论文标题
进化代数的自然家庭
Natural families in evolution algebras
论文作者
论文摘要
在本文中，我们介绍了进化等级的概念，并将演化代数的分解为其an灭者，加上具有进化等级的扩展进化子空间。这种分解可以用来证明在非分类进化代数中，任何天然和正交矢量的家族都可以扩展到自然基础。中心结果是那些线性独立向量的家族的表征，可以扩展到自然基础。 我们还考虑了完美进化代数的理想，并证明它们与基本理想相吻合。 通过仅查看结构矩阵，可以将第三顺序的nilpotent元素定位（以每个元素为正方形的磁场上的一个完美的进化代数）：任何消失的主小调都提供一个。相反，如果在任意字段上的完美进化代数具有第三顺序的nilpotent元素，则其结构矩阵具有消失的主辅助。 我们通过考虑伴随进化代数并将其属性与初始进化代数中的相应的属性联系起来结束。
In this paper we introduce the notion of evolution rank and give a decomposition of an evolution algebra into its annihilator plus extending evolution subspaces having evolution rank one. This decomposition can be used to prove that in non-degenerate evolution algebras, any family of natural and orthogonal vectors can be extended to a natural basis. Central results are the characterization of those families of orthogonal linearly independent vectors which can be extended to a natural basis. We also consider ideals in perfect evolution algebras and prove that they coincide with the basic ideals. Nilpotent elements of order three can be localized (in a perfect evolution algebra over a field in which every element is a square) by merely looking at the structure matrix: any vanishing principal minor provides one. Conversely, if a perfect evolution algebra over an arbitrary field has a nilpotent element of order three, then its structure matrix has a vanishing principal minor. We finish by considering the adjoint evolution algebra and relating its properties to the corresponding in the initial evolution algebra.
