Jonathan I. Hall, Yoav Segev, Sergey Shpectorov
In this note we give an overview of our knowledge regarding primitive axial algebras of Jordan type half and connections between $3$-transposition groups and Matsuo algebras. We also show that primitive axial algebras of Jordan type $η$ admit a Frobenius form, for any $η$.
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
Primitive 4-generated axial algebras of Jordan type
本文证明了 Jordan 型本原 4-生成轴代数的维数上界为 81 维，给出了该类代数的维数限制。
The notion of axial algebra is closely related to $3$-transposition groups, the Monster group and vertex operator algebras. In this work we continue our previous works and compete the proof that all algebras generated by a set of primitive axes not necessarily of the same type (see the definition in the body of the paper), are primitive axial algebras of Jordan type.
Axial algebras are a class of non-associative commutative algebras whose properties are defined in terms of a fusion law. When this fusion law is graded, the algebra has a naturally associated group of automorphisms and thus axial algebras are inherently related to group theory. Examples include most Jordan algebras and the Griess algebra for the Monster sporadic simple group. In this survey, we introduce axial algebras, discuss their structural properties and then concentrate on two specific classes: algebras of Jordan and Monster type, which are rich in examples related to simple groups.
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…
Solid lines in axial algebras of Jordan type $\tfrac{1}{2}$ and Jordan algebras
本文证明Jordan型1/2原始轴代数为Jordan代数当且仅当其所有二元生成子代数都是实心的，并刻画了实心性与结合子为导子的等价条件及非实心情形的轴数量。
An axial algebra over the field $\mathbb F$ is a commutative algebra generated by idempotents whose adjoint action has multiplicity-free minimal polynomial. For semisimple associative algebras this leads to sums of copies of $\mathbb F$. Here we consider the first nonassociative case, where adjoint minimal polynomials divide $(x-1)x(x-η)$ for fixed $0\neqη\neq 1$. Jordan algebras arise when $η=\frac{1}{2}$, but our motivating examples are certain Griess algebras of vertex operator algebras and the related Majorana algebras. We study a class of algebras, including these, for which axial automor…
On $3$-generated axial algebras of Jordan type $\frac{1}{2}$
本文针对二次闭域上的3-生成Jordan型1/2轴代数，在已有带4参数的泛代数构造基础上，完整刻画了所有这类半单代数，推进了非结合轴代数的分类研究。
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…
"Fusion rules" are laws of multiplication among eigenspaces of an idempotent. This terminology is relatively new and is closely related to primitive 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 earlier work we studied primitive axial algebras, not necessarily commutative, and showed that they all have Jordan type. In this paper, we show that all finitely generated primitive axial algebras are direct sums of specifically described flexible finite dimensional noncommu…
Axial algebras are non-associative algebras generated by semisimple idempotents whose adjoint actions obey a fusion law. Axial algebras that are generated by two such idempotents play a crucial role in the theory. We classify all primitive 2-generated axial algebras whose fusion laws have two eigenvalues and all graded primitive 2-generated axial algebras whose fusion laws have three eigenvalues. This represents a significant broadening in our understanding of axial algebras.
On Jordan algebras that are factors of Matsuo algebras
本研究刻画了对应Matsuo代数含非平凡Jordan代数因子的有限连通3-对合群，证明特征0域上存在无穷多非Matsuo代数因子的Jordan型1/2本原轴代数，并以例外Jordan代数为例验证。
Solid subalgebras in algebras of Jordan type half
该研究引入Jordan型代数的实心2-生成子代数概念，证明一般2-生成子代数为实心，特征零时例外值也成立，由此推出特征零下Jordan型代数要么有无穷多轴，要么是Matsuo代数或其商，支撑了相关分类猜想。
Skew Axial Algebras of Monster Type II
该研究延续此前关于2生成Monster型本原轴代数的工作，对带有更大斜轴集的这类代数进行分类，推进了该类代数的完整分类进程。
The Classification of the 2-generated Primitive Axial Algebras of Monster Type
本文研究了怪兽类型轴代数的2生成代数的分类问题，通过分情况讨论参数、子代数、轴集和轴维数，完成了对这些代数的完整分类，并提供了基底、乘法和相关信息。
In this paper we prove that $2$-generated primitive axial algebras of Monster type $(2β, β)$ over a ring $R$ in which $2$ and $β$ are invertible can be generated as $R$-module by $8$ vectors. We then completely classify $2$-generated primitive axial algebras of Monster type $(2β, β)$ over any field of characteristic other than $2$.
Let $A$ be a commutative, non-associative algebra over a field $\mathbb{F}$ of characteristic $\ne 2$. A half-axis in $A$ is an idempotent $e\in A$ such that $e$ satisfies the Peirce multiplication rules in a Jordan algebra, and, in addition, the $1$-eigenspace of ${\rm ad}_e$ (multiplication by $e$) is one dimensional. In this paper we consider the identities $(*)$ $x^2x^2=x^4$ and $x^3x^2=xx^4.$ We show that if identities $(*)$ hold strictly in $A,$ then one gets (very) interesting identities between elements in the eigenspaces of ${\rm ad}_e$ (note that if $|\mathbb{F}|>3$ and the identitie…
Axial algebras are a class of commutative non-associative algebras generated by idempotents, called axes, with adjoint action semi-simple and satisfying a prescribed fusion law. Axial algebras were introduced by Hall, Rehren and Shpectorov \cite{hrs,hrs1} as a broad generalisation of Majorana algebras of Ivanov, whose axioms were derived from the properties of the Griess algebra for the Monster sporadic simple group. The class of axial algebras of Monster type includes Majorana algebras for the Monster and many other sporadic simple groups, Jordan algebras for classical and some exceptional si…
In this paper, we describe the Galois module structure of $J=\mathbf{K}^{\times}/\mathbf{K}^{\times p}$, where $\mathbf{K}$ is an extension of a local field $\mathbf{k}$ containing a primitive $p$-th root of unity: for instance, if $\mathbf{K}/\mathbf{k}$ is a $p$-elementary abelian extension, we prove that $J$ is a module of constant Jordan type, with stable Jordan type $[1]^2$, which, in a way, extends the result of J. Mináč and J. Swallow. Also, we take profit from our proof by computing some invariants, which were previously introduced by A. Adem, W. Gao, D. B. Karageuzian and J. Mináč onl…
Skew Axial Algebras of Monster Type
本文首次构造出具有轴集 $X'(1+2)$ 的怪物型斜轴代数实例，并完成该类 2-生成本原轴代数的分类，同时为更大斜轴集的分类扩展提出建议。
An axial algebra $A$ is a commutative non-associative algebra generated by primitive idempotents, called axes, whose adjoint action on $A$ is semisimple and multiplication of eigenvectors is controlled by a certain fusion law. Different fusion laws define different classes of axial algebras. Axial algebras are inherently related to groups. Namely, when the fusion law is graded by an abelian group $T$, every axis $a$ leads to a subgroup of automorphisms $T_a$ of $A$. The group generated by all $T_a$ is called the Miyamoto group of the algebra. We describe a new algorithm for constructing axial …
An axial algebra is a commutative non-associative algebra generated by axes, that is, primitive, semisimple idempotents whose eigenvectors multiply according to a certain fusion law. The Griess algebra, whose automorphism group is the Monster, is an example of an axial algebra. We say an axial algebra is of Monster type if it has the same fusion law as the Griess algebra. The $2$-generated axial algebras of Monster type, called Norton-Sakuma algebras, have been fully classified and are one of nine isomorphism types. In this paper, we enumerate a subclass of $3$-generated axial algebras of Mons…
A code algebra $A_C$ is a non-associative commutative algebra defined via a binary linear code $C$. We study certain idempotents in code algebras, which we call small idempotents, that are determined by a single non-zero codeword. For a general code $C$, we show that small idempotents are primitive and semisimple and we calculate their fusion law. If $C$ is a projective code generated by a conjugacy class of codewords, we show that $A_C$ is generated by small idempotents and so is, in fact, an axial algebra. Furthermore, we classify when the fusion law is $\mathbb{Z}_2$-graded. In doing so, we…
"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…
Axial algebras are a class of commutative algebras generated by idempotents, with adjoint action semisimple and satisfying a prescribed fusion law. Axial algebras were introduced by Hall, Rehren, and Shpectorov in 2015 as a broad generalization of Majorana algebras of Ivanov, whose axioms were derived from the properties of the Griess algebra for the Monster group. The class of Matsuo algebras was introduced by Matsuo and later generalized by Hall, Rehren, and Shpectorov. A Matsuo algebra $M$ is built by a set of 3-transpositions $D$. Elements of $D$ are idempotents in $M$ and called axes. In …
2-generated axial algebras of Monster type
该研究为Monster型2生成轴代数的分类项目搭建基础框架，证明一般情形下这类代数维数不超8，完成了参数代数无关时的分类，并推广了Norton-Sakuma定理。
Axial algebras are a class of non-associative algebra with a strong link to finite (especially simple) groups which have recently received much attention. Of primary interest are the axial algebras of Monster type $(α, β)$, of which the Griess algebra (with the Monster as its automorphism group) is an important motivating example. In this paper, we complete the classification of the symmetric $2$-generated primitive axial algebras of Monster type $(α, β)$. By previous work of Yabe, and Franchi and Mainardis, any such algebra is either explicitly known, or is a quotient of the infinite-dimensio…
Motivated by Yabe's classification of symmetric $2$-generated axial algebras of Monster type, we introduce a large class of algebras of Monster type $(α, \frac{1}{2})$, generalising Yabe's $\mathrm{III}(α,\frac{1}{2}, δ)$ family. Our algebras bear a striking similarity with Jordan spin factor algebras with the difference being that we asymmetrically split the identity as a sum of two idempotents. We investigate the properties of this algebra, including the existence of a Frobenius form and ideals. In the $2$-generated case, where our algebra is isomorphic to one of Yabe's examples, we use our …
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。