A note on complete evolution algebras (2512.12418v1)

Abstract: This short note provides positive answers to two conjectures of Camacho, Khudoyberdiyev, and Omirov on the classification of complete evolution algebras. Our approach is based on analysing the solution set of a generic non-linear polynomial system of equations using elementary tools from algebraic geometry. We also obtain new results on subalgebras and idempotents of evolution algebras, and conclude by proposing a conjecture that may characterise solvable evolution algebras.

• The paper establishes the existence of nontrivial polynomial solutions that guarantee proper subalgebras and idempotents in evolution algebras.
• The paper classifies complete evolution algebras as direct sums of one-dimensional idempotent algebras, maximal nilpotent components, and zero algebras.
• The paper introduces a conjecture linking solvability with the absence of idempotents, suggesting a new structural framework for evolution algebras.

Complete Evolution Algebras: Resolution of Classification Conjectures

Introduction

Evolution algebras form a class of commutative, generally non-associative, algebras introduced to model genetic systems, and more specifically non-Mendelian inheritance phenomena. Their defining property is the existence of a natural basis $\{e_i\}$ with pairwise orthogonality under multiplication: $e_i e_j = 0$ for $i \neq j$. The ongoing classification problem for evolution algebras is challenging due to the lack of closure under subalgebra formation and the failure of the usual Wedderburn-Weil structure theory. This essay synthesizes the contributions of "A note on complete evolution algebras" (2512.12418), which establishes results on the structure and classification of complete evolution algebras, resolves two standing conjectures, and proposes a structural conjecture concerning solvability.

Algebraic-Geometric Framework for Existence Results

A critical technical component underpinning the classification results is the analysis of solutions to a specific nonlinear polynomial system parameterized by the structure matrix $A \in GL_n(\mathbb{C})$ of a regular evolution algebra. The system

$$x_i^2 = \sum_{j=1}^n a_{ij} x_j, \quad \text{for all } i$$

is reinterpreted in the language of algebraic geometry: the system describes the intersection of $n$ quadrics in $\mathbb{C}^n$. The authors prove that for any invertible $A$, there always exists a solution $(x_1, \ldots, x_n)$ with at least two nonzero components, using arguments based on the irreducibility of the associated projective hypersurfaces and degree counting via Bézout-type theorems. This is strictly weaker than the original conjecture—which demanded all coordinates nonzero—but, crucially, suffices for the structural consequences regarding subalgebras and idempotents. The role of this algebraic-geometric machinery is central: the existence of nontrivial solutions corresponds to the guarantee of existence of nontrivial subalgebras and idempotent elements in evolution algebras with invertible structure matrices.

Structural Consequences for Evolution Algebras

Nontrivial Subalgebras

The authors show that every complex evolution algebra, whether regular or not, contains a nontrivial proper subalgebra. For abelian (square-zero) algebras, every subspace is a subalgebra. For regular algebras ($E = E^2$), the existence of a nontrivial one-dimensional subalgebra is guaranteed by the existence of a nontrivial solution to the aforementioned polynomial system, leveraging the invertibility of the structure matrix. For intermediate cases, the proper nontrivial subalgebra $E^2$ always exists. This result demonstrates the impossibility of simple evolution algebras (those lacking any nontrivial proper ideal) also being simple as algebras, further clarifying the lattice of subalgebras in evolution algebras.

Classification of Complete Evolution Algebras

The concept of complete evolution algebras concerns the extendibility of natural bases: an algebra is complete if every subalgebra admits a natural basis that extends to one of the entire algebra. Addressing two conjectures posed by Camacho, Khudoyberdiyev, and Omirov, the authors establish:

• No complex regular evolution algebra of dimension $>1$ is complete.
• Every non-nilpotent complete evolution algebra is, up to isomorphism, a direct sum of a one-dimensional idempotent algebra, possibly an evolution algebra of maximal nilpotency, and a zero algebra:

$$\{e_1^2 = e_1\} \oplus \mathbb{C}^{n-1} \quad \text{or} \quad \{e_1^2 = e_1\} \oplus \tilde{E} \oplus \mathbb{C}^{n-s-1}$$

where $\tilde{E}$ has maximal index of nilpotency.

The proofs rely fundamentally on the algebraic-geometric existence theorem, primarily because nontrivial idempotents which cannot be part of a natural basis preclude completeness except in the explicitly characterized forms. Notably, it is sufficient that some solution to the structure equation has support of size at least two.

Collectively, these results yield a comprehensive classification of complete evolution algebras, confirming that they are necessarily constrained to being direct sums of simple one-dimensional idempotent subalgebras, maximal nilpotent components, and zero algebras.

Existence and Role of Idempotents

By analyzing the idempotency condition in terms of the structure matrix, the authors show that any regular evolution algebra always admits an idempotent element, as the existence of a nontrivial solution to the equation

$$M_B(E)^t (x_1^2, \ldots, x_n^2)^T = (x_1, \ldots, x_n)^T$$

corresponds to the existence of nontrivial idempotents. This connects the existence of idempotents tightly to the underlying field and the regularity of the algebra.

Conjecture on Solvability and the Structure of Idempotents

The paper closes with a conjecture proposing the equivalence of solvability (defined via vanishing derived sequence), absence of idempotents, and nonexistence of nontrivial solutions to the associated system for idempotents. The conjecture is verified for dimensions one and two by direct examination and classification. This conjecture, if established, would provide a structural characterization of solvable evolution algebras in terms of absence of idempotents, which is not in parallel with the available results for nilpotent evolution algebras.

Implications and Future Directions

These results significantly refine the structural theory surrounding complete evolution algebras and sharpen the boundary between regularity, subalgebra lattices, and completeness. The reliance on basic algebraic geometry for existential results points towards fruitful interactions between non-associative algebraic structures and classic algebraic geometry, especially in the context of idempotent subvarieties. The conjecture regarding the equivalence of solvability and the absence of idempotents, if resolved, would fill a notable gap in the landscape of evolution algebra theory, directly impacting the understanding of their representation theory and module categories. Extensions to fields of positive characteristic or non-algebraically closed fields remain open and present distinct phenomena, as highlighted by the explicit real counterexample given.

Conclusion

This work provides definitive resolutions of prior conjectures regarding the classification and structure of complete evolution algebras over $\mathbb{C}$, leveraging a careful analysis of nonlinear algebraic systems associated with their structure constants. The explicit link between the existence of subalgebras, idempotents, and algebraic solvability broadens foundational understanding and sets the stage for further structural and field-dependent investigations into evolution algebras.