$3$-generated axial algebras with a minimal Miyamoto group

Miyamoto involutions in axial algebras of Jordan type half

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…

On primitive axial decomposition algebras of Majorana type with degenerate eigenvalues

A class of axial decomposition algebras with Miyamoto group generated by two Miyamoto automorphisms and three eigenvalues $0,1$ and $η$ is introduced and classified in the case with $η\notin\{0,1,\frac{1}{2}\}$. This class includes specializations of 2-generated axial algebras of Majorana type $(ξ,η)$ to the case with $ξ=η$.

Primitive axial algebras of Jordan type

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…

An expansion algorithm for constructing axial algebras

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 …

轴向代数的自同构群

Automorphism groups of axial algebras

本文针对轴向代数中因维数重复出现而可能存在的意外同构问题，提出计算完全自同构群的方法，并在最高151维的若干例子上进行应用验证。

Decomposition algebras and axial algebras

We introduce decomposition algebras as a natural generalization of axial algebras, Majorana algebras and the Griess algebra. They remedy three limitations of axial algebras: (1) They separate fusion laws from specific values in a field, thereby allowing repetition of eigenvalues; (2) They allow for decompositions that do not arise from multiplication by idempotents; (3) They admit a natural notion of homomorphisms, making them into a nice category. We exploit these facts to strengthen the connection between axial algebras and groups. In particular, we provide a definition of a universal Miyamo…