Poisson-Nijenhuis Structure on Lie groupoids from the Invariance's Point of View

TL;DR

This paper explores the relationship between Poisson-Nijenhuis structures on Lie groupoids and their infinitesimal counterparts, providing a theoretical framework and examples from an invariance perspective.

Contribution

It introduces the concept of right- and left-invariant Poisson-Nijenhuis structures on Lie groupoids and establishes a correspondence with structures on Lie algebroids.

Findings

Mutual correspondence between $( abla , extbf{n})$-structures and Poisson-Nijenhuis structures.

Construction of Poisson-Nijenhuis pair groupoids as examples.

Framework for understanding invariance in Poisson-Nijenhuis structures.

Abstract

In this paper, we introduce right-invariant (similarly, left-invariant) Poisson-Nijenhuis Structures on Lie groupoids and their infinitesimal counterparts as called $(\Lambda , \mathbf{n})-$structures. We present a mutual correspondence between $(\Lambda ,\mathbf{n})-$structures on Lie algebroids with Poisson-Nijenhuis structures $(\Pi , \mathbf{N})$ on their Lie groupoids under some conditions. Also, we will construct Poisson-Nijenhuis pair groupoids as an example from the invariance's point of view.