Squares and associative representations of two-dimensional evolution algebras

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).