Right-invariant Poisson-Nijenhuis Structure on Lie groupoids

This paper focuses on the introduction of right-invariant Poisson-Nijenhuis structures on Lie groupoids and their infinitesimal counterparts, also known as structures. A Poisson-Nijenhuis structure refers to a combination of a Poisson structure and a Nijenhuis structure. The paper also presents a mutual correspondence between Poisson-Nijenhuis structures on Lie algebroids and Poisson-Nijenhuis structures on their corresponding Lie groupoids, under certain conditions. This correspondence demonstrates a connection between the infinitesimal and global structures. Finally, the paper concludes with an illustrative example to enhance the understanding of the concepts introduced.