Right invariant Poisson Nijenhuis structures on Lie groupoids Correspondence and Classification

TL;DR

This paper introduces right-invariant Poisson-Nijenhuis structures on Lie groupoids, establishes a correspondence with their Lie algebroid counterparts, and provides illustrative examples to demonstrate these concepts.

Contribution

It presents a novel framework for Poisson-Nijenhuis structures on Lie groupoids and their infinitesimal Lie algebroid structures, including a one-to-one correspondence.

Findings

Established a correspondence between structures on Lie groupoids and Lie algebroids.

Defined right-invariant Poisson-Nijenhuis structures on Lie groupoids.

Provided illustrative examples of these structures.

Abstract

In this paper, we introduce right-invariant Poisson-Nijenhuis Structures on Lie groupoids and their infinitesimal counterparts as called (Poisson bivector, Nijenhuis operator) structures. Also, we present a one-to-one correspondence between (Poisson bivector, Nijenhuis operator) structures on Lie algebroids with (Poisson, Nijenhuis) structures on their Lie groupoids under certian conditions. Also, we give some illustrative examples .