Derivations and automorphisms of nilpotent evolution algebras with maximal nilindex

Abstract

In this paper is devoted to nilpotent finite-dimensional evolution algebras E with dim E 2 = dim E − 1. We described Lie algebras associated with evolution algebras whose nilindex is maximal. Moreover, in terms of this Lie algebra we fully construct nilpotent evolution algebra with maximal index of nilpotency. Furthermore, this result allowed us fully characterize all local and 2-local derivations of the considered evolution algebras. All automorphisms and local automorphisms of the nilpotent evolution algebras with maximal nilindex are found.Mathematics Subject Classification: 46S10, 82B26, 12J12, 39A70, 47H10, 60K35.

Characterization theorems for the spaces of derivations of evolution algebras associated to graphs

“…This in turns implies 3d 11 = d 21 , and again from (16) we have 2d 33 = 4d 11 . Therefore, it should be d 11 = 0 which implies d 21 = 0.…”

“…In that work the authors consider K = C and they prove that the space of derivations of n-dimensional evolution algebras with non-singular matrices is zero; in addition, they describe the space of derivations of evolution algebras with matrices of rank n − 1. More recently, [16] describes the space of derivations of evolution algebras with maximal nilindex restricted to the case where char(K) = 0.…”

On evolution algebras and their derivations

“…This means a j−1j = 0 and a ij = 0 (i = j − 1) for all 4 j k 1 and (k + 1) 11 are inferred. Otherwise, the columns (k 1 + 1) th and (m 1 + 1) th become zero, which contradicts RankA = n − 2.…”

“…For any algebra, the space of all its derivations is Lie algebra with respect to the commutator multiplication. In the theory of non-associative algebras, particularly, in genetic algebras, the Lie algebra of derivations of a given algebra is one of the important tools for studying its structure [2,6,7,[11][12][13]. In this regard, the system of equations that describe the derivations of evolution algebras have been posed in [13].…”

The Derivations of Evolution Algebras under a Given Condition

“…Since then, the derivation of certain algebra is one of the most important tools that need to be verified in its structure. In addition, [2][3][4]13] investigated the derivations of a given algebras.…”