Evolution Algebras: Formal Deformations

• Evolution algebras are non-associative algebras with a natural basis where off-diagonal products vanish, enabling systematic formal deformations.
• Their deformations are introduced via power series expansions that incorporate cohomological obstructions, paralleling classical Gerstenhaber theory.
• Explicit low-dimensional examples show that every finite-dimensional evolution algebra admits nontrivial deformations, revealing a rich degeneration structure.

Evolution algebras are a class of non-associative, commutative algebras distinguished by the existence of a "natural basis" $\{e_1, \ldots, e_n\}$ such that for all $i \neq j$, the product $e_i e_j = 0$, and $e_i^2$ is an arbitrary linear combination of basis elements. This structure is motivated by connections to non-Mendelian genetics, as well as formal algebraic questions. The study of formal deformations—perturbations of the multiplication structure parameterized by a formal variable—reveals both the flexibility and geometric complexity inherent to evolution algebras. The systematic treatment of their formal deformations incorporates cohomological obstructions and provides a parallel with classical deformation theory as developed for associative and Lie algebras. Recent work provides a full deformation-theoretic and degeneration-theoretic framework for evolution algebras in finite dimension, with explicit results up to dimension four (Makhlouf et al., 7 Dec 2025).

1. Algebraic Structure and Formal Deformations

An evolution algebra $E = (V, \mu)$ over a field $\mathbb{K}$ is defined by a bilinear map $\mu: V \times V \to V$ for which there exists a basis $B = \{e_1, \ldots, e_n\}$ such that $\mu(e_i, e_j) = 0$ whenever $i \neq j$. In this basis, $$e_i^2 = \mu(e_i, e_i) = \sum_{j=1}^n \omega_{ij} e_j$$; the $\omega_{ij}$ are called the structure constants, and the matrix $M_B(E) = (\omega_{ij})$ is the structure matrix.

A formal deformation of $E$ is realized by considering the formal power series ring $\mathbb{K}[[t]]$ and extending $V$ to $V[[t]] = V \otimes_{\mathbb{K}} \mathbb{K}[[t]]$. The deformed multiplication is defined as a $\mathbb{K}[[t]]$-bilinear series: $\nu_t = \mu + \sum_{k \geq 1} t^k \nu_k,$ where each $\nu_k$ is a bilinear map with $\nu_k(e_i, e_j) = 0$ for $i \neq j$, ensuring the preservation of a natural basis at each order. This construction yields a family of evolution algebra structures over $\mathbb{K}[[t]]$ with the same basis $B$, providing a canonical setup for the study of deformation phenomena (Makhlouf et al., 7 Dec 2025).

2. Cohomological Framework and Infinitesimal Deformations

Classifying deformations involves analyzing the "infinitesimal" part $\nu_1$ of a deformation, which resides in

$$\mathcal{Z}^2(V) = \{ \theta \in \mathrm{Bil}_{\mathbb{K}}(V \times V, V) \mid \theta(e_i, e_j) = 0 \text{ for } i \neq j \}.$$

The space of 2-coboundaries is defined as

$$\mathcal{B}^2(E) = \operatorname{Im}(\delta_\mu) \cap \mathcal{Z}^2(V),$$

where

$$\delta_\mu \phi (u, v) = \phi(\mu(u, v)) - \mu(u, \phi(v)) - \mu(\phi(u), v), \quad \phi \in \operatorname{End}_\mathbb{K}(V).$$

The second cohomology group,

$$\mathcal{H}^2(E) = \mathcal{Z}^2(V) / \mathcal{B}^2(E),$$

classifies inequivalent infinitesimal deformations: two first-order deformations $\nu_t = \mu + t\nu_1 + \cdots$, $\lambda_t = \mu + t\lambda_1 + \cdots$ are equivalent if and only if $\lambda_1 - \nu_1 = \delta_\mu(\phi_1)$ for some $\phi_1$ (Makhlouf et al., 7 Dec 2025).

3. Rigidity, Existence, and Explicit Classification in Low Dimensions

A central result (Theorem 3.16) is that no finite-dimensional evolution algebra is formally rigid: every such algebra admits a nontrivial first-order deformation. This sharply contrasts with the rigidity detected in many classes of Lie and associative algebras.

Explicit cohomology computations for two-dimensional evolution algebras over $\mathbb{C}$ yield detailed descriptions of $\mathcal{B}^2(E)$ and all equivalence classes of infinitesimal deformations. For example, for the nilpotent algebra $E_1$ defined by $e_1^2 = e_1$, $e_2^2 = 0$: $$\mathcal{B}^2(E_1) = \operatorname{span} \left\{ \begin{bmatrix} 1 & 0 \ 0 & 0 \end{bmatrix},\, \begin{bmatrix} 0 & 1 \ 0 & 0 \end{bmatrix} \right\},$$ with inequivalent infinitesimal deformations: $$\operatorname{InfDef}(E_1) = \left\{ \begin{bmatrix} 1 & 0 \ 0 & 0 \end{bmatrix} + t \begin{bmatrix} 0 & 0 \ \alpha & \beta \end{bmatrix} : \alpha, \beta \in \mathbb{C} \right\}.$$ For the nilpotent type $\mu_{2,2}$ with $e_1^2=e_2,\, e_2^2=0$,

$$\operatorname{InfDef}(\mu_{2,2}) = \left\{ \begin{bmatrix} 0 & 1 \ 0 & 0 \end{bmatrix} + t \begin{bmatrix} \alpha & 0 \ \beta & \gamma \end{bmatrix} : \alpha, \beta, \gamma \in \mathbb{C} \right\}.$$

In dimension two, all higher-order obstructions vanish; thus, every infinitesimal deformation integrates to a full formal deformation. The classification in higher dimensions (3 and 4) is not fully resolved, but the existence of nontrivial deformations is established for all cases (Makhlouf et al., 7 Dec 2025).

4. Formal Degenerations and Partial Orders

Formal degenerations relate evolution algebras by contraction processes rather than deformations. Such a degeneration is realized by a continuous family $g_t \in GL(n, \mathbb{K})$ (for $t \in (0, 1]$) mapping natural bases to natural bases, with the limiting structure

$\lambda = \lim_{t \to 0} (g_t \cdot \mu)$

existing on all squares $e_i^2$. Every evolution algebra degenerates, for example, to the abelian zero-product algebra via $g_t = t^{-1} \mathrm{Id}$. A formal degeneration $\mu \to \lambda$ induces a relation on isomorphism classes, which in general is not transitive. The transitive closure, written $\mu \Rightarrow \lambda$, generates a poset structure among evolution algebras (Makhlouf et al., 7 Dec 2025).

Hasse Diagrams in Low Dimensions

The explicit degeneration relations among nilpotent evolution algebras of low dimension are represented as Hasse diagrams.

Dimension	Algebras (Isomorphism Classes)	Degeneration Relations
2	$\mu_{2,2}$ (rank 1), $\mu_{2,1}$ (0)	$\mu_{2,2} \to \mu_{2,1}$
3	$\mu_{3,1}, \ldots, \mu_{3,4}$	$$\mu_{3,4} \to \mu_{3,3} \to \mu_{3,2} \to \mu_{3,1}$$; $\mu_{3,4} \to \mu_{3,2}$
4	$\mu_{4,1},\ldots,\mu_{4,12}$	See Figure 1.1 (Makhlouf et al., 7 Dec 2025), with explicit chains among $\mu_{4,1}$ to $\mu_{4,7}$; $\mu_{4,12}$ is rigid

Each arrow in these diagrams is realized by an explicit $g_t$ whose diagonal or triangular form is constructed from monomial functions of $t$, and codifies an explicit contraction from one algebra to another.

5. Theoretical Implications and Contrast with Other Algebraic Classes

Unlike semisimple associative or Lie algebras, whose formal rigidity is a hallmark of their structure, evolution algebras are universally nonrigid: every finite-dimensional case admits nontrivial deformations. The cohomological classification via $\mathcal{H}^2(E)$ parallels the Gerstenhaber theory but with crucial differences in the types of obstructions and the structure of the deformation complex.

The systematic establishment of deformation and degeneration theories for evolution algebras elucidates the diversity of isomorphism classes, their interrelations, and their moduli. The poset structure revealed by Hasse diagrams underscores the rich geometric landscape, particularly in the classification of nilpotent types. The results provide a foundational toolkit for further investigations, especially in higher dimensions or in the context of genetic models informed by evolution algebraic structures (Makhlouf et al., 7 Dec 2025).

6. Prospects and Open Problems

The classification of inequivalent deformations in dimensions higher than two remains open. For $n=3,4$, the existence of nontrivial deformations is confirmed, but explicit descriptions—even of the second cohomology groups—are incomplete. The explicit construction and computation of Hasse diagrams for degenerations in higher dimensions suggest deep combinatorial and algebraic challenges. These developments open further possibilities for the application of evolution algebra deformation theory in the analysis of dynamical systems beyond the context of genetics.

Further directions include the extension of the cohomological framework, exploration of integrability obstructions in higher-order terms for $n>2$, and applications to the study of algebraic dynamical invariants under deformation and degeneration processes (Makhlouf et al., 7 Dec 2025).