On nilpotent evolution algebras

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.