Evolution algebras: a survey of results and a list of open problems

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.