Idempotent-free non-solvable evolution algebras over $\mathbb{C}$

Abstract

A recent conjecture states that a finite-dimensional complex evolution algebra is solvable if and only if it has no non-zero idempotents. We exhibit a three-dimensional counterexample over $\mathbb{C}$ whose isomorphism class already appears in the classification of three-dimensional complex evolution algebras. Since the conjecture is known in dimensions one and two, this counterexample has the smallest possible dimension. The algebra is defined over every field of characteristic different from $2$ and is the exceptional member of a one-parameter family of pairwise non-isomorphic non-solvable evolution algebras whose idempotents are determined explicitly. For the exceptional parameter, the derived series stabilises at a non-zero two-dimensional subalgebra whose only idempotent is zero. Direct sums with zero algebras give complex counterexamples in every dimension at least three. We also prove the conjectured equivalence whenever the stable term of the derived series is an evolution algebra.