Squares and associative representations of two-dimensional evolution algebras

Abstract

We associate a square to any two-dimensional evolution algebra. This geometric object is uniquely determined, does not depend on the basis and describes the structure and the behavior of the algebra. We determine the identities of degrees at most four, as well as derivations and automorphisms. We look at the group of automorphisms as an algebraic group, getting in this form a new algebraic invariant. The study of associative representations of evolution algebras is also started and we get faithful representations for most two-dimensional evolution algebras. In some cases, we prove that faithful commutative and associative representations do not exist, giving rise to the class of what could be termed as “exceptional” evolution algebras (in the sense of not admitting a monomorphism to an associative algebra with deformed product).

On new classes of chains of evolution algebras

“…Remark 2.3. We notice that the classification of the two-dimensional real evolution algebras consists of an alternative of the complex case [5] or the case [3]. E 5 only appears in the real case.…”

“…Different authors performed the classification of two-dimensional evolution algebras over several fields. In [5] for the field of complex numbers, in [2] over a field that is closed under all square and cubic roots, and in [3,9] without restrictions on the underlying field.…”

“…and therefore there is one more case. Moreover, the cases (f), (g) and (h) of [3,Theorem 3.3] correspond to E 6 (0, a 3 ) with a 3 ̸ = 0, E 6 (0, 0), and E 7 (0), respectively.…”

Chains in evolution algebras

“…0 , 1) ? Concerning the second question, by (8) we would have θ 3 (1) = 1. If we take the particular solution θ 2 = 0, then we get…”

“…Recently, a new branch of study has arisen in [18]. Its application in the setting of evolution algebras debuts in [8]: the determination of evolution algebras with a faithful associative and commutative representation. So far, this idea has been pursued only in dimension 2.…”