\section{Introduction}
\subsection{Reminders about quasi-triangular quasi-Hopf algebras}
A quasi-triangular quasi-Hopf algebra \cite{Drinfeld:QuasiHopf}
is an algebra $A$ together with a not-quite-cocommutative and
not-quite-coassociative coproduct $\Delta$, whose failure to be
cocommutative is ``controlled'' by some element $R\in A^{\otimes 2}$
and whose failure to be coassociative is ``controlled'' by some element
$\Phi\in A^{\otimes 3}$ (for more details, see~\cite{Drinfeld:QuasiHopf}
or~\cite{Kassel:Book, ShniderSternberg:Book}). For the representations of
$A$ to form a tensor category, $R$ and $\Phi$ have to obey the so-called
``pentagon'' $\pentagon\!$ and ``hexagon'' $\hexagon\!_\pm$ equations
(see section~\ref{ASS}). In~\cite{Drinfeld:QuasiHopf} Drinfel'd
finds a ``universal'' formula $(R_{\KZ},\Phi_{\KZ})$ for a solution
of $\pentagon\!$ and $\hexagon\!_\pm$ by considering holonomies of
the so-called Knizhnik-Zamolodchikov connection. The formula $R_{\KZ}$
is very simple --- $R_{\KZ}$ is in a clear sense ``an exponential''. The
formula $\Phi_{\KZ}$ is somewhat less satisfactory, as it requires
analysis --- differential equations and/or iterated integrals whose
values are most likely transcendental numbers~\cite{Drinfeld:QuasiHopf,
LeMurakami:Universal, Zagier:Values}. In~\cite{Drinfeld:GalQQ}
Drinfel'd proves that there is an iterative algebraic procedure for
finding a universal formula for a solution $(R,\Phi)$ of $\pentagon\!$,
$\hexagon\!_\pm$ (with $R=R_{\KZ}$), and that such a universal formula
(called {\em an associator}) can be found iteratively and over the
rationals.
Associators (and the iterative procedure for constructing them) are
important in the theory of finite-type invariants of knots (Vassiliev
invariants) \cite{Bar-Natan:VasBib, Bar-Natan:NAT, Cartier:Construction,
Kassel:Book, LeMurakami:Universal, Piunikhin:Combinatorial} and of
3-manifolds \cite{LeMurakamiOhtsuki:Universal, Le:UniversalIHS}.
Recently, Etingof and Kazhdan~\cite{EtingofKazhdan:Quantization,
EtingofKazhdan:Poisson} used associators to show that any Lie bialgebra
can be quantized. Their results become algorithmically computable once
we know that an associator can be found iteratively.\footnote{
For most applications of associators to finite-type invariants, it is
in fact sufficient to use a weaker but more complicated notion of an
associator for which an iterative construction
was given in~\cite{Bar-Natan:NAT}.
}
Unfortunately, Drinfel'd's paper is complicated and hard to read. It
involves the introduction, almost ``out of thin air'', of two groups,
$\GTc$ and $\GRTc$, that act on the set $\ASSc$ of all associators. Both
groups act simply transitively on $\ASSc$, with $\GTc$ acting on the right
and $\GRTc$ on the left, and the two actions commute. He then studies
these groups and their actions on $\ASSc$ to deduce the existence of
formulae better then $\Phi_{\KZ}$. Drinfel'd's ``{\bf G}rothendieck-{\bf
T}eichmuller'' group $\GTc$ is closely related to number theory and the
group $\text{Gal}(\bar{{\Bbb Q}}/{\Bbb Q})$. See~\cite{Drinfeld:GalQQ,
Schneps:DessinsdEnfants}. $\GRTc$ is in some sense a ``g{\bf R}aded''
version of $\GTc$, explaining why Drinfel'd inserted an R in the middle
of its name.
\subsection{What we do} The purpose of this paper is to present a
framework within which the set of associators $\ASSc$, the groups $\GTc$
and $\GRTc$, and the relevant facts about them are natural. In fact,
the mere fact that $\GTc$ and $\GRTc$ exist and act simply transitively
on the right (for $\GTc$) and on the left (for $\GRTc$), with the two
actions commuting, stems from the following basic principle (which I
learned from M.~Hutchings):
\begin{principle} \lbl{BasicPrinciple}
If $B$ is a mathematical structure (i.e., a set, a set
with a basepoint, an algebra, a category, etc.) and if $C$ is an
isomorphic mathematical structure, then on the set $A$ of all isomorphisms
$B\to C$ there are two commuting group actions, with both actions
simple and transitive:
\begin{itemize}
\item The group {\it GT} of (structure-preserving) automorphisms of $B$
acts on $A$ by composition on the right.
\item The group {\it GRT} of (structure-preserving) automorphisms of $C$
acts on $A$ by composition on the left.
\end{itemize}
\end{principle}
We apply this principle to a certain ``upgrade'' of the Kohno
isomorphism~\cite{Kohno:MonRep} (see also~\cite{KasselTuraev:Graphs})
between the unipotent completion $\widehat{PB}_n$ of the pure braid
group on $n$ strands and its associated graded algebra, which is a
certain completed algebra $\Apbc_n$ generated by symbols $t^{ij}$
with $1\leq i\neq j\leq n$.\footnote{In the language of Vassiliev
invariants, the Kohno isomorphism is a combination of three facts:
that the space of Vassiliev invariants of pure braids is the dual
of $\widehat{PB}_n$, that the associated graded space of Vassiliev
invariants of pure braids is dual to the algebra $\Apb_n$ of ``chord
diagrams'', and that the maps $\widehat{PB}_n\to\Apbc_n$ that we consider
are ``universal Vassiliev invariants''.} More specifically, in our case,
$B$ will be a certain category $\PB$ (defined in section~\ref{PBdef})
of {\em parenthesized braids}, and $C$ will be a certain category
$\PCD$ (defined in section~\ref{PCDdef}) of {\em parenthesized chord
diagrams}. On top of the category structure, both $\PB$ and $\PCD$ are
``fibered linear'', have natural ``basepoints'' (some specific morphisms
between some specific objects), natural ``coproducts'', and
natural ``extension'', ``cabling'', and ``strand removal''
operations, all defined in section~\ref{definitions}. Furthermore,
$\PB$ has a natural ``filtration'', $\PCD$ has a natural ``gradation''
(which induces a filtration as well), and these filtrations/gradations
(also defined in section~\ref{definitions}) respect all other structure
on $\PB$ and $\PCD$. In applying Principle~\ref{BasicPrinciple}, we will
only consider isomorphisms/automorphisms that respect all the additional
structure on $\PB$ and $\PCD$.
To be fair, we apply Principle~\ref{BasicPrinciple}
not to $B=\PB$ and $C=\PCD$, but rather
to their ``quotients'' $\PBm=\PB/\F{m+1}\PB$ and
$\PCDm=\PCD/\F{m+1}\PCD$ by their respective filtrations, or to
their ``completions'' $\PBc=\varprojlim_{m\to\infty}\PBm$ and
$\PCDc=\varprojlim_{m\to\infty}\PCDm$. In section~\ref{ASS}
we show that every isomorphism (invertible structure-preserving
functor) $\Zc:\PBc\rightarrow\PCDc$ is determined by its action
on some specific morphism $a$ in $\PBc$, and that $\Zc(a)$ can be
interpreted as an associator. We will thus identify the set of
all such $\Zc$'s with $\ASSc$, and get the two groups $\GTc$ and
$\GRTc$ (as well as their simple, transitive, and commuting actions)
entirely for free from Principle~\ref{BasicPrinciple}. Similarly, using
Principle~\ref{BasicPrinciple} with $B=\PBm$ and $C=\PCDm$, we get groups
$\GTm$ and $\GRTm$ that act on the set $\ASSm$ of all ``associators up
to degree $m$''.
In section~\ref{Main} we start by explaining why the surjectivity of
the natural map $\pi:\GRTm\to\GRTmm$ implies the surjectivity of the map
$\ASSm\to\ASSmm$, which implies that there exists an iterative procedure
for finding an associator, and that a rational associator exists.
We then turn to the proof of the surjectivity of $\pi$. To do this, we
first write the relations defining $\GRTc$ explicitly. These turn out
to be the ``pentagon'', the ``classical hexagon'', the ``semi-classical
hexagon'', and some technical relations of lesser interest. It turns out
that the only relation that could challenge the surjectivity of $\pi$ is
the semi-classical hexagon, and so we spend the rest of section~\ref{Main}
proving that the semi-classical hexagon follows from the classical
hexagon, the pentagon, and the lesser relations. This is done by using
a certain 12-face polyhedron to show that the failure $\psi$ of the
semi-classical hexagon to hold lies in the kernel of some differential,
and by studying the relevant cohomology of the corresponding complex.
Just for completeness, in section~\ref{Just} we display the defining
formulas of $\GTc$ and $\GRTc$ that are not needed in the main argument. A
future part II of this paper will contain some additional results,
following~\cite[section 6]{Drinfeld:GalQQ}.
It is worthwhile to note that all our arguments depend on the existence of
at least one associator. Otherwise, we do not know that $\PBc$ and $\PCDc$
are at all isomorphic. So in a sense, all that we do is to take the
Knizhnik-Zamolodchikov associator $\Phi_{\KZ}$ (constructed by Drinfel'd)
and ``improve'' it.
Almost everything that we do appears either explicitly or implicitly in
Drinfel'd's paper~\cite{Drinfeld:GalQQ}. The presentation of $\GTc$ as a
group of automorphisms of some braid-group-like objects is due to Lochak
and Schneps~\cite{LochakSchneps:Automorphisms, LochakSchneps:Tower} (who
work with a different completion than ours).
\subsection{Acknowledgement} This paper grew out of a course I gave at
Harvard University in the spring semester of 1995, titled ``Knot Theory as
an Excuse''. One of the advertised goals of that course was to ``attempt
to read together two papers by Drinfel'd [\cite{Drinfeld:QuasiHopf,
Drinfeld:GalQQ}]''\footnote{All quotes taken from the official course
description.}, where I admitted that ``I have read about 20\% of the
material in these papers, understood about 20\% of what I read, and
got a lot out of it''. The idea was then to have ``a discussion group
in which everybody holds copies of the papers and we jointly try to
understand them''. Courses like that are usually doomed to fail, but
due to the amazing group of participants I think we managed to meet the
target of ``get ourself up to about 50\% on both figures [of reading
and understanding]''. These participants were:
D.~D.~Ben-Zvi, % benzvi@math.harvard.edu
R.~Bott, % bott@math.harvard.edu
A.~D'Andrea, %
S.~Garoufalidis,% stavros@math.mit.edu
D.~J.~Goldberg, % dgoldbrg@phoenix.Princeton.EDU
E.~Haley, % haley@fas.harvard.edu
M.~Hutchings, % hutching@math.harvard.edu
D.~Kazhdan, % kazhdan@math.harvard.edu
A.~Kirillov, % kirillov@math.harvard.edu
T.~Kubo, % kubo@math.harvard.edu
S.~Majid, % smajid@math.harvard.edu
A.~Polishchuk, % apolish@math.harvard.edu
S.~Sternberg, % shlomo@math.harvard.edu
D.~P.~Thurston, % dpt@math.berkeley.edu
and
H.~L.~Wolfgang. % wolfgang@math.mit.edu
I wish to thank them all for the part they took in the joint effort
that led to this paper. In addition, I'd like to thank P.~Deligne,
E.~de-Shalit, and E.~Goren for teaching me some basic facts about
algebraic groups, and B.~Enriquez, A.~Haviv, D.~Grinberg, Yael~K., A.~Referee, E.~Rips, and
J.~D.~Stasheff for many useful comments.