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

In this paper, we introduce right-invariant (similarly, left-invariant) Poisson-Nijenhuis Structures on Lie groupoids and their infinitesimal counterparts as called $(\mathrm{\Lambda },\mathbf{n})-$structures. We present a mutual correspondence between $(\mathrm{\Lambda },\mathbf{n})-$structures on Lie algebroids with Poisson-Nijenhuis structures $(\mathrm{\Pi },\mathbf{N})$ 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.