 \documentclass[12pt]{iopart}
\usepackage[dvips]{epsfig}
\usepackage{amssymb}
\usepackage{amsfonts}
\usepackage{amscd}
\usepackage{pstricks}
\usepackage{rotating}
\usepackage[all,dvips]{xy}

\parindent=0pt

%%%%%%%%%%%%%%%%%%%%%%
% Fonts
%%%%%%%%%%%%%%%%%%%%%%
 \font\tensym=msbm10
 \font\sevensym=msbm7
 \font\fivesym=msbm5

 \font\twelvegoth=eufb10 at 12pt
 \font\tengoth=eufb10
 \font\sevengoth=eufb7
 \font\fivegoth=eufb5

\newfam\symfam
\textfont\symfam=\tensym \scriptfont\symfam=\sevensym
\scriptscriptfont\symfam=\fivesym
\def\sym{\fam\symfam\tensym}
\def\fns{\footnotesize}

\newfam\gothfam
\textfont\gothfam=\tengoth \scriptfont\gothfam=\sevengoth
\scriptscriptfont\gothfam=\fivegoth
\def\goth{\fam\gothfam\tengoth}

%%%%%%%%%%%%%%%%%%%%%%
% Jumps
%%%%%%%%%%%%%%%%%%%%%%

\def\hs{\hbox to 3mm{}}
\def\hhs{\hbox to 5cm{}}
\def\ss{\smallskip}
\def\ms{\smallskip}
\def\bs{\bigskip}
\def\bbss{\vskip 6mm}

%%%%%%%%%%%%%%%%%%%%%%
% Images
%%%%%%%%%%%%%%%%%%%%%%

\def\JPicScale{0.8}\ifx\JPicScale\undefined\def\JPicScale{1}\fi

%%%%%%%%%%%%%%%%%%%%%%
% Miscellaneous
%%%%%%%%%%%%%%%%%%%%%%

%%% Letters and abbreviations %%%

\def\A{\mathcal{A}}
\def\Aa{\mathbb{A}}
\def\U{\mathcal{U}}
\def\TBD{\times_{BD}}
\def\C{\mathbb{C}}
\def\H{\mathcal{H}}
\def\R{\mathbb{R}}
\def\N{\mathbb{N}}
\def\D{\mathcal{D}}
\def\DWS{\Delta_{WS}}
\def\DBS{\Delta_{BS}}
\def\E{{\sym E}}
\def\HWS{{\cal H}_{WS}}
\def\1HW{1_{HW}}
\def\L{\mathbb{L}}
\def\V{\mathbb{V}}
\def\X{\mathbb{X}}
\def\Y{\mathbb{Y}}
\def\Q{{\sym Q}}
\def\SG{{\frak S}}
\def\al{\alpha}
\def\be{\beta}
\def\ga{\gamma}
\def\ep{\epsilon}
\def\la{\lambda}
\def\Om{\Omega}
\def\om{\omega}
\def\si{\sigma}
\def\MQSym{{\bf MQSym}}
\def\red{\stackrel{\small red}{\longrightarrow}}
\def\bd{\bar\Delta}
\def\card{\mathrm{card}}
%\def\2m#1#2{\pmatrix{#1\cr #2}}
%\def\3m#1#2#3{\pmatrix{#1\cr #2\cr #3}}
\def\2m#1#2{(#2 #1)}
\def\3m#1#2#3{(#3 #2 #1)}
\def\inf{\uparrow}
\def\deg#1{\mathrm{deg}(#1)}

%%% Spaces %%%

\def\diag{\mathbf{diag}}
\def\ldiag{\mathbf{ldiag}}
\def\lbell{\mathbf{lbell}}
\def\LDIAG{\mathbf{LDIAG}}
\def\DIAG{\mathbf{DIAG}}
\def\BELL{\mathbf{BELL}}
\def\LBELL{\mathbf{LBELL}}
\def\MQS{\mathbf{MQSym}}
\def\LHW{\mathcal{L}_{HW}}
\def\Bq{\tilde{B}_{q}}

%%% Arrows %%%

\def\ra{\rightarrow}
\def\lar{\leftarrow}
\def\lra{\leftrightarrow}
\def\da{\downarrow}
\def\ua{\uparrow}
\def\sea{\searrow}
\def\swa{\swarrow}

%%% Signs %%%

\def\mod{{\rm\ mod\ }}
\def\quo{{\rm\ quo\ }}
\def\irem{{\rm\ rem\ }}
\def\peb#1{\lfloor #1\rfloor}
\def\peh#1{\lceil #1\rceil}
\def\adots{\mathinner{\mkern2mu\raise1pt\hbox{.}
\mkern3mu\raise4pt\hbox{.}\mkern1mu\raise7pt\hbox{.}}}
\def\binomial#1#2{\pmatrix{#1 \cr #2}}
\def\scc#1#2#3{(#1\ |\ #2\,,#3)}
\def\pointir{\unskip . --- \ignorespaces}
\def\quest{{\bf Q\ }}
\def\rep{{\bf R\ }}
\def\up#1{\raise 1ex\hbox{\footnotesize#1}}
\def\cs{$\clubsuit$}
\def\sps{$\spadesuit$}
\def\hes{$\heartsuit$}
\def\ds{$\diamondsuit$}
\def\mref#1{{\footnotesize ({\ref{#1}})}}
\def\ncp#1#2{#1\langle #2\rangle}


%%%%%%%%%%%%%%%%%%%%%%
% Statements
%%%%%%%%%%%%%%%%%%%%%%

\newtheorem{expl}{Example}[section]
\newtheorem{exercice}{Exercise}[section]
\newtheorem{nota}[expl]{Note}
\newtheorem{exo}[expl]{Exercisce}
\newtheorem{exm}[expl]{Exerc. Machine}
%\newtheorem{theo}[expl]{Th\'eor\`eme}
\newtheorem{theorem}[expl]{Theorem}
\newtheorem{corollaire}[expl]{Corollary}
\newtheorem{comm}[expl]{Comment}
\newtheorem{definition}[expl]{D\'efinition}
\newtheorem{proposition}[expl]{Proposition}
\newtheorem{prdef}[expl]{Proposition/D\'efinition}
\newtheorem{algorithme}[expl]{Algorithm}
\newtheorem{lemma}[expl]{Lemma}
\newtheorem{rem}[expl]{Remark}
\newtheorem{appli}[expl]{Applications}

\def\Proof{\medskip\noindent {\it Proof --- \ }}
\def\cqfd{\hfill $\Box$ \bigskip}

% Affichage date et l'heure courante %\isodayandtime%
%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%
\begingroup
\count0=\time \divide\count0by60 % Hour
\count2=\count0 \multiply\count2by-60 \advance\count2by\time
% Min
\def\2#1{\ifnum#1<10 0\fi\the#1}
\xdef\isodayandtime{
%\centerline
{\2\day-\2\month-\the\year\space\2{\count0}:%
\2{\count2}}}
\endgroup


\begin{document}


\title{The Hopf Algebra of Heisenberg-Weyl Diagrams}
\author{G H E Duchamp$^{a}$, P Blasiak$^{b}$, K A Penson$^{c}$, A Horzela$^{b}$, A I Solomon$^{c,d}$, \vspace{2mm}}

\address{$^a$ LIPN - UMR 7030\linebreak
CNRS - Universit\'e Paris 13\linebreak
F-93430 Villetaneuse, France\vspace{2mm}}

\address
{$^b$ H. Niewodnicza\'nski Institute of
Nuclear Physics, Polish Academy of Sciences\\
ul. Eliasza-Radzikowskiego 152,  PL 31342 Krak\'ow,
Poland\vspace{2mm}}

\address
{$^c$ Laboratoire de Physique Th\'eorique de la Mati\`{e}re Condens\'{e}e\\
Universit\'e Pierre et Marie Curie, CNRS UMR 7600\\
Tour 24 - 2i\`{e}me \'et., 4 pl. Jussieu, F 75252 Paris Cedex 05,
France\vspace{2mm}}

\address
{$^d$ The Open University, Physics and Astronomy Department\\
Milton Keynes MK7 6AA, United Kingdom\vspace{2mm}}

\eads{\linebreak\mailto{ ghed@lipn-univ.paris13.fr}, \mailto{
a.i.solomon@open.ac.uk} \mailto{ pawel.blasiak@ifj.edu.pl}, \mailto{
andrzej.horzela@ifj.edu.pl}, \mailto{penson@lptl.jussieu.fr}, }

\begin{abstract} It has been found by A. Varvak {\tt (TODO improve)} and others  that the structure 
constants of the Heisenberg-Weyl algebra ($HW$) could be computed through rook placement numbers.
We here give a diagrammatic realization of this formula in terms of graphs which constitute 
an algebra. It is also possible to keep track of the number cancellation in Wick's formula and 
this provides at once a $q$-analogue $\Bq$ of the previous algebra. It tuns out that the latter can be 
very naturally endowed with the structure of a Hopf algebra. Returning to $HW$, it is well known
that it has a 3-dimensional $\LHW$ Lie subalgebra which generates (as an algebra) $HW$. 
We construct a Hopf arrow between the enveloping algebra $\mathcal{U}(\LHW)$ and $\Bq$. It is there, 
in our sense, that resides the obstruction of $HW$ to have finite dimensional non-zero representation.\\
\isodayandtime
\end{abstract}

\tableofcontents

\newpage

\section{Introduction}

Since the advent of Hopf algebras in Combinatorial Physics, when facing an algebra, 
one always asks oneself whether its structure could be (more or less naturally) be enriched 
to a Hopf algebra one. For one of the simplest in Quantum Physics, the Heisenberg-Weyl (HW) algebra, 
the answer is ``no'' since the counit provides one with a one-dimensional representation HW and it is well-known 
(and easily checked) that this algebra has no non-zero finite dimensional representation (the ground field is supposed to that of the complex numbers or another field of zero characteristic).\\  
The paper is devoted ...

\begin{center}
$
\xymatrix{
& \tilde B_q \ar[ld]_{\varphi_3} \ar[rd]^{\varphi_4}  \\
\tilde B \ar[rr]_{\varphi_5} \ar[rd]_{\varphi_2} && \mathcal{U}(\LHW)\ar[ld]^{\varphi_1}\\
& HW
}
$
\end{center}

\section{Construction of the diagram algebras $\tilde B$ and $\tilde B_q$.}

\subsection{Heisenberg-Weyl graphs and their matching}

As usual in graph theory, a graph $\Gamma$ is given by the data of two sets $\Gamma^{(1)}$, the edges, $\Gamma^{(0)}$, the vertices and two functions (head and tail) $h,t\ :\ \Gamma^{(1)}\ra \Gamma^{(0)}$. We are intesrested here by graphs with some edges having a ```free tail'' or a ``free head'' but not both, so we allow the functions $h,t$ to be partially defined with the condition that 
\begin{equation}\label{HWcond}
dom(t)\cup dom(h)=\Gamma^{(1)}
\end{equation}

(every edge has at least a tail or a head maybe both). The edges $a$ with no origin i. e. such that $a\notin dom(t)$ (resp. with no endpoint i. e. such that $a\notin dom(h)$) will be called ``ingoing edges'' and denoted $\Gamma^-$ (resp. ``outgoing edges'', denoted $\Gamma^+$).\\ 
A cycle is a sequence of edges $a_1a_2\cdots a_n$ such that, forall $k<n,\ h(a_k)=t(a_{k+1})$ and $h(a_n)=t(a_{1})$. We will call $HW$-graph a finite graph with (possibly) ingoing and outgoing edges and no cycle.\\ 


{\tt Figure TODO : Heisenberg-Weyl Graphs}

One can then see that a $HW$-graph has a natural partition in three parts : 
\begin{itemize}
	\item the outgoing edges $dom(t)-dom(h)$ and its associated subgraph $\Gamma^+$ (with restriction of the tail mapping and void head mapping)
	\item the ingoing edges $dom(h)-dom(t)$ and its associated subgraph $\Gamma^-$ (with restriction of the head mapping and void tail mapping)
	\item the inner edges $dom(t)\cap dom(h)$ and its associated subgraph $\Gamma^{(in)}$ (with restriction of the tail and head mappings everywhere defined)	
\end{itemize}

Now, in order to define algebra structures on these graphs, one has to define what is a matching between two $HW$-graphs. In general set theory a \textit{matching} or one-to-one correspondence is a set of pairs 
\begin{equation}
	m\subset pr_1(m)\times pr_2(m)
\end{equation}
such that two different pairs have different components i. e. for $c_1,c_2\in m$ with $c_i=(x_i,y_i)$ one has 
\begin{equation}
c_1\not= c_2\Longrightarrow (x_1\not=x_2)\textrm{ and } (y_1\not=y_2)\ .
\end{equation}
If $U,V$ are finite sets $|U|=p,\ |V|=q$ the number of matchings $m\subset U\times V$ with $|m|=i$ is 
\begin{equation}
	\binomial{p}{i}\binomial{q}{i}i!\ .
\end{equation}
This formula will do the job when we construct the arrow $\varphi_2$.\\ 
For two $HW$-graphs $\Gamma_2,\Gamma_1$ with disjoint sets of vertices (i. e. $\Gamma_2^{(0)}\cap \Gamma_1^{(0)}=\emptyset$), we will call \textit{matching} of $\Gamma_1$ over $\Gamma_2$ any matching between 
$\Gamma_2^+$ and $\Gamma_1^-$, their set will be denoted $match(\Gamma_2,\Gamma_1)$. Any matching of $match(\Gamma_2,\Gamma_1)$ allows the ``pluging'' of $\Gamma_2$ into $\Gamma_1$ as follows

{\tt Figure TODO : a matching between two Heisenberg-Weyl Graphs}

The result will be denoted 
$$
\pmatrix{\Gamma_1\cr m\cr \Gamma_2}
$$
which can be formally defined by

\begin{equation}
\pmatrix{\Gamma_1\cr m\cr \Gamma_2}^{(1)}=\Gamma_1^{(1)}\cup \Gamma_2^{(1)}\cup m -(pr_1(m)\cup pr_2(m))\ .	
\end{equation}
Tail and head functions defined accordingly: remark that condition \mref{HWcond} implies that, 
\begin{equation}
\Gamma_2^{(0)}\cap \Gamma_1^{(0)}=\emptyset\Longrightarrow \Gamma_2^{(1)}\cap \Gamma_1^{(1)}=\emptyset	
\end{equation}
so, there is no ambiguity to define them on $\Gamma_1^{(1)}\cup \Gamma_2^{(1)}-(pr_1(m)\cup pr_2(m))$ and we define tail and head functions on $m$ (a subset, remind it, of $\Gamma_2^+\times \Gamma_1^-$) by 
\begin{equation}
t(a_2,a_1):=t(a_2)\ ;\	h(a_2,a_1):=h(a_1)\textrm{ where } (a_2,a_1)\in m\ .
\end{equation}
We have a sort of associativity of the matchings which will be useful later. Let $\Gamma_i,\ i=1,2,3$ be three disjoint $HW$-graphs. Let 
$$
m'\in match(\Gamma_2,\Gamma_1)\textrm{ and }m''\in match(\Gamma_3,\pmatrix{\Gamma_1\cr m'\cr \Gamma_2})
$$
then $m''$ can be partitioned in $m''_{3,2}=m''\cap (\Gamma_3^+\times \Gamma_2^-)$ and $m''_{3,1}=m''\cap (\Gamma_3^+\times \Gamma_1^-)$ and one has
\begin{equation}\label{preass}
\pmatrix{\pmatrix{\Gamma_1\cr m'\cr \Gamma_2}\cr m''\cr \Gamma_3}=\pmatrix{\Gamma_1\cr m'\cup m''_{31}\cr\pmatrix{\Gamma_2\cr m''_{32}\cr \Gamma_3}}
\end{equation}

\subsection{Multiplication of graphs and structure of the algebra $\tilde B_q$.}
    
Following the general conventions in graph theory, an isomorphism of graphs between $\Gamma_1=(\Gamma_1^{(1)},\Gamma_1^{(0)},t_1,h_1)$ and $\Gamma_2=(\Gamma_2^{(1)},\Gamma_2^{(0)},t_2,h_2)$ is the data of two one-to-one mappings $\alpha^{(1)}\ :\ \Gamma_1^{(1)}\ra \Gamma_2^{(1)}$ and $\alpha^{(0)}\ :\ \Gamma_1^{(0)}\ra \Gamma_2^{(0)}$ such that the following commutative diagrams of partially defined functions hold

\begin{equation}
\xymatrix{
\Gamma_1^{(1)} \ar[r]^{t_1}\ar[d]^{\alpha^{(1)}} &  \Gamma_1^{(0)} \ar[d]^{\alpha^{(0)}}  \\
\Gamma_2^{(1)} \ar[r]^{t_2} &  \Gamma_2^{(0)}
}
\hspace{1cm}\textrm{and}\hspace{1cm}
\xymatrix{
\Gamma_1^{(1)} \ar[r]^{h_1}\ar[d]^{\alpha^{(1)}} &  \Gamma_1^{(0)} \ar[d]^{\alpha^{(0)}}  \\
\Gamma_2^{(1)} \ar[r]^{h_2} &  \Gamma_2^{(0)} \ .
}
\end{equation}
It is straightforward to check that the class of $HW$-graphs is closed under isomorphisms. 
We will call $\mathcal{B}$, the set (it can be shown to be denumerable) of classes of $HW$-graphs ``up to isomorphisms''.

The algebra $\tilde B$ (resp. $\tilde B_q$) is the vector space $\C \mathcal{B}$ (resp. the module $\C[q] \mathcal{B}$). The multiplication table in $\tilde B_q$ is given by 
\begin{equation}
	\tilde\Gamma_1 *\tilde\Gamma_2=\sum_{m\in match(\Gamma_2,\Gamma_1)} q^{|m|}\widetilde{\pmatrix{\Gamma_1\cr m\cr \Gamma_2}} 
\end{equation}
where $\Gamma_1,\ \Gamma_2$ are representatives of the classes $\tilde\Gamma_1,\ \tilde\Gamma_2$ with disjoint set of vertices (i. e. $\Gamma_1^{(0)}\cap \Gamma_2^{(0)}=\emptyset$).\\ 
It can be checked easily that the result is independent of the choice of disjoint representatives.\\
It is straightforward that, for all $\Gamma$, $\tilde\Gamma *\tilde\emptyset=\tilde\emptyset*\tilde\Gamma=\tilde\Gamma$. Hence $\tilde\emptyset$ (which, in fact is $\{\emptyset\}$) is neutral for the law and will be denoted here $1_{\tilde{B_q}}$. Let us prove now that the law is associative. To this end, take three disjoint $HW$-graphs $\Gamma_i,\ i=1,2,3$ (we need two and a third which is disjoint of the results of the matchings of the two first). Then one has, using the denotations of formula \mref{preass}
%\begin{eqnarray}
%\hspace{-1cm}	(\tilde\Gamma_1 *\tilde\Gamma_2)*\tilde\Gamma_3=\sum_{m'\in match(\Gamma_2,\Gamma_1)}\sum_{m''\in 
%	match(\Gamma_3,{\tiny \pmatrix{\Gamma_1\cr m'\cr \Gamma_2}})} q^{|m'|}q^{|m''|}\widetilde{\overbrace{\pmatrix{\pmatrix{\Gamma_1\cr m'\cr \Gamma_2}\cr m''\cr \Gamma_3}}}=\cr
%	\sum_{m'\in match(\Gamma_2,\Gamma_1)}\sum_{m''\in 
%	match(\Gamma_3,{\tiny \pmatrix{\Gamma_1\cr m'\cr \Gamma_2}})} q^{|m'|}q^{|m''|}\pmatrix{\Gamma_1\cr m'\cup m''_{31}\cr\pmatrix{\Gamma_2\cr m''_{32}\cr \Gamma_3}} 
%\end{eqnarray}
\begin{eqnarray}\label{ass_proof}
\hspace{-1cm}	(\tilde\Gamma_1 *\tilde\Gamma_2)*\tilde\Gamma_3=\sum_{m'\in match(\Gamma_2,\Gamma_1)}\sum_{m''\in 
	match(\Gamma_3,{\tiny \pmatrix{\Gamma_1\cr m'\cr \Gamma_2}})} q^{|m'|}q^{|m''|}
	\pmatrix{\pmatrix{\Gamma_1\cr m'\cr \Gamma_2}\cr m''\cr \Gamma_3}^{\widetilde{\hspace{5mm}}}=\cr
	\sum_{m'\in match(\Gamma_2,\Gamma_1)}\sum_{m''\in 
	match(\Gamma_3,{\tiny \pmatrix{\Gamma_1\cr m'\cr \Gamma_2}})} q^{|m'|}q^{|m''|}\pmatrix{\Gamma_1\cr m'\cup m''_{31}\cr\pmatrix{\Gamma_2\cr m''_{32}\cr \Gamma_3}}^{\widetilde{\hspace{5mm}}}=\cr
	\sum_{m'\in match(\Gamma_2,\Gamma_1)}\sum_{m''\in 
	match(\Gamma_3,{\tiny \pmatrix{\Gamma_1\cr m'\cr \Gamma_2}})} q^{|m'\cup m''_{31}|}q^{|m''_{32}|}\pmatrix{\Gamma_1\cr m'\cup m''_{31}\cr\pmatrix{\Gamma_2\cr m''_{32}\cr \Gamma_3}}^{\widetilde{\hspace{5mm}}}
\end{eqnarray}
the result comes from the fact that the correspondence $(m',m'')\mapsto (m'\cup m''_{31},m''_{32})$ between pairs of matchings in $match(\Gamma_2,\Gamma_1)\times match(\Gamma_3,{\tiny \pmatrix{\Gamma_1\cr m'\cr \Gamma_2}})$ and pairs in $match({\tiny \pmatrix{\Gamma_2\cr m''_{32}\cr \Gamma_3}},\Gamma_1)\times match(\Gamma_3,\Gamma_2)$ is one-to-one. Then the sum amounts to 
\begin{eqnarray}
\hspace{-1cm}	\sum_{m''\in match(\Gamma_3,\Gamma_2)}\sum_{m'\in match({\tiny \pmatrix{\Gamma_2\cr m''\cr \Gamma_3}},\Gamma_1)} q^{|m''|}q^{|m'|}\pmatrix{\Gamma_1\cr m'\cr\pmatrix{\Gamma_2\cr m''\cr \Gamma_3}}^{\widetilde{\hspace{5mm}}}=\tilde\Gamma_1 *(\tilde\Gamma_2*\tilde\Gamma_3)\ .
\end{eqnarray}

\subsection{Specialization to $q=1$: the algebra $\tilde B$.}

Having constructed $\tilde B_q$, one can specialize $q$ to one (i. e. send each sum $\sum_{i\in I}p_i(q) \tilde\Gamma_i$ to $\sum_{i\in I}p_i(1) \tilde\Gamma_i$) this ($\C$-linear) correspondence (which will be called $\varphi_3$ below) ranges in $\tilde B$ and endows the latter with the structure of $\C$-AAU.

\section{Ado's theorem, the Lie algebra $\mathcal{L}_{HW}$ and its enveloping algebra.\\ 
TODO : This paragraph will be put as a remark. Replace it by physical motivations for central elements.}

It is well known that, in characteristic $0$, $HW$ admits no finite dimensional proper representation (its admits neither no representation in a Banach unital algebra). But one observes that the subspace $L=\C a\oplus \C a^+ \oplus \C 1_{HW}$ is closed for the Lie bracket and hence a Lie subalgebra of $(HW,[,])$. Now, by Ado theorem such a finite-dimensional Lie algebra admits finite dimensional faithful representations. As an example, one can give the following one (noted $\rho$)
\begin{equation}
\hspace{-1cm}	\rho(a)=\pmatrix{0&1&0 \cr 0&0&0 \cr 0&0&0}\ ;\  \rho(a^+)=\pmatrix{0&0&0 \cr 0&0&1 \cr 0&0&0}\ ;\ \rho(1_{HW})=\pmatrix{0&0&1 \cr 0&0&0 \cr 0&0&0}\ .
\end{equation}
One sees that the image of $1_{HW}$ is not a unity and, in order not to get confused in our forthcoming development, we will take a copy of the Lie algebra $L=\C a\oplus \C a^+ \oplus \C 1_{HW}$ as $\LHW=\C\{b,b^+,u\}$ with the multiplication table 

\begin{center}
\begin{tabular}{c|c|c|c}
$[\da,\ra]$ & $b^+$ & $b$ & u\\
\hline 
$b^+$ & $0$ & $-u$ & $0$\\
\hline
$b$ & $-u$ & $0$ & $0$\\
\hline
$u$ & $0$ & $0$ & $0$
\end{tabular}
\end{center}

Now, we have a morphism of Lie algebras $\phi_1:\LHW\ra HW$ given by 
\begin{equation}
\phi_1(b)=a\ ; \phi_1(b^+)=a^+\ ; \phi_1(u)=\1HW	
\end{equation}
and, by the universal propoerty of the envelopping algebras this can be extended uniquely as a morphism of AAU $\varphi_1 : \mathcal{U}(\LHW)\ra HW$. We can make it explicit, using the basis of Poincar\'e-Birkhoff-Witt of $\mathcal{U}(\LHW)$ which is given (using the order $b^+\prec b\prec u$ as in the multiplication table above) by the products $\{(b^+)^p\,b^q\,u^r\}_{p,q,r\in\N}$. We then have $\varphi((b^+)^p\,b^q\,u^r)=(a^+)^p\,a^q$. 

\section{The arrow $\varphi_4:\ \tilde{B}_q\ra \mathcal{U}(\LHW)$.}

The algebra $\tilde{B}_q$ is a $\C[q]$-AAU, but onee can restrict the scalars to $\C$. In this case, it is a $\C$-AAU with basis $q^n\tilde{\Gamma}$ ($n$ in $\N$ and $\Gamma$ choosen in a set of distinguished representatives). One sets 
\begin{equation}
	\varphi_4(q^n\tilde{\Gamma}):=(b^+)^{|\Gamma^+|}b^{|\Gamma^-|}u^n\ .
\end{equation}
In order to prove that $\varphi_4$ is a morphism of $\C$-AAU, one has to prove that 

\[
	\varphi_4(q^s\tilde{\Gamma_1}*q^t\tilde{\Gamma_2})=\varphi_4(q^s\tilde{\Gamma_1})\varphi_4(q^t\tilde{\Gamma_2})
\]
which can be rephrased as 
\[
	\varphi_4(\tilde{\Gamma_1}*\tilde{\Gamma_2})u^{s+t}=\varphi_4(\tilde{\Gamma_1})\varphi_4(\tilde{\Gamma_2})u^{s+t}
\]
so, one only has to prove the equality for $s=t=0$.\\
Suppose that two disjoint $HW$-graphs $\Gamma_i,i=1,2$ are given. For convenience, set 
$$
match(\Gamma_2,\Gamma_1;i)=\{m\in match(\Gamma_2,\Gamma_1)|\ |m|=i\}
$$
it is not difficult to get the following enumerations 
\begin{equation}
\hspace{-1cm} |match(\Gamma_2,\Gamma_1;i)|=
\left\{ \
\begin{array}{ccc}
{\small \binomial{|\Gamma_2^+|}{i}\binomial{|\Gamma_1^-|}{i}}i! &\textrm{ for }& i\leq min(|\Gamma_2^+|,|\Gamma_1^-|) \\
$0$																											&\textrm{ for } & i> min(|\Gamma_2^+|,|\Gamma_1^-|) 
\end{array}
\right.
\end{equation}

then, one has
\begin{eqnarray}\label{prod1}
\varphi_4(\tilde{\Gamma_1}*\tilde{\Gamma_2})=\varphi_4\Big(\sum_{m\ match(\Gamma_2,\Gamma_1)} 
q^{|m|}\widetilde{\pmatrix{\Gamma_1\cr m\cr \Gamma_2}}\Big)=\cr
\sum_{i=0}^{min(|\Gamma_2^+|,|\Gamma_1^-|)}u^i\sum_{m\ match(\Gamma_2,\Gamma_1;i)}\varphi_4\Big(\widetilde{\pmatrix{\Gamma_1\cr m\cr \Gamma_2}}\Big)=\cr
\sum_{i=0}^{min(|\Gamma_2^+|,|\Gamma_1^-|)}u^i\sum_{m\ match(\Gamma_2,\Gamma_1;i)} 
(b^+)^{\Big|{\tiny \pmatrix{\Gamma_1\cr m\cr \Gamma_2}^+}\Big|}\ b^{\Big|{\tiny \pmatrix{\Gamma_1\cr m\cr \Gamma_2}^-}\Big|}=\cr
\sum_{i=0}^{min(|\Gamma_2^+|,|\Gamma_1^-|)}u^i\sum_{m\ match(\Gamma_2,\Gamma_1;i)} 
(b^+)^{|\Gamma_1^+|+|\Gamma_2^+|-i}\ b^{|\Gamma_1^-|+|\Gamma_2^-|-i}\ .
\end{eqnarray}
Now it is a easy exercise to get, in $\mathcal{U}(\LHW)$, the formula
\begin{equation}
\hspace{-10mm}	(b^+)^{p_1}\, b^{q_1}\ (b^+)^{p_2}\, b^{q_2}=
	\sum_{i=0}^{min(q_1,p_2)} \binomial{q_1}{i}\binomial{p_2}{i}i!\, (b^+)^{p_1+p_2-i}\, b^{q_1+q_2-i}\, u^{i}
\end{equation}

which implies that the sum in eq \mref{prod1} is equal to 
\begin{equation}
(b^+)^{|\Gamma_1^+|}\, b^{|\Gamma_1^-|}\ (b^+)^{|\Gamma_2^+|}\, b^{|\Gamma_2^-|}=\varphi_4(\tilde{\Gamma_1})\varphi_4(\tilde{\Gamma_2})
\end{equation}

\section{Construction of the section $s_3:\ \tilde B\ra \tilde{B}_q$ and of $\varphi_5:\ \tilde B\ra \mathcal{U}(\LHW)$}

For a $HW$-diagram $\Gamma$, we note $\Gamma^{in}$ the set of inner edges i. e. $\Gamma^{in}=dom(t)\cap dom(h)$. Now, we construct an arrow 
$s_3:\ \tilde B\ra \tilde{B}_q$ by
\begin{equation}
	s_3(\Gamma)=q^{|\Gamma^{in}|}\Gamma
\end{equation}
and one checks at once that 

\begin{eqnarray}
\hspace{-2cm}	s_3(\Gamma_1*_1\Gamma_2)=\sum_{m\in match(\Gamma_2,\Gamma_1)} s_3(\widetilde{\pmatrix{\Gamma_1\cr m\cr \Gamma_2}})=
	\sum_{m\in match(\Gamma_2,\Gamma_1)} q^{|\Gamma_1^{in}|+|\Gamma_2^{in}|+|m|}\widetilde{\pmatrix{\Gamma_1\cr m\cr \Gamma_2}}=\cr
	\sum_{m\in match(\Gamma_2,\Gamma_1)} q^{|m|}\widetilde{\pmatrix{\Gamma_1\cr m\cr \Gamma_2}}=s_3(\Gamma_1)*s_3(\Gamma_2)\ .
\end{eqnarray}
From there, $\varphi_5$ is defined as $\varphi_5=\varphi_4\circ s_3$. It should be noted that $\varphi_5$ is onto.

\section{Diagram of algebras}

We have, so far, constructed the following diagram of algebras
\begin{center}
$
\xymatrix{
                                                 & \tilde B_q \ar@<-2pt>[ld]_{\varphi_3} \ar[rd]^{\varphi_4}  \\
\tilde B \ar[rr]_{\varphi_5}\ar[rd]_{\varphi_2}\ar@<-2pt>[ru]_{s_3} &                                  & \mathcal{U}(\LHW)\ar[ld]^{\varphi_1}\\
                                                 & HW                                                 &                 &\LHW \ar@{_{(}->}[ll]^{\phi_1} \ar[lu]_{nat} 
}
$
\end{center}

\section{Coproducts and Hopf Algebra structures.}
\subsection{Comultiplications and section coefficients.}

In a pioneering paper \cite{JR}, G-C Rota introduced the concept of ``section coefficients'' and gave many combinatorial examples (set partitions, classes of intervals, placements ...). \\
Let us take, for simplicity, the example of binomial coefficients but presented here as ``section coefficients''. One has, for $n,p,q\in\N$
\begin{equation}
	\binomial{n}{p,q}=
	\left\{ \ 
\begin{array}{rcl}

\displaystyle{\frac{n!}{p!\, q!}} & = & \binomial{n}{p}\ \textrm{if } p+q=n\\
0               && \textrm{otherwise} \\
\end{array}
\right.
\end{equation}

in this spirit, these coefficients count the number of ways of splitting a $n$-set into two disjoint subsets with, respectively $p$ and $q$ elements. Now consider the number of ways of splitting a $n$-set in three parts with respectively $p,q,r$ elements (of course it will be $0$ if $p+q+r\not= n$). This can be performed by first splitting the set in two parts and re-splitting the second part in two again or by first splitting the set in two parts and re-splitting the first part. These two procedures provide the ``trichotomy identity''
\begin{equation}\label{tricho}
	\sum_{m\in \N} \binomial{n}{p,m}\binomial{m}{q,r}=\sum_{m\in \N} \binomial{n}{m,r}\binomial{m}{p,q}
\end{equation}
which, in this case, is obvious as the two sums have only one term.\\
Now, one has also an interplay with the sum, called Van der Monde identity \cite{JR}
\begin{equation}\label{VdM}
	\binomial{n_1+n_2}{p\,,\,q}=\sum_{p_1+p_2=p\atop q_1+q_2=q} \binomial{n_1}{p_1,q_1}\binomial{n_2}{p_2,q_2}.
\end{equation}
In general, section coefficients $(i |j,k)$ count the number of ways of splitting an object $i$ into two sub-objects $j,k$. they may or may not fulfill identities \mref{tricho} and/or \mref{VdM} and have a deep relationship with modern algebra.\\
Let us give another example which generalizes the preceding. For a word $w=a_1a_2\cdots a_n$ and $I\subset \{i_1,i_2,\cdots i_k\}$ ($i_1<i_2<\cdots <i_k$) one defines the subword $w[I]$ as $w[I]=a_{i_1}a_{i_2}\cdots a_{i_k}$. For three words $u,v,w$, one can define the section coefficient 
\begin{equation}\label{subwords_sc}
	\scc{w}{u}{v}=\#\Big\{(I,J)\ |\ I+J=[1..|w|],\ w[I]=u,\ w[J]=v\Big\}
\end{equation}
for example, one has
$$
\hspace{-1cm}	\scc{abab}{aa}{bb}=1\ ;\ \scc{abab}{ba}{ba}=0\  ;\ \scc{abab}{ab}{ba}=1\ ;\ \scc{abab}{ab}{ab}=2.
$$
These coefficients can be considered as ``word functions'', they are a generalization of the binomial section coefficients as 
$$
\scc{a^n}{a^p}{a^q}=\binomial{n}{p,q}\ .
$$
For these coefficients, one has the trichotomy identity
\begin{equation}\label{trichow}
	\sum_{m\in A^*} \scc{w}{s}{m}\scc{m}{t}{u}=\sum_{m\in \N} \scc{w}{m}{u}\scc{m}{s}{t}
\end{equation}
and the Van der Monde one reads
\begin{equation}\label{VdM}
	\scc{w_1w_2}{u}{v}=\sum_{u=u_1u_2\atop v=v_1v_2} \scc{w_1}{u_1}{v_1}\scc{w_2}{u_2}{v_2}\ .
\end{equation}
The connection with modern algebra is done through the notion of a comultiplication. Taking $C$ the vector space freely generated by the objects under consideration (the basis will be denoted $(e_i)_{i\in I}$), one can consider the section coefficients as structure constants for a linear mapping (called comultiplication) 
\begin{equation}
\Delta\ :\ C\ra C\otimes C
\end{equation}
by 
\begin{equation}
\Delta(\sum_{i\in I}{\al_i e_i})=\sum_{i\in I} \al_i \Delta(e_i)=\sum_{i\in I}\sum_{j,k\in I} \al_i \scc{e_i}{e_j}{e_k} e_j\otimes e_k\ .
\end{equation}
(for the notion of structure constants of a comultiplication see \cite{GOF13}). Calling $Id_C$ the identity mapping $C\ra C$, the trichotomy identity reads
\begin{equation}
	(\Delta\otimes I)\circ \Delta=(I\otimes \Delta)\circ \Delta
\end{equation}
whereas the Van der Monde identity (providing $C$ be given an algebra law $\mu\ :\ C\otimes C\ra C$) reads
\begin{equation}
	\Delta\circ\mu=\mu^{\otimes\ 2}\circ\Delta
\end{equation}
where $\mu^{\otimes\ 2}$ is the law on $C\otimes C$ defined by
\begin{equation}
	\mu^{\otimes\ 2}(x_1\otimes y_1,x_2\otimes y_2)=\mu(x_1\otimes y_1)\otimes \mu(x_2\otimes y_2)\ .
\end{equation}

\begin{expl}
In the case of the ``subwords'' section coefficients (form. \mref{subwords_sc}), an alphabet $A$ and a field $K$ being given, the space ($C$ above) freely generated by the words is the algebra $\ncp{K}{A}$. It is easily seen that 
\begin{equation}\label{subwords_cm}
	\Delta(w)=\sum_{I+J=[1..|w|]} w[I]\otimes w[J]
\end{equation}
\end{expl}

\subsection{Comultiplication and Hopf Algebra structure on $\mathcal{U}(\LHW)$.}
As it is classical, the enveloping algebra $\mathcal{U}(\LHW)$ is endowed with a structure of Hopf Algebra by the extra structure given by 
\begin{itemize}
	\item a comultiplication 
	\item a counit
	\item an antipode.
\end{itemize}
The comultiplication $\Delta_\mathcal{U}$ is completely determined by the fact that the generators $b,b^+$ and $u$ are primitive i. e. 
\begin{equation}
	\Delta_\mathcal{U}(x)=x\otimes 1_\mathcal{U}+1_\mathcal{U}\otimes x\textrm{ for } x=b,b^+,u
\end{equation}
one then gets at once the coproduct of an element of the PBW basis

\begin{equation}
\hspace{-20mm}	\Delta_\mathcal{U}((b^+)^p\, b^q\, u^r)=\sum_{(\al,\be,\ga)\leq (p,q,r)} \binomial{p}{\al}\binomial{q}{\be}\binomial{r}{\ga} 
	(b^+)^\al\, b^\be\, u^\ga\otimes (b^+)^{p-\al}\, b^{q-\be}\, u^{r-\ga}\ .
\end{equation}
It should be noticed that this comultiplication is constructed on the model of \mref{subwords_cm} as, for a word $w=a^pb^qc^r$, on has
\begin{equation}
	\Delta(a^p\, b^q\, c^r)=\sum_{(\al,\be,\ga)\leq (p,q,r)} \binomial{p}{\al}\binomial{q}{\be}\binomial{r}{\ga} 
	a^\al\, b^\be\, c^\ga\otimes a^{p-\al}\, b^{q-\be}\, c^{r-\ga}\ .
\end{equation}

The counity is given by $\ep_\mathcal{U}((b^+)^p\, b^q\, u^r)=0$ unless $p=q=r=0$ (in which case it equals $1$). The antipode is given classically by 
$S(((b^+)^p\, b^q\, u^r))=(-1)^{p+q+r}u^r\,b^q\,((b^+)^p$. This is the standard structure of any enveloping algebra as a Hopf algebra. The question solved in this paper is a construction on $\tilde{B}$ of a structure of Hopf algebra such that $\varphi_5$ be a Hopf arrow.

\subsection{Partitions of a $HW$-graph and the comultplication.}

In order to get $\varphi_5$ as a Hopf arrow, we should use a formula of the type $\mref{subwords_cm}$ and define 
\begin{itemize}
	\item what is the support (where the subset $I$ would be taken)
	\item what is the notion of restriction in order to define $\Gamma[I]$ as we have defined $w[I]$.
\end{itemize}
 
Let $\Gamma$ be a $HW$-graph and $R\subset \Gamma^{(1)}$, one defines $\Gamma[R]$ as the subgraph with lines $R$, tail and head functions restricted to $R$ (call them $t_R,h_R$) and vertices $\Gamma[R]^{(0)}$ as $t_R(R)\cup h_R(R)$. It is easy to check that $\Gamma[R]$ is a $HW$-graph. It is also not difficult to check that, if $R_1\subset R_2\subset \Gamma^{(1)}$, on has the transitivity of restrictions i. e. 
\begin{equation}\label{trans_of_res}
\Gamma[R_2][R_1]=\Gamma[R_1]\ .
\end{equation}

We define a comultiplication on $\tilde{B}$ by 
\begin{equation}
	\Delta(\tilde{\Gamma})=\sum_{R+Y=\Gamma^{(1)}} \widetilde{\Gamma[R]}\otimes \widetilde{\Gamma[Y]}
\end{equation}

of course, one has to check that the result does not depend of the choice of the representative $\Gamma$ which is straightforward. The proof 
of coassociativity rests on \mref{trans_of_res}, i. e. 

\begin{eqnarray}
(\Delta\otimes I)\Delta(\tilde{\Gamma})=(\Delta\otimes I)\Big(\sum_{R+Y=\Gamma^{(1)}} \widetilde{\Gamma[R]}\otimes \widetilde{\Gamma[Y]}\Big)=\cr
\sum_{R+Y=\Gamma^{(1)}} \sum_{R_1+R_2=\Gamma[R]^{(1)}} \widetilde{\Gamma[R][R_1]}\otimes \widetilde{\Gamma[R][R_2]}\otimes \widetilde{\Gamma[Y]}
\end{eqnarray}

but, by construction $\Gamma[R]^{(1)}=R$ and then, using \mref{trans_of_res}, one has

\begin{eqnarray}
(\Delta\otimes I)\Delta(\tilde{\Gamma})=
\sum_{R+Y=\Gamma^{(1)}} \sum_{R_1+R_2=R} \widetilde{\Gamma[R_1]}\otimes \widetilde{\Gamma[R_2]}\otimes \widetilde{\Gamma[Y]}=\cr
\sum_{R_1+R_2+Y=R} \widetilde{\Gamma[R_1]}\otimes \widetilde{\Gamma[R_2]}\otimes \widetilde{\Gamma[Y]}
\end{eqnarray}

which is the same as $(I\otimes \Delta)\Delta(\tilde{\Gamma})$.\\
One has now to prove that $\Delta \tilde{B}\ra \tilde{B}\otimes \tilde{B}$ is a morphism of algebras. Let $\Gamma_i, i=1,2$ be two disjoint $HW$-graphs, one has

\begin{eqnarray}
\hspace{-22mm}\Delta(\tilde{\Gamma_1})*^{\otimes^2}\Delta(\tilde{\Gamma_2})=\Big(\sum_{R_1+Y_1=\Gamma_1^{(1)}} \widetilde{\Gamma_1[R_1]}\otimes \widetilde{\Gamma_1[Y_1]}\Big)*^{\otimes^2}\Big(\sum_{R_1+Y_1=\Gamma_1^{(1)}} \widetilde{\Gamma_1[R_1]}\otimes \widetilde{\Gamma_1[Y_1]}\Big)=\cr
\sum_{R_1+Y_1=\Gamma_1^{(1)}\atop R_2+Y_2=\Gamma_2^{(1)}} 
(\widetilde{\Gamma_1[R_1]}*\widetilde{\Gamma_2[R_2]})\otimes (\widetilde{\Gamma_1[Y_1]}*\widetilde{\Gamma_2[Y_2]})=\cr
\sum_{R_1+Y_1=\Gamma_1^{(1)}\atop R_2+Y_2=\Gamma_2^{(1)}}\sum_{m_R\in match(\Gamma_2[R_2],\Gamma_1[R_1])\atop m_Y\in match(\Gamma_2[Y_2],\Gamma_1[Y_1])} 
\widetilde{\pmatrix{\Gamma_1[R_1]\cr m_R\cr \Gamma_2[R_2]}}\otimes \widetilde{\pmatrix{\Gamma_1[Y_1]\cr m_R\cr \Gamma_2[Y_2]}}=\cr
\sum_{m\in match(\Gamma_2,\Gamma_1)} \sum_{R+Y=\pmatrix{\Gamma_1\cr m\cr \Gamma_2}^{(1)}} 
\pmatrix{\Gamma_1\cr m\cr \Gamma_2}[R]\otimes \pmatrix{\Gamma_1\cr m\cr \Gamma_2}[Y]
\end{eqnarray}

as the correspondence 
\begin{equation}
	(m,R,Y)\mapsto (R_1,Y_1,R_2,Y_2,m_R,m_Y)
\end{equation}

is one to one between the sets
\begin{equation}
\hspace{-1cm} \{ (m,R,Y)\textrm{ such that }(m\in match(\Gamma_2,\Gamma_1))\textrm{ and } (R+Y=\pmatrix{\Gamma_1\cr m\cr \Gamma_2}^{(1)})\}
\end{equation}

and

\begin{eqnarray}
\{(R_1,R_2,Y_1,Y_2,m_R,m_Y)\textrm{ such that } (R_1+Y_1=\Gamma_1^{(1)},\ R_2+Y_2=\Gamma_2^{(1)})\cr 
(m_R\in match(\Gamma_2[R_2],\Gamma_1[R_1]),\ m_Y\in match(\Gamma_2[Y_2],\Gamma_1[Y_1]))\}
\end{eqnarray}

\section{Endnotes}
Formula \mref{ass_proof} : Strictly speaking, the (double) sum in the LHS of this formula is indexed by the set
\begin{equation}
I_1:=\bigcup_{m'\in match(\Gamma_2,\Gamma_1)} \{m'\}\times match(\Gamma_3,{\tiny \pmatrix{\Gamma_1\cr m'\cr \Gamma_2}})
\end{equation}
and the RHS of this formula is indexed by the set
\begin{equation}
I_2:=\bigcup_{m''\in match(\Gamma_3,\Gamma_2)} match(\pmatrix{\Gamma_2\cr m''\cr \Gamma_3},\Gamma_1)\times \{m''\}
\end{equation}
the (one-to-one) correspondence $I_1\ra I_2$ is given by $(m',m'')\mapsto (m'\cup m''_{31},m''_{32})$ and its inverse by 
$(m',m'')\mapsto (m'_{21},m''\cup m'_{31})$ with $m_{ij}=m\cap (\Gamma_i^+ \times \Gamma_j^-)$.



\section*{References}
\begin{thebibliography}{ABC}
%
\bibitem{BFFLS} {\sc Bayen (F.), Flato (M.), Fronsdal (C.), Lichnerowicz (A.) and Sternheimer (D.)}, {\it Deformation and Quantization}, Ann. of Phys. 111 (1978), pp. 61-151.
%
\bibitem{BR} {\sc J. Berstel, C. Reutenauer}, {\it Rational series and their
languages EATCS Monographs on Theoretical Computer Science},
Springer (1988).
%
\bibitem{BBM} {\sc C. M. Bender, D. C. Brody, and B. K. Meister},
Quantum field theory of partitions, J. Math. Phys. Vol 40 (1999)
%
\bibitem{GOF7} {\sc P. Blasiak, A. Horzela, K. A. Penson, G. H. E. Duchamp,
A.I. Solomon}, {\it Boson normal ordering via substitutions and
Sheffer-Type Polynomials}, Phys. Lett. A {\bf 338} (2005) 108
%
\bibitem{GOF8} {\sc P. Blasiak,  K. A. Penson, A.I. Solomon,
A. Horzela, G. H. E. Duchamp}, {\it Some useful formula for bosonic
operators}, Jour. Math. Phys.
 {\bf 46} 052110 (2005).
%
\bibitem{B_Alg_III} {\sc Bourbaki N.}, {\it Algebra, chapter III}, Springer
%
\bibitem{B_Alg_VI} {\sc Bourbaki N.}, {\it Algebra, chapter VI}, Springer
%
\bibitem{B_ST} {\sc Bourbaki N.}, {\it Theory of sets}, Springer
%
\bibitem{Ca} {\sc Cartier P.}, {\it S\'eminaire ``Sophus Lie'', 2\`eme ann\'ee}, Facult\'e des Sciences de Paris (1955-56)
%
\bibitem{Ca2} \textsc{P. Cartier}, {\it Fonctions polylogarithmes, nombres polyzeta et groupes pro-unipotents, S\'eminaire Bourbaki, Asterisque n. 282 (2002)}
%
\bibitem{Ca3} \textsc{P. Cartier}, {\it A primer of Hopf algebras}, Septembre (2006), IHES preprint IHES/M/06/40.
%
\bibitem{CP} {\sc V. Chari, A. Pressley}, A guide to quantum groups.
Cambridge (1994).
%
\bibitem{GOF4} {\sc G. H. E. Duchamp, P. Blasiak, A. Horzela, K. A. Penson, A. I. Solomon, Feynman graphs and related Hopf algebras}, Journal of Physics: Conference Series, SSPCM'05, Myczkowce, Poland.
arXiv : cs.SC/0510041
%
\bibitem{DFLL} {\sc Duchamp G., Flouret M., Laugerotte E., Luque J-G.},
{\it Direct and dual laws for automata with multiplicities}\\
{\tt arXiv} : math.CO\/0607412
%
\bibitem{DL} {\sc Duchamp G., Luque J-G.},
{\it Congruences Compatible with the Shuffle Product}\\
{\tt arXiv} : math.CO\/0607419
%
\bibitem{DHT} {\sc G. Duchamp, F. Hivert, J. Y. Thibon}, {\it Non commutative
functions VI: Free quasi-symmetric functions and related algebras},
International Journal of Algebra and Computation Vol 12, No 5
(2002).
%
\bibitem{DKPTT} {\sc G. H. E. Duchamp, G. Koshevoy, K. A. Penson, C. Tollu, F. Toumazet}, {\it Geometric combinatorial twisting and shifting}, S\'eminaire Lotharingien (in preparation).
%
\bibitem{FPSAC07} {\sc G. H. E. Duchamp, J. -G. Luque, J. -C. Novelli, C. Tollu, F. Toumazet}, {\it Hopf algebras of diagrams}, FPSAC07.
%
\bibitem{OPG} {\sc G. Duchamp, A.I. Solomon, K.A. Penson, A. Horzela
and P. B lasiak}, {\it One-parameter groups and combinatorial
physics}, Proceedings of the Symposium Third International Workshop
on Contemporary Problems in Mathematical Physics (COPROMAPH3)
(Porto-Novo, Benin, Nov. 2003),  J. Govaerts, M. N. Hounkonnou and A. Z. Msezane (eds.), p.436 (World Scientific Publishing 2004)\\
arXiv: {\tt quant-ph/04011262}
%
\bibitem{GOF13} \textsc{G. H. E. Duchamp, P. Blasiak, A. Horzela, K. A. Penson, A. I Solomon}, \textit{Hopf Algebras in General and in Combinatorial Physics: a practical introduction}
\texttt{arXiv:0802.0249v1} (to be published).
%
\bibitem{Fo1} {\sc L. Foissy}, {\it Isomorphisme entre l'alg\`ebre des fonctions quasi-sym\'etriques libres et une alg\`ebre de Hopf des arbres enracin\'es d\'ecor\'es plans}, personal communication.
%
\bibitem{Fo2} {\sc L. Foissy}, {\it Les alg\`ebres de Hopf des arbres enracin\'es decor\'es}, PhD Memoir, Reims University (2002).
%
\bibitem{Fo3} {\sc L. Foissy}, {\it Personal Communication.}
%
\bibitem{Ha} {\sc M. Hazewinkel}, {Hopf algebras of endomorphisms of Hopf algebras}, (Oct 2004)
{\tt ArXiv : math.QA/0410364}
%
\bibitem{Ho} {\sc M. E. Hoffman}, {\it Quasi-shuffle products}, J. Algebraic Combin. (2000), 49-68
%
\bibitem{GOF6} {\sc A. Horzela, P. Blasiak, G. Duchamp, K. A. Penson
and A.I. Solomon}, {\it A product formula and combinatorial field
theory}, Proceedings of the XI International Conference on Symmetry
Methods
in Physics (SYMPHYS-11) (Prague, Czech Republic, June 2004), C. Burdik, O. Navratil, and S. Posta (eds.) (JINR Publishers, Dubna)\\
arXiv:{\tt quant-ph/0409152}
%
\bibitem{Kr} {\sc D. Kreimer}, {\it Knots and Feynman Diagrams}, Cambridge Lecture Notes in Physics (2000).
%
\bibitem{Oc} {\sc P. Ochsenschl\"ager}, Binomialkoeffitzenten und Shuffle-Zahlen, Technischer
Bericht, Fachbereich Informatik, T. H. Darmstadt,1981.
%
\bibitem{Re} {\sc Reutenauer C.}, {\it Free Lie algebras}, Oxford University Press (1993).
%
\bibitem{Ro} {\sc M. Rosso}, {\it Quantum groups and quantum shuffles}, Inventiones Math. 133 (1998).
%
\bibitem{JR} {\sc S. A. Joni, G.-C. Rota}, {\it Coalgebras and Bialgebras in Combinatorics}, Studies in Applied Mathematics {\bf 61} (1979)
%
\bibitem{GOF2} {\sc A.I. Solomon, P. Blasiak, G. Duchamp, A. Horzela and K.A. Penson}, {\it Combinatorial Physics,
Normal Order and Model Feynman Graphs}, Proceedings of the Symposium
'Symmetries in Science XIII', Bregenz, Austria, 2003, B. Gruber, G.
Marmo and N. Yoshinaga (eds.), p.527 (Kluwer Academic
Publishers 2004)\\
arXiv : {\tt quant-ph/0310174}
%
\bibitem{GOF3} {\sc A.I. Solomon, G. Duchamp, P. Blasiak, A. Horzela and K.A. Penson},
{\it Normal Order: Combinatorial Graphs Quantum Theory and
Symmetries}, Proceedings of the 3rd International Symposium P.C.
Argyres, T.J. Hodges, F. Mansouri, J.J. Scanio, P. Suranyi, and
L.C.R.
Wijewardhana (eds.), p.398 (World Scientific Publishing 2004)\\
arXiv:{\tt quant-ph/0402082}
%
\bibitem{GOF5} {\sc A.I. Solomon, G. Duchamp, P. Blasiak, A. Horzela
and K. A. Penson}, {\it Partition functions and graphs: A
combinatorial approach}, Proceedings of the XI International
Conference on Symmetry Methods
in Physics (SYMPHYS-11) (Prague, Czech Republic, June 2004), C. Burdik, O. Navratil, and S. Posta (eds.) (JINR Publishers, Dubna, 2004)\\
arXiv:{\tt quant-ph/0409082}
%
\bibitem{GOF9} {\sc A. I. Solomon, G. Duchamp, P. Blasiak, A. Horzela  and K. A. Penson}, {\it Hopf algebra structure of a model quantum field theory}, Proceedings of the 26th International Colloquium on Group Theoretical Methods in Physics, New York 2006, Editor: S. Catto (2007).
%
\bibitem{GOF10} {\sc G. Duchamp, A. I. Solomon, P. Blasiak, A. Horzela  and K. A. Penson}, {\it A multipurpose Hopf deformation of the algebra of Feynman-like diagrams}, Proceedings of the 26th International Colloquium on Group Theoretical Methods in Physics, New York 2006, Editor: S. Catto (2007).
%

\end{thebibliography}
\end{document}
