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\begin{document}
\begin{opening}

\title{\bf{Hopf algebra and Lie group structures inherent in combinatorial field theories}}
\author{Gerard  \surname{Duchamp}$^{\S}$}
\author{Karol A. \surname{Penson}$^{\dag}$}
\author{Allan I. \surname{Solomon}$^{\dag\S\S}$}
\author{Pawel \surname{Blasiak}$^{\dag\ddag}$}
\author{Andrzej \surname{Horzela}$^{\ddag}$}

\institute{\vspace{10pt}$^{\dag}$Universit\'{e} Pierre et Marie Curie\\
Laboratoire   de  Physique   Th\'{e}orique  des  Liquides, CNRS UMR 7600\\
Tour 16, $5^{i\grave{e}me}$ \'{e}tage, 4, place Jussieu, F 75252
Paris, Cedex 05, France\\
e-mail: penson@lptl.jussieu.fr\\\vspace{5pt}
$^{\ddag}$H. Niewodnicza{\'n}ski Institute of Nuclear Physics, Polish
Academy of Sciences\\
ul. Radzikowskiego 152, PL 31-342 Krak{\'o}w, Poland
\\e-mail: pawel.blasiak@ifj.edu.pl, andrzej.horzela@ifj.edu.pl \\\vspace{5pt}
$^{\S}$LIFAR, Universit\'{e} de Rouen\\
76821 Mont-Saint Aignan Cedex,
France\\
e-mail: gduchamp2@free.fr\\\vspace{5pt}
$^{\S\S}$The Open University\\
Physics and Astronomy Department\\
Milton Keynes MK7 6AA\\
e-mail: a.i.solomon@open.ac.uk\\\vspace{5pt}
}
\begin{abstract}

We consider two aspects of the product formula for formal power series 
applied to combinatorial field theories. Firstly, we remark that the 
case when the functions involved in the product formula have a constant 
term  is of special interest as often these functions give rise to 
substitutional groups. The groups arising from the normal ordering 
problem of boson strings are naturally associated with explicit vector 
fields, or their conjugates, in the case when there is only one 
annihilation operator. We also consider one-parameter groups of 
operators when several annihilators are present. 
Secondly, we discuss the Feynman-type graph representation resulting 
from the product formula. We show that there is a correspondence 
between the packed integer matrices of the theory of noncommutative 
symmetric functions and these Feynman-type graphs. 
In particular, we obtain a new Hopf algebra structure over the space 
of matrix quasi-symmetric functions and a natural commutative Hopf 
algebra structure on the space of diagrams themselves. 
It is shown that this Hopf algebra originates from the formal 
doubling of variables in the product formula.

\end{abstract}
\end{opening}

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