\def\selecta{n}
\def\draft{n}
\documentclass[12pt]{amsart} % AMSLaTeX
\usepackage{amssymb,epic,eepic,mathtools}
\usepackage[dvipsnames]{xcolor}
\usepackage[breaklinks=true,pagebackref]{hyperref}\hypersetup{colorlinks,
linkcolor={green!50!black},
citecolor={green!50!black},
urlcolor=blue
}
\input macros.tex
\input defs.tex
\begin{document}
\title[Associators and the Grothendieck-Teichmuller Group]
{On Associators and the Grothendieck-Teichmuller Group I}
\author{Dror Bar-Natan}
\address{Institute of Mathematics\\
The Hebrew University\\
Giv'at-Ram, Jerusalem 91904\\
Israel}
\curraddr{
Department of Mathematics\\
University of Toronto\\
Toronto Ontario M5S 2E4\\
Canada
}
\email{drorbn@math.toronto.edu}
\urladdr{\url{http://www.math.toronto.edu/~drorbn}}
\thanks{This article is available electronically at
{\tt \url{http://www.math.toronto.edu/~drorbn}}, and at
\arXiv{q-alg/9606021}. Source files at
\url{http://drorbn.net/AcademicPensieve/Projects/GT1/}.
}
\ifselecta{}{\dedicatory{
Modified from the version at
Selecta Mathematica, New Series {\bf 4} (1998) 183--212
}}
\ifselecta{}{\date{This edition: \today; \ \ First edition: June 21, 1996.}}
\begin{abstract}
We present a formalism within which the relationship (discovered
by Drinfel'd in~\cite{Drinfeld:QuasiHopf, Drinfeld:GalQQ}) between
associators (for quasi-triangular quasi-Hopf algebras) and (a variant of)
the Grothendieck-Teichmuller group becomes simple and natural, leading
to a simplification of Drinfel'd's original work. In particular, we
reprove that rational associators exist and can be constructed
iteratively, though the proof itself still depends on the apriori
knowledge that a not-necessarily-rational associator exists.
\end{abstract}
\maketitle
\ifselecta{}{\tableofcontents}
\input{intro}
\section{The basic definitions} \lbl{definitions}
In this section we introduce the two mathematical structures $\PB$ and
$\PCD$ on which we will apply Principle~\ref{BasicPrinciple}. Let $A$ be
some fixed commutative associative ${\Bbb Q}$-algebra with unit (typically
${\Bbb C}$ or ${\Bbb Q}$). Most objects that we will define below ``have
coefficients'' in $A$. We will mostly suppress $A$ from the notation,
except in the few places where it matters.
\subsection{Parenthesized braids and $\protect\GT$} \lbl{PBdef}
A {\em parenthesized braid} is a braid (whose ends are points ordered
along a line) together with a parenthesization of its bottom end (the
{\em domain}) and its top end (the {\em range}). A {\em parenthesization}
of a sequence of points is a specification of a way of ``multiplying'' them
as if they were elements in a non-associative algebra. Rather then giving a
formal definition, Figure~\ref{ParenthesizedBraids} contains some
examples.
\begin{figure}[htpb]
\def\b{\bullet}
\[ \eepic{ParenthesizedBraids}{0.6} \]
\caption{A parenthesized braid whose domain is $((\b\b)\b)$ and whose
range is $(\b(\b\b))$ (left), and a parenthesized braid whose domain
is $(((\b\b)\b)\b)$ and whose range is $((\b\b)(\b\b))$ (right). Notice
that by convention we draw ``inner multiplications'' as closer endpoints,
and ``outer multiplications'' as farther endpoints. Below we will not
bother to specify the parenthesizations at the ends explicitly, as this
information can be read from the distance scales appearing in the way
we draw the ends.}
\lbl{ParenthesizedBraids}
\end{figure}
\par\noindent\par\noindent
\parbox{3.9in}{\setlength{\parindent}{\globalparindent}
Parenthesized braids form a category in an obvious way. The objects of
this category are parenthesizations, the morphisms are the parenthesized
braids themselves, and composition is the operation of putting two
parenthesized braid on top of each other, as on the right (provided the
range of the first is the domain of the second).
} \qquad $
{\def\arg{B_1}\eepic{BoxMorphism}{0.5}}\!\!\circ
{\def\arg{B_2}\eepic{BoxMorphism}{0.5}} =
\begin{array}{c}
{\def\arg{B_2}\eepic{BoxMorphism}{0.5}} \\
\vspace{-9mm} \\
{\def\arg{B_1}\eepic{BoxMorphism}{0.5}}
\end{array}
$
Furthermore, there are some naturally defined operations on
parenthesized braids. If $B$ is such a braid with $n$ strands, these
operations are:
\begin{itemize}
\item {\em Extension operations:} Let $d_0B=d^n_0B$ ($d_{n+1}B=d^n_{n+1}
B$) be $B$ with one straight strand added on the left (right), with ends
regarded as outer-most:
\[ d_0\left(\,\eepic{extend1}{0.5}\right)=\eepic{extend2}{0.5}
\quad;\qquad
d_3\left(\,\eepic{extend1}{0.5}\right)=\eepic{extend3}{0.5}.
\]
\item {\em Cabling operations:} Let $d_iB=d^n_iB$ for $1\leq i\leq n$ be
the parenthesized braid obtained from $B$ by doubling its $i$th strand
(counting at the bottom), taking the ends of the resulting ``daughter
strands'' as an inner-most product:
\[ d_2\left(\,\eepic{doubling1}{0.5}\right)=\eepic{doubling2}{0.5}. \]
\item {\em Strand removal operations\footnote{The strand removal
operations (and all other $s_i$'s below) are important in the
applications, but play no crucial role in this paper and can be
systematically removed with no change to the end results.}:} Let
$s_iB=s^n_iB$ for $1\leq i\leq n$ be the parenthesized braid obtained
from $B$ by removing its $i$th strand (counting at the bottom):
\[ s_2\left(\,\eepic{skeleton1}{0.5}\right)=\eepic{removal2}{0.5}. \]
\end{itemize}
The {\em skeleton} ${\bold S}B$ of a parenthesized braid $B$ is the map
that it induces from the points of its domain to the points of its range,
taken together with the domain and range:
\begin{equation} \lbl{skeleton}
{\bold S}\left(\,\eepic{skeleton1}{0.5}\right) = \eepic{skeleton2}{0.5}.
\end{equation}
More precisely, the skeleton ${\bold S}$ is a functor on the category of
parenthesized braids whose image is in the category $\PP$ of parenthesized
permutations, whose definition should be clear from its name and a simple
inspection of the example in \eqref{skeleton}. There are naturally defined
operations $d_i$ and $s_i$ on $\PP$ as in the case of parenthesized
braids, and the skeleton functor ${\bold S}$ intertwines the $d_i$'s
and the $s_i$'s acting on parenthesized braids and on parenthesized
permutations.
The category that we really need is a category of formal linear
combinations of parenthesized braids sharing the same skeleton:
\begin{definition} Let $\PB(A)=\PB$ (for {\bf Pa}renthesized {\bf B}raids)
be the category whose objects are parenthesizations and whose morphisms
are pairs $(P,\sum_{j=1}^k \beta_j B_j)$, where $P$ is a morphism in the
category of parenthesized permutations, the $B_j$'s are parenthesized
braids whose skeleton is $P$, and the $\beta_j$'s are coefficients in
the ground algebra $A$. The composition law in $\PB$ is the bilinear
extension of the composition law of parenthesized braids. There is a
natural forgetful ``skeleton'' functor ${\bold S}:\PB\to\PP$. If the sum
$\sum \beta_j B_j$ is not the empty sum, we usually suppress $P$ from the
notation, as it can be inferred from the $B_j$'s. See Figure~\ref{LinComb}.
\end{definition}
\begin{figure}[htpb]
\[ \eepic{LinComb}{0.75} \]
\caption{A morphism $B$ in $\protect\PB$ and its skeleton ${\bold S}(B)$
in $\protect\PP$.}
\lbl{LinComb}
\end{figure}
\subsubsection{Fibered linear categories}
The category $\PB$ together with the functor ${\bold S}:\PB\to\PP$ is
an example of a fibered linear category. Let ${\bold P}$ be a category
``of skeletons''. A {\em fibered linear category over ${\bold P}$} is a
pair $({\bold B},{\bold S}:{\bold B}\to{\bold P})$ of the form (category,
functor into ${\bold P}$), in which ${\bold B}$ has the same objects as
${\bold P}$, the ``skeleton'' functor ${\bold S}$ is the identity on
objects, the inverse image ${\bold S}^{-1}(P)$ of every morphism $P$ in
${\bold P}$ is a linear space, and so the composition maps in ${\bold
B}$ are bilinear in the natural sense. Many notions from the theory of
algebras have analogs for fibered linear categories, with the composition
of morphisms replacing the multiplication of elements. Let us list the
few such notions that we will use, without giving precise definitions:
\begin{myitemize}
\item A {\em subcategory} of a fibered linear category $({\bold B},{\bold
S}:{\bold B}\to{\bold P})$ is a choice of a linear subspace in each
``space of morphisms with a fixed skeleton'' ${\bold S}^{-1}(P)$, so
that the system of subspaces thus chosen is closed under composition.
\item An {\em ideal} in $({\bold B},{\bold S}:{\bold B}\to{\bold P})$ is a
subcategory ${\bold I}$ so that if at least one of the two composable
morphisms $B_1$ and $B_2$ in ${\bold B}$ is actually in ${\bold I}$, then
the composition $B_1\circ B_2$ is also in ${\bold I}$.
\item One can take {\em powers} of ideals --- The morphisms of ${\bold
I}^m$ will be all the morphisms in ${\bold B}$ that can be presented
as compositions of $m$ morphisms in ${\bold I}$. The power ${\bold
I}^m$ is also an ideal in ${\bold B}$.
\item One can form the {\em quotient} ${\bold B}/{\bold I}$ of a fibered
linear category ${\bold B}$ by an ideal ${\bold I}$ in it, and the
result is again a fibered linear category.
\item {\em Direct sums} of fibered linear categories that are fibered over
the same skeleton category can be formed.
\item One can define {\em filtered} and {\em graded} fibered linear
categories. One can talk about the {\em associated graded} fibered linear
category of a given filtered fibered linear category.
\item One can take the {\em inverse limit} of an inverse system of
fibered linear categories (fibered in a compatible way over the same
category of skeletons). In particular, if ${\bold I}$ is an ideal
in a fibered linear category ${\bold B}$, one can form ``the ${\bold
I}$-adic completion $\hat{{\bold B}}=\varprojlim_{m\to\infty}{\bold
B}/{\bold I}^m$. The ${\bold I}$-adic completion is a filtered fibered
linear category.
\item {\em Tensor powers} of a fibered linear category $({\bold
B},{\bold S}:{\bold B}\to{\bold P})$ can be defined. For example,
${\bold B}\otimes{\bold B}$ will have the same set of objects as
${\bold B}$, and for any two such objects $O_1$ and $O_2$, we set
\[ \mor_{{\bold B}\otimes{\bold B}}(O_1,O_2)=
\coprod_{P\in\mor_{{\bold P}}(O_1,O_2)}
{\bold S}^{-1}(P)\otimes{\bold S}^{-1}(P).
\]
${\bold B}\otimes{\bold B}$ is again a fibered linear category.
\item The notion of a {\em coproduct functor}
${\bold \Box}:{\bold B}\to{\bold B}\otimes{\bold B}$ makes sense.
\end{myitemize}
\subsubsection{Back to parenthesized braids}
We can now introduce some more structure on $\PB$, and specify completely
the mathematical structures that will play the role of $B$ in
Principle~\ref{BasicPrinciple}.
\begin{definition} Let ${\bold \Box}:\PB\to\PB\otimes\PB$ be the coproduct
functor defined by setting each individual parenthesized braid $B$ to
be {\em group-like}, that is, by setting ${\bold \Box}(B)=B\otimes B$.
\end{definition}
Let ${\bold I}$ be the {\em augmentation ideal} of $\PB$, the ideal of
all pairs $(P,\sum\beta_j B_j)$ in which $\sum\beta_j=0$. Powers of this
ideal define the {\em unipotent filtration} of $\PB$: $\F{m}\PB={\bold
I}^{m+1}$.
\begin{definition} Let $\PBm=\PB/\F{m}\PB=\PB/{\bold I}^{m+1}$ be the $m$th
{\em unipotent quotient}\footnote{If you are familiar with Vassiliev
invariants, notice that $\PBm$ is simply $\PB$ moded out by
``$(m+1)$-singular parenthesized braids''.} of $\PB$, and let
$\PBc=\varprojlim_{m\to\infty}\PBm$ be the {\em unipotent completion}
of $\PB$.
\end{definition}
Let $\sigma$ be the parenthesized braid $\eepic{sigma}{0.25}$.
The fibered linear categories $\PBm$ and $\PBc$ inherit the operations
$d_i$ and $s_i$ from parenthesized braids, and a coproduct ${\bold
\Box}$ and a filtration $\F{\star}$ from $\PB$.\footnote{%
Added April 2016: To define $\Box$ one must first understand the relevant
monoidal structures, namely $\PBc\hat{\otimes}\PBc$ and
$\PBm\otimes^{(m)}\PBm$, and some subtelty occurrs. In the case of
$\PBc\hat{\otimes}\PBc$, the tensor product must be understood in the sense of
completed filtered objects: in general, if $A$ and $B$ are filtered
objects then their ordinary tensor product $C\coloneqq A\otimes B$ is
again filtered with $\F{m}C\coloneqq\sum_{i+j=m}\F{i}A\otimes\F{j}B$,
and one may set $A\hat{\otimes}B\coloneqq\varprojlim_{m\to\infty}C/\F{m}C$.
The case of $\otimes^{(m)}$ is even more subtle, as it must be understood
in the sense of ``filtered objects with $\F{m+1}=0$''. Namely, if $A$ and
$B$ are filtered objects with $\F{m+1}A=\F{m+1}B=0$ then
$A\otimes^{(m)}B\coloneqq A\otimes
B\left/\sum_{i+j>m}\F{i}A\otimes\F{j}B\right.$.
}
The specific parenthesized braid $\sigma$ can be regarded as a morphism
in any of these categories.
\begin{definition} \label{def:GT}
Let $\GTm$ and $\GTc$ (really, $\GTm(A)$ and $\GTc(A)$)
be the groups of structure preserving automorphisms of $\PBm$ and
$\PBc$, respectively. That is, the groups of all functors $\PBm\to\PBm$
(or $\PBc\to\PBc$) that cover the skeleton functor, intertwine $d_i$,
$s_i$ and ${\bold \Box}$ and fix $\sigma$. In short, let
\begin{eqnarray*}
B^{(m)} & = & \left(
\PBm,{\bold S}:\PBm\to\PP,d_i,s_i,{\bold \Box},\sigma \right); \\
\hat{B} & = & \left(
\PBc,{\bold S}:\PBc\to\PP,d_i,s_i,{\bold \Box},\sigma \right);
\end{eqnarray*}
\[ \GTm=\Aut B^{(m)}; \qquad \GTc=\Aut\hat{B}. \]
\end{definition}
\begin{remark} One easily sees that elements of $\GTm$ ($\GTc$)
automatically preserve the filtration $\F{\star}$.\footnote{%
Added May 2016: Given the delicacy of the proof of this remark, it would
have probably been more natural to include the filtration $\F{\star}$
within the list of ``items to preserve'' in Definitions~\ref{def:GT}
and~\ref{def:GRT}.
Anyway, here's a sketch of the proof (partially
following~\cite{Grinberg:MO235835}): For simplicity let us argue in
a bialgebra $B$ rather than in the more complicated structure $\PB$.
In $B$ the kernel $I$ of the counit $\epsilon$ is automatically invariant
under automorphisms $\varphi$ respecting the coproduct $\Box$, for indeed
$\epsilon\circ\varphi = ((\epsilon\circ\varphi)\otimes\epsilon)\circ\Box =
((\epsilon\circ\varphi)\otimes(\epsilon\circ\varphi^{-1}\circ\varphi)\circ\Box
= (\epsilon\otimes(\epsilon\circ\varphi^{-1}))\circ\Box\circ\varphi =
\epsilon\circ\varphi^{-1}\circ\varphi = \epsilon$. Hence the filtration of
$B$ by $\{I^p\}$ is preserved by automorphisms of $B$. Applying this to
each of $B^{(m)}$ and $\hat{B}$ we see that their filtrations defined by
the powers of their respective augmentation ideals $I_{(m)}$ and $\hat{I}$
are preserved by their respective automorphism groups. It remains to show
that the filtrations that $B^{(m)}$ and $\hat{B}$ inherit from $B$ are the
same as their filtrations by the powers of their augmentation ideals. In
the case of $B^{(m)}$, this amounts to the statement $I^p/I^m=(I/I^m)^p$
(for $p\geq m$), which is trivial. In the case of $\hat{B}$ the statement
is $\varprojlim_{m\geq p}(I^p/I^m)=(\varprojlim_mI/I^m)^p$. The
latter is probably false in general, but it is true if $B$ is graded.
In our ``braids'' case, $\hat{B}$ is isomorphic to its associated graded
as is shown later in this paper in a manner which is independent of this
footnote, so the required statement is true.
}
\end{remark}
\begin{claim} $\PB$ is generated by $a^{\pm 1}$, $\sigma^{\pm 1}$, and
their various images by repeated applications of the $d_i$'s, where
\[ a=\eepic{a}{0.5}, \qquad \sigma=\eepic{sigma}{0.5}. \]
\end{claim}
\begin{proof} (sketch) The main point is that any of the standard generators
of the braid group can be written in terms of $a^{\pm 1}$ and $\sigma^{\pm
1}$ and their images. For example,
\[ \eepic{BraidGenerator}{0.5}=\eepic{BGRewritten}{0.5}
= d_0a^{-1}\circ d_0d_3\sigma\circ d_0a.
\]
\vskip -1cm
\end{proof}
\vskip 1cm
\subsection{Parenthesized chord diagrams and $\protect\GRT$} \lbl{PCDdef}
The category $\PCD$, the main ingredient of the mathematical object
$C$ on which we will apply Principle~\ref{BasicPrinciple}, can be
viewed as natural in two (equivalent) ways. First, $\PCD$ is natural
because it is the associated graded of $\PB$, as will be proven in
section~\ref{ASS}. $\PCD$ can also be viewed as the category of ``chord
diagrams for finite-type (Vassiliev) invariants \cite{Bar-Natan:Vassiliev,
Bar-Natan:Braids, Birman:Bulletin, BirmanLin:Vassiliev, Goussarov:New,
Goussarov:nEquivalence, Kontsevich:Vassiliev, Vassiliev:CohKnot,
Vassiliev:Book} of parenthesized braids'', and all the operations that we
will define on $\PCD$ are inherited from their parallels on parenthesized
braids, that were defined in section~\ref{PBdef}. I prefer not to make
more than a few comments about the latter viewpoint below. Saying more
requires repeating well known facts about finite-type invariants, and
these can easily be found in the literature. If you already know about
Vassiliev invariants and chord diagrams, you'll find the relation between
them and the definitions below rather clear. Unfortunately, if
finite-type invariants are not mentioned, we have to start with some
unmotivated definitions.
\begin{definition} \lbl{def:Apb}
Let $\Apb_n=\Apb_n(A)$ be the algebra (over the ground algebra $A$)
generated by symbols $t^{ij}$ for $1\leq i\neq j\leq n$, subject to the
relations $t^{ij}=t^{ji}$, $[t^{ij},t^{kl}]=0$ if $|\{i,j,k,l\}|=4$,
and $[t^{jk},t^{ij}+t^{ik}]=0$ if $|\{i,j,k\}|=3$. The algebra
$\Apb_n$ is graded by setting $\deg t^{ij}=1$; let $\G{m}\Apb_n$
be the degree $m$ piece of $\Apb_n$, let $\F{\star}\Apb_n$ be the
filtration defined by $\F{m}\Apb_n=\bigoplus_{m'>m}\G{m'}\Apb_n$,
let $\Apbm_n$ be $\Apb_n/\F{m}\Apb_n$, and let $\Apbc_n$ be the
graded completion $\varprojlim_{m\to\infty}\Apbm_n$ of $\Apb_n$. We
call elements of $\Apb_n$ {\em chord diagrams}, and draw them as in
Figure~\ref{ChordDiagram}. (In the language of finite-type invariants,
$\Apb_n$ is the algebra of chord diagrams for $n$-strand pure braids,
and the last relation is the ``$4T$'' relation.)
\end{definition}
\begin{figure}[htpb]
\[ t^{13}t^{13}t^{12}t^{23} \quad\longleftrightarrow\quad
\eepic{ChordDiagram}{0.5}; \qquad
4T:\quad\eepic{4T}{0.5}
\]
\caption{Elements of $\Apb_3$ are presented as chord diagrams made of $3$
vertical strands and some number of horizontal chords connecting them. A
chord connecting the $i$th strand to the $j$th strand represents $t^{ij}$,
and products are read from the bottom to the top of the diagram.}
\lbl{ChordDiagram}
\end{figure}
\begin{definition} \lbl{PermutationAction} There is an action of the
symmetric group ${\mathcal S}_n$ on $\Apb_n$ by ``permuting the vertical
strands'', denoted by $(\tau,\Psi)\mapsto\Psi^\tau$:
\[ \Psi=\eepic{Psi}{0.5}\mapsto\Psi^{231}=\eepic{Psi231}{0.5}. \]
\end{definition}
\begin{definition} \lbl{Action} Let $d_i=d^n_i:\Apb_n\to\Apb_{n+1}$
for $0\leq i\leq n+1$ and $s_i=s^n_i:\Apb_n\to\Apb_{n-1}$ for $1\leq i\leq
n$ be the algebra morphisms defined by their action on the generators
$t^{jk}$ (with $jm}\G{m'}\PCD$. $\PCD$ inherits a coproduct
${\bold \Box}:\PCD\to\PCD\otimes\PCD$ from the coproduct $\Box$ of
$\Apb_n$.
\end{definition}
\begin{figure}[htpb]
\[
\left(t^{12}\cdot\silenteepic{Comp1}{0.5}\right)\circ
\left(t^{23}t^{12}\cdot\silenteepic{Comp2}{0.5}\right)
\to \silenteepic{Comp3}{0.5}\circ\silenteepic{Comp4}{0.5}
\to \silenteepic{Comp5}{0.5}
\to \silenteepic{Comp6}{0.5}
\to t^{12}t^{23}t^{13}\cdot\silenteepic{Comp7}{0.5}
\]
\caption{The composition of a morphism in
$\protect\mor_{\protect\PCD}
((\bullet(\bullet\bullet)),(\bullet(\bullet\bullet)))$
with a morphism in
$\protect\mor_{\protect\PCD}
((\bullet(\bullet\bullet)),((\bullet\bullet)\bullet))$.
} \lbl{CompositionLaw}
\end{figure}
\begin{definition} \lbl{PCDOperators}
As in the case of $\PB$, there are some naturally defined
operations on $\PCD$. If $D\cdot P$ is a parenthesized chord diagram on
$n$ strands, set $d_i(D\cdot P)=d^n_i(D\cdot P)=d^n_iD\cdot d^n_iP$,
and similarly for $s_i=s^n_i$. These operations are:
\begin{itemize}
\item {\em Extension operations:} $d_0$ ($d_{n+1}$) adds a far-away
independent strand on the left (right).
\item {\em Cabling operations:} $d_i B$ with $1\leq i\leq n$ doubles
the $i$th strand and sums all possible ways of lifting the chords that
were connected to the $i$th strand to the two daughter strands.
\item {\em Strand removal operations:} $s_i$ removes the $i$th strand and
maps everything to $0$ if there was any chord connected to the $i$th
strand.
\end{itemize}
\end{definition}
\begin{definition} Let $\PCDm$ be the category $\PCD/\F{m}\PCD$ of
parenthesized chord diagrams of degree up to $m$, and let $\PCDc$ be the
category $\varprojlim_{m\to\infty}\PCDm$ of formal power series of
parenthesized chord diagrams. The fibered linear categories $\PCDm$
and $\PCDc$ inherit the operations $d_i$ and $s_i$, the coproduct
${\bold \Box}$ and the filtration $\F{\star}$ from $\PCD$.
\end{definition}
Let $X$ and $H$ be the parenthesized chord diagrams $\eepic{X}{0.25}$
and $\eepic{H}{0.25}$ respectively, and let $\tilde{R}$ be the formal
exponential $\tilde{R}=\exp\left(\frac{1}{2}H\right)\cdot X$, regarded
a morphism in $\PCDm$ or $\PCDc$.
\begin{definition} \label{def:GRT} Let $\GRTm$ and $\GRTc$ (really,
$\GRTm(A)$ and $\GRTc(A)$) be the groups of structure preserving
automorphisms of $\PCDm$ and $\PCDc$, respectively. That is, the groups
of all functors $\PCDm\to\PCDm$ (or $\PCDc\to\PCDc$) that cover the
skeleton functor, intertwine $d_i$, $s_i$ and ${\bold \Box}$ and fix
$\tilde{R}$. In short, let
\begin{eqnarray*} C^{(m)} & = & \left(
\PCDm,{\bold S}:\PCDm\to\PP,d_i,s_i,{\bold \Box},\tilde{R}
\right); \\
\hat{C} & = & \left(
\PCDc,{\bold S}:\PCDc\to\PP,d_i,s_i,{\bold \Box},\tilde{R}
\right);
\end{eqnarray*}
\[ \GRTm=\Aut C^{(m)}; \qquad \GRTc=\Aut\hat{C}. \]
\end{definition}
\begin{remark} \lbl{FixXH} Elements of $\GRTm$ ($\GRTc$) fix each of $X$
and $H$ individually. Indeed, $\tilde{R}^2=\exp H$ and hence $\exp H$
and thus $H$ are fixed. But then $X=\exp(-\frac{1}{2}H)\tilde{R}$ is
fixed too.
\end{remark}
\begin{claim} \lbl{PaCDGens} $\PCD$ is generated by $a^{\pm 1}$, $X$, $H$,
and their various images by repeated applications of the $d_i$'s, where
\[ a=\eepic{a}{0.5}, \qquad X=\eepic{X}{0.5}, \qquad H=\eepic{H}{0.5}. \]
(Notice that the symbol ``$a$'' plays a double role, as a generator of $\PB$
and as a generator of $\PCD$).
\end{claim}
\begin{proof} (sketch) Perhaps one illustrative example will suffice:
\[ \eepic{t13}{0.5}=\eepic{t13Rewritten}{0.5}
= d_0X\circ a^{-1}\circ d_3H\circ a\circ d_0X.
\]
\vskip -8mm
\end{proof}
\begin{remark} Remark~\ref{FixXH} and claim~\ref{PaCDGens} imply that
elements of $\GRTm$ ($\GRTc$) automatically preserve the filtration
$\F{\star}$.
\end{remark}
\section{Isomorphisms and associators} \lbl{ASS}
In this section we make the key observation that makes
Principle~\ref{BasicPrinciple} useful in our case: The fact that
the set of all associators \`a la Drinfel'd~\cite{Drinfeld:QuasiHopf,
Drinfeld:GalQQ} can be identified with the set of all structure-preserving
functors $\Zc:\hat{B}\to\hat{C}$. Recall that $A$ is some fixed commutative
associative ${\Bbb Q}$-algebra with unit.
\begin{definition} An {\em associator} is an invertible element $\Phi$
of $\Apbc_3(A)$ satisfying the following axioms:
\begin{itemize}
\item The {\em pentagon} axiom holds in $\Apbc_4$:
\begin{equation}
d_4\Phi\cdot d_2\Phi\cdot d_0\Phi
= d_1\Phi\cdot d_3\Phi.
\tag{$\pentagon$}
\end{equation}
\item The {\em hexagon} axioms hold in $\Apbc_3$:
\begin{equation}
d_1\exp\left(\pm\frac{1}{2}t^{12}\right) =
\Phi\cdot\exp\left(\pm\frac{1}{2}t^{23}\right)\cdot(\Phi^{-1})^{132}
\cdot\exp\left(\pm\frac{1}{2}t^{13}\right)\cdot\Phi^{312}.
\tag{$\hexagon\!_\pm$}
\end{equation}
\item $\Phi$ is {\em non-degenerate}: $s_1\Phi=s_2\Phi=s_3\Phi=1$.
\item $\Phi$ is {\em group-like}: $\Box\Phi=\Phi\otimes\Phi$.
\end{itemize}
Apart from the different conventions, this definition is equivalent to
Drinfel'd's~\cite{Drinfeld:GalQQ} definition of an $Fr(A,B)$-valued
$\varphi$,\footnote{Precisely, our $\Phi$ is Drinfel'd's $\varphi^{-1}$.} and practically equivalent to the definition of an $\text{\bf
AP}^{\text{\it hor}}$-valued $\Phi$ in~\cite{Bar-Natan:NAT}.
\end{definition}
\begin{definition} Let $\ASSc=\ASSc(A)$ be the set of associators
$\Phi\in\Apbc_3(A)$. Similarly, if we mod out by degrees higher than $m$,
we can define {\em associators up to degree $m$} and the set $\ASSm$.
\end{definition}
\begin{remark} The hexagon axiom for $\Phi\in\ASSc$ or $\Phi\in\ASSm$
implies that $\Phi=1+$(higher degree terms).
\end{remark}
By the definition of $\hat{B}$ and $\hat{C}$, a structure-preserving
functor $\Zc:\hat{B}\to\hat{C}$ carries $\sigma$ to $\tilde{R}$, and thus
it is determined by its value $\Zc(a)$ on the remaining generator of $\PB$.
As $\Zc$ must cover the skeleton functor, $\Zc(a)$ must be of the form
$\Phi_\Zc\cdot a$, for some $\Phi_\Zc\in\Apbc_3$.
\begin{proposition} \lbl{KeyObservation}
If $\Zc$ is a structure preserving functor $\hat{B}\to\hat{C}$, then
$\Phi_\Zc$ is an associator, and the map $\Zc\mapsto\Phi_\Zc$ is
a bijection between the set of all structure-preserving functors
$\Zc:\hat{B}\to\hat{C}$ and the set $\ASSc$ of all associators
$\Phi\in\Apbc_3$. A similar construction can be made in the case
of $B^{(m)}$, $C^{(m)}$ and $\ASSm$, and the same statements hold.
\end{proposition}
Before we can prove Proposition~\ref{KeyObservation}, we need a bit more
insight about the structure of $\Apb_n$.
\begin{lemma} \lbl{Locality} The following two relations hold in $\Apb_n$:
\begin{enumerate}
\item {\em Locality in space:} For any $k\leq n$, the subalgebra of
$\Apb_n$ generated by $\{t^{ij}:i,j\leq k\}$ commutes with the subalgebra
generated by $\{t^{ij}:i,j>k\}$. In pictures, we see that elements that
live in ``different parts of space'' commute:
\[ \eepic{ApbLocSpace}{0.5} \]
\item {\em Locality in scale} Elements that live in ``different scales''
commute. This is best explained by a picture, with notation as in
Definition~\ref{Action}:
\[ \eepic{ApbLocScale}{0.7} \]
(We think of the part $A$ as ``local'', as it involves only the ``local''
group of strands, and of the rest as ``global'', as it regards the
``local'' group of strands as ``equal''.)
\end{enumerate}
(A similar statement is \cite[Lemma 3.4]{Bar-Natan:NAT}.)
\end{lemma}
\begin{proof}[Proof of Lemma~\ref{Locality}]
Locality in space follows from repeated application of the relation
$t^{ij}t^{kl}=t^{kl}t^{ij}$ with $i2$ then $\psi$ must be in $\tilde{d}^1{\mathcal G}_k\Apb_2$. That
is, it must be a multiple of $\chi=\tilde{d}^1(t^{12})^k$. But as
$\Gamma$ is group-like, $\psi$ must be primitive: $\Box\psi=\psi\otimes
1+1\otimes\psi$. One easily verifies that $\chi$ is not primitive,
and hence $\psi=0$ as required. If $k=2$, equation~\eqref{P20eq}
and Proposition~\ref{CohomResult} tell us that $\psi$ is of the form
$c_1\tilde{d}^1(t^{12})^2+c_2[t^{13},t^{23}]$. A routine verification
shows that if the semi-classical hexagon relation is pre-multiplied
by $d_3 X$ and post-multiplied by $d_0 X$, then modulo the other
relations, it does not change. This means that $\psi^{213}=\psi$ (this
identity follows more easily from the cabling relation), and thus
$c_2=0$. But then the primitivity of $\psi$ implies that $c_1$ vanishes as
well, and thus $\psi=0$ as required.
}\end{proof}
\section{Just for completeness} \lbl{Just}
For completeness, this section contains a description of the group
law of $\GRTc$, a description of its action on $\ASSc$, and similar
descriptions for the group $\GTc$. This information is not needed in the
main part of this paper. Throughout this section one can replace unipotent
completions by unipotent quotients ($\GRTm$, $\ASSm$, $\Apbm$, etc.)
with no change to the results.
\begin{proposition} \lbl{GRTGroupLaw}
The group law $\times$ of $\GRTc$ is expressed in terms of the $\Gamma$'s
(of Proposition~\ref{GammaEquations}) as
\begin{equation} \lbl{GammaProduct}
\Gamma_1\times\Gamma_2=\Gamma_1\cdot\left(\left.\Gamma_2\right|_{
t^{12}\to\Gamma_1^{-1}t^{12}\Gamma_1,\
t^{13}\to(\Gamma_1^{-1})^{132}t^{13}\Gamma_1^{132},\
t^{23}\to t^{23}
}\right),
\end{equation}
where ``$\cdot$'' is the product of $\Apbc$, $\Gamma_1^{-1}$ is
interpreted in $\Apbc$, and the substitution above means: replace every
occurrence of $t^{12}$ in $\Gamma_2$ by $\Gamma_1^{-1}t^{12}\Gamma_1$,
etc.\ (In particular, we claim that this substitution is well defined
on $\Apbc$).
\end{proposition}
\begin{proof} $\Apbc_3$ can be identified with the algebra of
self-morphisms in $\PCDc$ of the object $(\bullet(\bullet\bullet))$. Let
$\overline\Gamma$ denote the self-morphism corresponding to a
$\Gamma\in\Apbc_3$. We have $\Gamma\cdot a=a\circ\overline\Gamma$, and
hence (with $\Gamma\mapsto G_\Gamma$ denoting the identification in
Proposition~\ref{GammaEquations})
\begin{equation} \lbl{GammaComposition}
a\!\circ\!\overline{\Gamma_1\!\times\!\Gamma_2}
= G_{\Gamma_1\!\times\!\Gamma_2}(a)
= G_{\Gamma_1}(G_{\Gamma_2}(a))
= G_{\Gamma_1}(a\!\circ\!\overline{\Gamma_2})
= G_{\Gamma_1}(a)\!\circ\! G_{\Gamma_1}(\overline{\Gamma_2})
= a\!\circ\!\overline{\Gamma_1}\!\circ\! G_{\Gamma_1}(\overline{\Gamma_2}).
\end{equation}
To compute $G_{\Gamma_1}(\overline{\Gamma_2})$ we need to write
$\overline{\Gamma_2}$ in terms of the generators of $\PCDc$. This
we do by replacing every $t^{12}$ appearing in $\Gamma_2$ by
$\overline{t^{12}}=a^{-1}\circ d_3H\circ a$, every $t^{13}$ by
$\overline{t^{13}}=d_0X\circ a^{-1}\circ d_3H\circ a \circ
d_0X$, and every $t^{23}$ by $\overline{t^{23}}=d_0H$. By the
definition of the action of $G_{\Gamma_1}$ on the generators
of $\PCDc$, we find that it maps $\overline{t^{12}}$ to
$\overline{\Gamma_1^{-1}t^{12}\Gamma_1}$, $\overline{t^{13}}$
to $\overline{(\Gamma_1^{-1})^{132}t^{13}\Gamma_1^{132}}$ and
$\overline{t^{23}}$ to $\overline{t^{23}}$. Combining this
and~\eqref{GammaComposition} we get~\eqref{GammaProduct}.
\end{proof}
Similar reasoning leads to the following:
\begin{proposition} The action of $\GRTc$ on $\ASSc$, written in terms of
$\Gamma$'s and $\Phi$'s, is given by
\[ \Gamma(\Phi)=\Gamma\cdot\left(\left.\Phi\right|_{
t^{12}\to\Gamma_1^{-1}t^{12}\Gamma_1,\
t^{13}\to(\Gamma_1^{-1})^{132}t^{13}\Gamma_1^{132},\
t^{23}\to t^{23}
}\right),
\]
with products and inverses taken in $\Apbc_3$. \qed
\end{proposition}
The group $\GTc$ admits a similar description. Any element of $\GTc$
maps $a$ to a limit of formal sums of parenthesized braids whose
skeleton is $a$. Such a limit is of the form $a\circ\Sigma$, where
$\Sigma$ is a self-morphism whose skeleton is the identity of the
object $(\bullet(\bullet\bullet))$ of $\PBc$, regarded as an element of
$\widehat{PB}_3$. Let $\sigma_1$ and $\sigma_2$ be the standard generators
$\silenteepic{sigma1}{0.33}$ and $\silenteepic{sigma2}{0.33}$ of the
(non-pure) braid group $B_3$ on 3 strands. Every $\Sigma\in\widehat{PB}_3$
is a limit of formal sums of combinations of $\sigma_{1,2}$.
\begin{proposition}
\begin{enumerate}
\item $\GTc$ can be identified as the group of all group-like
non-degenerate $\Sigma\in\widehat{PB}_3$ satisfying:
\begin{itemize}
\item The pentagon for pure braids, in $\widehat{PB}_4$:
\[ d_4\Sigma\cdot d_2\Sigma\cdot d_0\Sigma
= d_1\Sigma\cdot d_3\Sigma
\]
(with the obvious interpretation for the $d_i$'s).
\item The hexagons for pure braids, in $\hat{B}_3$, the unipotent
completion of $B_3$:
\[ \sigma_2\sigma_1=\Sigma\cdot\sigma_2\cdot\Sigma^{-1}\cdot
\sigma_1\cdot\Sigma.
\]
\end{itemize}
\item The group law is given by
\[ \Sigma_1\times\Sigma_2 = \Sigma_1\cdot\left(\left.\Sigma_2\right|_{
\sigma_1\to\Sigma^{-1}\sigma_1\Sigma,\
\sigma_2\to\sigma_2
}\right),
\]
with products and inverses taken in $\hat{B}_3$.
\item The action on $\ASSc$ is given by
\[ (\Phi,\Sigma)\mapsto\Phi^\Sigma
= \Phi\cdot\left(\left.\Sigma\right|_{
\sigma_1\to\Phi^{-1}e^{t^{12}/2}X_1\Phi,\
\sigma_2\to e^{t^{23}/2}X_2
}\right).
\]
This formula makes sense in $\Apbc_3\rtimes S_3$, with $X_1=(12)$ and
$X_2=(23)$ the standard generators of the permutation group $S_3$ which
acts on $\Apbc_3$ as in Definition~\ref{PermutationAction}. Implicitly
we claim that this formula is well defined and valued in
$\Apbc_3\subset\Apbc_3\rtimes S_3$. \qed
\end{enumerate}
\end{proposition}
\input{refs}
\end{document}
\endinput