Abstracts of talks
Tensor functors and operads
Marcelo Aguiar (Texas A&M)
We study functors between tensor categories that preserve various types of algebraic structures, particularly bialgebras. The prototypical example is afforded by the Alexander-Whitney and Eilenberg-Zilber maps in the context of simplicial sets and chain complexes. We discuss an analogous construction in the context of species and graded vector spaces that is the basis for our applications to combinatorial Hopf algebras in the second lecture.
Hopf monoids in species and associated Hopf algebras
Marcelo Aguiar (Texas A&M)
We continue the study of tensor functors from species to graded vector spaces and explain how they can be used to construct "combinatorial" Hopf algebras in a unified manner. This relates to recent work of Patras, Livernet, Reutenauer, and Schocker. We explain the role played by the various operations on the category of species such as the Hadamard, Dirichlet, and substitution products, in constructing Hopf monoids in species. The categorical approach yields uniform deformations and higher dimensional generalizations of all these objects.
The Kashiwara-Vergne conjecture and the eulerian idempotent
Emily Burgunder (Montpellier, France)
By using the interplay of the Eulerian idempotent and the Dynkin idempotnet, we construct explicitly a particular symmetric solution (F,G) of the first equation of the Kashiwara-Vergne conjecture:
x+y-log(exp(y)exp(x))=(1-exp(-adx))F(x,y)+(exp(ady)-1)G(x,y). Then, we explicit all the solutions of the equation in the completion of the free Lie algebra generated by two indeterminates x and y thanks to the kernel of the Dynkin idempotent.
Pre-Lie algebras
Muriel Livernet (Paris 13, France)
We give the definition of preLie algebras and some examples. We describe the underlying operad in terms of rooted trees and link the free preLie algebra with the Connes-Kreimer Hopf algebra. We'll explain Foissy's technics which prove that some Lie algebras of primitive elements are free. We explore the homology theory for preLie algebras and prove that the operad defining preLie algebras is Koszul. We also give a rigidity theorem for preLie algebras and introduce the notion of nonassociative permutative algebras.
Combinatorial "multi" Hopf algebras
Thomas Lam (Harvard University) and Pavlo Pylyavskyy (MIT)
In his study of the K-theory of the Grassmannian, Buch used set-valued tableaux to define a Hopf algebra the associated graded of which is the Hopf algebra Sym. Inspired by his work, we define and study such "multi" analogues of the Hopf algebras MPR, QSym and NSym.
Operads and props
Jean-Louis Loday (Strasbourg, France)
We introduce the basic notions "type of algebras", nonsymmetric operad, cyclic operad, prop, properad.
Koszul duality
Jean-Louis Loday (Strasbourg, France)
Conceptual treatment of Koszul duality of quadratic algebras, putting algebra and coalgebra on the same footing. Generalization to operads.
Generalized bialgebras
Jean-Louis Loday (Strasbourg, France)
A generalized bialgebra is a vector space equipped with some algebra structure, some coalgebra structure entwined by some compatibility relation. We prove a structure theorem for generalized bialgebra types. It says that a connected generalized bialgebra is completely determined by its primitive part. This structure theorem covers several known cases (PBW and CMM theorems in the classical case of cocommutative-associative bialgebras, for instance) and many new ones. Several of these examples lead to combinatorial Hopf algebras. (Reference: ArXiv: math.QA/0611885)
Wheeled operads and props
Serguei Merkulov (Stockholm, Sweden)
We discuss wheeled PROPs, an extension of the theory of PROPs which can treat traces and, in particular, solutions to the master equations which involve divergence operators. One of the motivating examples is a dg free wheeled PROP whose representations are in one-to-one correspondence with formal germs of SP-manifolds, key geometric objects in the theory of Batalin-Vilkovisky quantization. Another non-trivial examples include new minimal wheeled resolutions of classical operads Com and Ass which are rather non-obvious extensions of Com∞ and As∞, involving, e.g., a mysterious mixture of associahedra with cyclohedra.
Formality and quantization of Lie bialgebras
Serguei Merkulov (Stockholm, Sweden)
Using theory of props we prove a formality theorem associated with universal quantizations of (strongly homotopy) Lie bialgebras.
Representations of towers of algebras and combinatorial Hopf algebras
Jean-Christophe Novelli (Marne-la-Vallée, France)
On the moduli space of extensions of algebras
Michael Penkava (Eau Claire)
Differential combinatorial Hopf algebras
Maria Ronco (Valparaiso, Chile)
We consider the boundary morphism on the space spanned by the set of faces of the permutahedra, which we denote P∞. We give the relationship of the boundary with the elementary operations which define the product and the coproduct on P∞, and we look at the restriction to the space spanned by planar rooted trees.
In particular, we describe the behaviour of the boundary morphism on the primitive space of the associated combinatorial Hopf algebras. In the particular case of some tridendriform trialgebras we get a structure of weak G∞-algebra on the primitive elements.
Lie idempotnets
Jean-Yves Thibon (Marne-la-Vallee, France)
Lie idempotents are symmetrizers which project the tensor algebra onto the free Lie algebra. Almost all known examples turn out to belong to the descent algebras of symmetric groups. This makes it possible to analyze them in terms of noncommutative symmetric functions. By extending various classical techniques of the theory of ordinary symmetric functions, it is then possible to produce many new examples, and in some sense, to classify all the possibilities.
Koszul duality for operads and algebraic combinatorics
Bruno Vallette (Nice, France)
We make Koszul duality of operads explicit and we describe the relationship with the theory of posets. For instance, we explain conceptually the form of the homology of the partition poset. We show how to prove that an operad is Koszul with the help of posets and we compute the homology of partition types posets in terms of Koszul dual (co)operads. We also give applications to generating series.
Prop(erad)s and types of bialgebras
Bruno Vallette (Nice, France)
A properad is a direct generalization of the notion of operad and is used to model categories of algebraic structures with multiple inputs and multiple outputs. We extend Koszul duality to the category of properads. We use it to introduce a cohomology theory for generalized bialgebras of a given type and to construct the type of bialgebras up to homotopy.
Mapping class groups operads
Nathalie Wahl (Copenhagen, Denmark)
A surprising result of Tillmann says that the mapping class groups of surfaces give rise to an operad that detects infinite loop spaces without being an E∞ operad. After reviewing the dimension 2 case, I will describe anologous operads in dimension 3 and explore the consequences of Tillmann's result in that case.
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