Natural families in evolution algebras

Abstract

In this paper we introduce the notion of evolution rank and give a decomposition of an evolution algebra into its annihilator plus extending evolution subspaces having evolution rank one. This decomposition can be used to prove that in non-degenerate evolution algebras, any family of natural and orthogonal vectors can be extended to a natural basis. Central results are the characterization of those families of orthogonal linearly independent vectors which can be extended to a natural basis.We also consider ideals in perfect evolution algebras and prove that they coincide with the basic ideals.Nilpotent elements of order three can be localized (in a perfect evolution algebra over a field in which every element is a square) by merely looking at the structure matrix: any vanishing principal minor provides one. Conversely, if a perfect evolution algebra over an arbitrary field has a nilpotent element of order three, then its structure matrix has a vanishing principal minor.We finish by considering the adjoint evolution algebra and relating its properties to the corresponding in the initial evolution algebra.

Chains in evolution algebras

“…A useful criterion for an element z of A with z 2 = 0 to be natural (i.e. embeddable in a natural basis) is that dim(span({e 2 i : i ∈ supp(z)})) = 1 (see [1,Teorema 3.3]). We recall that an evolution algebra A is non-degenerate if there exists a natural basis {e i } such that e 2 i = 0 for every i.…”

“…With this in mind, we have Proposition 2.4. Let A be a non-degenerate evolution algebra, then (1) Let B = {e i } i∈Λ be a natural basis of A. If z, z ′ are different elements such that zz ′ = 0, then supp(z) ∩ supp(z ′ ) has cardinal different from 1 (supports relative to B).…”

“…are isomorphic if and only if M and M ′ are in the same orbit under the action of (K × ) 2 × Z 2 described in Definition 6.4 (1). A moduli set for this class is…”

“…Furthermore, ssi (A) = 1 because, as Soc (A) = I and Soc (A/ Soc (A)) = Soc(Ke 3 ) = 0, we have Soc 2 (A)/ Soc(A) = 0. (a) If ω 13 ω 23 = 0, v can be chosen to be (scaling the basis if necessary) 1 1 . Now, we will see that an evolution algebra with structure matrix is isomorphic to any algebra with structure matrix…”

“…Roughly speaking the study of three-dimensional evolution algebras falls into two disjoint classes: those with nonzero annihilator (sections 3 and 4); and the ones with zero annihilator (sections 5 and 6). The classification of algebras A with nonzero annihilator fall into 5 disjoint cases depending on two parameters: (1) the values of the annihilating stabilizing index (abbreviated asi(A)) and ( 2) the value of dim(ann(A)). In the case asi(A) = dim(ann(A)) = 1 all the algebras come from a construction Adj 1 (B, α) for a suitable evolution algebra B of dimension two (see Theorem 4.2).…”