arXiv 0811.37022008-11-22math.RA

Pseudo-euclidean Jordan algebras

Amir Baklouti, Said Benayadi

备注：: 39 pages

英文摘要：

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 pseudo-euclidean Jordan algebras is provided and finally we describe a particular class of these algebras namely the class of symplectic Jordan-Manin Algebras. In addition to these descriptions, this paper demonstrates that these last two classes are identical and provides several information on the structure of pseudo-euclidean Jordan algebras.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

Lattice-like subsets of Euclidean Jordan algebras

While studying some properties of linear operators in a Euclidean Jordan algebra, Gowda, Sznajder and Tao have introduced generalized lattice operations based on the projection onto the cone of squares. In two recent papers of the authors of the present paper it has been shown that these lattice-like operators and their generalizations are important tools in establishing the isotonicity of the metric projection onto some closed convex sets. The results of this kind are motivated by metods for proving the existence of solutions of variational inequalities and methods for finding these solutions…

2014-02-05math.RA

Infinite dimensional Jordan algebras and symmetric cones

A celebrated result of Koecher and Vinberg asserts the one-one correspondence between the finite dimensional formally real Jordan algebras and Euclidean symmetric cones. We extend this result to the infinite dimensional setting.

2017-07-12math.RA

Commutation principles in Euclidean Jordan algebras and normal decomposition systems

The commutation principle of Ramirez, Seeger, and Sossa \cite{ramirez-seeger-sossa} proved in the setting of Euclidean Jordan algebras says that when the sum of a Fréchet differentiable function $Θ(x)$ and a spectral function $F(x)$ is minimized over a spectral set $Ω$, any local minimizer $a$ operator commutes with the Fréchet derivative $Θ^{\prime}(a)$. In this paper, we extend this result to sets and functions which are (just) invariant under algebra automorphisms. We also consider a similar principle in the setting of normal decomposition systems.

2016-04-15math.OCmath.RA

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

University of Coimbra（科英布拉大学）；Université de Lorraine（洛林大学）；New York University Abu Dhabi（纽约大学阿布扎比分校）

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。

2026-08-04math.RA

欧几里得Jordan代数上双曲多项式的行列式优超

Determinant majorization for hyperbolic polynomials on Euclidean Jordan algebras

Beijing International Center for Mathematical Research, Peking University（北京大学北京国际数学研究中心）

在锥包含和中心射线假设下，证明了欧几里得Jordan代数上双曲多项式的行列式优超结果，推广了Harvey和Lawson的定理，并给出了新的σ₂优超结果，同时构造了反例表明σₖ优超对k≥3不成立。

2026-06-16math.AGmath.APmath.FA

Algèbres de Jordan et théorie des invariants

If V is a simple complex euclidean Jordan algebra and G the subgroup of GL(V) fixing the determinant of V, we give a unified description of the invariant algebras C[pV]^G, for p not greater than three.

2011-03-13math.RAmath.RT

更多相关论文收起更多论文

伪欧几里得Novikov超代数：结构与性质

Pseudo-Euclidean Novikov Superalgebras: Structure and Properties

Université de Lorraine（洛林大学）；New York University Abu Dhabi（纽约大学阿布扎比分校）；Université Cadi-Ayyad（卡迪·阿雅德大学）

本文研究伪欧几里得Novikov超代数的结构与性质，引入了Milnor超代数这一特殊子类，并证明了所有两面理想非退化的伪欧几里得Novikov超代数都属于此类，同时给出了构造此类超代数的方法，以及通过双扩展过程将其分类。

2026-05-20math.RA

Cohomology and deformations of Jacobi-Jordan algebras

In this paper, we define and develop a cohomology and deformation theories of Jacobi-Jordan algebras. We construct a cohomology based on two operators, called zigzag cohomology, and detail the low degree cohomology spaces. We describe the relationships between first and second cohomology groups with extensions and deformations. Moreover, we consider cohomology properties of pseudo-euclidean Jacobi-Jordan algebras and provide a deformation theory that fits with our zigzag cohomology of Jacobi-Jordan algebras. Furthermore, the paper includes several examples and applications.

2021-09-25math.RAmath.RT

关于平坦伪欧几里得可解Malcev代数

On flat pseudo-Euclidean solvable Malcev algebras

Université Cadi-Ayyad（卡迪·阿亚德大学）

本文为伪欧几里得Malcev代数引入曲率算子，定义平坦性，发展平坦双扩张构造，证明平坦洛伦兹幂零Malcev代数必为Lie代数且平坦。

2026-09-15math.RA

平坦伪欧几里得莱布尼茨超代数

Flat pseudo-Euclidean Leibniz superalgebras

本文引入预李和预莱布尼茨超代数，定义平坦伪欧几里得左莱布尼茨超代数，证明二次情形下平坦等价于对称且2步幂零，并利用双扩张刻画所有此类超代数。

2025-10-17math.RA

伪欧几里得 Hom-交错超代数与 Hom-post-交错超代数

Pseudo-Euclidean Hom-alternative superalgebras and Hom-post-alternative superalgebras

本文研究伪欧几里得与辛 Hom-交错超代数的性质和构造，引入任意权 Rota-Baxter 算子及 Hom-post-交错超代数，并建立二者间的自然诱导关系。

2023-07-14math.RA

Hom-Jordan-Malcev-Poisson algebras

The purpose of this paper is to provide and study a Hom-type generalization of Jordan-Malcev-Poisson algebras, called Hom-Jordan-Malcev-Poisson algebras. We show that they are closed under twisting by suitable self-maps and give a characterization of admissible Hom-Jordan-Malcev-Poisson algebras. In addition, we introduce the notion of pseudo-Euclidian Hom-Jordan-Malcev-Poisson algebras and describe its $T^*$-extension. Finally, we generalize the notion of Lie-Jordan-Poisson triple system to the Hom setting and establish its relationships with Hom-Jordan-Malcev-Poisson algebras.

2020-10-25math-phmath.MPmath.RA

伪欧几里得mock-Lie超代数的扩张

Extensions of pseudo-euclidean mock-Lie superalgebras

本文引入mock-Lie超代数及其表示，定义伪欧几里得mock-Lie超代数，并研究其双重扩张与广义双重扩张及等距。

2025-10-03math.RAmath.RT

信息几何、Jordan代数与类余伴随轨道构造

Information Geometry, Jordan Algebras, and a Coadjoint Orbit-Like Construction

本文在实Jordan代数中构建了类似余伴随轨道的叶结构并诱导出伪黎曼度量，揭示了其与经典及量子信息几何中Fisher-Rao和Bures-Helstrom度量的直接联系。

2023-10-20math-phmath.DGmath.MP

Exceptional quantum geometry and particle physics

Based on an interpretation of the quark-lepton symmetry in terms of the unimodularity of the color group $SU(3)$ and on the existence of 3 generations, we develop an argumentation suggesting that the "finite quantum space" corresponding to the exceptional real Jordan algebra of dimension 27 (the Euclidean Albert algebra) is relevant for the description of internal spaces in the theory of particles. In particular, the triality which corresponds to the 3 off-diagonal octonionic elements of the exceptional algebra is associated to the 3 generations of the Standard Model while the representation o…

2016-05-01gr-qchep-thmath-ph

Realization of Frobenius manifolds as submanifolds in pseudo-Euclidean spaces

We introduce a class of k-potential submanifolds in pseudo-Euclidean spaces and prove that for an arbitrary positive integer k and an arbitrary nonnegative integer p, each N-dimensional Frobenius manifold can always be locally realized as an N-dimensional k-potential submanifold in ((k + 1) N + p)-dimensional pseudo-Euclidean spaces of certain signatures. For k = 1 this construction was proposed by the present author in a previous paper (2006). The realization of concrete Frobenius manifolds is reduced to solving a consistent linear system of second-order partial differential equations.

2009-11-22hep-thmath-phmath.AG