The relationship between local derivations and local automorphisms of some associative algebras

TL;DR

This paper investigates local derivations and automorphisms of specific five-dimensional nilpotent associative algebras, revealing their structures, relationships, and positive solutions to related algebraic problems.

Contribution

It characterizes the forms of local derivations and automorphisms for these algebras and establishes their Lie algebra structures, advancing understanding of local automorphism and derivation properties.

Findings

Local automorphisms and derivations differ from global automorphisms and derivations.

Sets of local derivations form Lie algebras with respect to Lie brackets.

Positive solutions to several algebraic problems related to local derivations are provided.

Abstract

In the present paper, local derivations and local automorphisms of five-dimensional naturally graded nilpotent associative algebras are studied. Namely, a general form of the matrices of local derivations and local automorphisms of algebras $\pi_2$ and $\pi_3$ is clarified. It turns out that the general form of the matrix of an automorphism (derivation) on these algebras does not coincide with the local automorphism's (resp. local derivation's) matrix's general form on these algebras. Therefore, these associative algebras have local automorphisms (resp. local derivations) that are not automorphisms (resp. derivations). We also establish a relationship between local automorphisms and local derivations via an exponential expression. We prove that the sets of local derivations of algebras $\pi_2$ and $\pi_3$ form Lie algebras with respect to the Lie brackets. Thus, we show that the Lie algebra problem from the Ayupov-Eldique-Kudaybergenov problems for local derivations of the algebras under consideration has a positive solution. The remaining problems from the Ayupov-Eldique-Kudaybergenov problems also have a positive solution for algebras $\pi_2$ and $\pi_3$.