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
Résumés :
- 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.