Deformations of Nijenhuis Lie algebras and Nijenhuis Lie algebroids

This paper is the second in a series dedicated to the operadic study of Nijenhuis structures, focusing on Nijenhuis Lie algebras and Nijenhuis geometry. We introduce the concept of homotopy Nijenhuis Lie algebras and establish that the differential graded (=dg) operad $\mathfrak{NjL}_{\infty}$ governing these structures serves as the minimal model of the operad $\mathfrak{NjL}$ for Nijenhuis Lie algebras. We construct an $L_\infty$-algebra that encodes the simultaneous deformations of Lie brackets and Nijenhuis operators, leading to the deformation cochain complex and an associated cohomology theory for Nijenhuis Lie algebras. Extending these ideas to geometry, we investigate the deformations of geometric Nijenhuis structures. We introduce the notion of a Nijenhuis Lie algebroid-a Lie algebroid equipped with a Nijenhuis structure, which generalizes the classical Nijenhuis structure on vector fields of manifolds. Using the framework of dg manifolds, we construct an $L_\infty$-algebra that governs the simultaneous deformations of Lie algebroid structures and Nijenhuis operators. As a computational application, we prove that a certain class of Nijenhuis operators satisfies the Poincar\'e Lemma, meaning its cohomology vanishes, which confirms a conjecture by Bolsinov and Konyaev.