论文标题
在原始$ 3 $生成的约旦类型的轴向代数
On primitive $3$-generated axial algebras of Jordan type
论文作者
论文摘要
约旦型$η$的轴向代数是由iDempotents产生的可交换代数，其伴随操作员具有最小的多项式分隔$（x-1）x（x-- h）$，其中$η\ in \ in \ in \ in \ {0,1 \} $是固定的，具有限制性的乘数乘数。这些属性概括了约旦代数中的diDempotents的Pierce分解，其中$ \ frac {1} {2} $被$η$代替。特别是，由IDEMPOTENTS生成的Jordan代数是Jordan类型$ \ frac {1} {2} $的轴向代数。如果$η\ neq \ frac {1} {2} $，则众所周知，约旦型$η$的轴向代数是所谓的MATSUO代数的因素，该代数对应于3个反序组。 我们称为生成dempotents {\ it axes}，并说轴是{\ it rigitive}，如果其伴随运算符具有1维1-eigenspace。众所周知，由两个原始轴产生的亚词法最多具有三个。到目前为止，这三个生成的案例已经开放。我们证明，由三个原始轴产生的约旦类型的任何轴向代数最多都具有九个尺寸。如果尺寸为九，$η= \ frac {1} {2} $，则我们要么显示如何在该代数中找到适当的理想，要么证明代数对某些Jordan矩阵代数是同构的。
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 corresponding to 3-transposition groups. We call the generating idempotents {\it axes} and say that an axis is {\it primitive} if its adjoint operator has 1-dimensional 1-eigenspace. It is known that a subalgebra generated by two primitive axes has dimension at most three. The 3-generated case has been opened so far. We prove that any axial algebra of Jordan type generated by three primitive axes has dimension at most nine. If the dimension is nine and $η=\frac{1}{2}$ then we either show how to find a proper ideal in this algebra or prove that the algebra is isomorphic to certain Jordan matrix algebras.
