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 $i\mathbf{III}{\lambda_1,\ldots,\lambda_n}^{p,q}$ 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.