On nilpotent evolution algebras

TL;DR

This paper introduces invariants and constructs specific families of nilpotent evolution algebras to classify them up to dimension five over algebraically closed fields.

Contribution

It defines new invariants related to the upper annihilating series and explicitly constructs families of nilpotent evolution algebras for classification.

Findings

Invariants can be computed from any natural basis.

Families of algebras are explicitly constructed using bilinear forms and endomorphisms.

Complete classification of nilpotent evolution algebras up to dimension five.

Abstract

The type and several invariant subspaces related to the upper annihilating series of finite-dimensional nilpotent evolution algebras are introduced. These invariants can be easily computed from any natural basis. Some families of nilpotent evolution algebras, defined in terms of a nondegenerate symmetric bilinear form and some commuting symmetric diagonalizable endomorphisms relative to the form, are explicitly constructed. Both the invariants and these families are used to review and complete the classification of nilpotent evolution algebras up to dimension five over algebraically closed fields.