On nilpotent index and dibaricity of evolution algebras

An evolution algebra corresponds to a quadratic matrix A of structural constants. It is known the equivalence between nil, right nilpotent evolution algebras and evolution algebras which are defined by upper triangular matrices A. We establish a criterion for an n-dimensional nilpotent evolution algebra to be with maximal nilpotent index 2n-1+1. We give the classification of finite-dimensional complex evolution algebras with maximal nilpotent index. Moreover, for any s=1,…,n-1 we construct a wide class of n-dimensional evolution algebras with nilpotent index 2n-s+1. We show that nilpotent evolution algebras are not dibaric and establish a criterion for two-dimensional real evolution algebras to be dibaric.