Conservative algebras of $2$-dimensional algebras, III

TL;DR

This paper proves that all local and 2-local derivations and automorphisms on conservative 2-dimensional algebras are actually derivations and automorphisms, respectively, strengthening the understanding of their structural properties.

Contribution

It establishes that local and 2-local derivations and automorphisms coincide with derivations and automorphisms in conservative 2-dimensional algebras, providing new insights into their algebraic structure.

Findings

All local and 2-local derivations are derivations.

All local and 2-local automorphisms are automorphisms.

Results apply specifically to conservative algebras of 2-dimensional algebras.

Abstract

In the present paper we prove that every local and $2$-local derivation on conservative algebras of $2$-dimensional algebras are derivations. Also, we prove that every local and $2$-local automorphism on conservative algebras of $2$-dimensional algebras are automorphisms.