nilpotent Jordan algebras over any arbitrary fields by the second cohomolgy of nilpotent Jordan algebras of low dimension "analogue of Skjelbred-Sund method", we see that every nilpotent Jordan algebras can be constructed by the second cohomolgy of nilpotent Jordan algebras of low dimension. We use this method to classify nilpotent Jordan algebras up to dimension three over any field and nilpotent Jordan algebras of dimension four over an algebraic closed field of characteristic not 2 and over the real field R. Also commutative nilpotent a…

半Jordan型拟定轴代数

Quasi-definite axial algebras of Jordan type half

该研究针对半Jordan型轴代数，从弗罗贝尼乌斯形式和幂等元性质出发，找到了其为幺正的充分条件，推进了轴代数结构的相关理论研究。

Flexible idempotents in nonassociative algebras

``Fusion rules'' are laws of multiplication among eigenspaces of an idempotent. We establish fusion rules for flexible power-associative algebras, following Albert. We define the notion of an axis in the noncommutative setting (compare with [HRS]) and accumulate information about pairs of axes. We also describe a class of noncommutative examples of flexible power-associative algebras.

The universal Associative envelope of the anti-Jordan triple system of $n \times n$ matrices

We show that the universal associative enveloping algebra of the simple anti-Jordan triple system of all $n \times n$ matrices $(n \ge 2)$ over an algebraically closed field of characteristic 0 is finite dimensional. We investigate the structure of the universal envelope and focus on the monomial basis, the structure constants, and the center. We explicitly determine the decomposition of the universal envelope into matrix algebras. We classify all finite dimensional irreducible representations of the simple anti-Jordan triple system, and show that the universal envelope is semisimple. We also …

Degenerations of complex associative algebras of dimension three via Lie and Jordan algebras

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…

从带有自同态算子、微分算子或左平均算子的结合代数构造一些非结合代数

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。

Primitive orthogonal idempotents of Brandt semigroup algebras

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.

有限维单 Jordan 代数中的幂等词

Idempotent words in finite-dimensional simple Jordan algebras

本文研究特征非2域上有限维单Jordan代数中幂等元的左嵌套积，分类了可表示元素并确定最小词长，发现最大词长随矩阵规模对数增长。

导出约尔旦和双交换代数的代数与几何分类

The algebraic and geometric classification of derived Jordan and bicommutative algebras

本文提出新方法分类n维导出约尔旦代数，并应用于3维情况，同时得到3维metabelian交换代数和导出交换结合代数的代数分类，进而分类双交换代数，并探讨3维metabelian交换、导出交换结合、导出约尔旦和双交换代数的几何分类。

Two-Local derivations on associative and Jordan matrix rings over commutative rings

In the present paper we prove that every 2-local inner derivation on the matrix ring over a commutative ring is an inner derivation and every derivation on an associative ring has an extension to a derivation on the matrix ring over this associative ring. We also develop a Jordan analog of the above method and prove that every 2-local inner derivation on the Jordan matrix ring over a commutative ring is a derivation.

非结合代数的几何分类：综述

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。

Non-associative central extensions of null-filiform 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.

On a class of Algebras Satisfying polynomial identity of degree six

In this paper we study the structure of a class of algebras satisfying a polynomial identity of degree 6. We show, assuming the existence of a non-zero idempotent, that if an algebra satisfies such an identity, it admits a Peirce decomposition related to this idempotent. We studied the algebraic structure and highlighted the connections of the algebras of this class with Bernstein algebras, train algebras, Jordan algebras and power associative algebras. Keywords: Peirce decomposition, Bernstein algebra, Jordan algebra, Power associative algebra, train algebra,polynomial identity, idempotent.

三维反树形代数的分类

Classification of three-dimensional anti-dendriform algebras

本文分类了与结合性相关的反树形代数，重点处理零幂零结合代数及三维代数情形，并证明了含非零幂等元的结合代数不存在相容的反树形代数结构。

On primitive $3$-generated axial algebras of Jordan type

Axial algebras of Jordan type $η$ are commutative algebras generated by idempotents whose adjoint operators have the minimal polynomial dividing $(x-1)x(x-η)$, where $η\not\in\{0,1\}$ is fixed, with restrictive multiplication rules. These properties generalize the Pierce decompositions for idempotents in Jordan algebras, where $\frac{1}{2}$ is replaced with $η$. In particular, Jordan algebras generated by idempotents are axial algebras of Jordan type $\frac{1}{2}$. If $η\neq\frac{1}{2}$ then it is known that axial algebras of Jordan type $η$ are factors of the so-called Matsuo algebras corresp…

Generalization of Jordan-Lie of Finite Dimensional Associative Algebras

We generalize Baranov and Shlaka's results about bar-minimal Jordan-Lie and regular inner ideals of finite dimensional associative algebras. Let A be a finite dimensional 1-perfect associative algebras A over an algebraically closed field \mathbb{F} of characteristic p\ge0 and let A' be a subalgebra of A. We prove that for any bar-minimal Jordan-Lie inner ideal B' a A', there is a bar-minimal Jordan-Lie inner ideal of A that contains B' and if B' is regular, then is a regular inner ideal of A that contains B'. We also prove that for any strict orthogonal pair (e',f') in A', there is a strict o…

Jordan algebras admitting derivations with invertible values

The notion of derivation with invertible values as a derivation of ring with unity that only takes multiplicatively invertible or zero values appeared in a paper of Bergen, Herstein and Lanski, in which they determined the structure of associative rings that admit derivations with invertible values. Later, the results of this paper were generalized in many cases, for example, for generalized derivations, associative super-algebras, alternative algebras and many others. The present work is dedicated to description of all Jordan algebras admitting derivations with invertible values.

结合三系与标准嵌入

Associative triple trisystems and standard embeddings

本文在 Jordan 三双系与 Leibniz 三系基础上提出结合三系理论，建立其与 Jordan、Lie 三系的经典关系，并利用模的双自同态构造任意结合三系的标准嵌入。

Okubo algebras: automorphisms, derivations and idempotents

A survey of some properties of Okubo algebras is presented. Emphasis is put on automorphisms and derivations of these algebras, especially in characteristic three, where the situation is more involved and interesting. In this case, the Okubo algebra is closely related to a nodal noncommutative Jordan algebra. Dedicated to Helmut Strade on the occasion of his 70th birthday.

On the filtration of a free algebra by its associative lower central series

This paper concerns the associative lower central series ideals $M_i$ of the free algebra $A_n$ on $n$ generators. Namely, we study the successive quotients $N_i=M_i/M_{i+1}$, which admit an action of the Lie algebra $W_n$ of vector fields on $\Bbb C^n$. We bound the degree $|λ|$ of tensor field modules $F_λ$ appearing in the Jordan-Hölder series of each $N_i$, confirming a recent conjecture of Arbesfeld and Jordan. As an application, we compute these decompositions for small $n$ and $i$.

关于反左不变和左不变伪欧几里得近结合代数

On anti-left-invariant and left-invariant pseudo-Euclidean nearly associative algebras

该研究引入两类伪欧几里得近结合代数，建立其与李、若当结构的对应，通过双扩张程序实现递归构造，为两类代数的描述与分类提供统一框架。

关于与一个幂等且非结合代数结构相关的极限几何性质的注记

Remarks on some Limit Geometric Properties related to an Idempotent and Non-Associative Algebraic Structure

本文分析扩展 Max-Times 半环的幂等非结合代数结构的几何性质，引入直角与三角概念证明毕达哥拉斯关系类比，并定义过两点的直线为广义直线序列的极限，揭示平行线可共享无穷多点的独特性质。

Pseudo-euclidean Jordan algebras

A Jordan algebra J is said to be pseudo-euclidean if J is endowed with an associative non-degenerate symmetric bilinear form B. B is said an associative scalar product on J. First, we provide a description of the pseudo-euclidean Jordan K-algebras in terms of double extensions and generalized double extensions. In particular, we shall use this description to construct all pseudo-euclidean Jordan algebras of dimension less than or equal to 5. And then, from one of these algebras, we shall construct a twelve dimension Lie algebra by the "TKK" construction. Second, a description of symplectic pse…

Gelfand-Kirillov dimension and Jordan algebras

Let A be any associative algebra graded by a finite abelian group G, then if we denote by GKdim_k(A) and GKdim^G_k (A) the Gelfand-Kirillov dimension of its relatively free algebra and its relatively free G-graded algebra in k variables respectively, then GKdim_k(A)\leq GKdim^G_k (A). We show a counterexample of the previous result for Jordan algebras (hence non-associative). In particular, there exists a $Z_2$-grading on $UJ_n$, the Jordan algebra of $n\times n$ upper triangular matrices, n equal to 2 or 3, such that the previous inequality does not hold.

Local derivations on associative and Jordan matrix algebras

In the present paper we prove that every additive (not necessarily homogenous) local inner derivation on the algebra of matrices over an arbitrary field is an inner derivation, and every local inner derivation on the ring of matrices over a finite ring generated by the identity element or the ring of integers is an inner derivation. We also prove that every additive local inner derivation on the Jordan algebra of symmetric matrices over an arbitrary field is a derivation, and every local inner derivation on the Jordan ring of symmetric matrices over a finite ring generated by the identity elem…

Fractal nil graded Lie, associative, Poisson, and Jordan superalgebras

We construct a just infinite fractal 3-generated Lie superalgebra $\mathbf Q$ over arbitrary field, which gives rise to an associative hull $\mathbf A$, a Poisson superalgebra $\mathbf P$, and two Jordan superalgebras $\mathbf J$, $\mathbf K$. One has a natural filtration for $\mathbf A$ which associated graded algebra has a structure of a Poisson superalgebra and $\mathrm{gr} \mathbf A\cong\mathbf P$, also $\mathbf P$ admits an algebraic quantization. The Lie superalgebra $\mathbf Q$ is finely $\mathbb Z^3$-graded by multidegree in the generators, $\mathbf A$, $\mathbf P$ are $\mathbb Z^3$-gr…

Bimodules over Hom-Jordan and Hom-alternative algebras

In this paper, bimodules over Hom-Jordan algebras and the ones over Hom-alternative algebras are defined. It is shown that bimodules over Jordan and alternative algebras are twisted into bimodules over Hom-Jordan and Hom-alternative algebras via endomorphisms respectively. Some relations between bimodules over Hom-associative, Hom-Jordan and Hom-alternative algebras, are given.

轴心恒等式

Axial identities

本文引入轴心代数中的轴心恒等式，以研究固态子代数的性质，并给出了一个既非约当代数也非Matsuo代数同态像的轴心代数例子。

*Jordan型映射在交替*代数上的研究

*-Jordan-type maps on alternative *-algebras

本文研究了交替*代数上乘法*Jordan型映射的特征化问题，探讨了非平凡对称幂等元在其中的作用。

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.

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。

三元幂等Γ半环、不可约性与高阶路径代数

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。

Algebres non associatives normees de division. Classification des algebres reelles de Jordan non commutatives de division lineaire de dimension 8

In this work we are interested in the general problem of the determination of the normed division algebras. Our fundamental results are obtained in the particular subclass of those 8-dimensional quadratic flexible real division algebras. We give a new process which generalizes that of Cayley-Dickson and which allows the obtaining of a new family of eight-dimensional quadratic flexible real division algebras. We give examples of 8-dimensional quadratic flexible real division algebras which cannot be obtained by this first process of duplication and by means of a second process, which consists i…

自由结合代数中Jordan元素的多项式判据

A polynomial criterion for Jordan elements in a free associative algebra

本文为自由结合代数中的Jordan元素提出一个多项式判据，通过构造$U_n$和$q_n$，将元素属于Jordan子代数等价于单一方程$a\\,q_n(U_n)=0$，并给出显式投影及低度维数。

Jordan algebras and 3-transposition groups

An idempotent in a Jordan algebra induces a Peirce decomposition of the algebra into subspaces whose pairwise multiplication satisfies certain fusion rules $Φ(\frac{1}{2})$. On the other hand, $3$-transposition groups $(G,D)$ can be algebraically characterised as Matsuo algebras $M_α(G,D)$ with idempotents satisfying the fusion rules $Φ(α)$ for some $α$. We classify the Jordan algebras $J$ which are isomorphic to a Matsuo algebra $M_{1/2}(G,D)$, in which case $(G,D)$ is a subgroup of the (algebraic) automorphism group of $J$; the only possibilities are $G = \operatorname{Sym}(n)$ and $G = 3^2:…

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.

指数为4的约当 nil 潜力群环

Jordan Nilpotent Group Rings of index $4$

本文研究了指数为4的约当 nil 潜力群环的条件，探讨了群环的结构特性及其在非交换环上的应用。

On von Neumann regularity of algebras of quotients of Jordan algebras

We address a Jordan version of Johnson theorem on (associative) algebras of quotients, namely whether a strongly nonsingular (the Jordan version of nonsingularity) has a von Neumann regular algebra of quotients. Although the answer is negative in general, we prove that the result holds in the important case of algebras satisfying a polynomial identity.

Construtions and bimodules of BiHom-alternative and BiHom-Jordan algebras

The aim of this paper is to give some construction results and examples of BiHom-alternative and BiHom-Jordan algebras. Next, we define BiHom-alternative and BiHom-Jordan (bi)modules and we prove that from a given BiHom-alternative and BiHom-Jordan bimodules, a sequence of this kind of bimodules can be constructed. Some relations between Bihom-associative, BiHom-alternative and BiHom-Jordan bimodules are given.

Degenerations of Jordan Algebras and ''Marginal'' Algebras

We describe all degenerations of the variety $\mathfrak{Jord}_3$ of Jordan algebras of dimension three over $\mathbb{C}.$ In particular, we describe all irreducible components in $\mathfrak{Jord}_3.$ For every $n$ we define an $n$-dimensional rigid ''marginal'' Jordan algebra of level one. Also, we discuss ''marginal'' algebras in associative, alternative, left alternative, non-commutative Jordan, Leibniz, and anticommutative cases.

Hom-alternative algebras and Hom-Jordan algebras

The purpose of this paper is to introduce Hom-alternative algebras and Hom-Jordan algebras. We discuss some of their properties and provide construction procedures using ordinary alternative algebras or Jordan algebras. Also, we show that a polarization of Hom-associative algebra leads to Hom-Jordan algebra.

Jordan Triple Disystems

We take an algorithmic and computational approach to a basic problem in abstract algebra: determining the correct generalization to dialgebras of a given variety of nonassociative algebras. We give a simplified statement of the KP algorithm introduced by Kolesnikov and Pozhidaev for extending polynomial identities for algebras to corresponding identities for dialgebras. We apply the KP algorithm to the defining identities for Jordan triple systems to obtain a new variety of nonassociative triple systems, called Jordan triple disystems. We give a generalized statement of the BSO algorithm intro…

Axes of Jordan type in non-commutative algebras

The Peirce decomposition of a Jordan algebra with respect to an idempotent is well known. This decomposition was taken one step further and generalized recently by Hall, Rehren and Shpectorov, withtheir introduction of {\it axial algebras}, and in particular {\it primitive axial algebras of Jordan type} (PJs for short). It turns out that these notions are closely related to $3$-transposition groups and vertex operator algebras. De Medts, Peacock, Shpectorov, and M. Van Couwenberghe generalized axial algebrasto {\it decomposition algebras} which, in particular, are not necessarily commutative. …

八元数代数的子代数

Subalgebras of Octonion Algebras

本文确定了任意八元数代数的所有子代数，揭示了其结合性与维数的关系，并证明了同构子代数在自同构下的轨道性质。

Right Hom-alternative algebras

It is shown that every multiplicative right Hom-alternative algebra is both Hom-power associative and Hom-Jordan admissible. Multiplicative right Hom-alternative algebras admit Albert-type decompositions with respect to idempotents. Multiplication operators defined by idempotents in right Hom-alternative algebras are studied. Hom-versions of some well-known identities in right alternative algebras are proved.

On the variety of two dimensional real associative algebras

This paper consists of a description of the variety of two dimensional associative algebras within the framework of Nonstandard Analysis. By decomposing each algebra in A^2 as sum of a Jordan algebra and a Lie algebra, we calculate the isomorphism classes of two dimensional real associative algebras over the field of real numbers and determine the components and the contractions of the variety.

Classification of five-dimensional nilpotent Jordan algebras

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…

代数的幂等前自同态

Idempotent Pre-Endomorphisms of Algebras

本文将 pre-Lie 代数中的 pre-morphism 推广至任意代数，探讨其对偶形式，并研究幂等前自同态与子伴随反交换代数半直积分解的关系。