Complete Evolution Algebras Overview

• Complete evolution algebras are commutative, non-associative algebras defined by natural bases that satisfy eₖeₗ = 0 for k ≠ l, modeling hereditary dynamics.
• They employ a nonlinear polynomial system and structure matrix analysis to detect one-dimensional subalgebras and resolve key classification conjectures.
• Their rigorous decomposition into fundamental types provides actionable insights on idempotents, solvability, and applications in mathematical genetics.

A complete evolution algebra is a special class of commutative, generally non-associative, algebras introduced to capture specific structural and hereditary extension properties with respect to natural bases and subalgebras. This concept is central in the algebraic study of systems with discrete, independently evolving components, notably in mathematical genetics and related dynamical frameworks. The modern understanding of complete evolution algebras draws from rigorous algebraic geometry and systemic classification efforts, culminating in their exhaustive characterisation and the resolution of key conjectures about their structure.

1. Foundational Definitions and Characterisation

Let $\K$ be a field and $E$ a commutative, possibly non-associative, $\K$-algebra. An evolution algebra is defined by the existence of a distinguished basis

$B = \{ e_1,\ldots,e_n \}$

(called a natural basis) such that

$e_i e_j = 0 \quad \text{for all}\ i\ne j.$

Each basis vector satisfies

$e_i^2 = \sum_{j=1}^n a_{ij}e_j,\quad a_{ij}\in\K,$

and the matrix $M_B(E) = (a_{ij})_{1\le i,j\le n}$ is termed the structure matrix. A subalgebra $F\subseteq E$ is said to admit a natural basis if some subset of $B$ spans $F$, and admits a natural basis that extends to one of $E$ if this subset extends to a full natural basis of $E$. The algebra $E$ is complete if every subalgebra of $E$ admits a natural basis which extends to a natural basis of $E$ (García-Martínez et al., 13 Dec 2025).

2. Obstructions and a Polynomial System

Completeness of a regular evolution algebra (i.e., $E=E^2$ and $A=M_B(E)\in\GL_n(\K)$) is governed by the existence of one-dimensional subalgebras which do not extend to a natural basis. For $u = x_1e_1+\cdots+x_ne_n$, the requirement that $u$ spans a subalgebra translates algebraically to $u^2 = \lambda u$ for some $\lambda\in\K$. Explicitly,

$$u^2 = \sum_{i=1}^n x_i^2 e_i^2 = \sum_{i=1}^n x_i^2\sum_{j=1}^n a_{ij}e_j,$$

reduces (up to scaling) to the nonlinear polynomial system

$$\begin{pmatrix} x_1^2 \ x_2^2 \ \vdots \ x_n^2 \end{pmatrix} = A \begin{pmatrix} x_1 \ x_2 \ \vdots \ x_n \end{pmatrix}.$$

The existence of a solution $(x_1,\ldots,x_n)\in\K^n$ with all $x_i\ne0$ identifies a one-dimensional subalgebra not contained in a natural basis extension, certifying incompleteness (García-Martínez et al., 13 Dec 2025).

3. Structure Theorems and Resolution of CKO Conjectures

The classification of complete evolution algebras was consolidated by García-Martínez and Pérez-Rodríguez through analysis of the polynomial system delineated above. Two conjectures of Camacho, Khudoyberdiyev, and Omirov (2019) are resolved as follows:

• Conjecture 5.2: Every regular, complex evolution algebra of dimension $>1$ is not complete. This follows because the system above always admits a solution with at least two nonzero coordinates when $A\in\GL_n(\C)$ and $n>1$. This solution corresponds to a one-dimensional subalgebra that cannot be extended to a natural basis, obstructing completeness.
• Conjecture 5.3: The only non-nilpotent, complete $n$-dimensional evolution algebras over $\C$ are isomorphic to either

$\{e_1^2 = e_1\} \oplus \C^{n-1}$

or

$\{e_1^2 = e_1\} \oplus \widetilde E \oplus \C^{n-s-1},$

where $\widetilde E$ is an $s$-dimensional evolution algebra of maximal nilpotency index and $\C^k$ denotes the $k$-dimensional zero algebra. This characterisation hinges on the algebraic-geometric arguments showing that the existence of a nontrivial solution with two or more nonzero coordinates is generic (García-Martínez et al., 13 Dec 2025).

The general classification, including nilpotent cases, is thus:

$\{e_1^2 = e_1\},\quad \widetilde E \oplus \C^{n-s},\quad \{e_1^2 = e_1\} \oplus \C^{n-1},\quad \{e_1^2 = e_1\} \oplus \widetilde E \oplus \C^{n-s-1}$

where $\widetilde E$ has maximal nilpotency index.

4. Subalgebras, Extension, and Decomposition

Subalgebras of a complete evolution algebra inherit completeness if and only if they are direct sums of complete coordinate subalgebras. For a complete evolution algebra $E$ with unique decomposition

$E \cong \left(\{e_1^2 = e_1\}\right)^{\epsilon_1} \oplus \widetilde E^{\epsilon_2} \oplus \C^{\epsilon_3},\quad \epsilon_i\in\{0,1\},$

any subalgebra $F\subseteq E$ is complete if and only if $F = F_1\oplus F_2\oplus F_3$, with each $F_i$ equal to either $0$ or the full corresponding summand. Consequently, every subalgebra of a complete evolution algebra decomposes as a direct sum of coordinate subalgebras of the four fundamental types (García-Martínez et al., 13 Dec 2025).

5. Idempotents and Regularity

An element $u\in E$ is an idempotent ($u^2 = u$) if and only if for $u = \sum x_i e_i$, the equation

$$A^t \begin{pmatrix} x_1^2 \ \vdots \ x_n^2 \end{pmatrix} = \begin{pmatrix} x_1 \ \vdots \ x_n \end{pmatrix}$$

is satisfied. In the regular case ($A\in\GL_n$), this coincides structurally with the system characterising one-dimensional subalgebras. Hence, every finite-dimensional regular complex evolution algebra admits a nonzero idempotent, establishing that no such algebra of dimension $>1$ can be idempotent-free (García-Martínez et al., 13 Dec 2025).

6. Solvable Evolution Algebras: Conjectural Characterisation

An evolution algebra $E$ is solvable if its derived series

$$E^{(1)}=E,\quad E^{(k+1)}=E^{(k)}E^{(k)},\quad \exists N:\ E^{(N)}=0$$

eventually vanishes. A necessary condition for solvability is the absence of nonzero idempotents, since any idempotent persists through all derived products. The authors propose the converse:

• For a finite-dimensional complex evolution algebra $E$, the following are equivalent:
  1. $E$ is solvable.
  2. $E$ admits no nonzero idempotents.
  3. The equation for idempotents

  $$A^t \begin{pmatrix} x_1^2 \ \vdots \ x_n^2 \end{pmatrix} = \begin{pmatrix} x_1 \ \vdots \ x_n \end{pmatrix}$$

  has only the trivial solution $(0,\ldots,0)$.

Verification holds in dimension $1$ trivially and in dimension $2$ follows from explicit classification. Should this conjecture be validated, it would represent the first purely algebraic or polynomial-equation criterion for solvability in complex evolution algebras (García-Martínez et al., 13 Dec 2025).

7. Classification in Low Dimensions

For real (i.e., $\K = \R$), two-dimensional evolution algebras, full classification up to isomorphism is achieved with seven distinct types. Structure constants, automorphism groups, and derivation algebras are catalogued explicitly (Bekbaev, 2017). The tabulation below summarizes the nontrivial two-dimensional real evolution algebras and key invariants.

Algebra Family	Multiplicative Structure	Automorphism Group
$E_1(b,c)\ (bc\ne1)$	$e_1^2 = e_1 + b e_2$, $e_2^2 = c e_1 + e_2$	$\{I\}$; $\{I, P\}$ if $b=c$
$E_2(b)$	$e_1^2 = e_1 + b e_2$, $e_2^2 = e_2$	$\{I\}$ or matrices if $b=0$
$E_3$	$e_1^2 = e_1 - e_2$, $e_2^2 = e_1 + e_2$	$\{I\}$
$E_4$	$e_1^2 = e_1$, $e_2^2 = 0$	$\operatorname{diag}(1, t)$
$E_5$	$e_1^2 = e_1 - e_2$, $e_2^2 = -e_1$	$\{I\}$
$E_6$	$e_1^2 = e_2$, $e_2^2 = 0$	Parametrized family (see above)
$E_7$	$e_1^2 = 0$, $e_2^2 = e_1$	Parametrized family (see above)

No two families are isomorphic, and the exhaustive listing affirms the diversity of possible local structures (Bekbaev, 2017).

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The theoretical landscape of complete evolution algebras is now sharply delimited: their existence is severely constrained under regularity (invertibility of the structure matrix), and their structure is rigidly specified by a handful of explicit types. This structural rigidity underpins current understanding and further highlights the role of properly formulated algebraic and geometric criteria in classifying and analysing hereditary algebraic systems.