# 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.,
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 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 associated to an algebra *A*. We prove then that for any connected Hopf algebra , there exists a canonical injective morphism from *H* to . This morphism induces an action of on so that the BRB decomposition in is determined by the action of a unique (universal) element of .

### 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

Germany

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