Power-Associative Evolution Algebras

Abstract

The paper is devoted to the study of evolution algebras that are power-associative algebras. We give the Wedderburn decomposition of evolution algebras that are power-associative algebras and we prove that these algebras are Jordan algebras. Finally, we use this decomposition to classify these algebras up to dimension six.

Evolution train algebras

“…Let E = F e 1 ⊕ ker ω be a finite-dimensional power-associative evolution train algebra, where e 2 1 = e 1 + z with z ∈ ker ω, z ∈ ann(ker ω) and e 1 ker ω = 0. Then e = e 2 1 is the unique idempotent of E and the sub-algebra generated by e have the extension property [7,Lemme 5]. This result is also verified by direct calculation.…”

“…22 )e 4 , σ(e 2 ) = σ 22 e 2 + σ 24 e 4 , σ(e 3 ) = σ 2 22 e 3 , σ(e 4 ) = σ 4 22 e 4 with a 13 (1 + σ 22 ) = 0. In ( [7]), the authors give the classification of evolution nil-algebras of nil-index 2, 3 and 4. In the particular case of nil-index 4, they give the classification only for the power-associative algebras.…”

“…Evolution algebras are commutative ( [8]) and they are associative if and only if e 2 i e j = 0 for all 1 ≤ i = j ≤ n ( [7]).…”

A Characterization of Associative Evolution Algebras

“…Regarding the associativity of the algebra and as already mentioned in the Introduction, evolution algebras are not associative in general and only a small number of them have this property. This might be the reason why the topic of the associativity of the algebras of evolution has been scarcely dealt with by researchers until now, since there are quite few references in the literature about it: [19] and [24] are the most representative. In the first of them, authors deal with power-associative evolution algebras, which are those evolution algebras verifying the property (x•x)•(x•x) = ((x•x)•x)•x, ∀x ∈ E (and the same occurs performing e i by itself several times).…”

“…Examples of usual topics in the literature, which have proven to be a very convenient approach, are the study of derivation spaces [11][12][13][14][15] and the classification of some family of evolution algebras sharing interesting properties. For example, nilpotent evolution algebras are characterized in [16][17][18] and power-associative evolution algebras are classified in [19], up to dimension six. Three dimensional real evolution algebras with condition dim(E 2 ) = 1 are analyzed in [20], while in [21] the authors study the case of four dimensional perfect non-simple evolution algebras.…”