Cohomologies of Reynolds operators on n-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 -Lie algebras and Reynolds 3-Lie algebras from Reynolds n-Lie algebras and Reynolds commutative associative algebras respectively.