A Peirce decomposition for generalized Jordan triple systems of second order
Issai Kantor, Noriaki Kamiya
二阶广义三元系的三幂等元定义了由10个分支组成的空间的分解。给出了一般由4个分支组成的Jordan三元系的Peirce分解n的推广。
A tripotent of a generalized triple system of second order defines a decomposition of the space of the space which consist of 10 components. It gives a generalization of the Peirce decomposition n for the Jordan triple system which consists in general of 4 components.
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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.
Commutative algebras satisfying univariate identities with vanishing Peirce polynomial
本文引入并研究(2,3)-palintropic代数，这类交换代数满足恒等式(x^3)^2 - (x^2)^3 = 0，具有平凡的Peirce多项式，并证明幂等元具有良好融合规则及代数同态性质，最后给出具体例子及其在交换多项式映射中的应用。
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. …
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:…
(-1,-1)-Balanced Freudenthal Kantor triple systems and noncommutative Jordan algebras
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Jordan Triple Elementary Maps on Rings
证明了含有非平凡幂等元的酉环上的Jordan三重初等满射是自动可加的。
We find a class of hermitian generalized Jordan triple systems (HGJTSs) and hermitian $(ε, δ)$-Freudenthal-Kantor triple systems (HFKTSs). We apply one of the most simple HGJTSs which we find to a field theory, and obtain a typical u(N) Chern-Simons gauge theory with a fundamental matter.
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Dendriform analogues of Lie and Jordan triple systems
本文用计算机代数分类了自由树状二重代数中 pre-Lie 与 pre-Jordan 三重乘积在次数不超过 7 的多重线性多项式恒等式，给出各次数的独立恒等式数量并证明 pre-Lie 情形次数 7 恒等式均可由低次推出。
Jordan triple product homomorphisms on Hermitian matrices to and from dimension one
本文完整刻画了n×n复厄米矩阵与一维数域（复数、实数、非负实数）之间两个方向上的所有Jordan三重积同态映射。
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。
Automatic additivity for injective Jordan semi-triple maps on structural matrix rings over division rings
本文研究除环上结构矩阵环中的单射Jordan半三重映射，给出其自动可加性的判定准则，并刻画了两种障碍情形及可加情况下的映射形式。
We determine explicit orbit representatives of reducible Jordan algebras and of their corresponding Freudenthal triple systems. This work has direct application to the classification of extremal black hole solutions of N = 2, 4 locally supersymmetric theories of gravity coupled to an arbitrary number of Abelian vector multiplets in D = 4, 5 space-time dimensions.
Particle models from special Jordan backgrounds and spectral triples
该研究基于若尔当坐标代数定义谱三元组与代数背景，提出涨落狄拉克算子的规范不变玻色构型空间并完成计算，指出其不等同于常规结合非交换几何，且若尔当情形下规范场恒为幺模，解决了非交换几何的长期难题。
Let $\mathcal A$ and $\mathcal B$ be unital rings and $\mathcal M$ be a $(\mathcal A, \mathcal B)$-bimodule, which is faithful as a left $\mathcal A$-module and also as a right $\mathcal B$-module. Let ${\mathcal U}={\rm Tri}(\mathcal A, \mathcal M, \mathcal B)$ be the associated triangular ring. It is shown that every additive generalized Jordan (triple) higher derivation on $\mathcal U$ is a generalized higher derivation.
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本文研究多项式代数上由偏导数定义的双导子结构，刻画其二次齐次部分与对称矩阵特殊 Jordan 代数的同构、各齐次空间的双模性质、权空间分解、自同构群及特征零下的单性。
Lie-Yamaguti algebras (or generalized Lie triple systems) are binary-ternary algebras intimately related to reductive homogeneous spaces. The Lie-Yamaguti algebras which are irreducible as modules over their inner derivation algebras are the algebraic counterpart of the isotropy irreducible homogeneous spaces. These systems splits into three disjoint types: adjoint type, non-simple type and generic type. The systems of the first two types were classified in a previous paper through a generalized Tits Construction of Lie algebras. In this paper, the Lie-Yamaguti algebras of generic type are cla…
Additivity of Maps on Triangular Algebras
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Exceptional symmetric domains
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The aim of this paper is to offer an overview of the most important applications of Jordan structures inside mathematics and also to physics, up-dated references being included. For a more detailed treatment of this topic see - especially - the recent book Iordanescu [364w], where sugestions for further developments are given through many open problems, comments and remarks pointed out throughout the text. Nowadays, mathematics becomes more and more nonassociative and my prediction is that in few years nonassociativity will govern mathematics and applied sciences. Keywords: Jordan algebra, Jor…
We classify two classes of B_2-graded Lie algebras which have a second compatible grading by an abelian group A: (a) graded-simple Lie algebras for A torsion-free and (b) division-A-graded Lie algebras. Our results describe the centreless cores of a class of affine reflection Lie algebras, hence apply in particular to the centreless cores of extended affine Lie algebras, the so-called Lie tori, for which we recover results of Allison-Gao and Faulkner. Our classification (b) extends a recent result of Benkart-Yoshii. Both classifications are consequences of a new description of Jordan algebras …
Jordan geometries are defined as spaces equipped with point reflections depending on triples of points, exchanging two of the points and fixing the third. In a similar way, symmetric spaces have been defined by Loos (Symmetric Spaces I, 1969) as spaces equipped with point reflections depending on a point and fixing this point; therefore the theories of Jordan geometries and of symmetric spaces are closely related to each other -- in order to describe this link, the notion of symmetry actions of torsors and of symmetric spaces is introduced. Jordan geometries give rise both to symmetry actions …
Generalized conformal realizations of Kac-Moody algebras
给出了一个构造，它将一个正整数n标记的卡茨-穆迪代数的无穷序列与一个Jordan代数相关联.对于n=1，这简化为众所周知的Kantor-Koecher-Tits构造。我们的推广利用了不同的广义约旦三元系之间的一个新的关系，连同他们已知的连接约旦和李代数。应用于除代数R，C，H，O上的Hermitian 3 × 3矩阵的Jordan代数，给出了n=2的例外李代数f4，e6，e7，e8.此外，我们得到了它们的无限维扩展n大于或等于3。在2x2矩阵的情况下，所得到的李代数的形式为so（p+n，q+n），并且伴随的非线性实现推广了签名为（p，q）的时空中的共形变换。
Symmetric matrices, orthogonal Lie algebras, and Lie-Yamaguti algebras
本文利用对称矩阵上的 Jordan 与三元系结构构造了一族以正交李代数 so(n,K) 为导子代数的不可约 Lie-Yamaguti 代数，并用计算机代数研究了 n<5 时的低次多项式恒等式。
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