N-derivations for finitely generated graded Lie algebras

TL;DR

This paper investigates conditions under which N-derivations of finitely generated graded Lie algebras are equivalent to derivations, with applications to various important algebraic structures like Schrödinger-Virasoro and Kac-Moody algebras.

Contribution

It provides a sufficient condition for N-derivations to coincide with derivations in finitely generated graded Lie algebras, extending understanding of their algebraic structure.

Findings

N-derivations of Schrödinger-Virasoro algebra are derivations.

N-derivations of generalized Witt algebras are derivations.

N-derivations of Kac-Moody algebras are derivations.

Abstract

$N$-derivation is the natural generalization of derivation and triple derivation. Let ${\cal L}$ be a finitely generated Lie algebra graded by a finite dimensional Cartan subalgebra. In this paper, a sufficient condition for Lie $N$-derivation algebra of ${\cal L}$ coinciding with Lie derivation algebra of ${\cal L}$ is given. As applications, any $N$-derivation of Schr\"{o}dinger-Virasoro algebra, generalized Witt algebras, Kac-Moody algebras and their Borel subalgebras, is a derivation.