\documentclass[proceedings,submission]{dmtcs}
%
% if you want to use pdftex, uncomment the following line
%,pdftex%
% if you have difficulties with hyperref uncomment the following line
%,nohyperref%
% if you have difficulties with fonts uncomment the following line
%,notimes%

\usepackage[latin1]{inputenc}
\usepackage{subfigure}
\usepackage{amsmath}

% graphicx is now loaded automatically no need to put this in here anymore.
% 
%\usepackage{graphicx}

% just comment this out if you don't have natbib
%\usepackage[round]{natbib}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%% Environments%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newtheorem{thm}{Theorem}
\newtheorem{lemma}[thm]{Lemma}
\newtheorem{cor}[thm]{Corollary}
\newtheorem{prop}[thm]{Proposition}
\newtheorem{defn}[thm]{Definition}
\newtheorem{rem}[thm]{Remark}
\newtheorem{example}[thm]{Example}
\newtheorem{counterexample}[thm]{Counter-example}
\newtheorem{note}[thm]{Note(s)}
%%%%%%%%%%%%%%%%%%%%%%
% Jumps
%%%%%%%%%%%%%%%%%%%%%%

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

\def\ra{\rightarrow}
\def\da{\downarrow}
\def\sw{\swarrow}
\def\se{\searrow}
\def\ne{\nearrow}
\def\ua{\uparrow}
\def\lra{\leftrightarrow}
\def\Lra{\Longrightarrow}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%Signes discursifs%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\def\mref#1{{\footnotesize ({\ref{#1}})}}
\def\bu{\bullet}
\def\Proof{\medskip\noindent {\it Proof --- \ }}
\def\cqfd{\hfill $\Box$ \bigskip}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Letters
%%%%%%%%%%%%%%%%%%%%%%

\def\A{\mathcal{A}}
\def\D{\mathcal{D}}
\def\B{{\mathbb B}}
\def\E{{\mathbb E}}
\def\F{{\mathbb F}}
\def\N{{\mathbb N}}
\def\C{{\mathbb C}}
\def\R{{\mathbb R}}
\def\Z{{\mathbb Z}}
\def\Q{{\mathbb Q}}
\def\X{{\mathbb X}}
\def\al{\alpha}
\def\be{\beta}
\def\ga{\gamma}
\def\ep{\epsilon}
\def\om{\omega}
\def\S{{\Sigma}}
\def\Q{{\mathbb Q}}
\def\SG{{\goth S}}
\def\L{\mathbb{L}}
\def\V{\mathbb{V}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%% Operators, space and laws %%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\def\scal#1#2{\langle #1 | #2 \rangle}
\def\ncp#1#2{#1\langle #2\rangle}
\def\shuffle{{\mathop{_{^{\sqcup\!\sqcup}}}}}
%
\gdef\stuffle{\;% 
  \setlength{\unitlength}{0.0125cm}% 
  \begin{picture}(20,10)(220,580) 
  \thinlines 
  \put(220,592){\line( 0,-1){ 10}} 
  \put(220,582){\line( 1, 0){ 20}} 
  \put(240,582){\line( 0, 1){ 10}} 
  \put(230,592){\line( 0,-1){ 10}} 
  \put(225,587){\line( 1, 0){ 10}} 
  \end{picture}\; 
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\author{Gérard H. E. Duchamp\addressmark{1}\thanks{The authors wish to acknowledge support from Agence Nationale de la Recherche (Paris, France) under Program No. ANR-08-BLAN-0243-2.} \and Hoang Ngoc Minh\addressmark{1}}
\title[Radford theorem for generalized deformations of Hoffmann type]{Radford theorem for generalized deformations of Hoffmann type}
\address{\addressmark{1}LIPN - UMR 7030, 
CNRS - Universit\'e Paris 13, 93430 Villetaneuse,
France}
\keywords{Comultiplicatioon, Hopf algebra}

\begin{document}

\maketitle

\begin{abstract}
\paragraph{Abstract.}
\paragraph{R\'esum\'e.}
\end{abstract}

%For long papers, you may include a table of contents, like the
%following. But in most cases this is not necessary. In addition, PDF
%documents will have an internal table of contents. Usually this table
%is shown by PDF browsers in a different panel than the text itself.

%\clearpage
%\tableofcontents

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%% Corps de l'article %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Introduction}\label{section-introduction}

With the advent of Quantum theories emerged the need for deforming not only parameters or formulas but structures and laws (see \cite{CP}). On the other hand, computer science provided us with laws the revealed later to be better understood as dual as the shuffle product recursively defined by
\begin{equation}
	au\shuffle bv = 	a(u\shuffle bv) +	b(au\shuffle v)
\end{equation}
and the $q$-infiltration product \cite{DFLL} by
\begin{equation}
	au\ua bv = 	a(u\ua bv) +	b(au\ua v) + q \delta_{a,b} a(u\ua v). 
\end{equation}
In fact all these laws are particular cases of diagonal deformations \cite{DT} and such deformations can be applied to stuffles.

.../...

Dual laws of theoretical computer science : shuffle, infiltration, $q$-infiltration (work with Luque), $LDIAG(q_c,_qs)$ and explanationn of the different natures of the parameters.

\section{Background}

Twisted products of graded algebras, diagonal deformations.

\section{Colour factors and products}

Colours factors were introduced by \cite{Ree} and the theory was developped or used in \cite{ncsf3,D15,MZ,JZ}.

Let $\displaystyle{\mathcal A={\oplus}_{\alpha\in \mathcal
D}\mathcal A_\alpha}$ and $\displaystyle{\mathcal
B=\oplus_{\beta\in \mathcal D}\mathcal B_\beta}$ be two $\mathcal
D$-graded associative algebras ($\mathcal D$ is a commutative semigroup whose law is denoted additively). 
Readers that are not familiar with graded algebras can think of $\mathcal D=\N^{(X)}$, the free commutative monoid over $X$ and $\displaystyle{\mathcal A_\alpha=K[X]_\alpha}$, the space of homogeneous polynomials of multidegree $\alpha$.


We suppose given a mapping $\chi :\mathcal D\times\mathcal D\longrightarrow K$
and define a law of algebra on $\mathcal{A}\otimes \mathcal{B}$ by

\begin{eqnarray}\label{prod_tens_color}
  (x_1\otimes y_1)(x_2\otimes y_2) &=& \chi(\beta_1,\alpha_2)(x_1x_2\otimes
  y_1y_2)
\end{eqnarray}

for $\displaystyle{(x_i)\in\mathcal A_{\al_i}}$ and $\displaystyle{(y_i)\in\mathcal B_{\be_i}}$ ($i=1,2$).\\
The computations of $\left((x_1\otimes y_1)(x_2\otimes y_2)\right)(x_3\otimes y_3)$ and 
$(x_1\otimes y_1)\left((x_2\otimes y_2)(x_3\otimes y_3)\right)$ using \mref{prod_tens_color} both lead to the following 
proposition the second part -- converse -- of which relies on the existence of free elements.

\begin{prop}\label{colour_cond}\cite{JZ} Let $\chi :\mathcal D\times\mathcal D\longrightarrow K$. The following are
equivalent\\
i) For $\mathcal A$, $\mathcal B$ $\mathcal D$-graded
    associative algebras, the product defined by
    (\ref{prod_tens_color}) is associative.\\
ii) $(\forall \alpha_1,\alpha_2,\alpha_3,\beta_1,\beta_2,\beta_3)\in\mathcal D$
\begin{equation}\label{cocycle}
      \chi (\beta_1,\alpha_2)\chi (\beta_1+\beta_2 ,\alpha_3) = \chi (\beta_2,\alpha_3)\chi (\beta_1,\alpha_2+\alpha_3)
\end{equation}
\end{prop}

\begin{defn} Every mapping $\chi :\mathcal D\times\mathcal D\longrightarrow K$ which fulfills the equivalent conditions of proposition \mref{colour_cond} will be called a colour factor.
\end{defn}

\begin{rem} i) If $\chi$ is bilinear, which means in this context that the following equations are satisfied (for all $\al,\al',\be,\be'\in\mathcal{D}$)
\begin{eqnarray}\label{bichar}
      \chi (\alpha+\alpha ',\beta) &=& \chi (\alpha ,\beta)\chi (\alpha' ,\beta) \cr
      \chi (\alpha ,\beta+\beta ') &=& \chi (\alpha ,\beta)\chi
      (\alpha ,\beta')
\end{eqnarray}
then, the two members of (\ref{prod_tens_color}) amount to
    \begin{eqnarray}
      \chi (\beta_1, \alpha_2)\chi (\beta_1,\alpha_3)\chi (\beta_2,\alpha_3) &=& \prod_{1\leq i< j\leq 3} \chi (\beta_i,\alpha_j)
    \end{eqnarray}
and hence $\chi$ is a colour factor. But the full class of colour factors is much larger than solutions of Eq. \mref{bichar}. Just observe that Eq.(\ref{cocycle}) is 
homogeneous in the classical sense {\em i.e.} for all $\lambda\in k$, if $\chi$ fulfills \mref{cocycle} then $\lambda\chi$ still does. Hence, for example, any constant function on $\mathcal{D}\times \mathcal{D}$ is a colour factor. This shows the existence of colour factors that are not bilinear.\\
ii) It may seem that one could generalize \mref{bichar} to the case when $\D$ is noncommutative but, in fact, there is no gain of generality because, as $K$ is commutative, the bicharacter factorizes through $\mathcal{D}^{ab}$ (the quotient of $\mathcal{D}$ by the finest congruence $\equiv$ such that $\mathcal{D}/\equiv$ is abelian).
\end{rem}

\begin{note}\label{ass_and_morph}
i) The colour product of two algebras $\displaystyle{\mathcal A={\oplus}_{\alpha\in \mathcal D}\mathcal A_\alpha}$ and 
$\displaystyle{\mathcal B=\oplus_{\beta\in \mathcal D}\mathcal B_\beta}$ comes also as a graded algebra by 
\begin{equation}
	(\mathcal A\otimes \mathcal B)_\ga=\oplus_{\al+\be=\ga}\ \mathcal A_\al\otimes \mathcal B_\be .
\end{equation}

The usual identification 

\begin{equation}
	(\mathcal A\otimes \mathcal B)\otimes \mathcal C\simeq \mathcal A\otimes (\mathcal B\otimes \mathcal C)
\end{equation}
holds for the coloured products.\\
ii) Moreover, if $\mathcal A \stackrel{f}{\ra}\mathcal A'$ (resp. $\mathcal B \stackrel{g}{\ra}\mathcal B'$) are two morphisms of (graded) algebras, then 
$\mathcal A\otimes \mathcal B \stackrel{f\otimes g}{\longrightarrow}\mathcal B\otimes \mathcal B'$ is a morphism of algebras (colour products).
\end{note}


\section{Special classes of laws}
\subsection{Dual laws}
\subsubsection{Algebras and coalgebras in duality}\label{dual_laws}

An algebra $(\A,\mu)$ and a coalgebra $(C,\Delta)$ are called in duality iff there is a non-degenerate pairing $\scal{-}{-}$ such that for all $x,y\in\A,\ z\in C$

\begin{equation}
	\scal{\mu(x,y)}{z}=\scal{x\otimes y}{\Delta(z)}^{\otimes 2}
\end{equation}

In the following, we will call \textit{dual law} a law $\ncp{K}{A}\otimes \ncp{K}{A}\stackrel{*}{\longrightarrow} \ncp{K}{A}$ on the free algebra which is the dual of a comultiplication, the pairing being given on the basis of words by $\scal{u}{v}=\delta_{u,v}$.

\bs
Our first examples are essential in modern and not-so-modern research (\cite{Ro,Oc}). Firstly, we have the dual of the Cauchy product
\begin{equation}\label{coCauchy}
	\Delta_{Cauchy}(w)=\sum_{uv=w} u\otimes v\ .
\end{equation}
Contrary to this one \mref{coCauchy}, which is not a morphism of algebras\footnote{Unless $A=\emptyset$.} 

\begin{equation}\label{free_alg}
\ncp{K}{A}\longrightarrow \ncp{K}{A}\otimes \ncp{K}{A}\ ,
\end{equation}

one has three very well-known examples being so, namely duals of the shuffle $\shuffle$, the Hadamard $\odot$ and the infiltration product $\ua$. As they are morphisms between the algebras \mref{free_alg}, they are well defined by their values on the letters. Respectively 

\begin{equation}
\Delta_\shuffle(x)=x\otimes 1+ 1\otimes x\ ;\ \Delta_\odot(x)=x\otimes x\ ;\ 
\Delta_\ua(x)=x\otimes 1+ 1\otimes x + x\otimes x\ .
\end{equation}

One can prove that the deformations $\Delta_q=\Delta_\shuffle(x)+ q \Delta_\odot(x)$ are also co-associative and that they are the unique solutions of the problem of bialgebra comultiplications on $\ncp{K}{A}$ that are compatible with subalphabets \cite{DFLL}.

In the sequel, we will make use several time of the following lemma the proof of which is left to the reader.

\begin{lemma}\label{alg_coalg} Let $\mathcal{A}$ be an algebra and $\mathcal{C}$ be a coalgebra in (non-degenerate) duality, then $\mathcal{A}$ is associative iff $\mathcal{C}$ is coassociative.
\end{lemma}

\subsubsection{Duality between grouplike elements and unities}\label{dual_unit_grouplike}

Let $(\mathcal{C},\Delta)$ be a coalgebra with counit $\ep$. We call \textit{group-like} an element $u$ such that 
\begin{equation}
	\ep(u)=1\ ;\ \Delta(u)=u\otimes u\ .
\end{equation}

One then has $\mathcal{C}=ker(\ep)\oplus K.u$ and 
\begin{equation}\label{delta+}
\Delta(y)=\Delta^+(y)+y\otimes u+u\otimes y-\ep(y)u\otimes u\ .
\end{equation}
where $\Delta^+$ is a comultiplication on $\mathcal{C}$ for which $ker(\ep)=\mathcal{C}^+$ is a subcoalgebra (i. e. $\Delta^+(\mathcal{C}^+)\subset \mathcal{C}^+\otimes \mathcal{C}^+$) \cite{B_Lie_I_III}.

\begin{prop} Let $(\mathcal{C},\Delta,\ep)$ be a coalgebra with counit, $u$ a group-like element in $\mathcal{C}$ and $(\mathcal{C}^+,\Delta^+)$ be as in \mref{delta+}. On the other hand, let $\A$ be an algebra and $\A^{(1)}=\A\oplus K.v$ be the algebra with unit constructed from $\A$ by adjunction of the unity $v$. Then, if $\mathcal{C}^+$ and $\A$ are in duality by $\scal{\ }{\ }$, so are $\mathcal{C}$ and $\A^{(1)}$ by $\scal{\ }{\ }_\bu$ defined as follows
\begin{equation}
	\scal{x+\al v}{y+\be u}_\bu=\scal{x}{y}+\be\al\ .
\end{equation}
\end{prop}
\Proof Let 
\begin{eqnarray}\label{eq14}
	\scal{(x_1+\al_1 v)\otimes (x_2+\al_2 v)}{\Delta(y+\be u)}_\bu^{\otimes 2}=\cr
	\scal{(x_1+\al_1 v)\otimes (x_2+\al_2 v)}{\Delta^+(y)+ y\otimes u+u\otimes y+\be u\otimes u)}_\bu^{\otimes 2}
\end{eqnarray}
but, according to the fact that 
$$
\scal{x_i}{u}=\scal{x_1\otimes v}{\Delta^+(y)}=\scal{v\otimes x_2}{\Delta^+(y)}=\scal{v\otimes v}{\Delta^+(y)}=\scal{v}{y}=0
$$
 one has from \mref{eq14}
\begin{eqnarray}
&&\scal{(x_1+\al_1 v)\otimes (x_2+\al_2 v)}{\Delta(y+\be u)}_\bu^{\otimes 2}=\cr
	&&\scal{x_1\otimes x_2}{\Delta^+(y)}^{\otimes 2} +\al_2 \scal{x_1}{y}+ 
	\al_1 \scal{x_2}{y}+\al_1\al_2\be=\cr
	&&\scal{x_1x_2+\al_2 x_1+\al_1 x_2+\al_1\al_2 v}{y+\be u}_\bu=\scal{(x_1+\al_1 v)(x_2+\al_2 v)}{y+\be u}_\bu
\end{eqnarray}
which proves the claim.\cqfd

\subsection{Deformed laws}\label{Deformed laws}

Let $S$ be a semigroup graded on a semigroup of degrees $\mathcal{D}$ and $\A=K[S]$ its algebra. A colour factor 
$\chi : \mathcal{D}\times \mathcal{D}\ra K$ being given, we endow the algebra $\A\otimes \A$ with the coloured tensor product structure. Notice that the diagonal subspace $D_S=\oplus_{x\in S}K x\otimes x$ is a subalgebra as 
\begin{equation}\label{eq13}
(x\otimes x)(y\otimes y)=\chi(|x|,|y|) xy\otimes xy\ .
\end{equation}
Carrying \mref{eq13} back to $\A$ by means of the isomorphism of vector spaces, $\A\ra D_S$, on sees immediately that the deformed product on $\A$ given by 
\begin{equation}\label{def_prod}
x._\chi y=\chi(|x|,|y|) xy
\end{equation}
is associative.\\
If $\A$ is endowed with the scalar product for which the basis 
$(s)_{s\in S}$ is orthonormal, the pairing is non-degenerate and the dual comultiplucation is given by 
\begin{equation}
\Delta(z)=\sum_{xy=z}\chi(|x|,|y|)x\otimes y\ .
\end{equation}
The construction together with lemma \mref{alg_coalg} proves that this comultiplication on $\A$ is coassociative.



\section{Generalized shuffles and stuffles}

Discussion of the commutation factor

\section{Conclusion}

Grading polyzeta values

\begin{thebibliography}{ABC}
%
\bibitem{BouAlg} N.~Bourbaki. \newblock \emph{\'El\'ements de Math\'ematique - Alg\`ebre, chap. 1 \`a 3}, 2nd edition, Springer, 2007.
%
\bibitem{BouAlgCom} N.~Bourbaki. \newblock \emph{\'El\'ements de Math\'ematique - Alg\`ebre commutative, chap. 1 \`a 4}, Springer, 2006.  
%
\bibitem{CP} V. Chari, A. Pressley, {\it A guide to quantum groups.} Cambridge Univ. Press, 1994.
%
\bibitem{D15} \textsc{D\'esarm\'enien J., Duchamp G., Krob D., Melan{\c c}on G}, 
{\it Quelques remarques sur les superalg\`ebres de Lie libres} C.R.A.S. 
 n$^0$5 (1994).
%
\bibitem{DFLL} G. Duchamp, M. Flouret, \'E. Laugerotte, J.-G. Luque, {\it Direct and dual laws for automata with multiplicities}, Theoret. Comp. Sci. {\bf 267} (2001) 105-120.
%
\bibitem{ncsf3} \textsc{G H E Duchamp, A Klyachko, D Krob, J Y Thibon}, {\it Noncommutative symmetric functions
III: Deformations of Cauchy and convolution algebras}. Discrete Mathematics and Theoretical Computer Science
Vol. {\bf 2} (1998).
%
\bibitem{DT}  \textsc{Duchamp G. H. E., Tollu C.}, \textit{Sweedler's duals and Sch\"utzenberger's calculus}
\texttt{arXiv:0712.0125} (to be published).
%
\bibitem{Eil74} S.~Eilenberg. \newblock \emph{Automata, Languages and Machines, vol. A.}, Academic Press, 1974. 
%
\bibitem{MZ} \textsc{A. A. Mikhalev and A. A. Zolotykh}, \textit{Combinatorial Aspects of Lie Superalgebras.} 
CRC Press, Boca Raton, New York, 1995.
%
\bibitem{Ree} {\sc R. Ree}, {\it Generalized Lie elements}, Canad.
J.  Math. {\bf 12} (1960), 493--502.
%
\bibitem{JZ} \textsc{J. Zhou}, {\it Combinatoire des derivations}, Th\`ese de l'Universit\'e de Marne-la-Vall\'ee (1996).
%
\end{thebibliography}

\end{document}


\end{document}
