Abstract. In this paper and its follow-up [MV08], we study the deformation theory of morphisms of properads and props thereby extending Quillen’s deformation theory for commutative rings to a non-linear framework. The associated chain complex is endowed with an L∞-algebra structure. Its Maurer-Cartan elements correspond to deformed structures, which allows us to give a geometric interpretation of these results.

Abstract. The aim of this note, which raises more questions than it answers, is to study natural operations acting on the cohomology of various types of algebras. It contains a lot of very surprising partial results and examples.

This paper provides an explicit cofibrant resolution of the operad encoding Batalin-Vilkovisky algebras. Thus it defines the notion of homotopy Batalin-Vilkovisky algebras with the required homotopy properties. To define this resolution we extend the theory of Koszul duality to operads and properads that are defined by quadratic and linear relations. The operad encoding Batalin– Vilkovisky algebras is shown to be Koszul in this sense. This allows us to prove a Poincaré-Birkhoff-Witt Theorem for such an operad and to give an explicit small quasi-free resolution for it. This particular resolution enables us to describe the deformation theory and homotopy theory of BV-algebras and of homotopy BV-algebras. We show that any topological conformal field theory carries a homotopy BV-algebra structure which lifts the BV-algebra structure on homology. The same result is proved for the singular chain complex of the double loop space of a topological space endowed with an action of the circle. We also prove the cyclic Deligne conjecture with this cofibrant resolution of the operad BV. We develop the general obstruction theory for algebras over the Koszul resolution of a properad and apply it to extend a conjecture of Lian–Zuckerman, showing that certain vertex algebras have an explicit homotopy BV-algebra structure.

Abstract. This paper is the follow-up of [MV08].

The aim of this note, which raises more questions than it answers, is to study natural operations acting on the cohomology of various types of algebras. It contains a lot of very surprising partial results and examples.

Abstract. In this paper, we define the notion of strong homotopy properads and prove that this structure transfers over left homotopy inverses. We give explicit formulae for the induced structure. 1.

Abstract. It is shown that for any morphism, φ: g → h, of Lie algebras the vector space underlying the Lie algebra h is canonically a g-homogeneous formal manifold with the action of g being highly nonlinear and twisted by Bernoulli numbers. This fact is obtained from a study of the 2-coloured operad of formal homogeneous spaces whose minimal resolution gives a new conceptual explanation of both Ziv Ran’s Jacobi-Bernoulli complex and Fiorenza-Manetti’s L∞algebra structure on the mapping cone of a morphism of two Lie algebras. All these constructions are iteratively extended to the case of a morphism of arbitrary L∞-algebras. 1.

Abstract. Lada introduced strong homotopy algebras to describe the structures on a deformation retract of an algebra in topological spaces. However, there is no satisfactory general definition of a morphism of strong homotopy (s.h.) algebras. Given a monad ⊤ on a simplicial category C, we instead show how s.h. ⊤-algebras over C naturally form a Segal space. Given a distributive monad-comonad pair (⊤, ⊥), the same is true for s.h. (⊤, ⊥)-bialgebras over C; in particular this yields the homotopy theory of s.h. sheaves of s.h. rings. There are similar statements for quasi-monads and quasi-comonads. We also show how the structures arising are related to derived connections on bundles.