Generalized Reynolds operators on 3-Lie algebras and NS-3-Lie algebras

Abstract

In this paper, first, we introduce the notion of a generalized Reynolds operator on a [Formula: see text]-Lie algebra [Formula: see text] with a representation on [Formula: see text]. We show that a generalized Reynolds operator induces a 3-Lie algebra structure on [Formula: see text], which represents on [Formula: see text]. By this fact, we define the cohomology of a generalized Reynolds operator and study infinitesimal deformations of a generalized Reynolds operator using the second cohomology group. Then we introduce the notion of an NS-[Formula: see text]-Lie algebra, which produces a 3-Lie algebra with a representation on itself. We show that a generalized Reynolds operator induces an NS-[Formula: see text]-Lie algebra naturally. Thus NS-[Formula: see text]-Lie algebras can be viewed as the underlying algebraic structures of generalized Reynolds operators on [Formula: see text]-Lie algebras. Finally, we show that a Nijenhuis operator on a 3-Lie algebra gives rise to a representation of the deformed 3-Lie algebra and a 2-cocycle. Consequently, the identity map will be a generalized Reynolds operator on the deformed 3-Lie algebra. We also introduce the notion of a Reynolds operator on a [Formula: see text]-Lie algebra, which can serve as a special case of generalized Reynolds operators on 3-Lie algebras.