Squares and associative representations of two dimensional evolution algebras

TL;DR

This paper introduces a geometric square associated with two-dimensional evolution algebras, providing new invariants and exploring associative representations, including cases where such representations do not exist.

Contribution

It defines a unique geometric square invariant for 2D evolution algebras, analyzes their automorphisms, derivations, and associative representations, and identifies exceptional cases lacking faithful associative embeddings.

Findings

The square invariant uniquely characterizes the algebra's structure.

Automorphism groups are described as algebraic groups, providing new invariants.

Some 2D evolution algebras do not admit faithful associative representations.

Abstract

We associate an 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 behaviour 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 raise 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).