On simple evolution algebras of dimension two and three. Constructing simple and semisimple evolution algebras

Abstract

This work classifies three-dimensional simple evolution algebras over arbitrary fields. For this purpose, we use tools such as the associated directed graph, the moduli set, inductive limit group, Zariski topology and the dimension of the diagonal subspace. Explicitly, in the three-dimensional case, we construct some models ${}_i{\mathbf{III}}_{{\lambda }_1,\dots ,{\lambda }_n}^{p,q}$ of such algebras with $1\le i\le 4$, ${\lambda }_i\in {\mathbb{K}}^{\times }$, $p,q\in \mathbb{N}$, such that any algebra is isomorphic to one (and only one) of the given in the models and we further investigate the isomorphic question within each one. Moreover, we show how to construct simple evolution algebras of higher-order from known simple evolution algebras of smaller size.

Keywords:

• Evolution algebra
• simple algebra
• directed graph
• strongly connected
• tensor product
• moduli set

2020 Mathematics Subject Classifications:

• 17A60
• 17D92
• 05C25

Disclosure statement

No potential conflict of interest was reported by the author(s).

Data Availability Statement

The authors confirm that the data supporting the findings of this study are available within the article.

Additional information

Funding

The authors are supported by the Spanish Ministerio de Ciencia e Innovación through project PID2019-104236GB-I00/AEI/10.13039/501100011033 and by the Junta de Andalucía through projects FQM-336 and UMA18-FEDERJA-119, all of them with FEDER funds.