Evolution algebras: a survey of results and a list of open problems
Ivan Kaygorodov, Cándido Martín González, Antonio Viruel
math.RA
Oct 15, 2026 · v1
TL;DR
New observations about evolution algebras (automorphism groups, idempotent counts) were formalized and proved in Lean 4 with Mathlib, assisted by the Aristotle AI system.
Abstract
Evolution algebras, introduced by J. P. Tian as an algebraic model of non- Mendelian inheritance, are commutative algebras with a basis whose distinct elements multiply to zero. The first part of the paper surveys the theory: low-dimensional classifica- tions, structure theory, nilpotency, automorphisms and derivations, directed graphs and random walks, infinite-dimensional and dynamical aspects, and embedding theorems. The second part is a unified list of fifty open questions and eleven new conjectures, each accompanied by the available evidence; two of the conjectures and one of the questions are settled here. We also prove several new observations, most of them formally verified in the proof assistant Lean (see Section 2.1.11): every finite group is the automorphism group of a simple evolution algebra over every field, and | Aut(A)|/ dim A takes every positive ratio- nal value on simple evolution algebras over every field; infinitude of the automorphism group of nilpotent algebras, the possible numbers of idempotents of real perfect evolution algebras, a Galois-theoretic criterion for the existence of idempotents in dimension two, the average number of idempotents over finite fields, and a perfect evolution algebra whose automorphism group is a non-split extension.
Problem
Evolution algebras are commutative algebras with a natural basis whose distinct elements multiply to zero. The theory is scattered across the literature, and many questions about these algebras remain open.
Approach
The paper surveys low-dimensional classifications, structure theory, nilpotency, automorphisms, graphs and random walks, and embedding theorems. It then compiles fifty open questions and eleven conjectures, each with the available evidence. Several new observations are proved, and most of them are formalized in Lean 4 with Mathlib, with the AI system Aristotle assisting.
Results
Every finite group is the automorphism group of a simple evolution algebra over every field, and |Aut(A)|/dim A takes every positive rational value on such algebras. Further results cover idempotent counts, a Galois-theoretic criterion for idempotents in dimension two, and a non-split automorphism-group extension. Two conjectures and one question are settled.
