Groupe de travail sur les Prop et les opérades

Montpellier, les 22 et 23 mai 2008

Programme

Exposés :

Jeudi 22 mai
• 11h15 : I. Moerdijk (Utrecht) : A Milnor-Moore theorem for Lie-Rinehart algebras
• 14h10 : H. Hoffbeck (Lille) : Un critère de Poincaré-Birkhoff-Witt pour les opérades
• 15h00 : J. Granaaker (Stockholm) : Unimodular L-infinity algebras
• 16h30 : B. Vallette (Nice) : Deformation theory of morphisms of props

Vendredi 23 mai
• 10h00 : I. Moerdijk (Utrecht) : The Boardman-Vogt resolution of operads
• 10h50 : H. Strohmayer (Stockholm) : Prop profile of bi-Hamiltonian structures
• 14h00 : J. Milles (Nice) : André-Quillen cohomology of algebras over an operad
• 15h30 : S. Merkulov (Stockholm) : Wheeled pro(p)file of Batalin-Vilkovisky quantization

• J. Granaaker (Stockholm) : Unimodular L-infinity algebras: We consider the wheeled operad of unimodular Lie algebras and give an explicit minimal resolution, representations of which we call unimodular L-infinity algebras. Geometrically, such an algebra corresponds to a homological vector field together with an invariant measure. We also give explicit formulae for homotopy transfer, define the deformation complex, and give a cohomological obstruction to extending a finite dimensional L-infinity algebra to a unimodular L-infinity algebra.
• H. Hoffbeck (Lille) : Un critère de Poincaré-Birkhoff-Witt pour les opérades: Le but de l'exposé est de donner un critère effectif pour montrer qu'une opérade est de Koszul. Ce critère généralise celui introduit par Priddy pour les algèbres. Il définit ce qu'il appelle une base de Poincaré-Birkhoff-Witt : une base monomiale possédant certaines propriétés par rappport à une relation d'ordre, assurant qu'une algèbre est de Koszul. On adapte la notion de base PBW aux opérades.
• S. Merkulov (Stockholm) : Wheeled pro(p)file of Batalin-Vilkovisky quantization: Using technique of wheeled props we establish a correspondence between the homotopy theory of unimodular Lie 1-bialgebras and the famous Batalin-Vilkovisky formalism. Solutions of the so called quantum master equation satisfying certain boundary conditions are proven to be in 1-1 correspondence with representations of a wheeled dg prop which, on the one hand, is isomorphic to the cobar construction of the prop of unimodular Lie 1-bialgebras and, on the other hand, is quasi-isomorphic to the dg wheeled prop of unimodular Poisson structures. These results allow us to apply properadic methods for computing formulae for a homotopy transfer of a unimodular Lie 1-bialgebra structure on an arbitrary complex to the associated quantum master function on its cohomology. We extend Losev-Mnev's quantum simplicial BF theory to the case of unimodular Lie 1-bialgebras (and, eventually, to the case of unimodular Poisson structures) and show that the Feynman integrals computing the effective action of this new BF theory describe precisely homotopy transfer formulae obtained within the wheeled properadic approach to the quantum master equation. Quantum corrections (which are present in our BF model to all orders of the Planck constant) correspond precisely to what are often called ``higher Massey products" in the homological algebra.
• J. Milles (Nice) : André-Quillen cohomology of algebras over an operad: Following the ideas of Quillen and by means of model category structures, Goerss and Hopkins developed a cohomology theory for (simplicial) algebras over a (simplicial) operad. Using Koszul duality theory, we make explicit these theories in the differential graded framework. We recover some known theories (Hochschild cohomology for associative algebras, Chevalley-Eilenberg cohomology for Lie algebras, ...) and we describe new ones (homotopy associative algebras and homotopy Lie algebras for instance). We will also study the general properties of such cohomology theories. They will be related to the properties of an important object: the cotangent complex.
• I. Moerdijk (Utrecht) : A Milnor-Moore theorem for Lie-Rinehart algebras: Lie-Rinehart algebras arise naturally as the algebraic counterpart of Lie algebroids (which are the infinitesimal structures related to Lie groupoids). I will discuss to what extent the enveloping algebra of a Lie-Rinehart algebra carries a structure like that of a Hopf algebra, and discuss a Milnor-Moore type theorem for these structures. (The talk is based on a joint paper with J. Mrcun, available on the ArXiv.)
• I. Moerdijk (Utrecht) : The Boardman-Vogt resolution of operads:I will discuss how to extend a classical resolution for topological operads, due to Boardman and Vogt, to operads in an arbitrary monoidal model category. This extended resolution is cofibrant in the model structure on operads in such a monoidal model category. Many know resolutions, such as the Godement resolution and the bar-cobar resolution, arise as special cases of this generalised Boardman-Vogt construction. (The talk is based on a joint paper with C. Berger, Topology 45 (2006), 807-849.)
• H. Strohmayer (Stockholm) : Prop profile of bi-Hamiltonian structures: Recently S.~Merkulov established a new link between differential geometry and algebraic topology by giving a propic description of several differential geometric structures. In particular he described Poisson geometry using Lie^1Bi-algebras, i.e. Lie-bialgebras with the degree of the bracket and cobracket differing by one. We will briefly describe the link in this case. A bi-Hamiltonian structure is a manifold endowed with two compatible Poisson structures. We will construct a prop from two Lie^1Bi-algebras and show that representations of its minimal model are in one-to-one correspondence with bi-Hamiltonian structures.
• B. Vallette (Nice) : Théorie de déformation des morphismes de props: Pour tout morphisme de props, nous définirons, à la Quillen, un complexe de chaines qui mesure les déformations de ce morphisme. Lorsque le prop but est le prop des endomorphismes d'un module A, ceci définit la théorie homologique des déformations de A comme (bi)gèbre sur le prop source. Nous retrouvons de cette manière les différents complexes de chaines de la littérature : (co)homologie de Hochschild des algèbres associatives, de Chevalley-Eilenberg des algèbres de Lie, de Harrison des algèbres commutatives, de Lecomte-Roger des bigèbres de Lie, de Gerstenhaber-Schack des bigèbres associatives. Grâce à ce point de vue, nous monterons que ce complexe de chaines est toujours une algèbre de Lie à homotopie près (stricte dans le cas Koszul). Les solutions de Maurer-Cartan généralisées correspondent alors aux structures déformées de (bi)gèbre sur A. De plus, nous construirons des opérations supérieures (non binaires) agissant sur ce complexe qui généralisent les opérations braces du complexe de Hochschild. Ceci nous permettra de montrer une version généralisée de la conjecture de Deligne.