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

• Published in: Linear & multilinear algebra 2025-02, Vol.73 (3), p.507-524
• Main Authors: Casado, Yolanda Cabrera, Barquero, Dolores Martín, González, Cándido Martín, Tocino, Alicia
• Format: Article
• Language: English
• Subjects: directed graph; Evolution algebra; moduli set; simple algebra; strongly connected; tensor product
• ISSN: 0308-1087; 1563-5139
• DOI: 10.1080/03081087.2024.2352452

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,\ldots,\lambda _n}^{p,q} $ i III λ 1 , ... , λ n p , q of such algebras with $ 1\le i\le 4 $ 1 ≤ i ≤ 4 , $ \lambda _i\in {\mathbb {K}}^\times $ λ i ∈ K × , $ p,q\in {\mathbb {N}} $ p , q ∈ 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.