Bruno Vallette

Homology of generalized partition posets

Abstract We define a family of posets of partitions associated to an operad. We prove that the operad is Koszul if and only if the posets are Cohen–Macaulay. On the one hand, this characterization allows us to compute completely the homology of the posets. The homology groups are isomorphic to the Koszul dual cooperad. On the other hand, we get new methods for proving that an operad is Koszul.

Deformation theory of representations of prop(erad)s II

Abstract In this paper and its follow-up [Merkulov and Vallette, J. reine angew. Math.], we study the deformation theory of morphisms of properads and props thereby extending Quillens 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. To do so, we endow the category of prop(erad)s with a model category structure. We provide a complete study of models for prop(erad)s. A new effective method to make minimal models explicit, that extends the Koszul duality theory, is introduced and the associated notion is called homotopy Koszul. As a corollary, we obtain the (co)homology theories of (al)gebras over a prop(erad) and of homotopy (al)gebras as well. Their underlying chain complex is endowed with an L ∞-algebra structure in general and a Lie algebra structure only in the Koszul case. In particular, we make the deformation complex of morphisms from the properad of associative bialgebras explicit. For any minimal model of this properad, the boundary map of this chain complex is shown to be the one defined by Gerstenhaber and Schack. As a corollary, this paper provides a complete proof of the existence of an L ∞-algebra structure on the Gerstenhaber-Schack bicomplex associated to the deformations of associative bialgebras.

A Koszul duality for props

The notion of PROP models the operations with multiple inputs and multiple outputs, acting on some algebraic structures like the bialgebras or the Lie bialgebras. We prove a Koszul duality theory for PROPs generalizing the one for associative algebras and for operads.

Homotopy Batalin-Vilkovisky Algebras

This paper provides an explicit cobrant resolution of the operad encoding Batalin{Vilkovisky algebras. Thus it denes the notion of homotopy Batalin{Vilkovisky algebras with the required homotopy properties. To dene this resolution, we extend the theory of Koszul duality to operads and properads that are dened 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 Poincar e{Birkho{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 eld 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 cobrant 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.

Manin products, Koszul duality, Loday algebras and Deligne conjecture

Abstract In this article we give a conceptual definition of Manin products in any category endowed with two coherent monoidal products. This construction can be applied to associative algebras, non-symmetric operads, operads, colored operads, and properads presented by generators and relations. These two products, called black and white, are dual to each other under Koszul duality functor. We study their properties and compute several examples of black and white products for operads. These products allow us to define natural operations on the chain complex defining cohomology theories. With these operations, we are able to prove that Delignes conjecture holds for a general class of operads and is not specific to the case of associative algebras. Finally, we prove generalized versions of a few conjectures raised by M. Aguiar and J.-L. Loday related to the Koszul property of operads defined by black products. These operads provide infinitely many examples for this generalized Delignes conjecture.

De Rham cohomology and homotopy Frobenius manifolds

We endow the de Rham cohomology of any Poisson or Jacobi manifold with a natural homotopy Frobenius manifold structure. This result relies on a minimal model theorem for multicomplexes and a new kind of a Hodge degeneration condition.

Free Monoid in Monoidal Abelian Categories

We give an explicit construction of the free monoid in monoidal abelian categories when the monoidal product does not necessarily preserve coproducts. Then we apply it to several new monoidal categories that appeared recently in the theory of Koszul duality for operads and props. This gives a conceptual explanation of the form of the free operad, free dioperad and free properad.

Givental group action on topological field theories and homotopy Batalin–Vilkovisky algebras

Abstract In this paper, we initiate the study of the Givental group action on Cohomological Field Theories in terms of homotopical algebra. More precisely, we show that the stabilisers of Topological Field Theories in genus 0 (respectively in genera 0 and 1) are in one-to-one correspondence with commutative homotopy Batalin–Vilkovisky algebras (respectively wheeled commutative homotopy BV-algebras).

Homotopy Operadic Algebras

When a chain complex is equipped with some compatible algebraic structure, its homology inherits this algebraic structure. The purpose of this chapter is to show that there is some hidden algebraic structure behind the scene. More precisely if the chain complex contains a smaller chain complex, which is a deformation retract, then there is a finer algebraic structure on this small complex. Moreover, the small complex with this new algebraic structure is homotopy equivalent to the starting data. The operadic framework with the Koszul duality theory enables us to state explicitly this transfer of structure result.

Methods to Prove Koszulity of an Operad

This chapter extends to the operadic level the various methods, obtained in Chap. 4, to prove that algebras are Koszul. They rely either on rewriting systems, PBW and Grobner bases, distributive laws (Diamond Lemma), or combinatorics (partition poset method). The notion of shuffle operad plays a key role in this respect. We also introduce the Manin products constructions for operads.