Nijenhuis operators on $n$-Lie algebras

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.

arXiv comment19 pages

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• Left counital Hopf algebra structures on free commutative Nijenhuis algebras
• Deformations and extensions of modified $\lambda$-differential $3$-Lie Algebras
• Reynolds $n$-Lie algebras and NS-$n$-Lie algebras
• Deformations and their controlling cohomologies of $\mathcal{O}$-operators
• Lie n-algebras and cohomologies of relative Rota-Baxter operators on n-Lie algebras