Cohomologies of [formula omitted] pairs and compatible structures on Poisson algebras

Abstract

In this paper, first we give the cohomology theory of a Poisson algebra with a module (called a PoiMod pair) and study the linear deformation theory of a PoiMod pair. We introduce the notion of a Nijenhuis structure on a PoiMod pair, which gives a trivial linear deformation. Then by adding compatibility conditions between Nijenhuis structures and O-operators, we introduce the notion of an ON-structure on a PoiMod pair and show that an ON-structure gives rise to a hierarchy of pairwise compatible O-operators. Finally we introduce the notions of PN- and ΩN-structures on Poisson algebras and study their relations.