Evolution Algebra Idempotents

• Evolution algebras are non-associative structures defined by a natural basis where cross-products vanish and idempotents satisfy a coupled quadratic system.
• Permutation symmetries allow the idempotent system to decouple into cycle-specific subsystems, enabling explicit analysis in low dimensions.
• Applications in genetics and population dynamics use idempotents as fixed points to indicate equilibrium states and bifurcation phenomena in time-evolving models.

An evolution algebra is a commutative, generally non-associative algebra equipped with a basis (the “natural” or “evolution basis”) such that all cross-products of basis elements vanish and each square $e_i^2$ expands as a linear combination of basis elements. Idempotent elements—those $x$ satisfying $x^2 = x$—play a central role in the structure theory of evolution algebras, serving as fixed points of the nonlinear evolution operator and as key invariants in both algebraic and dynamical investigations. Their study provides insight into genetic, dynamical, and combinatorial properties encoded by various classes of evolution algebras.

1. Definition and Characterization of Idempotents

Given an evolution algebra $E$ over a field $K$, with basis $\{e_1,\ldots,e_n\}$ and structure constants $a_{ij}$ defined by $e_i^2 = \sum_{j=1}^n a_{ij} e_j$, an arbitrary element $x = \sum_{i=1}^n x_i e_i$ is idempotent if and only if

$$x_i = \sum_{j=1}^n a_{ji} x_j^2, \qquad i=1,\dots,n.$$

This reduces to a coupled system of quadratic equations known as the idempotent system. In the case of permutation-type evolution algebras determined by permutations $\sigma,\tau\in S_n$, the equations specialize to

$$a_{\sigma^{-1}(k),\,k} \, x_{\sigma^{-1}(k)}^2 \;+\; a_{\tau^{-1}(k),\,k} \, x_{\tau^{-1}(k)}^2 = x_k,\qquad k=1,\dots, n.$$

This system is always satisfied by the zero vector (the trivial idempotent), and potentially by nonzero solutions dependent on the specifics of the structure constants and the permutation cycles (Narkuziyev, 2020).

2. Decomposition via Cycle Structure and Direct Sums

A principal simplification arises in the context of permutation-based evolution algebras. If the pair $(\sigma, \tau)$ decomposes simultaneously into disjoint cycles (after possible conjugation), the entire algebra splits as a direct sum of subalgebras, each supported on the indices of a single cycle: $$E_n(\sigma, \tau) \cong \bigoplus_{i=1}^r E_{k_i}(\sigma_i, \tau_i),$$ where $(\sigma_i, \tau_i)$ act simultaneously on cycle subsets of length $k_i$. The idempotent system for $E_n(\sigma, \tau)$ then decouples, and every idempotent element is represented as a direct sum of idempotents from each cycle component (Narkuziyev, 2020, Narkuziyev, 2018).

The practical implication is that the complex $n$-variable quadratic system is reduced to independent polynomial subsystems of size equal to each cycle length, greatly facilitating explicit analysis in small dimensions.

3. Explicit Solutions in Low Dimensions and Special Cases

For small $n$, explicit classification is possible:

• One-dimensional case: The only nontrivial idempotent is $e_1$ itself when $e_1^2 = e_1$. If $e_1^2=0$, $e_1$ is absolute nilpotent, not idempotent (Khudoyberdiyev et al., 2013).
• Two-dimensional permutation algebra: Let $\sigma=(12), \tau=(12)$, and $a=a_{1,2}, b=a_{2,1}$, then the idempotent equations become:

$$\begin{cases} a x_1^2 + b x_2^2 = x_2 \ b x_1^2 + a x_2^2 = x_1 \end{cases}$$

This decouples into a quartic equation for $x_1$, whose real root structure is controlled by a discriminant; the count of idempotents can be $1$ (trivial), $2$, $3$, or $4$ depending on the values of $a$, $b$ via the cubic discriminant analysis (Narkuziyev, 2020).

• General cycle of length $p$: In permutation-based evolution algebras each $p$-cycle reduces to a $p$-variate system with a similar iterative structure, e.g., in 3-cycles with uniform structure constants, symmetric solutions such as $x_1 = x_2 = x_3 = 0$ (trivial) and $x_1 = x_2 = x_3 = 1/2$ (nonnilpotent) emerge (Narkuziyev, 2020).
• Evolution algebra of a “chicken” population (EACP): For an EACP with $n$ female types and one male type, the idempotent system reduces to $n+1$ equations. In low dimensions, e.g., $\{h_1,r\}$ with $h_1 r = h_1 + r$, exactly two idempotents appear: $0$ and $h_1 + r$ (Ladra et al., 2013).

4. Idempotents in Chains and Dynamical Evolution Algebras

For time-parametrized or dynamic evolution algebras (“chains of evolution algebras”), the idempotent set may evolve as algebra parameters change, leading to bifurcation phenomena:

• In two-dimensional chains, the idempotent set can exhibit critical times $t_c$ at which the number of real idempotents changes—e.g., dropping from four to two as real roots of the idempotent equations coalesce or vanish, with exact thresholds determined by discriminant calculations (e.g., $t_c = \frac{\ln 2}{\ln(\lambda/\mu)}$ in certain parameterizations) (Casas et al., 2010).
• In three-dimensional Chapman-Kolmogorov chains, the bifurcation set for idempotent number is completely controlled by “controller functions” appearing in the structure matrix; generally two idempotents (zero and one nontrivial) appear or disappear at loci where these controller functions vanish (Imomkulov, 2020).
• In indicator-function or periodic chains, idempotents may be created or annihilated along specific loci in time-parameter space, sometimes leading to continuous families at “resonant” parameter values (Rozikov et al., 2012).

5. Classification and Enumeration over Finite Fields

For fields of finite characteristic, the isomorphism classes of idempotent evolution algebras are controlled combinatorially. An $n$-dimensional evolution algebra over $\mathbb{F}_q$ is idempotent if its structure matrix is invertible. Isomorphism types correspond to $G$-orbits under the semidirect product $G = S_n \ltimes T_n$ (monomial group), with $T_n = (\mathbb{F}_q^*)^n$ the torus of admissible basis scalings (Wei et al., 2023): $$N_n(q) = \frac{1}{|G|} \sum_{g \in G} |\operatorname{Fix}_X(g)|,$$ where $X=GL_n(\mathbb{F}_q)$. Closed formulas for $N_n(q)$, the number of isomorphism classes, are derived explicitly for $n=2,3,4$. For $n=2$,

$$N_2(q) = \begin{cases} 2q^2 + 3q + 3 & p=2,\;3\mid(q-1)\ 2q^2 + 3q + 4 & p\neq 2,\;3\mid(q-1)\ q^2 + 2q - 1 & p=2,\;3\nmid(q-1)\ q^2 + 2q & p\neq 2,\;3\nmid(q-1) \end{cases}$$

with analogous explicit structures for higher $n$ (Wei et al., 2023).

6. Idempotents in Constrained and Structured Evolution Algebras

In constrained evolution algebras inspired by population genetics (e.g., bisexual population models with “type 1” preference), the complete set of idempotents can be classified explicitly by analysis of the quadratic operator fixed points. For “hard-constrained” ($0,1$ structure constants) bisexual models, all idempotents are on coordinate faces: $$\mathrm{Idemp} = \left\{ (1,0,\dots,0; y_1,\dots, y_\nu)\ \big|\ \sum_k y_k = 1 \right\} \cup \left\{ (x_1,\dots, x_n; 1,0,\dots, 0)\ \big|\ \sum_i x_i=1 \right\}.$$ In $2\times 2$ constrained settings with general parameters, the idempotent set (fixed points) is determined by solving two quadratic equations in the simplex, which may yield zero, one, two, or infinitely many solutions depending on parameter relations (Dzhumadil'daev et al., 2015).

7. Significance and Algebraic Consequences

• The decomposition into direct sums means that the idempotent element analysis is tractable in many special cases and under permutational symmetries.
• Existence and multiplicity of nontrivial idempotents are highly sensitive to the structure matrix parameters, and bifurcations are generically signaled by vanishing discriminants.
• In algebraic genetics, idempotents correspond to stable type distributions (fixed points of the evolution operator) and may encode equilibrium genetic states or stationary population structures.
• The general coupled quadratic system remains unsolved in full generality for arbitrary structure constants and arbitrary $n$, with explicit classifications beyond $n=2$ generally unavailable, except in the presence of high symmetry or strong algebraic constraints (Narkuziyev, 2020, Khudoyberdiyev et al., 2013).
• In dynamical contexts (chains of evolution algebras), the study of idempotent evolution and bifurcation structure provides a framework for modeling transitions in population dynamics and other time-dependent systems (Imomkulov, 2020, Casas et al., 2010).

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Table: Idempotent System by Algebra Type

Algebra Type	Idempotent Characterization	Key Solution Features
General evolution algebra	$\sum_{i=1}^n a_{ij} x_i^2 = x_j,\ \forall j$	Coupled degree-2 polynomial system
Permutation-type (with cycles $\sigma,\tau$)	Sums involve only cycle-coordinates	System splits per cycle block
EACP ("chicken" population)	$u\sum_i a_{ij}x_i = x_j$, $u\sum_i b_ix_i = u$	Linear in $x$ for fixed $u$
Constrained bisexual populations	Structured quadratic fixed-point system	Often leads to idempotents on coordinate faces

Tables serve as a taxonomy of the recurring algebraic patterns and solution spaces for idempotents across the principal classes of evolution algebras discussed in the literature.