Recently S.A. Merkulov, building on work of Losev and Mnev established a connection between the homotopy theory of unimodular Lie 1-bialgebras and Batalin-Vilkovisky formalism via the theory of wheeled properads. In this paper we extend this connection to unimodular quasi-Lie 1-bialgebras, thus widening the class of models treatable with the methods of wheeled properads.

Recently, Willwacher showed that the Grothendieck-Teichmuller group GRT acts by L-infinity-automorphisms on the Schouten algebra of polyvector fields T_poly(R^d) on affine space R^d. In this article, we prove that a large class of L-infinity-automorphisms on the Schouten algebra, including Willwacher's, can be globalized. That is, given an L-infinity-automorphism of T_poly(R^d) and a general smooth manifold M with the choice of a torsion-free connection, we give an explicit construction of an L-infinity-automorphism of the Schouten algebra T_poly(M) on the manifold M, depending on the chosen connection. The method we use is the Fedosov trick.

Recently, Willwacher showed that the Grothendieck-Teichmuller group GRT acts by L-infinity-automorphisms on the Schouten algebra of polyvector fields T-poly(R-d) on affine space R-d. In this article, we prove that a large class of L-infinity-automorphisms on T-poly(R-d), including Willwacher's, can be globalized. That is, given an L-infinity-automorphism of T-poly(R-d) and a general smooth manifold M with the choice of a torsion-free connection, we give an explicit construction of an L-infinity-automorphism of the Schouten algebra T-poly(M) on the manifold M, depending on the chosen connection. The method we use is the Fedosov trick.

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 Poincare-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.