Classification of nilpotent evolution algebras and extensions of their derivations

This article is devoted to n-dimensional nilpotent finite-dimensional evolution algebras (Formula presented.) with maximal index of nilpotency. We are able to find full invariants of these algebras which allows us to classify them. Furthermore, we explicitly describe its associative enveloping algebras. Moreover, we give an affirmative answer to the question: is it possible to extend a given derivation of the nilpotent evolution algebra to its enveloping algebra.