Deformation cohomology of Nijenhuis algebras and applications to extensions, inducibility of automorphisms and homotopy algebras

Our primary aim in this paper is to introduce and study the cohomology of a Nijenhuis operator and of a Nijenhuis algebra. Our cohomology of a Nijenhuis algebra controls the simultaneous deformations of the underlying associative structure and the Nijenhuis operator. We interpret the second cohomology group as the space of all isomorphism classes of abelian extensions. Then we study the inducibility of a pair of Nijenhuis algebra automorphisms in a given abelian extension and show that the corresponding obstruction can be seen as the image of a suitable Wells-type map. We also consider skeletal and strict $2$-term homotopy Nijenhuis algebras and characterize them by third cocycles of Nijenhuis algebras and crossed modules of Nijenhuis algebras, respectively. Finally, we introduce strict homotopy Nijenhuis operators on $A_\infty$-algebras and show that they induce $NS_\infty$-algebras.