Hilbert演化代数、加权有向图与幂零性
Hilbert evolution algebras, weighted digraphs, and nilpotency
AI总结：
本文研究无限维Hilbert演化代数，通过加权有向图表示获得新性质，证明其nil与幂零性不等价，并给出代数为nil或幂零的充要条件。
摘要：
Hilbert演化代数通过Hilbert空间的框架推广了演化代数。在本文中，我们关注无限维Hilbert演化代数及其通过适当定义的加权有向图的表示。通过研究这种有向图，我们为这些结构获得了新性质，扩展了与有限维演化代数幂零性相关的著名结果。我们表明，与有限维演化代数中发生的情况不同，对于Hilbert演化代数，nil与幂零性的概念并不等价。此外，我们展示了给定Hilbert演化代数为nil或幂零的充分必要条件。我们的方法包含说明性示例。
英文摘要：
Hilbert evolution algebras generalize evolution algebras through a framework of Hilbert spaces. In this work we focus on infinite-dimensional Hilbert evolution algebras and their representation through a suitably defined weighted digraph. By means of studying such a digraph we obtain new properties for these structures extending well-known results related to the nilpotency of finite dimensional evolution algebras. We show that differently from what happens for the finite dimensional evolution algebras, the notions of nil and nilpotency are not equivalent for Hilbert evolution algebras. Furthermore, we exhibit necessary and sufficient conditions under which a given Hilbert evolution algebra is nil or nilpotent. Our approach includes illustrative examples.
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