Reynolds $n$-Lie algebras and NS-$n$-Lie algebras

TL;DR

This paper introduces Reynolds operators and NS-$n$-Lie algebras, explores their properties, cohomology, and deformations, and constructs higher-dimensional Reynolds $n$-Lie algebras from existing structures.

Contribution

It defines Reynolds operators on $n$-Lie algebras, introduces NS-$n$-Lie algebras as generalizations, and constructs new Reynolds $n$-Lie algebras from known ones and associative algebras.

Findings

Reynolds operators relate to derivations on $n$-Lie algebras.

Cohomology theory for Reynolds operators is developed.

NS-$n$-Lie algebras generalize $n$-Lie and $n$-pre-Lie algebras.

Abstract

In this paper, first we introduce the notion of a Reynolds operator on an $n$-Lie algebra and illustrate the relationship between Reynolds operators and derivations on an $n$-Lie algebra. We give the cohomology theory of Reynolds operators on an $n$-Lie algebra and study infinitesimal deformations of Reynolds operators using the second cohomology group. Then we introduce the notion of NS-$n$-Lie algebras, which are generalizations of both $n$-Lie algebras and $n$-pre-Lie algebras. We show that an NS-$n$-Lie algebra gives rise to an $n$-Lie algebra together with a representation on itself. Reynolds operators and Nijenhuis operators on an $n$-Lie algebra naturally induce NS-$n$-Lie algebra structures. Finally, we construct Reynolds $(n+1)$-Lie algebras and Reynolds $3$-Lie algebras from Reynolds $n$-Lie algebras and Reynolds commutative associative algebras respectively.