Hopf Algebras, Combinatorics, and Quantum Field Theory

Abstracts for the talks

Hopf monoids in Species and associated Hopf algebras

Marcelo Aguiar  (Texas A&M University, USA)
Wednesday, September 27, 2006
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.

Coloring QSym and invariants of Fuss-Catalan Algebras

Nantel Bergeron  (York University, Canada)
Wednesday, September 27, 2006
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.

Quantum field theory on bialgebras

Christian Brouder  (Université Pierre et Marie Curie, France)
Tuesday, September 26, 2006
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.

Locality, the beta function and the Dynkin idempotent

Kurusch Ebrahimi-Fard  (IHES, France, and Universität Bonn, Germany)
Wednesday, September 27, 2006
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.

Bidendriform Bialgebras

Loic Foissy  (Université de Reimes, France)
Monday, September 25, 2006
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).

Generalized Euler constants associated to divergent polyzêtas

Vincel Hoang Ngoc Minh  (University Lille 2, France)
Thursday, September 28, 2006
In order to extract the constant part of polyzêtas, some results "à l'Abel" dealing with noncommutative generating series of polylogarithms and multiple harmonic sums are established by using techniques "à la Hopf.

Quasi-symmetric functions and multiple zeta values

Michael Hoffman  (U.S. Naval Academy, Annapolis, MD, USA)
Thursday, September 28, 2006
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.

Coassociative magmatic bialgebras and the Fine numbers

Ralf Holtkamp  (Ruhr-Universität Bochum, Germany)
Tuesday, September 26, 2006
There are many interpretations of Fine's sequence 1, 0, 1, 2, 6, 18, 57, 186, 622, 2120, 7338, 25724, .... E.g., formula7 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.

Branching rules, plethysms and Hopf algebras - some surprises

Ron King  (University of Southampton, United Kingdom)
Monday, September 25, 2006
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.

Beyond Hopf algebras

Jean Louis Loday  (CNRS, Strasbourg, France)
Wednesday, September 27, 2006
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).

Formulas for Birkhoff-(Rota-Baxter) decompositions related to connected bialgebra

Frederic Menous  (Université Paris Sud, France)
Wednesday, September 27, 2006
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 formula20 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 formula26 associated to an algebra A. We prove then that for any connected Hopf algebra formula30, there exists a canonical injective morphism from H to formula34. This morphism induces an action of formula36 on formula20 so that the BRB decomposition in formula20 is determined by the action of a unique (universal) element of formula36.

Combinatorial Hopf Algebras: an outsider's survey

Arun Ram  (University of Wisconsin-Madison, USA)
Monday, September 25, 2006
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?

The twisted descent algebra and related Hopf algebras

Manfred Schocker  (University of Wales Swansea, United Kingdom)
Tuesday, September 26, 2006
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.

Combinatorial Hopf algebras in the theory of symmetric functions

Jean-Yves Thibon  (Université de Marne-la-Vallée, France)
Monday, September 25, 2006
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.

Date and Location

September 25 - 28, 2006
Max Planck Institute for Mathematics in the Sciences
Inselstraße 22
04103 Leipzig
see travel instructions

Scientific Organizers

Bertfried Fauser
Max Planck Institute for Mathematics in the Sciences

Alessandra Frabetti
Université Claude Bernard Lyon 1

Frank Sottile
Texas A&M University (TAMU)

Administrative Contact

Bertfried Fauser
Max Planck Institute for Mathematics in the Sciences

Regine Lübke
Max Planck Institute for Mathematics in the Sciences
Contact by Email

17.10.2019, 11:59