Operad profiles of Nijenhuis structures

Recently S. Merkulov [S.A. Merkulov, Operads, deformation theory and F-manifolds, in: Frobenius manifolds, in: Aspects Math., vol. E36, Vieweg, Wiesbaden, 2004, pp. 213-251; S.A. Merkulov, Nijenhuis infinity and contractible differential graded manifolds, Compos. Math. 141 (5) (2005) 1238-1254; S.A. Merkulov, Prop profile of Poisson geometry, Comm. Math. Phys. 262 (2006) 117-135] established a new link between differential geometry and homological algebra by giving descriptions of several differential geometric structures in terms of algebraic operads and props. In particular he described Nijenhuis structures as corresponding to representations of the cobar construction on the Koszul dual of a certain quadratic operad. In this paper we prove, using the PBW-basis method of E. Hoffbeck [E. Hoffbeck, A Poincaré-Birkhoff-Witt criterion for Koszul operads, arXiv:0709.2286v3 [math.AT], 2008], that the operad governing Nijenhuis structures is Koszul, thereby showing that Nijenhuis structures correspond to representations of the minimal resolution of this operad. We also construct an operad such that representations of its minimal resolution in a vector space V are in one-to-one correspondence with pairs of compatible Nijenhuis structures on the formal manifold associated to V.