EVOLUTION ALGEBRAS SATISFYING DEGREE FOUR IDENTITIES NOT IMPLIED BY COMMUTATIVITY AND WITH NO UNIT ELEMENT

The aim of this paper is to study evolution algebras satisfying identity \left\{(x y)^2-x^2 y^2\right\}-2\left\{((x y) x) y+((x y) y) x-\left(y^2 x\right) x-\left(x^2 y\right) y\right\}=0 not implied by commutativity and with no unit element. We prove that the class of power-associative evolution algebras is contained in the given class, and that the latter also admits one and only one finite-dimensional non-nil indecomposable evolution algebra without idempotent. We then present a weighting criterion and a classification, up to isomorphism, in dimension up to three. Finally, we conclude with a derivation study of the aforementioned algebras.