Nijenhuis operators on $n$-Lie algebras

TL;DR

This paper introduces Nijenhuis operators on $n$-Lie algebras, explores their properties, and provides methods for constructing such operators, contributing to the understanding of algebra deformations.

Contribution

It defines Nijenhuis operators on $n$-Lie algebras, proves their polynomial stability, and offers new construction techniques and examples.

Findings

Polynomial of a Nijenhuis operator remains a Nijenhuis operator

Nijenhuis operators induce trivial deformations of $n$-Lie algebras

Various constructions and examples of Nijenhuis operators are provided

Abstract

In this paper, we study $(n-1)$-order deformations of an $n$-Lie algebra and introduce the notion of a Nijenhuis operator on an $n$-Lie algebra, which could give rise to trivial deformations. We prove that a polynomial of a Nijenhuis operator is still a Nijenhuis operator. Finally, we give various constructions of Nijenhuis operators and some examples.