The intention of the meeting is to collect people working on the very recent subject of noncommutative symmetric functions, and combinatorial Hopf algebras, which special emphasis on applications in mathematical physics, that is in quantum field theory. The meeting shall gather people from the combinatorics and physics communities to provide an exchange of techniques and ideas. Many interesting mathematical results wait for applications and some techniques of combinatorial quantum field theory are currently under development. It is expected that a wider range of people working on combinatorial Hopf algebras also is addressed. The core of the meeting is formed by 4 lecture courses delivered by outstanding scientists of the particular field. These courses shall start from an medium level and may end at the recent research frontier, perhaps posing key problems to be solved in near future. The lecturers are:

Marcelo Aguiar, Texas A&M University

Jean Louis Loday, IRMA, Strasbourg

Arun Ram, University of Wisconsin-Madison, Wisconsin

The lectures may also allow postgraduate students to become introduced to this fast developing and interesting research field. Furthermore, we intend to offer a limited amount (maximally 12) of slots for contributed talks. These talks should address the mathematics/physics frontier and are intended to present recent research results. However, problems and algorithms are also welcome.

The goal of this course is to survey the field of Combinatorial Hopf algebras from the point of view of an outsider who is not working in the field: what are the main results in the field, the main techniques, the main problems, and the primary applications?

Symmetric functions form a commutative self-dual Hopf algebra based on the set of integer partitions. Its study leads naturally to the introduction of a wealth of Hopf algebras based on many kinds of combinatorial objects: compositions, permutations, tableaux, trees, parking functions, and many others. These algebras are often the same as those encountered in other fields, such as the theory of operads, or renormalization problems in quantum field theory. However, for applications to symetric functions, we need to realize these algebras in terms of an auxiliary set of variables. This leads to different constructions, relying upon analogues of the Robinson-Schensted correspondence and of the plactic monoid.

Each of the classical groups GL(n-1), SO(n) and Sp(n), with n even, may be thought of as subroups of GL(n) that preserve some invariant -a vector, a 2nd rank symmetric tensor and a 2nd rank antisymmetric tensor, respectively. In each case the branching rules from GL(n) to the subgroup are determined by certain series of Schur functions defined by means of generating functions or plethysms. The rules for decomposing tensor products for each of the subgroups are well known. It is shown that each may be derived using the outer Hopf algebra of the ring of symmetric functions. Indeed they are determined by the coproduct of the relevant Schur function series. Other subgroups of GL(n) may be defined as those leaving invariant higher rank tensors of specified symmetry. The corresponding branching rules are once again determined by new series of Schur functions defined by means of plethysms, and it is shown that the decomposition of tensor products is again governed by a coproducts of these series. Amongst the surprises are the fact that these new subgroups may be finite or non-reductive.

Bidendriform bialgebras are bialgebras such that both product and coproduct can be split into two parts satisfying good compatibilities. For example, the Malvenuto-Reutenauer Hopf algebra (also known under the name of Hopf algebra of free quasi symmetric functions) and the non-commutative Connes-Kreimer Hopf algebras of planar decorated rooted trees are bidendriform bialgebras. We prove that all connected bidendriform bialgebras are generated by their primitive elements as a dendriform algebra (bidendriform Milnor-Moore theorem) and then is isomorphic to a Connes-Kreimer Hopf algebra. As a corollary, the Hopf algebra of Malvenuto-Reutenauer is isomorphic to the Connes-kreimer Hopf algebra of planar rooted trees decorated by a certain set. We deduce that the Lie algebra of its primitive elements is free in characteristic zero (G. Duchamp, F. Hivert and J.-Y. Thibon conjecture).

I am going to report on joint work with Frederic Patras on the (free) twisted descent algebra. It has basis indexed by set compositions. Such set compositions are in 1-1 correspondence with monomials in non-commuting variables, with increasing trees, with faces of permutahedra, etc. These combinatorial correspondences extend to the level of Hopf algebras and provide links to the work of various authors during the past years, including Bergeron/Zabrocki, Chapoton and Novelli/Thibon.

There are many interpretations of Fine's sequence 1, 0, 1, 2, 6, 18, 57, 186, 622, 2120, 7338, 25724, .... E.g., $F_n$ counts planar trees with a root of even out-degree and $n$ edges.We consider vector spaces equipped with a binary operation and a binary cooperation. We don't suppose that one is a morphism for the other (the Hopf case), but we suppose that they satisfy the unital infinitesimal relation. Studying the structure of such generalized bialgebras, we obtain a new interpretation of Fine's sequence, and we identify their primitive parts as MagFine-algebras. In the terminology of J.L.Loday, the triple of operads (As, Mag, MagFine) is a good triple of operads.This is joint work with Jean-Louis Loday and Maria Ronco.

Several constructions of quantum field theory can be easily generalized to functors on cocommutative coalgebras or bialgebras. With a little more work we can obtain functors on general bialgebras. As examples, we discuss the connected chronological product and the renormalisation functors.

In the theory of Hopf algebras there is a structure theorem which is very useful. It says that, in characteristic zero, a connected cocommutative Hopf algebra is cofree as a coalgebra and is, as an algebra, isomorphic to the universal enveloping algebra of a Lie algebra. This structure theorem is essentially equivalent to the union of the Poincaré-Birkhoff-Witt theorem with the Cartier-Milnor- Moore theorem. It involves three types of algebras, that is three operads: Com for the coalgebra structure, As for the algebra structure, and Lie for the structure of the primitive part. The purpose of this series of talks is to show that there are numerous other examples of this form, many of them already in the literature. We give elementary conditions on a triple of operads (C, A, P) so that there is a structure theorem for C^c-A-bialgebras, the primitive part being a P-algebra. Then it is called a good triple. The paradigm is (Com, As, Lie).
In many cases the C^c-A-bialgebras are, in fact, combinatorial Hopf algebras (with more structure).

I plan to give an overview of joint work in progress with Swapneel Mahajan.The first lecture will be on category theory, the second on species and Hopf algebras, the third on deformations and higher dimensional generalizations.We study the tensor category of species and relate it to the tensor category of graded vector spaces by means of bilax tensor functors. A substantial theory of abstract bilax tensor functors is developed first and then applied in this context. Constructions of Stover of graded Hopf algebras from Hopf monoids in species are then derived from the general theory. Deformations and higher dimensional generalizations of these constructions are prompted by the categorical approach. We study several specific examples of Hopf monoids in species and the graded Hopf algebras that correspond to them under the bilax tensor functors. We use the geometry and combinatorics of the Coxeter complex of type A to construct Hopf monoids and understand their interconnections. The corresponding Hopf algebras include those of symmetric functions, quasisymmetric functions, noncommutative symmetric functions, other Hopf algebras of prominence in the recent literature, and new ones. This relates to recent interesting work of Patras with Livernet, Reutenauer, and Schocker.

Descent compositions yield to the remarkable and well-studied Hopf subalgebras NSym of the Malvenuto-Reutenauer Hopf algebra SSym. These algebras can be obtained by a nice combinatorial construction: the standardized permutation of a word yields to a realization into words of SSym. Letting the variables be commutative gives a morphism from SSym to QSym. This is the core of the theory of noncommutative symmetric functions. When restricted to finitely many variables, QSym[x1,...,xn], can be understood as polynomial invariants/coinvariants of the Temperley-Lieb algebras. This was the work of Hivert on one part and Aval-Bergeron(s) on the other.

In this talk we review briefly Connes-Kreimer's Hopf algebra approach to perturbative renormalization. We present direct proofs of the main combinatorial properties of the renormalization procedures, using the properties of the Dynkin idempotent, one of the fundamental Lie idempotents in the theory of free Lie algebras, and Hopf algebras and their associated descent algebras. This joint work with F. Patras and J.M. Gracia-Bondia.

In recent years, The BPHZ algorithm for renormalization in quantum field theory has been interpreted, after dimensional regularization, as the Birkhoff-(Rota-Baxter) decomposition (BRB) of characters on the Hopf algebra of Feynmann graphs, with values in a Rota-Baxter algebra.We give in this paper formulas for the BRB decomposition in the group $\mathcal{C}( H, A )$ of characters on a connected Hopf algebra $H$, with values in a Rota-Baxter (commutative) algebra $A$.To do so we first define the stuffle (or quasi-shuffle) Hopf algebra $A^{\tmop{st}}$ associated to an algebra $A$. We prove then that for any connected Hopf algebra $H = k 1_H \oplus H'$, there exists a canonical injective morphism from $H$ to $H'^{\tmop{st}}$. This morphism induces an action of $\mathcal{C}( A^{\tmop{st}}, A )$ on $\mathcal{C}( H, A )$ so that the BRB decomposition in $\mathcal{C}( H, A )$ is determined by the action of a unique (universal) element of $\mathcal{C}( A^{\tmop{st}}, A )$.

We consider an interesting character on the quasi-symmetric functions that can be defined using multiple zeta values. We discuss some of its uses, and its even/odd decomposition in the sense of Aguiar et. al.

In order to extract the constant part of polyzêtas, some results "` l'Abel" dealing with noncommutative generating series of polylogarithms and multiple harmonic sums are established by using techniques à la Hopf.

Participants

Marcelo Aguiar

Texas A&M University, USA

Nantel Bergeron

York University, Canada

Riccardo Biagioli

Université Lyon I, France

Christian Brouder

Université Pierre et Marie Curie, France

Emily Burgunder

Université Montpellier 2, France

Frederic Chapoton

Université Lyon I, France

Kurusch Ebrahimi-Fard

IHES, France

Edward G. Effros

UCLA, USA

Bertfried Fauser

Max Planck Institute for Mathematics in the Sciences, Germany

Loic Foissy

Université de Reimes, France

Alessandra Frabetti

Université Lyon I, France

Partha Guha

Max Planck Institute for Mathematics in the Sciences, Germany

Istvan Heckenberger

University Leipzig, Germany

Vincel Hoang Ngoc Minh

University Lille 2, France

Michael Hoffman

U.S. Naval Academy, Annapolis, MD, USA

Ralf Holtkamp

Ruhr-Universität Bochum, Germany

Saifullah Khalid

School of Mathematical Sciences, GC University, Pakistan

Ron King

University of Southampton, United Kingdom

Govind Krishnaswami

Utrecht University, Netherlands

Jean Louis Loday

CNRS, Strasbourg, France

Mitja Mastnak

Ludwig-Maximilians-Universität München, Germany

Frederic Menous

Université Paris Sud, France

Robert Oeckl

Universidad Nacional Autónoma de México (UNAM), Mexico

Mario Paschke

Max Planck Institute for Mathematics in the Sciences, Germany

Frederic Patras

Université Nice, France

Arun Ram

University of Wisconsin-Madison, USA

Francesco Regonati

Università di Bologna, Italy

Maria Ronco

Universidad de Valparaiso, Chile

William Schmitt

Washington University, USA

Manfred Schocker

University of Wales Swansea, United Kingdom

Jean-Yves Thibon

Université de Marne-la-Vallée, France

Dmitri Vassilevich

University of Sao Paulo, Brazil

Rainer Verch

University of Leipzig, Germany

Eberhard Zeidler

Max Planck Institute for Mathematics in the Sciences, Germany

Scientific Organizers

Bertfried Fauser

Max-Planck-Institut für Mathematik in den Naturwissenschaften

Alessandra Frabetti

Université Claude Bernard Lyon 1

Frank Sottile

Texas A&M University (TAMU)

Administrative Contact

Bertfried Fauser

Max-Planck-Institut für Mathematik in den Naturwissenschaften
Contact via Mail

Regine Lübke

Max-Planck-Institut für Mathematik in den Naturwissenschaften
Contact via Mail