Hierarchies and compatibility on Courant algebroids

TL;DR

This paper generalizes hierarchies from Poisson-Nijenhuis manifolds to Courant algebroids, introducing new notions like Poisson-Nijenhuis and Nijenhuis pairs, and demonstrates how hierarchies arise through successive deformations of these structures.

Contribution

It extends the concept of hierarchies from Poisson-Nijenhuis manifolds to Courant algebroids by defining new structures and showing their natural hierarchies via deformation.

Findings

Introduction of Poisson-Nijenhuis, deformation-Nijenhuis, and Nijenhuis pairs for Courant algebroids

Construction of natural hierarchies through successive deformation by $(1,1)$-tensors

Unified framework for structures on Courant algebroids extending classical notions

Abstract

We extend to the context of Courant algebroids several hierarchies that can be constructed on Poisson-Nijenhuis manifolds. More precisely, we introduce several notions (Poisson-Nijenhuis, deformation-Nijenhuis and Nijenhuis pairs) that extend to Courant algebroids the notion of a Poisson-Nijenhuis manifold, by using the idea that both the Poisson and the Nijenhuis structures, although they seem to be different in nature when considered on manifolds, are just $(1,1)$-tensors on the usual Courant algebroid $TM \oplus T^*M $ satisfying several constraints. For each of the generalizations mentioned, we show that there are natural hierarchies obtained by successive deformation by one of the $(1,1)$-tensor.