Louis Rowen, Yoav Segev
``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.
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Let $F$ be a field, char$(F)\neq 2$. Then every finite-dimensional $F$-algebra has either an idempotent or an absolute nilpotent if and only if over $F$ every polynomial of odd degree has a root in $F$. This is also necessary and sufficient for existence of eigenvectors for all quadratic operators in finite-dimensional spaces over $F$.
In this paper, we study the variety of all nonassociative (NA) algebras from the idempotent point of view. We are interested, in particular, in the spectral properties of idempotents when algebra is generic, i.e. idempotents are in general position. Our main result states that in this case, there exist at least $n-1$ nontrivial obstructions (syzygies) on the Peirce spectrum of a generic NA algebra of dimension $n$. We also discuss the exceptionality of the eigenvalue $λ=\frac12$ which appears in the spectrum of idempotents in many classical examples of NA algebras and characterize its extremal…
There exists two types of nonassociative algebras whose associator satisfies a symmetric relation associated with a 1-dimensional invariant vector space with respect to the natural action of the symmetric group on three elements. The first one corresponds to the Lie-admissible algebras that we studied in a previous paper. Here we are interested by the second one corresponding to the third power associative algebras.
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。
Matsuo algebras are an algebraic incarnation of 3-transposition groups with a parameter $α$, where idempotents takes the role of the transpositions. We show that a large class of idempotents in Matsuo algebras satisfy the Seress property, making these nonassociative algebras well-behaved analogously to associative algebras, Jordan algebras and vertex (operator) algebras. We calculate eigenvalues in the Matsuo algebra of ${\rm Sym}(n)$ for any $α$, generalising some vertex algebra results for which $α=\frac{1}{4}$. Finally, in the Matsuo algebra of the root system ${\rm D}_n$, we show $n-3$ con…
The purpose of this paper is to give a systematic study of two new classes of commutative nonassociative algebras, the so-called isospectral and medial algebras. An isospectral algebra $\mathbb{A}$ is a generic commutative nonassociative algebra whose idempotents have the same Peirce spectrum. A medial algebra is algebra with identity $(xy)(zw)=(xz)(yw)$. We show that these two classes are essentially coincide. We also prove that any medial spectral algebra is isomorphic to a certain isotopic deformation of the commutative associative quotient algebra $\mathbf{K}[z]/(z^n-1)$.
The aim of this paper is to give a survey of nonassociative Hom-algebra and Hom-superalgebra structures. The main feature of these algebras is that the identities defining the structures are twisted by homomorphisms. We discuss Hom-associative algebras, Hom-Flexible algebras, Hom-Lie algebras, $G$-hom-associative algebras, Hom-Poisson algebras, Hom-alternative algebras and Hom-Jordan algebras and $\mathbb{Z}_2$-graded versions. We give an overview of the development of Hom-algebras structures which have been intensively investigated recently.
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。
If $V$ is a finite-dimensional unital commutative (maybe nonassociative) algebra carrying an associative positive definite bilinear form then there exist a nonzero idempotent $c\ne e$ ($e$ being the algebra unit) of the shortest possible length $|c|^2$. In particular, $|c|^2\le \frac12|e|^2$. We prove that the equality holds exactly when $V$ is a Jordan algebra of Clifford type.
Nonassociative algebras of anti-biderivation-type
本文研究一类乘积为反双导子的Jacobi-Jordan-容许代数（$\mathcal A{\rm BD}$-代数），刻画其性质，证明包含若干已知代数类，引入post-Jacobi-Jordan结构，并给出复数3维情形的代数分类。
The commutative nonassociative algebra of metric curvature tensors
本文研究欧几里得向量空间上度量曲率张量构成的交换非结合代数，刻画其低维结构，并证明Weyl子代数在维数大于四时单性，反自对偶Weyl张量构成五维单理想。
Nonassociative commutative algebras $A$ generated by idempotents $e$ whose adjoint operators ${\rm ad}_e\colon A \rightarrow A$, given by $x \mapsto xe$, are diagonalizable and have few eigenvalues are of recent interest. When certain fusion (multiplication) rules between the associated eigenspaces are imposed, the structure of these algebras remains rich yet rather rigid. For example vertex operator algebras give rise to such algebras. The connection between the Monster algebra and Monster group extends to many axial algebras which then have interesting groups of automorphisms. Axial algebras…
Hsiang cubics as spherical Chebyshev polynomials: equioscillation and idempotent quadrature
本文将Hsiang三次式与球面上的切比雪夫多项式关联，研究其等振荡性质、平稳性，证明Cartan等参三次式在特定维度下是最小偏差问题的全局极小元，还给出了相关求积恒等式与球面设计的结果。
A Majorana algebra is a commutative nonassociative real algebra generated by a finite set of idempotents, called Majorana axes, that satisfy some of the properties of the $2A$-axes of the Monster Griess algebra. The term was introduced by A. A. Ivanov in 2009 inspired by the work of S. Sakuma and M. Miyamoto. In the present paper, we revisit Mayer and Neutsch's theorem on associative subalgebras of the Griess algebra in the context of Majorana theory. We apply this result to determine all the maximal associative subalgebras of some low-dimensional Majorana algebras; namely, the Majorana algebr…
On the Peirce spectral rigidity of decorated incidence algebras
本文研究由带符号装饰的部分Steiner三元系确定的代数，证明其幂等元谱仅依赖于负Pasch构型数，并给出显式矩阵模型展示刚性，联系极代数与对称Clifford系统。
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。
In the first half of this paper, we define axial algebras: nonassociative commutative algebras generated by axes, that is, semisimple idempotents---the prototypical example of which is Griess' algebra [C85] for the Monster group. When multiplication of eigenspaces of axes is controlled by fusion rules, the structure of the axial algebra is determined to a large degree. We give a construction of the universal Frobenius axial algebra on $n$ generators with a specified fusion rules, of which all $n$-generated Frobenius axial algebras with the same fusion rules are quotients. In the second half, w…
Hom-unitality and hom-associative structures
本文研究了非结合代数上的同态结合结构，探讨了一侧和双侧单位代数的特性，提出新的特征化方法，并应用于多种非结合代数类。
We consider an evolution algebra which corresponds to a bisexual population with a set of females partitioned into finitely many different types and the males having only one type. We study basic properties of the algebra. This algebra is commutative (and hence flexible), not associative and not necessarily power associative, in general. Moreover it is not unital. A condition is found on the structural constants of the algebra under which the algebra is associative, alternative, power associative, nilpotent, satisfies Jacobi and Jordan identities. In a general case, we describe the full set of…
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