Cogroups in algebras over an operad

The cogroup objects of the category are the structures associated to the representable functors GR(-)=Mor(R,-) with values in the category of groups. The classical commutative Hopf algebras can be interpreted as cogroup objects in the category of commutative algebras. I have proved that the main results of the theory of commutative algebras in characteristic zero extends to the cogroup objects in the categories of algebras over an operad. More specifically, I have established: a structure theorem, which asserts that every (graded, complete) algebra over an operad P equipped with a cogroup structure necessarily forms a free P-algebra (see [1998a,1998b]); a Lie theorem, which gives an equivalence between the category of cogroups in algebras over P and the category of algebras over an operad P! associated to P (see [1998d]). For the classical operads P = Com, As, Lie, associated to the categories of commutative, associative and Lie algebras, we respectively have Com! = Lie, As = As!, Lie! = Com. For P = Com, I retrieve with these statements the result of the Poincaré-Birkhoff Witt theorem and of the Milnor-Moore theorem of commutative Hopf algebras.

Cohomology of Poisson algebras

A Poisson variety is a (differential, algebraic) variety X whose algebra of regular functions OX is equipped with an operation {-,-} : OX ⊗ OX → OX that generalizes the Lie bracket of Hamiltonian vector fields on a symplectic manifold. The classical Lichnerowicz-Poisson cohomology is well suited for the study of Poisson structures and their deformations on smooth varieties, but certain constructions, such as the quotient X/G of a Poisson variety X under an action of a finite group G, naturally return Poisson varieties with singularities. By using the theory of operads, I have defined a Poisson cohomology theory HPois(OX) (as well as a Poisson homology theory), which reduces to the Lichnerowicz-Poisson cohomology for smooth Poisson varieties, but which is still well suited for the study of Poisson structures on singular varieties because this cohomology embodies cohomological information that reflects the existence of deformations of the variety itself in the presence of singularities (see [1998c,2006a]). By using operadic methods, I have also obtained a normal form for the Poisson structures that are associated to complete intersections with isolated singularities (see [2002b]).

Operads and homotopical algebra

The Quillen homotopical algebra gives a conceptual framework to apply the methods of algebraic topology in problems of algebra and, in the converse direction, to construct algebraic models of the homotopy of spaces and hence to work out questions of topology through algebra methods. The discovery of the Koszul duality of operads by Ginzburg and Kapranov gives a way to apply methods of homotopical algebra effectively in the operad framework. A significant outcome of the Koszul duality theory is the definition of new homology theories associated to operads. A significant defect of the original theory is that it can only be used in the characteristic zero setting. I have undertaken a series of research to work out the problems which one encounters outside this context. I have obtained general results which prove: how the divided power operations and other classical polynomial operations, such as the Frobenius operation of restricted Lie algebras, have a unified interpretation in the framework of operads [1997a,2000a]; how the Koszul duality extends to operads defined over an arbitrary ground ring [2004a] (see also [2009b] for applications); and how one can apply the methods of homotopical algebra to modules over operads in order to define effective models of functors between algebra categories [2009a]. I have also studied the applications of methods of homotopical algebra for the categories of algebras over a PROP [2010a] (the PROPs are structures, more general than operads, which are used to model categories of bialgebras).

Operads and models of the homotopy of spaces

I have also studied the applications of operads in positive characteristic p>0 for the definition of models of the p-complete homotopy type of topological spaces and for the definition of generalizations of the Quillen and Sullivan models of the rational homotopy theory. This research involve the notion of an En operad, n=1,2,…,∞, which is used to describe a hierarchy of homotopy commutative structures. In a first step, I have defined, in a series of work with Clemens Berger [2002a,2004b], an explicit E algebra structure on the cochain algebra of spaces C*(X) (see also [2016a] for an application of this result in general homotopical algebra). In a second step, I have proved in a series of work [2003a,2010b,2011c] (see also [2014a]) that the cohomology of the iterated loop spaces ΩnX (the spaces of based maps f : Sn → X with the sphere Sn as domain) can be determined by applying to this cochain algebra C*(X) a natural cohomology theory associated with En operads. I have also studied a version of this cohomology theory with coefficients in a joint work with Stephanie Ziegenhagen [2016b].

E_n operads and their applications

The class of En operads is formally defined as the class of operads that are homotopy equivalent to the operads of little n-discs when we work in the category of topological spaces, and to the class of operads that are quasi-isomorphic to the operads of singular chains on the little n-discs spaces when we work in the category of chain complexes. My most recent research concerns the homotopy of these objects (see [2019a] for a survey).

I have notably proved, after building a rational homotopy theory of operads (see also [2018a] for this subject), that the group of homotopy automorphism classes of an E2 operads is isomorphic, in the rational framework, to the rational Grothendieck–Teichmüller group GT(Q) (the group introduced by Drinfeld in the theory of quantum groups, see [2017a]). I have also obtained a graph complex description of the rational homotopy of the space of maps from an Em operad to an En operad, for all n≥m≥1, as well as an intrinsic formality theorem which proves that the En operads are determined by their homology up to rational homotopy equivalence (collaborations with Victor Turchin and Thomas Willwacher [2017pre,2018b,2020a], see also [2011b] for other computations of operadic mapping spaces, and [2011a,2013a] for applications of the Koszul duality theory for the study of En operads).