We introduce and study wheeled PROPs, an extension of the theory of PROPs which can treat traces and, in particular, solutions to the master equations which involve divergence operators. We construct a dg free wheeled PROP whose representations are in one-to-one correspondence with formal germs of SP-manifolds, key geometric objects in the theory of Batalin-Vilkovisky quantization. We also construct minimal wheeled resolutions of classical operads Com and As s as rather non-obvious extensions of Com ∞ and As s∞, involving, e.g., a mysterious mixture of associahedra with cyclohedra. Finally, we apply the above results to a computation of cohomology of a directed version of Kontsevich’s complex of ribbon graphs.

We find a minimal differential graded (dg) operad whose generic representations in R n are in one-to-one correspondence with formal germs of those endomorphisms of the tangent bundle to R n which satisfy the Nijenhuis integrability condition. This operad is of a surprisingly simple origin — it is the cobar construction on the quadratic operad of homologically trivial dg Lie algebras. As a by product we obtain a strong homotopy generalization of this geometric structure and show its homotopy equivalence to the structure of contractible dg manifold. 1. Introduction. Extended

We review definitions and basic properties of operads, PROPs and algebras over these structures.

We show how the machine invented by S. Merkulov [19, 20, 22] can be used to study and classify natural operators in differential geometry. We also give an interpretation of graph complexes arising in this context in terms of representation theory. As application, we prove several results on classification of natural operators acting on vector fields and connections.

This paper is dedicated to Jean-Louis Loday on the ocasion of his 60th birthday with admiration and gratitude

This paper is dedicated to Jean-Louis Loday on the occasion of his 60th birthday with admiration and gratitude Structures of Lie algebras, Lie coalgebras, Lie bialgebras and Lie quasibialgebras are presented as solutions of Maurer-Cartan equations on corresponding governing differential graded Lie algebras using the big bracket construction of Kosmann-Schwarzbach. This approach provides a definition of an L∞-(quasi)bialgebra (strongly homotopy Lie (quasi)bialgebra). We recover an L∞-algebra structure as a particular case of our construction. The formal geometry interpretation leads to a definition of an L ∞ (quasi)bialgebra structure on V as a differential operator Q on V, self-commuting with respect to the big bracket. Finally, we establish an L∞-version of a Manin (quasi) triple and get a correspondence theorem with L∞-(quasi) bialgebras. 1. Introduction. Algebraic structures are often defined as certain maps which must satisfy quadratic relations. One of the examples is a Lie algebra structure: a Lie bracket satisfies the Jacobi identity (indeed the Jacobi identity is a quadratic relation since the bracket appears twice in each summand). Other examples include an associative multiplication (the associativity condition is quadratic), L ∞ and