Local and $2$-local derivations of simple $n$-ary algebras

TL;DR

This paper proves that all local and 2-local derivations of complex finite-dimensional simple Filippov algebras are derivations, providing a complete description and identifying unique cases like the Malcev algebra $M_8$.

Contribution

It establishes that local and 2-local derivations coincide with derivations in simple Filippov algebras and describes all such derivations, including a unique example in Malcev algebra.

Findings

All local and 2-local derivations of simple Filippov algebras are derivations.

Description of local and 2-local derivations of semisimple Filippov algebras.

Identification of the Malcev algebra $M_8$ as the first example with pure local derivations.

Abstract

In the present paper, we prove that every local and $2$-local derivation of the complex finite-dimensional simple Filippov algebra is a derivation. As a corollary we have the description of all local and $2$-local derivations of complex finite-dimensional semisimple Filippov algebras. All local derivations of the ternary Malcev algebra $M_8$ are described. It is the first example of a finite-dimensional simple algebra that admits pure local derivations, i.e. algebra admits a local derivation which is not a derivation.