On Evolution Algebras

Abstract

The structural constants of an evolution algebra are given by a quadratic matrix. In this work we establish an equivalence between nil, right nilpotent evolution algebras and evolution algebras defined by upper triangular matrices. The classification of 2-dimensional complex evolution algebras is obtained. For an evolution algebra with a special form of the matrix, we describe all its isomorphisms and their compositions. We construct an algorithm running under Mathematica which decides if two finite dimensional evolution algebras are isomorphic.

Power-Associative Evolution Algebras

“…Type Associative N 5,1 = N 4,1 ⊕ N 1,1 e 2 1 = e 2 2 = e 2 3 = e 2 4 = e 2 5 = 0 [5] Yes N 5,2 = N 4,2 ⊕ N 1,1 e 2 1 = e 2 , e 2 2 = e 2 3 = e 2 4 = e 2 5 = 0 [4,1] Yes N 5,3 (α) = N 4,3 (α) ⊕ N 1,1 e 2 1 = e 3 , e 2 2 = αe 3 , e 2 3 = e 2 4 = e 2 5 = 0 with α ∈ F * [3,2] Yes N 5,4 = N 4,4 ⊕ N 1,1 e 2 1 = e 2 , e 2 2 = 0, e 2 3 = e 4 , e 2 4 = e 2 5 = 0 [3,2] Yes N 5,5 (α, β) = N 4,5 (α, β) ⊕ N 1,1 e 2 1 = e 4 , e 2 2 = αe 4 , e 2 3 = βe 4 , e 2 4 = e 2 5 = 0 with α, β ∈ F * [2,3] Yes N 5,6 (α) = N 3,3 (α) ⊕ N 2,2 e 2 1 = e 3 , e 2 2 = αe 3 , e 2 3 = 0, e 2 4 = e 5 , e 2 5 = 0 with α ∈ F * [2,3] Yes N 5,7 = N 4,6 ⊕ N 1,1 e 2 1 = e 2 + e 3 , e 2 2 = e 4 , e 2 3 = −e 4 , e 2 4 = e 2 5 = 0 [2,2,1] No N 5,8 (α, β, γ) e 2 1 = e 5 , e 2 2 = αe 5 , e 2 3 = βe 5 , e 2 4 = γe 5 , e 2 5 = 0 with α, β, γ ∈ F * [1,4] Yes N 5,9 (α, β) e 2 1 = e 4 , e 2 2 = αe 4 + βe 5 , e 2 3 = e 5 , e 2 4 = e 2 5 = 0 with α, β ∈ F * [2,3] Yes N 5,10 (α) e 2 1 = e 2 + e 3 , e 2 2 = e 5 , e 2 3 = −e 5 , e 2 4 = αe 5 , e 2 5 = 0 with α ∈ F * [1,3,1] No . Let E be a power-associative evolution algebra of dimension 6.…”

“…Associative N 6,1 = N 5,1 ⊕ N 1,1 e 2 1 = e 2 2 = e 2 3 = e 2 4 = e 2 5 = e 2 6 = 0 [6] Yes N 6,2 = N 5,2 ⊕ N 1,1 e 2 1 = e 2 , e 2 2 = e 2 3 = e 2 4 = e 2 5 = e 2 6 = 0 [5,1] Yes N 6,3 (α) = N 5,3 (α) ⊕ N 1,1 e 2 1 = e 3 , e 2 2 = αe 3 , e 2 3 = e 2 4 = e 2 5 = e 2 6 = 0 with α ∈ F * [4,2] Yes N 6,4 = N 5,4 ⊕ N 1,1 e 2 1 = e 2 , e 2 2 = 0, e 2 3 = e 4 , e 2 4 = e 2 5 = e 2 6 = 0 [4,2] Yes N 6,5 (α, β) = N 5,5 (α, β) ⊕ N 1,1 e 2 1 = e 4 , e 2 2 = αe 4 , e 2 3 = βe 4 , e 2 4 = e 2 5 = e 2 6 = 0 with α, β ∈ F * [3,3] Yes N 6,6 (α) = N 5,6 (α) ⊕ N 1,1 e 2 1 = e 3 , e 2 2 = αe 3 , e 2 3 = 0, e 2 4 = e 5 , e 2 5 = e 2 6 = 0 with α ∈ F * [3,3] Yes N 6,7 (α, β, γ) = N 5,8 (α, β, γ) ⊕ N 1,1 e 2 1 = e 5 , e 2 2 = αe 5 , e 2 3 = βe 5 , e 2 4 = γe 5 , e 2 5 = e 2 6 = 0 with α, β, γ ∈ F * [2,4] Yes E…”

“…Associative N 6,8 (α, β) = N 5,9 (α, β) ⊕ N 1,1 e 2 1 = e 4 , e 2 2 = αe 4 + βe 5 , e 2 3 = e 5 , e 2 4 = e 2 5 = e 2 6 = 0 with α, β ∈ F * [3,3] Yes N 6,9 (α, β) = N 4,5 (α, β) ⊕ N 2,2 e 2 1 = e 4 , e 2 2 = αe 4 , e 2 3 = βe 4 , e 2 4 = 0, e 2 5 = e 6 , e 2 6 = 0 with α, β ∈ F * [2,4] Yes N 6,10 (α, β) = N 3,3 (α) ⊕ N 3,3 (β) e 2 1 = e 3 , e 2 2 = αe 3 , e 2 3 = 0, e 2 4 = e 6 , e 2 5 = βe 6 , e 2 6 = 0 with α, β ∈ F * [2,4] Yes N 6,11 = N 5,7 ⊕ N 1,1 e 2 1 = e 2 + e 3 , e 2 2 = e 4 , e 2 3 = −e 4 , e 2 4 = e 2 5 = e 2 6 = 0 [3,2,1] No N 6,12 (α) = N 5,10 (α) ⊕ N 1,1 e 2 1 = e 2 + e 3 , e 2 2 = e 5 , e 2 3 = −e 5 , e 2 4 = αe 5 , e 2 5 = e 2 6 = 0 with α ∈ F * [2, 3,1] No N 6,13 (α) = N 5,11 (α) ⊕ N 1,1 e 2 1 = e 2 +e 3 , e 2 2 = e 5 , e 2 3 = −e 5 , e 2 4 = α(e 2 +e 3 ), e 2 5 = e 2 6 = 0 with α ∈ F * [2,2,2] No N 6,14 (α, β) = N 5,12 (α, β) ⊕ N 1,1 e 2 1 = e 2 + e 3 , e 2 2 = e 5 , e 2 3 = −e 5 , e 2 4 = α(e 2 + e 3 ) + βe 5 , e 2 5 = e 2 6 = 0 with α, β ∈ F * [2,2,2] No N 6,15 = N 4,6 ⊕ N 2,2 e 2 1 = e 2 +e 3 , e 2 2 = e 4 , e 2 3 = −e 4 , e 2 4 = 0, e 2 5 = e 6 , e 2 6 = 0 [2,3,1] No N 6,16 (α, β, γ, δ) e 2 1 = e 6 , e 2 2 = αe 6 , e 2 3 = βe 6 , e 2 4 = γe 6 , e 2 5 = δe 6 , e 2 6 = 0 with α, β, γ, δ ∈ F * [1,5] Yes N 6,17 (α, β, γ) e 2 1 = e 5 , e 2 2 = αe 5 + βe 6 , e 2 3 = γe 6 , e 2 4 = e 6 , e 2 5 = e 2 6 = 0 with αβγ = 0 [2,4] Yes N 6,18 (α, β, γ, δ) e 2 1 = e 5 , e 2 2 = αe 5 + βe 6 , e 2 3 = γe 5 + δe 6 , e 2 4 = e 6 , e 2 5 = e 2 6 = 0 with αβ = 0, γδ = 0 and αδ − βγ = 0 [2,4] Yes N 6,19 (α, β) e 2 1 = e 2 + e 3 , e 2 2 = e 6 , e 2 3 = −e 6 , e 2 4 = αe 6 , e 2 5 = βe 6 , e 2 6 = 0 with α, β ∈ F * [1,4,1] No N 6,20 (α, β) e 2 1 = e 2 +e 3 , e 2 2 = e 6 , e 2 3 = −e 6 , e 2 4 = α(e 2 +e 3 ), e 2 5 = βe 6 , e 2 6 = 0 with α, β ∈ F * [1,3,2] No N...…”

A class of nilpotent evolution algebras

“…In [3] (see also [4]) two-dimensional evolution algebras over the complex numbers were classified. For the classification of two-dimensional evolution algebras over the real numbers see [9].…”

The Jacobson radical of an evolution algebra

“…In [18], the evolution algebras have been used to describe the inheritance of a bisexual population and, in this setting, the existence of non-trivial homomorphisms onto the sex differentiation algebra have been studied in [19]. Algebraic notions such as nilpotency and solvability may be interpreted biologically as the fact that some of the original gametes (or generators) become extinct after certain generations, and these algebraic properties have been studied in [8,6,30,36,9,17,12]. Moreover evolution algebras associated with function spaces defined by Gibbs measures on a graph are considered in [29], to provide a natural introduction of thermodynamics in the studying of several systems in Biology, Physics and Mathematics.…”