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\begin{document}
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\title{{Finite Type Invariants of w-Knotted Objects IV: Some Computations}}
\author{Dror~Bar-Natan}
\address{
Department of Mathematics\\
University of Toronto\\
Toronto Ontario M5S 2E4\\
Canada
}
\email{drorbn@math.toronto.edu}
\urladdr{http://www.math.toronto.edu/~drorbn}
\date{First edition Nov.\ 15, 2015, this edition \today. Electronic version
and related files at~\cite{WKO4}, \url{\web}. The \arXiv{1511.05624}
edition may be older}
\subjclass[2010]{57M25}
\keywords{
w-knots,
w-tangles,
Kashiwara-Vergne,
associators,
double tree,
Mathematica,
free Lie algebras%
}
\thanks{This work was partially supported by NSERC grant RGPIN 262178.}
\begin{abstract}
\input abstract.tex
\end{abstract}
\maketitle
\setcounter{tocdepth}{3}
\tableofcontents
\input intro.tex
\draftcut
\needspace{3\baselineskip}
\section{Group-like elements in $\calA^w$} \label{sec:Aw}
\subsection{A brief review of $\calA^w$} \label{subsec:Aw} Let
$\glosm{S}{S} = \{\glosm{a}{a}_1,a_2,\dots\}$\footnoteT{Yellow highlighting
corresponds to the \yellowt{glossary}, Section~\ref{sec:glossary}.}
be a finite set of ``strand labels''. The space
$\glosm{calAw}{\calA^w}(S)$ is the completed
graded vector space\footnoteT{For simplicity we always work over $\bbQ$.}
of diagrams made of (vertical) ``strands'' labelled by the elements of $S$,
and ``arrows'' as summarized by the following picture:
\[
\def\comments{{\parbox{2.28in}{\scriptsize
\begin{myitemize}
\item Diagrams are connected.
\item Vertices are 2-in 1-out.
\item Vertices are oriented.
\item Degree is half the number of trivalent vertices.
\item The ``skeleton'' is a union of vertical strands labelled by the
elements of $S$.
\end{myitemize}
}}}
\label{g:wRels}
\input{figs/AwSummary.pstex_t}
\]
When $S=\{1,2,\ldots,n\}$ we abbreviate
$\calA^w(\glosm{uparrow}{\uparrow_n})\coloneqq\calA^w(S)$.
\TopLieAT In topology, elements of $\calA^w(S)$ are closely related
to (finite type invariants of) simply knotted 2-dimensional tubes in
$\bbR^4$ (\cite{WKO1}--\cite{WKO3}, \cite{KBH}). In Lie theory, they
represent ``universal'' $\glosm{frakg}{\frakg}$-invariant tensors in
$\glosm{calU}{\calU}(I\frakg)^{\otimes S}$, where $\glosm{Ifrakg}{I\frakg}
\coloneqq \frakg\ltimes\frakg^\ast$\footnoteT{In earlier papers we have
used the order $I\frakg=\frakg^\ast\rtimes\frakg$.} and $\frakg$ is some
finite dimensional Lie algebra (\cite{WKO1}--\cite{WKO3}). Readers of
Alekseev and Torossian~\cite{AT} may care about $\calA^w$ because
using notation from~\cite{AT}, $\calA^w(\uparrow_n)$ is the completed
universal enveloping algebra of $(\fraka_n\oplus\attder_n)\ltimes\attr_n$
(see~\cite{WKO2}), and hence much of the \cite{AT} story can be told
within $\calA^w$. Several significant Lie theoretic problems (e.g., the
Kashiwara-Vergne problem,~\cite{KashiwaraVergne:Conjecture, AT, WKO2})
can be interpreted as problems about $\calA^w(\uparrow_n)$.
\begin{comment} \label{com:SortedForm} Using the $\aSTU_2$ relation
one may sort the skeleton vertices in every $\glosm{D}{D}\in\calA^w(S)$
so that along every skeleton component all arrow heads appear ahead
of all arrow tails, and by a diagrammatic analogue of the PBW theorem
(compare~\cite[Theorem~8]{Bar-Natan:OnVassiliev}), this sorted form is
unique modulo $\aSTU_1$, $TC$, $\aAS$ and $\aIHX$ relations.
\end{comment}
\begin{definition} \label{def:Operations}
A number of operations are defined on elements of the $\calA^w(S)$
spaces:
\vskip \topsep
\begin{enumerate}[leftmargin=*,labelindent=0pt,topsep=0pt]
%\setlength{\parskip}{12pt}
%\setlength{\itemsep}{0pt plus 0.3ex}
\parpic[r]{\input{figs/Union.pstex_t}}
\item If $S_1$ and $S_2$ are disjoint, then given
$D_1\in\calA^w({S_1})$ and $D_2\in\calA^w({S_2})$,
their union $D_1D_2=D_1\glosm{sqcup}{\sqcup} D_2\in\calA^w(S)$, where
$S=S_1\sqcup S_2$,
is obtained by placing them side by side as illustrated on the right.
\vskip 1mm \TopLie In topology, $\sqcup$ corresponds to the disjoint
union of 2-tangles\footnoteT{To be clear, the ``$2$'' in ``2-tangles''
refers to the dimension of the things being knotted, and not to
the number of components.}. In Lie theory, it corresponds to the
map $\calU(I\frakg)^{\otimes S_1}\otimes\calU(I\frakg)^{\otimes
S_2}\to\calU(I\frakg)^{\otimes(S_1\sqcup S_2)}$.
\vskip 2mm
\item Given $D_1\in\calA^w(S)$ and
$D_2\in\calA^w(S)$, their product
$D_1\glosm{ast}{\ast} D_2\in\calA^w(S)$ is obtained by ``stacking $D_2$
on top of $D_1$'':
\begin{equation} \label{eq:TubeProduct}
(D_1,D_2)=
\begin{array}{c}\input{figs/Stacking.pstex_t}\end{array}
=D_1\ast D_2.
\end{equation}
\TopLieAT In topology, the stacking product corresponds to the
concatenation operation on knotted tubes, akin to the standard
stacking product of tangles. In Lie theory, it comes from the
algebra structure of $\calU(I\frakg)^{\otimes S}$. In~\cite{AT},
it is the product of the completed universal enveloping algebra
$\hat{\calU}((\fraka_n\oplus\attder_n)\ltimes\attr_n)$.
\setcounter{tunnel}{\theenumi}
\end{enumerate}
\noindent Note that below and throughout this paper we use
$\glosm{act}{\act}$ for postfix operator application and for ``composition
done right''. Meaning that $x\act f$ is equivalent to $f(x)$ and $f\act g$
is $g\circ f$ is ``do $f$ then do $g$''.
\begin{enumerate}[leftmargin=*,labelindent=0pt]\setcounter{enumi}{\thetunnel}
\vskip 2mm
\parpic[r]{
\def\eps{{$\act(d\eta^1,d\eta^2,d\eta^3)$}}
\input{figs/depsilon.pstex_t}
}
\item Given $D\in\calA^w(S)$ and $a\in S$, $D\act
\glosm{deta}{d\eta}^a$ is the result of deleting strand $a$ from $D$
and mapping it to $0$ if any arrow connects to $a$, as illustrated on
the right.
\vskip 1mm
\TopLie In topology, $d\eta^a$ is the removal of one component from a
2-tangle. In Lie theory it corresponds to the co-unit
$\glosm{eta}{\eta}\colon\calU(I\frakg)\to\bbQ$.
\Needspace{2cm}
\setcounter{tunnel}{\theenumi}
\end{enumerate}
\begin{enumerate}[leftmargin=*,labelindent=0pt]\setcounter{enumi}{\thetunnel}
\vskip 2mm
\parpic[r]{
\def\dAAA{{$\act(\dA^1,\dA^2,\dA^3)$}}
\input{figs/dA.pstex_t}
}
\item Given $D\in\calA^w(S)$ and $a\in S$, $D\act
\glosm{dA}{\dA}^a$ is the result of ``flipping over stand $a$ and
multiplying by a $(-)$ sign for each arrow whose head connects to $a$'', as
illustrated above. We denote by $\dA$ the operation of likewise flipping
(with signs) {\em all} strands: $\dA=\dA^S\coloneqq\prod_{a\in S}\dA^a$.
\vskip 1mm
\TopLie In topology, $\dA^a$ is the reversal of the 1D orientation
of a knotted tube \cite{WKO2}. In Lie theory, it is the antipode of
$\calU(I\frakg)$ combined with the sign reversal $\varphi\to-\varphi$
acting on the $\frakg^\ast$ factor of $I\frakg$. When elements of
$\calU(I\frakg)^{\otimes S}$ are interpreted as differential operators
acting on functions on $\frakg^S$, $\dA$ corresponds to the $L^2$ adjoint.
\Needspace{3cm}
\setcounter{tunnel}{\theenumi}
\end{enumerate}
\begin{enumerate}[leftmargin=*,labelindent=0pt]\setcounter{enumi}{\thetunnel}
\vskip 2mm
\parpic[r]{
\def\dSSS{{$\act(\dS^1,\dS^2,\dS^3)$}}
\input{figs/dS.pstex_t}
}
\item Similarly, $D\act \glosm{dS}{\dS}^a$ is the result of ``flipping over
stand $a$ and multiplying by a $(-)$ sign for each arrow head or tail
that connects to $a$'', as illustrated above\footnoteT{The letter $S$
is used here for both ``a set of strands'' and ``an operation similar
to an antipode''. Hopefully no confusion will arise.}.
\vskip 1mm
\TopLie In topology, $\dS^a$ is the reversal of both the 1D and the 2D
orientation of a knotted tube \cite{WKO2}. In Lie theory, it is the
antipode of $\calU(I\frakg)$.
\vskip 2mm
\parpic[r]{
\def\dm{{$\act dm^{23}_2$}}
\input{figs/dm.pstex_t}
}
\item Given $D\in\calA^w(S)$, given $a,b\in S$, and
given $c\not\in S\remove\{a,b\}$, $D\act \glosm{dm}{dm}^{ab}_c$ is the
result of ``stitching strands $a$ and $b$ and calling the resulting
strand $c$'', as illustrated on the right.
\vskip 1mm
\TopLie In topology, $dm^{ab}_c$ is the ``internal stitching''
of two tubes within a single 2-link, akin to the ``stitching''
operation that combines two strands of an ordinary tangle into a
single ``longer'' one. In Lie theory, it is an ``internal product''
$\calU(I\frakg)^{\otimes n}\to\calU(I\frakg)^{\otimes(n-1)}$ which
``merges'' two factors within $\calU(I\frakg)^{\otimes n}$.
\vskip 2mm
\parpic[r]{
\def\dD{{$\act d\Delta^2_{2'2''}$}}
\input{figs/dD.pstex_t}
}
\item Given $D\in\calA^w(S)$, given $a\in S$, and
given $b,c\not\in S\remove a$, $D\act\glosm{dDelta}{d\Delta}^a_{bc}$ is
the result of ``doubling'' strand $a$, calling the resulting ``daughter
strands'' $b$ and $c$, and summing over all ways of lifting the arrows
that were connected to $a$ to either $b$ or $c$ (so if there are $k$
arrows connected to $a$, $D\act d\Delta^a_{bc}$ is a sum of $2^k$
diagrams).
\vskip 1mm
\TopLieAT In topology, $d\Delta$ is the operation of ``doubling''
one component in a 2-link. In Lie theory, it is the co-product
$\glosm{Delta}{\Delta}\colon\calU(I\frakg)\to\calU(I\frakg)^{\otimes 2}$
acting on the $a$ factor in $\calU(I\frakg)^{\otimes S}$, extended by
the identity acting on all other factors. In~\cite{AT}, it is the coface
maps of \cite[Example~3.14]{AT}.
\setcounter{tunnel}{\theenumi}
\end{enumerate}
\begin{enumerate}[leftmargin=*,labelindent=0pt]\setcounter{enumi}{\thetunnel}
\item Finally, the operation $\glosm{dsigma}{d\sigma}^a_b\colon
\calA(S)\to \calA({S\remove\{a\}\sqcup\{b\}})$ does
nothing but renaming the strand $a$ to $b$ (assuming $a\in S$ and
$b\not\in S\remove\{a\}$). \endpar{def:Operations}
\end{enumerate}
\end{definition}
We note that the product operation $(D_1,D_2)\mapsto D_1\ast D_2$ can be
implemented using the union operation $\sqcup$, the stitching
operation $dm$, and some renaming --- namely, if $\bar{S}=\{\bar{a}\colon
a\in S\}$ is some set of ``temporary'' labels disjoint from $S$ but in
a bijection with $S$, then
\begin{equation} \label{eq:multiplem} D_1\ast D_2 =
\left(
D_1\sqcup\left(D_2\act\prod_ad\sigma^a_{\bar{a}}\right)
\right)\act\prod_adm^{a\bar{a}}_a.
\end{equation}
Therefore below we will sometimes omit the implementation of
$(D_1,D_2)\mapsto D_1D_2$ provided all other operations are implemented.
We note that $\calA^w(S)$ is a co-algebra, with the co-product
$\glosm{Box}{\Box}(D)$, for a diagram $D$ representing an element of
$\calA^w(S)$, being the sum of all ways of dividing $D$ between a ``left
co-factor'' and a ``right co-factor'' so that connected components of
$D\setminus(\uparrow\!\!\times S)$ ($D$ with its skeleton removed) are
kept intact (compare with~\cite[Definition~3.7]{Bar-Natan:OnVassiliev}).
\begin{definition} \label{def:GroupLike} An element $Z$ of
$\calA^w(S)$ is ``group-like'' if $\Box(Z)=Z\otimes Z$. We
denote the set of group-like elements in $\calA^w(S)$ by
$\glosm{calAwexp}{\calA^w_{\exp}}(S)$.
\end{definition}
We leave it for the reader to verify that all the operations defined above
restrict to operations $\calA^w_{\exp}\to\calA^w_{\exp}$.
\Topology In topology, $\Box$ is the operation of ``cloning'' an entire
2-link. It is not to be confused with $d\Delta$; one dimension down and
with just one component, the pictures are:
\[ \input{figs/BoxVsDelta.pstex_t} \]
\AT In~\cite{AT}, $\Box$ is the co-product of
$\hat{\calU}((\fraka\oplus\attder)\ltimes\attr)$ and moding out by
wheels, $\calA^w_{\exp}$ is $\TAut$.
\Lie In Lie theory, $\Box$ is {\em not} the co-product
$\Delta\colon\calU(I\frakg)\to\calU(I\frakg)^{\otimes 2}$. Rather, given
two finite dimensional Lie algebras $\frakg_1$ and $\frakg_1$, $\Box$
corresponds to the map
\[ \Box\colon
\calU(I(\frakg_1\oplus\frakg_2))^{\otimes S}
\to \calU(I\frakg_1)^{\otimes S}\otimes\calU(I\frakg_2)^{\otimes S}.
\]
\picskip{0}
\begin{discussion} \label{disc:Primitives} We seek to have efficient
descriptions of the elements of $\calA^w_{\exp}(S)$ and efficient
means of computing the above operations on such elements.
Let $\glosm{Aprimw}{\calA_{\text{prim}}^w}(S)$\footnoteT{$\calA_{\text{prim}}^w$
is elsewhere denoted $\calP^w$.} denote the set of primitives
of $\calA^w(S)$: these are the elements
$\zeta\in\calA^w(S)$ satisfying $\Box(\zeta)=\zeta\otimes
1+1\otimes\zeta$. Let $\glosm{FL}{\FL}(S)$ denote the degree-completed free Lie
algebra with generators $S$, and let $\glosm{CW}{\CW}(S)$ denote the degree-completed
vector space spanned by non-empty cyclic words on the alphabet $S$.
In~\cite[Proposition~\ref{2-prop:Pnses}]{WKO2} we have shown that there
is a short exact sequence of vector spaces
\begin{equation} \label{eq:Primitives}
0\to\CW(S)\to\calA_{\text{prim}}^w(S)\to\FL(S)^S\to 0,
\end{equation}
where $\FL(S)^S$ denotes the set of all functions $S\to\FL(S)$. Hence
$\calA_{\text{prim}}^w(S)\simeq\FL(S)^S\oplus\CW(S)$ (not canonically!). Often
in bi-algebras there is a bijection given by $\zeta\mapsto
e^\zeta$ between primitive elements $\zeta$ and group-like elements
$e^\zeta$. Hence we may expect to be able to present elements of
${\calA^w_{\exp}(S)}$ as formal exponentials of combinations
of ``trees'' (elements of $\FL(S)^S$) and ``wheels'' (elements of
$\CW(S)$)\footnoteT{
We use the set-theoretic notation ``$\times$'' rather than the
linear-algebraic ``$\oplus$'' in Equation~\eqref{eq:expectation} to
emphasize that the two sides of that equation are only expected to be
set-theoretically isomorphic. The left-hand-side, in fact, is not even
a linear space in a natural way.
}:
\begin{equation} \label{eq:expectation}
\calA^w_{\exp}(S) \sim \yellowm{\TW}(S)
\coloneqq \FL(S)^S\times\CW(S)
=\left\{
(\yellowm{\lambda};\,\yellowm{\omega})\colon\begin{array}{c}
\lambda=\{a\to\lambda_a\}_{a\in S},\,\lambda_a\in\FL(S) \\
\omega\in\CW(S)
\end{array}
\right\}.
\end{equation}
\glosm{expectation}{\null}We implement Equation~\eqref{eq:expectation}
in a more-or-less straightforward way in Section~\ref{subsec:AT} and in a
less straightforward but somewhat stronger way in Section~\ref{subsec:Ef}.
\endpar{disc:Primitives}
\end{discussion}
\begin{discussion} \label{disc:WhyTwo}
Why are there two presentations for elements of $\calA^w_{\exp}$?
Because as we shall see, $\calA^w$ is a bi-algebra in two ways, using two
different products, yet with the same co-product $\Box$. In $\calA^w$, the
notions ``primitive'' and ``group-like'', whose definition involves only
$\Box$, are canonical. Yet the bijection between primitive and group-like
elements, $\zeta\leftrightarrow e^\zeta$, depends also on the product
used within the power-series interpretation of $e^\zeta$. Thus there are
two different ways to describe the group-like elements $\calA^w_{\exp}$
of $\calA^w$ in terms of its primitives $\TW$.
The first product on $\calA^w$ is the stacking product of
Equation~\eqref{eq:TubeProduct}. The second will be introduced later,
in Equations~\eqref{eq:AHTStacking} and~\eqref{eq:FProduct}.
\Topology Very roughly speaking, $\calA^w$ is a combinatorial
model of ``$\pi_1\ltimes\pi_2$'' (with homotopies replaced by isotopies;
see~\cite{KBH}). The other product on $\calA^w$ is the one coming from the
direct product ``$\pi_1\times\pi_2$''.
\Lie Very roughly speaking, $\calA^w$ is a combinatorial model of
(tensor powers of a completion of) $\calU(I\frakg)$. By PBW,
$\calU(I\frakg)\simeq\calU(\frakg)\otimes\glosm{calS}{\calS}(\frakg^\ast)$
as co-algebras but not as algebras. The other product on
$\calA^w$ is the one corresponding to the natural product on
$\calU(\frakg)\otimes\calS(\frakg^\ast)$. The reality is a bit more
delicate, though. $\calA^w$ is only a model of (a small part of)
the $\frakg$-invariant part of $\calU(I\frakg)$, and the co-product
$\Box$ of $\calA^w$ does not correspond to the co-product $\Delta$
of $\calU(I\frakg)$. \endpar{disc:WhyTwo}
\end{discussion}
\draftcut\subsection{Some preliminaries about free Lie algebras and cyclic
words} \label{subsec:FL}
It should be clear from Discussion~\ref{disc:Primitives} that free Lie
algebras and cyclic words play a prominent role in this paper. For the
convenience of our readers we collect in this section some preliminaries
about about these topics. Almost everything in this section comes either
from Alekseev-Torossian's~\cite{AT}, or from~\cite{WKO2,KBH}, and the
detailed proofs of the assertions made here can be found in these papers.
\NotLie Note that Lie algebras appear in two distinct
roles in this paper. {\em Free} Lie algebras $\FL$ appear
along with cyclic words $\CW$ as the primitives of $\calA^w$
(Equation~\eqref{eq:Primitives}). {\em Finite dimensional} Lie algebras
$\frakg$ appear only as motivational comments, always marked with a
\raisebox{-4pt}{\scalebox{0.75}{\input{figs/Lie.pstex_t}}} symbol. As
already indicated, elements in $\calA^w$, and hence elements of $\FL$
and of $\CW$ can represent ``universal'' formulas that make sense in any
finite dimensional Lie algebra $\frakg$. Hence part of our discussion
of $\FL$ and $\CW$ is a discussion of things that make sense universally
for all finite dimensional Lie algebras.
Recall that $\FL(S)$ denotes the graded completion of the
free Lie algebra over a set of generators $S$, all considered
to have degree $1$. In the case when $S=\{x_1,\ldots,x_n\}$,
Alekseev and Torossian~\cite{AT}
denote this space $\glosm{lie}{\lie}_n$.\footnoteC{
In computer talk, generators of $\FL(S)$ are always
single-character ``Lyndon words'' (e.g.~\cite{Reutenauer:FreeLie});
in our case we set $x$ and $y$ to be the single-character words ``$x$'' and
``$y$'', and then $\alpha$, $\beta$, and $\gamma$ to be the Lie series
$x+[x,y]$, $y-[x,[x,y]]$, and $x+y-2[x,y]$ (elements of $\FL$ are infinite
series, in general, but these examples are finite):
\shortdialoginclude{alphabetagamma}
Note that as we requested earlier, our example series are printed
to degree 4. Note also that they are printed using ``top bracket''
$\glosm{ob}{\ob{xy}}\coloneqq [x,y]$ notation, which is easier to read
when many brackets are nested.
We then compute $[\alpha,\beta]$ and verify the Jacobi identity for
$\alpha$, $\beta$, and $\gamma$:
\shortdialoginclude{BracketExample}
}
A noteworthy element of $\FL(x,y)$ is the Baker-Campbell-Hausdorff
series,\footnoteC{In computer talk:
\shortdialogincludewithlink{bch}
Just to show that we can, here are the lexicographically
middle three of the 2,181 terms of the BCH series in degree 16, along
with the time in seconds it took my humble laptop to compute it:
\shortdialoginclude{bch16}
(In a few hours my laptop computed the BCH series to degree 23; in as much
as I know, the farthest it was ever computed. See~\cite{Bar-Natan:BCH,
CasasMurua:EfficientBCH}.)
}
\[ \glosm{BCH}{\BCH}(x,y)\coloneqq\log(e^xe^y)
= x+y+\frac{[x,y]}{2} +\frac{[x,[x,y]] + [[x,y],y]}{12}
+\ldots.
\]
Recall also that $\CW(S)$ ($\glosm{attr}{\attr}_n$,
in~\cite{AT}) denotes the graded completion
of the vector space spanned by non-empty cyclic words in the alphabet
$S$. Our convention is to crown cyclic words with an ``arch''; thus
$\glosm{wideparen}{\wideparen{uvw}} = \wideparen{vwu}$\footnoteC{Cyclic
words in computer talk:
\shortdialoginclude{omegas}
}. Note that there is a map $\CW(\FL(S))\to\CW(S)$ by interpreting brackets
within elements of $\FL(S)$ as commutators and then mapping ``long'' words
to cyclic words. E.g., $\wideparen{u[v,w]} = \wideparen{uvw} -
\wideparen{uwv}$.
We denote by $\glosm{hdeg}{h^{\deg}}$ the operations $\FL\to\FL$ and
$\CW\to\CW$ which multiply any degree $k$ element by $h^k$. In particular,
$\glosm{mdeg}{(-1)^{\deg}}$ acts on $\FL/\CW$ as the identity in even
degrees and as minus the identity in odd degrees.\footnoteC{In computer
talk:
\shortdialoginclude{DegreeScale}
}
Let $\glosm{der}{\der}_S$ denote the Lie algebra of all derivations of $\FL(S)$
($\glosm{atder}{\atder}_n$ in~\cite{AT}). There
is a linear map $\glosm{partial}{\partial}\colon\FL(S)^S\to\der_S$
which assigns to every $\lambda=(\lambda_a)_{a\in
S}\in\FL(S)^S$ the unique derivation $\partial_\lambda$ for
which $\partial_\lambda(a)=[a,\lambda_a]$ for every $a\in
S$.\footnotemarkT\,\footnotemarkC\ The image of $\partial$ is
a subalgebra of $\der_S$ denoted $\glosm{tder}{\tder}_S$
($\glosm{attder}{\attder}_n$ in~\cite{AT}); the elements of $\tder_S$
are called ``tangential derivations''. The kernel of $\partial$ can
be identified as the Abelian Lie algebra $\glosm{A}{A}_S$ generated by $S$
($\glosm{fraka}{\fraka}_n$ in~\cite{AT}), which is linearly embedded
in $\FL(S)^S$ as the set of all sequences $\lambda\colon S\to\FL(S)$ for
which $\lambda_a$ is a scalar multiple of $a$ for every $a\in S$. Thus
we have a short exact sequence of vector spaces
\begin{equation} \label{eq:FLisAtder}
0 \rightarrow A_S\rightarrow \FL(S)^S\xrightarrow{\partial}
\tder_S\rightarrow 0.
\end{equation}
The map $\FL(S)^S\ni\lambda=(\lambda_a)\mapsto\sum_a\langle\lambda_a,
a\rangle a\in A_S$, where $\langle\lambda_a, a\rangle$ is the
coefficient of $a$ in $\lambda_a$ is a splitting of the above sequence,
and hence $\FL(S)^S\simeq A_S\oplus\tder_S$ in a canonical manner.
\footnotetextC{An example:
\shortdialoginclude{TangentialDerivative}
}
There is a unique Lie bracket $\glosm{tb}{[\cdot,\cdot]_{tb}}$ (the
``tangential bracket'') on $\FL(S)^S$ which makes \eqref{eq:FLisAtder}
a split exact sequence of Lie algebras, and hence
$(\FL(S)^S,[,]_{tb})\simeq A_S\oplus\tder_S$ as Lie algebras. With
$[\cdot,\cdot]$ denoting the ordinary direct-sum bracket on $\FL(S)^S$
and with the action of $\partial_\lambda$ extended to
$\partial_\lambda\colon\FL(S)^S\to\FL(S)^S$ in the obvious manner,
we have\footnotemarkC
\[ [\lambda_1,\lambda_2]_{tb}
=[\lambda_1,\lambda_2]
+\partial_{\lambda_1}\lambda_2
-\partial_{\lambda_2}\lambda_1.
\]
\footnotetextT{Using the notation of~\cite{KBH}, $\partial_\lambda =
-\sum_{a\in S}\ad_a^{\lambda_a} = -\sum_{a\in S}\ad_a\{\lambda_a\}$. I
apologize for the minus sign which stems from a bad choice made
in~\cite{KBH}.}
\footnotetextC{For example:
\shortdialoginclude{tb}
}
The $\lambda\mapsto\partial_\lambda$ action of $(\FL(S)^S,[,]_{tb})$
on $\FL(S)$ extends to an action on the universal enveloping
algebra of $\FL(S)$, the free associative algebra $\FA(S)$ on $S$
generators, and then descends to the vector-space quotient of
$\FA(S)$ by commutators, namely to cyclic words. Leaving aside
the empty word, we find that $(\FL(S)^S,[,]_{tb})$ acts on
$\CW(S)$, and hence also on $\TW(S)$.\footnoteC{ We check that
up to degree 8, $\partial_{[\lambda_1,\lambda_2]_{tb}}(\omega_1) =
[\partial_{\lambda_1},\partial_{\lambda_2}](\omega_1)$ (for our choice
of $\lambda_1$, $\lambda_2$, and $\omega_1$, both sides vanish below
degree 8):
\shortdialoginclude{tb2}
Note that the comparison operator $\equiv$ returns a ``Boolean Sequence''
({\tt BS}) rather than a single {\tt True}/{\tt False} value, as the
computer has no way of knowing whether two series are equal without
computing them up to a given degree. In our case, we've asked for the
comparison of {\tt lhs} with {\tt rhs} up to degree 8, and the output,
including degree 0, is a sequence of 9 affirmations, summarized as ``{\tt 9
True}''.
}
There are two ways to assign an automorphism of the free Lie algebra
$\FL(S)$ to an element $\lambda\in\FL(S)^S$:
\begin{enumerate}
\item One may exponentiate the derivation $\partial_\lambda$ to get
$e^{\partial_\lambda}\colon\FL(S)\to\FL(S)$.
\item One may define an automorphism
$\glosm{C}{C^\lambda}\colon\FL(S)\to\FL(S)$ by setting
its values on the generators by $C^\lambda(a)\coloneqq
e^{\lambda_a}ae^{-\lambda_a}=e^{\ad\lambda_a}a$. We denote the inverse
of $C^\lambda$ by $\glosm{RC}{RC^{-\lambda}}$ and note that it is {\em not}
$C^{-\lambda}$.
\end{enumerate}
\AT In~\cite{AT},~(1) corresponds to the presentation of elements of the
automorphism group $\glosm{TAut}{\TAut_n}$ as exponentials of elements of
its Lie algebra $\tder_n$, while~(2) corresponds to its presentation in
terms of ``basis conjugating automorphisms'' $x_i\mapsto g_i^{-1}x_ig_i$
where $g_i=e^{-\lambda_i}$. Compare with \cite[Section~5.1]{AT}.
The following pair of propositions, which we could not find elsewhere, relates
these two automorphisms:
\begin{proposition} \label{prop:Gamma}
Given $\lambda\in\FL(S)^S$, let $t$ be a scalar-valued
formal variable and let $\glosm{Gammat}{\Gamma_t(\lambda)}\in\FL(S)^S$
be the (unique) solution of the ordinary differential equation
\begin{equation} \label{eq:GammaODE}
\Gamma_0(\lambda)=0
\qquad\text{and}\qquad
\frac{d\Gamma_t(\lambda)}{dt} = \lambda \act e^{-t\partial_\lambda}
\act \frac{\ad\Gamma_t(\lambda)}{e^{\ad\Gamma_t(\lambda)}-1}.
\end{equation}
\begin{flalign} \label{eq:Gamma}
& \text{Then} & e^{-t\partial_\lambda}=C^{\Gamma_t(\lambda)}.\footnotemarkC &&
\end{flalign}
\end{proposition}
\footnotetextC{
We verify that the computer-calculated $\Gamma_t(\lambda)$ satisfies the
ODE in~\eqref{eq:GammaODE} and then that the operator equality~\eqref{eq:Gamma}
holds, at least when evaluated on ``our'' $\gamma$:
\shortdialoginclude{TestingGammaODE}
\shortdialoginclude{TestingGamma}
}
\begin{proof} The two sides $L_t$ and $R_t$ of Equation~\eqref{eq:Gamma}
are power-series perturbations of the identity automorphism of
$\FL(S)$. More fully, $L_t$ can be written $L_t=\sum_{d\geq 0}t^dL(d)$
where $L(d)\colon\FL(S)\to\FL(S)$ raises degrees by at least $d$
(and so the sum converges), and where $L(0)$ is the identity. $R_t$
can be written in a similar way. We claim that it is enough to prove that
\begin{equation} \label{eq:AB}
A_t\coloneqq(\frac{dL_t}{dt})\act L_t^{-1}
= (\frac{dR_t}{dt})\act R_t^{-1} \eqqcolon B_t.
\end{equation}
Indeed, if otherwise $L_t\neq R_t$, consider the minimal $d$ for which
$L(d)\neq R(d)$. Then $d>0$ and the least-degree term in $A_t-B_t$
is the degree $d-1$ term, which equals $dt^{d-1}L(d)\act L_t^{-1} -
dt^{d-1}R(d)\act R_t^{-1} = dt^{d-1}(L(d)-R(d))\act L_t^{-1} \neq 0$
(the last equality is because $L_t^{-1}=R_t^{-1}$ to degree $d$),
contradicting Equation~\eqref{eq:AB}. Note that in fact we have shown
that if $A_t=B_t$ to degree $d$ in $t$, then Equation~\eqref{eq:Gamma}
holds to degree $d+1$.
To compute $B_t$ we need the differential of
$C^\mu$ (at $\mu=\Gamma_t(\lambda)$) and the chain rule. The differential
of $C^\mu$ is quite difficult; fortunately, we have computed it in the
case where $\mu=(u\to\gamma)$ is supported on just one $u\in S$,
in~\cite[Lemma~\ref{KBH-lem:dC}]{KBH}. Both the result and its proof
generalize simply, and so we have
\[ \delta C^\mu = -\partial\left\{
\delta\mu \act \frac{e^{\ad\mu}-1}{\ad\mu} \act RC^{-\mu}
\right\}\act C^\mu,
\]
where we have written $\partial\{\text{mess}\}$ instead of
$\partial_{\text{mess}}$ because $\text{mess}$ is too big to fit as
a subscript. Hence by the chain rule and then by Equation~\eqref{eq:GammaODE},
\[ B_t
= -\partial\left.\left\{
\frac{d\Gamma_t(\lambda)}{dt} \act \frac{e^{\ad\mu}-1}{\ad\mu} \act RC^{-\mu}
\right\}\right|_{\mu=\Gamma_t(\lambda)}
= -\partial\left\{
\lambda \act e^{-t\partial_\lambda} \act RC^{-\Gamma_t(\lambda)}
\right\}
= -\partial_{\lambda \act e^{-t\partial_\lambda} \act RC^{-\Gamma_t(\lambda)}}.
\]
On the other hand, computing $A_t$ is a simple differentiation, and
we get that $A_t=-\partial_\lambda$. Comparing with the line above,
we find that if Equation~\eqref{eq:Gamma} holds to degree $d$, then
Equation~\eqref{eq:AB} also holds to degree $d$. But then as we
noted,~\eqref{eq:Gamma} holds to degree $d+1$. As
Equation~\eqref{eq:Gamma} clearly holds at $t=0$, we find that it holds
to all orders. \qed
\end{proof}
\begin{comment} It is easier (though insufficient) to assume that there is
a solution $\Gamma_t(\lambda)$ to Equation~\eqref{eq:Gamma} and deduce that
it must satisfy the differential equation~\eqref{eq:GammaODE}: simply
differentiate~\eqref{eq:Gamma} with respect to $t$ and simplify as much as
you can allowing yourself to use~\eqref{eq:Gamma} as needed within the
simplification process. The result is~\eqref{eq:GammaODE}, and the steps
follow the computational steps of the above proof rather closely. The
actual proof is a bit harder because if we cannot assume~\eqref{eq:Gamma}
while deriving it, so we have to resort to an inductive process.
\end{comment}
\begin{proposition} \label{prop:Lambda} As in the previous proposition,
let $\glosm{Lambdat}{\Lambda_t(\lambda)}$ be the (unique) solution of
\begin{equation} \label{eq:LambdaODE}
\Lambda_0(\lambda)=0
\qquad\text{and}\qquad
\frac{d\Lambda_t(\lambda)}{dt} =
\lambda \act e^{\partial_{\Lambda_t(\lambda)}}
\act \frac{\ad_{tb}\Lambda_t(\lambda)}{e^{\ad_{tb}\Lambda_t(\lambda)}-1}.
\end{equation}
\begin{flalign} \label{eq:Lambda}
& \text{Then} & C^{t\lambda}=e^{-\partial_{\Lambda_t(\lambda)}}. &&
\end{flalign}
\end{proposition}
The proof of this proposition is very similar and not even a tiny bit
nicer than the proof of the previous one. So we skip it and
instead include a computer verification.\footnoteC{
We verify that the computer-calculated $\Lambda_t(\lambda)$
satisfies the ODE in~\eqref{eq:LambdaODE} and then that
the operator equality~\eqref{eq:Lambda} holds, at least when
evaluated on ``our'' $\gamma$:
\shortdialoginclude{TestingLambdaODE}
\shortdialoginclude{TestingLambda}
}
As special cases, we denote $\Gamma_1(\lambda)$ by
$\glosm{Gamma}{\Gamma(\lambda)}$ and $\Lambda_1(\lambda)$ by
$\glosm{Lambda}{\Lambda(\lambda)}$.
One special case of $C^\lambda$ deserves to be named:
\begin{definition} \label{def:CRC}
(Compare~\cite[Section~\ref{KBH-subsec:FLSuccess}]{KBH}) Given $u\in
S$ and $\gamma\in\FL(S)$ let $\glosm{Cu}{C_u^{\gamma}}$ denote the
automorphism of $\FL(S)$ defined by mapping the generator $u$ to its
``conjugate'' $e^{\gamma}ue^{-\gamma}=e^{-\ad\gamma}(u)$ (this is simply
$C^\lambda$, where $\lambda$ is the length $1$ sequence $(u\to\gamma)$).
Let $\glosm{RCu}{RC_u^{-\gamma}}$ be the inverse of $C_u^{\gamma}$
(which is {\em not} $C_u^{-\gamma}$).\footnoteC{Just testing:
\shortdialoginclude{CCAndRC}
}
\end{definition}
\Needspace{2in}
Last we define/recall a number of functionals $\FL(S)\to\CW(S)$:
\vskip 1mm
{
\makeatletter\def\thm@space@setup{%
\thm@preskip=0cm plus 0cm minus 0cm %\thm@postskip=\thm@preskip
}\makeatother
\parpic[r]{\input{figs/tru.pstex_t}}
\begin{definition} \label{def:tru}
For $u\in S$ we let $\glosm{tru}{\tr_u}\colon\FL(S)\to\CW(S)$
be the sum of all ways of connecting the head of $\gamma$ to any
of its $u$-labelled tails and regarding the result as an element
of $\CW(\FL(S))\to\CW(S)$. The example on the right corresponds
to the specific computation $\tr_u[[v,u],u] = \wideparen{[v,u]} +
\wideparen{v(-u)} = -\wideparen{uv}$\footnoteC{In computer talk, and
using a temporary value for $\gamma$, so as not to interfere with its
existing value:
\shortdialoginclude{tru}
}
\end{definition}
}
\vskip 1mm
{
\makeatletter\def\thm@space@setup{%
\thm@preskip=0cm plus 0cm minus 0cm %\thm@postskip=\thm@preskip
}\makeatother
\parpic[r]{\input{figs/divu.pstex_t}}
\begin{definition} \label{def:J}
(Compare~\cite[Section~\ref{KBH-subsec:divJ}]{KBH})
For $u\in S$ we let $\glosm{atdivu}{\atdiv_u}\colon\FL(S)\to\CW(S)$
be the functional defined schematically by the picture on the right,
which corresponds to the specific computation $\atdiv_u[[v,u],u] =
\wideparen{u[v,u]} + \wideparen{uv(-u)} = -\wideparen{uuv}$\footnoteC{In
computer talk:
\shortdialoginclude{divu}
}
(more details in~\cite{KBH}). Given also $\gamma\in\FL(S)$, set
\[ \glosm{Ju}{J_u}(\gamma) := \int_0^1ds\,\atdiv_u\!\left(
\gamma \sslash RC_u^{s\gamma}
\right) \sslash C_u^{-s\gamma}.\footnotemarkC
\]
\end{definition}
}
\footnotetextC{We quote the implementation of $J$ in
\href{\web/FreeLie.m}{\tt FreeLie.m} (\href{\web/FreeLie.m}{\tt FL}) and,
reverting to the ``old'' $\gamma$, compute $J_1(\gamma)$:
\FLQuote{JDef}
\shortdialoginclude{Ju}
}
\picskip{0}
\begin{definition} \label{def:j} Let
$\glosm{atdiv}{\atdiv}\colon\FL(S)\to\CW(S)$ be the Alekseev-Torossian
``divergence'' functional, as in~\cite[Section~5.1]{AT}, but extended
by $0$ on $A_S$. In our language, $\atdiv\lambda=\sum_{u\in
S}\atdiv_u\lambda$. Let $\glosm{j}{j}\colon\FL(S)\to\CW(S)$ is
the Alekseev-Torossian ``logarithm of the Jacobian'':
$j(\lambda) = \frac{e^{\partial_\lambda}-1}{\partial_\lambda}
(\atdiv\lambda)$.\footnoteC{A quote of the computer-definition, and then
$\atdiv\lambda$ and $j(\lambda)$, computed to degree 5:
\FLQuote{divDef}
\shortdialoginclude{j}
}
\end{definition}
Alekseev and Torossian prove in~\cite{AT} that $j$ is the unique functional
$j\colon\FL(S)\to\CW(S)$ satisfying the ``cocycle condition''
$j\left(\glosm{BCHtb}{\BCH_{tb}}(\lambda_1,\lambda_2)\right) =
j(\lambda_1)+e^{\partial_{\lambda_1}}j(\lambda_2)$, where $\BCH_{tb}$
stands for the $\BCH$ formula using the tangential bracket
$[\cdot,\cdot]_{tb}$ on $\FL(S)^S$:
\[ \BCH_{tb}(\lambda_1,\lambda_2)
= \lambda_1+\lambda_1+\frac12[\lambda_1,\lambda_2]_{tb}
+\ldots,
\]
and the ``initial condition''
$\frac{\partial}{\partial\epsilon}j(\epsilon\lambda) =
\atdiv\lambda$.\footnoteC{We verify the cocycle condition and the initial
condition. For the latter, we first declare $\epsilon$ to be ``an
infinitesimal'' by declaring that $\epsilon^2=0$, and then we verify that
$j(\epsilon\lambda) = \epsilon\atdiv\lambda$:
\shortdialoginclude{cocycle4j}
\shortdialoginclude{dj}
}
\input El.tex
\draftcut
\parpic[r]{\input{figs/AwDiagram.pstex_t}}
\subsection{The factored presentation $E_f$ of $\calA^w_{\exp}$ and its
stronger precursor $E_s$} \label{subsec:Ef}
Following~\cite{KBH}, in the ``factored'' presentation $E_f$
of $\calA^w_{\exp}$ arrow heads are treated separately from arrow
tails in diagrams such as the one on the right. This presentation of
$\calA^w_{\exp}$ is more complicated than the previous one, yet it is also
more powerful, and in some sense, it is made of simpler ingredients. We
first enlarge the collection of spaces $\{\calA^w(S)\}$ to a somewhat
bigger collection $\{\calA^w(H;T)\}$ on which a larger class of operations
act. The new operations are more ``atomic'' than the old ones, in the
sense that each of the operations of Definition~\ref{def:Operations} is
a composition of 2-3 of the new operations. The advantage is that the
new operations all have reasonably simple descriptions as operations
on the group-like subsets $\{\calA^w_{\exp}(H;T)\}$ (the ``split''
presentation $E_s$ below), and hence even the few operations whose description
in the $E_l$ presentation was omitted in Definition-Proposition~\ref{dp:ElOps}
can be fully described and computed in the $E_f$ presentation.
A sketch of our route is as follows: In Section~\ref{sssec:Family},
right below, we describe the spaces $\{\calA^w(H;T)\}$. In
Section~\ref{sssec:AHTOperations} we describe the zoo of operations acting
on $\{\calA^w(H;T)\}$. Section~\ref{sssec:AHTExp} is the tofu of the matter
--- we describe the operations of the previous section in
terms of spaces $\{\TW_s(H;T)\}$ of trees and wheels, whose elements are in a
bijection $E_s$ with the group like elements of
$\{\calA^w(H;T)\}$. Finally in Section~\ref{sssec:Inclusion} we explain
how the system of spaces $\{\calA^w(S)\}$ includes into the system
$\{\calA^w(H;T)\}$ and how the operations of the former are expressed
in terms of the latter, concluding the description of $E_f$.
\subsubsection{The family $\{\calA^w(H;T)\}$} \label{sssec:Family}
Let $\glosm{H}{H} = \{\glosm{h}{h_1},h_2,\ldots\}$ be some finite set of
``head labels'' and let $\glosm{T}{T} = \{\glosm{t}{t_1},t_2,\ldots\}$
be some finite set of ``tail labels'' (these sets need not be of the
same cardinality). Let $\glosm{calAwHT}{\calA^w(H;T)}$ be
$\calA^w({H\sqcup T})$\footnotemarkT\ moded out by the following
further relations:
\footnotetextT{\label{foot:BruteDisjoint} We will often use sets of labels
$H$ and $T$ that are {\em not} disjoint. The notation
``$H\glosm{Brutesqcup}{\sqcup}T$'' stands for the union of $H$ and
$T$, made disjoint by brute force; for example, by setting $H\sqcup
T\coloneqq(\{h\}\times H)\cup(\{t\}\times T)$, where $h$ and $t$ are two
distinct labels chosen in advance to indicate ``heads'' and ``tails''. In
practise we will keep referring to the images of the elements of $H$
within $H\sqcup T$ as $h_i$ rather than $(h,h_i)$, and likewise for
the $t_i$'s. We will mostly avoid the confusion that may arise when
$H\cap T\neq\emptyset$ by labelling operations as ``head operations'' which
will always refer to labels in $H\hookrightarrow H\sqcup T$ or as ``tail
operations'', when referring to labels in $T\hookrightarrow H\sqcup T$.}
\parbox[t]{4in}{\begin{myitemize}
\item If an arrow tail lands anywhere on a head strand ($\ast1$ on the
right), the whole diagram is zero.
\item The $\glosm{CP}{CP}$ relation: If an arrow head is the lowest vertex
on a tail strand ($\ast2$ on the right), the whole diagram is zero. (As
on the right, we indicate the bottom ends of tail strands with bullets
``$\bullet$'').
\end{myitemize}}
\hfill\imagetop{\input{figs/HeadTailRels.pstex_t}}
\Needspace{2in}
\vskip 1mm
{
\makeatletter\def\thm@space@setup{%
\thm@preskip=0cm plus 0cm minus 0cm %\thm@postskip=\thm@preskip
}\makeatother
\parpic[r]{\input{figs/TypicalAHT.pstex_t}}
\begin{comment} \label{com:PureForm}
Using these two relations one may show that $\calA^w(H;T)$
is isomorphic to the set of arrow diagrams in which only arrow heads
land on the head strands (obvious, by the first relation) and in which
only arrow tails meet the tail strands (use $\aSTU_2$ to slide any arrow
head on a tail strand until it's near the bottom, then use the second
relation; see also Comment~\ref{com:SortedForm}), still modulo $\aAS$,
$\aIHX$, $\aSTU_1$ and $TC$. Thus a typical element of $\calA^w(H;T)$
is shown on the right.
\end{comment}
}
\Topology In topology (see~\cite{KBH}), head strands correspond to
``hoops'', or based knotted circles, and tail strands correspond to
balloons, or based knotted spheres. The two relations and the isomorphism
above are also meaningful~\cite{KBH}.
\Lie In Lie theory head strands represent $\calU(\frakg)$ and tail strands
represent the (right) Verma module
$\calU(I\frakg)/\frakg\calU(I\frakg) \simeq \calU(\frakg^\ast) \simeq
\calS(\frakg^\ast)$. The evaluation $\frakg^\ast\to 0$ induces a surjection
of $\calU(I\frakg)$ onto the first of these spaces whose kernel is ``any
word containing a letter in $\frakg^\ast$'', explaining the first relation
above. The second relation is the definition of the Verma module.
\Needspace{2cm}
\subsubsection{Operations on $\{\calA^w(H;T)\}$.}
\label{sssec:AHTOperations}
\begin{definition} \label{def:AHTOperations}
Just as in Definition~\ref{def:Operations}, there are several
operations that are defined on $\calA^w(H;T)$. In brief, these are:
\begin{enumerate}[leftmargin=*,labelindent=0pt]
\item A union operation $\glosm{htsqcup}{\sqcup}\colon \calA^w(H_1;T_1)
\otimes \calA^w(H_2;T_2) \to \calA^w(H_1\sqcup H_2;T_1\sqcup T_2)$,
defined when $H_1\cap H_2=T_1\cap T_2=\emptyset$, with obvious topological
(compare with ``$\ast$'' of~\cite[Figure~\ref{KBH-fig:ConnectedSum}]{KBH})
and Lie theoretic meanings. (The symbol $\sqcup$ is sometimes omitted:
$D_1D_2\coloneqq D_1\sqcup D_2$).
\item A ``stacking'' product $\glosm{jail}{\#}$ can be defined on
$\calA^w(H;T)$ by stitching all pairs of equally-labelled head strands
and then merging all pairs of equally-labelled tail strands in a
pair of diagrams $D_1,D_2\in\calA^w(H;T)$. The ``merging'' of tail
strands is described in more detail as the operation $tm$ below. In
fact, it may be better to define $\#$ using a formula similar to
Equation~\eqref{eq:multiplem} and the operations $hm$, $tm$, $h\sigma$,
and $t\sigma$ defined below:
\begin{equation} \label{eq:AHTStacking}
D_1\#D_2 = \left(
D_1\sqcup\left(D_2
\act\prod_{x\in H}h\sigma^x_{\bar{x}}
\act\prod_{u\in T}t\sigma^u_{\bar{u}}
\right)
\right)
\act\prod_{x\in H}hm^{x\bar{x}}_x
\act\prod_{u\in T}tm^{u\bar{u}}_u.
\end{equation}
\TopLie In topology, $\#$ is the stitching of hoops followed by the
merging of balloons; this is not the same as the stitching of knotted
tubes. In Lie theory, $\#$ corresponds to the componentwise product of
$\calU(\frakg)^{\otimes H}\otimes\calS(\frakg^\ast)^{\otimes T}$. Even when
$H$ and $T$ are both singletons, this is not the same as the product of
$\calU(I\frakg)$, even though linearly
$\calU(I\frakg)\simeq\calU(\frakg)\otimes\calS(\frakg^\ast)$.
\setcounter{tunnel}{\theenumi}
\end{enumerate}
\begin{enumerate}[leftmargin=*,labelindent=0pt]\setcounter{enumi}{\thetunnel}
\item If $\glosm{xinH}{x}\in H$ and $\glosm{uinT}{u}\in T$, the operations
$\glosm{heta}{h\eta}^x$ and $\glosm{teta}{t\eta}^u$ drop the head-strand
$x$ or the tail-strand $u$ similarly to the operation $d\eta^a$ of
Definition~\ref{def:Operations}.
\item $\glosm{hA}{hA}^x$ reverses the head-strand $x$ while multiplying
by a $(-1)$ factor for every arrow head on $x$. $\glosm{tA}{tA}^u$
is the identity.
\item $\glosm{hS}{hS}^x=hA^x$ while $\glosm{tS}{tS}^u$ multiplies by a
factor of $(-1)$ for every arrow tail on $u$ (by $TC$, there's no need
to reverse $u$).
\item The operation $\glosm{hm}{hm}^{xy}_z$ is defined similarly
to $dm^{ab}_c$ of Definition~\ref{def:Operations}. Likewise for
$\glosm{tm}{tm}^{uv}_w$, except in this case, the tail-strands $u$
and $v$ must first be cleared of all arrow-heads using the process of
Comment~\ref{com:PureForm}. Once $u$ and $v$ carry only arrow-tails,
all these tail can be put on a new tail-strand $w$ in some arbitrary
order (which doesn't matter, by $TC$). Note that $tm^{uv}_w=tm^{vu}_w$,
so $tm$ is ``meta-commutative''.
\Topology In topology, $tm^{uv}_w$ is the ``merging of balloons''
operation of \cite[Section~\ref{KBH-subsec:MMAOperations}]{KBH}, which
in itself is analogues to the (commutative) multiplication of $\pi_2$.
\Lie In Lie theory, $tm^{uv}_w$ is the product of $\calS(\frakg^\ast)$.
Note that tail strands more closely represent the Verma module
$\calU(I\frakg)/\frakg\calU(I\frakg)$ whose isomorphism with
$\calS(\frakg^\ast)$ involves ``sliding all $\frakg$-letters in a
$\calU(I\frakg)$-word to the left and then cancelling them''. This is
analogous to the process of cancelling arrow-heads which is a
pre-requisite to the definition of $tm^{uv}_w$.
\setcounter{tunnel}{\theenumi}
\end{enumerate}
\begin{enumerate}[leftmargin=*,labelindent=0pt]\setcounter{enumi}{\thetunnel}
\item $\glosm{hDelta}{h\Delta}^x_{yz}$ and
$\glosm{tDelta}{t\Delta}^u_{vw}$ are defined similarly to
$d\Delta^a_{bc}$.
\item $\glosm{hsigma}{h\sigma}^x_y$ and $\glosm{tsigma}{t\sigma}^u_v$
are defined similarly to $d\sigma^a_b$.
\item \textbf{New!} Given a tail $u\in T$, a ``new'' tail
label $v\not\in T\remove u$ and a head $x\in H$ the operation
$\glosm{thm}{thm}^{ux}_v\colon\calA^w(H;T)\to\calA^w(H\remove x;(T\remove
u)\sqcup\{v\})$ is the obvious ``tail-strand head-strand stitching''
--- similarly to $dm^{ab}_c$, stitch the strand $u$ to the strand $x$
putting $u$ before $x$, and call the resulting ``new'' strand $v$. Note
that for this to be well defined, $v$ must be a tail strand.\footnoteT{
Note also that the analogous operation $htm^{xu}_v$ ``put $x$ before $u$
to get a tail $v$'' is $0$ and hence we can safely ignore it, and that
$thm^{xu}_y$ and $htm^{xu}_y$, defined in the same way as $thm^{ux}_v$
and $htm^{xu}_v$ except to produce a head strand $y$, are not well
defined because they do not respect the $CP$ relation.}
In practise, $thm^{ux}_v$ is never used on its own, but the combination
$h\Delta^x_{xx'}\act thm^{ux'}_u$ (where $x'$ is a temporary label) is
very useful. Hence we set
$\glosm{tha}{tha}^{ux}\colon\calA^w(H;T)\to\calA^w(H;T)$ (``tail by head
action on $u$ by $x$'') to be that combination. In words, this is ``double
the strand $x$ and put one of the copies on top of $u$''.\footnoteT{Note
that $thm^{ux}_v=tha^{ux}\act h\eta^x\act t\sigma^u_v$ so we lose no
generality by considering $tha^{ux}$ instead of $thm^{ux}_v$.}
\TopLie In topology, $tha$ is the action of hoops on balloons as
in~\cite[Section~\ref{KBH-subsec:MMAOperations}]{KBH}, which is similar to
the action of $\pi_1$ on $\pi_2$. In Lie theory, it is the right action of
$\calU(\frakg)$ on the Verma module $\calU(I\frakg)/\frakg\calU(I\frakg)$,
or better, the action of $\calU(\frakg)$ on $\calS(\frakg^\ast)$ induced
from the co-adjoint action of $\frakg$ on $\frakg^\ast$.
\endpar{def:AHTOperations}
\end{enumerate}
\end{definition}
\TopLie {\em Exercise \refstepcounter{theorem}\thetheorem.} In the cases
when we did not state the topological or Lie theoretical meaning of an
operation in Definition~\ref{def:AHTOperations}, find what it is.
\subsubsection{Group-like elements in $\{\calA^w(H;T)\}$.}
\label{sssec:AHTExp}
For any fixed finite sets $H$ and $T$ there is a co-product
$\glosm{htBox}{\Box}\colon\calA^w(H;T)\otimes\calA^w(H;T)$ defined just
as in the case of $\calA^w(S)$ (Definition~\ref{def:GroupLike}),
and along with the product $\#$ (and obvious units and co-units),
$\calA^w(H;T)$ is a graded connected co-commutative bi-algebra. Hence
it makes sense to speak of the group-like elements $\glosm{AwHTexp}{\calA^w_{\exp}(H;T)}$
within $\calA^w(H;T)$, and they are all $\#$-exponentials of primitives in
$\calA^w(H;T)$. The primitives $\glosm{AprimwHT}{\calA_{\text{prim}}^w(H;T)}$ in
$\calA^w(H;T)$ are connected diagrams and hence they are trees and
wheels. As in Comment~\ref{com:PureForm}, the trees must have their
roots on head strands and their leafs on tail strands, and the
wheels must have all their ``legs'' on tail strands. As tails commute, we
may think of the trees as abstract trees with leafs labelled by labels in
$T$ and roots in $H$, and the wheels are abstract cyclic words with letters
in $T$. Hence canonically $\calA_{\text{prim}}^w(H;T)\simeq\FL(T)^H\oplus\CW(T)$ and
hence there is a bijection (called ``the split presentation $E_s$'')\glosm{Es}{$\null$}
\begin{equation} \label{eq:EsHT}
\yellowm{E_s}\colon\yellowm{\TW_s(H;T)}\coloneqq\FL(T)^H\oplus\CW(T)
\overset{\sim}{\longrightarrow} \calA^w_{\exp}(H;T)
\end{equation}
defined on an ordered pair $\glosm{parens}{(\lambda;\,\omega)_s}$ in
$\TW_s(H;T)$ by
\begin{equation} \label{eq:esHT}
(\lambda;\,\omega)_s\mapsto
\exp_\#\left(e_s(\lambda;\omega)\right),
\end{equation}
\Needspace{2in}
\parpic[r]{\parbox{1.75in}{
\centering{\input{figs/Es.pstex_t}}
\begin{figcap} \label{fig:Es} $E_s(\lambda;\,\omega)_s$. \end{figcap}
}}
\noindent where $\glosm{es}{e_s}(\lambda;\omega)_s$ is the sum over $x\in
H$ of planting $\lambda_x$ with its root on strand $x$ and its leafs on
the strands in $T$ so that the labels match but at an arbitrary order
on any $T$ strand, plus the result of planting $\omega$ on just the $T$
strands so that the labels match but at an arbitrary order on any $T$
strand. A pictorial representation of $E_s(\lambda;\,\omega)_s$, using
the same visual language as in Figure~\ref{fig:El}, appears on the right.
It is easy to verify that the
operations in Definition~\ref{def:AHTOperations} intertwine $\Box$ and
hence map group-like elements to group-like elements and hence they induce
operations on $\TW_s(H;T)$. These are summarized within the following
definition-proposition.
\begin{defprop} \label{dp:EsOps} The bijection $E_s$ intertwines the
operations defined below with the operations in
Definition~\ref{def:AHTOperations}:\footnoteT{Here
we no longer state conditions such as $H_1\cap H_2=\emptyset$,
$u\in T$, $x\in H$. They are the same as in
Definition~\ref{def:AHTOperations}, and more importantly, they are ``what
makes sense''.}\,\footnoteC{We quote from
\href{\web/AwCalculus.m}{\tt AwCalculus.m} only the most interesting
implementations --- of $\sqcup$~\eqref{eq:EsCup}, of $hm$~\eqref{eq:Eshm},
of $tm$~\eqref{eq:Estm}, and of $tha$~\eqref{eq:Estha}. Then we set
the values of two ``sample'' elements in the $E_s$ presentation
(on the computer we represent $(\lambda;\,\omega)_s$ as {\tt
Es[$\lambda$,$\omega$]}):
\ACQuote{EsSampleDefs}
\shortdialoginclude{EsSetup1}
\shortdialoginclude{EsSetup2}
(Note that the second of sample elements was set to be a random series,
with a seed of $0$. It is printed only to degree $2$, but it extends
indefinitely as a random series.)
}
\begin{enumerate}[leftmargin=*,labelindent=0pt]
\item
$\ds
(\lambda_1;\,\omega_1)_s(\lambda_2;\,\omega_2)_s
= (\lambda_1;\,\omega_1)_s\glosm{ssqcup}{\sqcup}(\lambda_2;\,\omega_2)_s
\coloneqq (\lambda_1\sqcup\lambda_2;\,\omega_1+\omega_2)_s
$\hfill\inlineeq\label{eq:EsCup}
\item
$\ds
(\lambda_1;\,\omega_1)_s\glosm{sjail}{\#}(\lambda_2;\,\omega_2)_s
\coloneqq \left(
(x\to\BCH(\lambda_{1x},\lambda_{2x}))_{x\in H}
;\,
\omega_1+\omega_2
\right)_s
$\hfill\inlineeq\label{eq:EsProduct}
\item
$\ds
(\lambda;\,\omega)_s \act \glosm{sheta}{h\eta}^x
\coloneqq (\lambda\remove x;\,\omega)_s
$\hfill\inlineeq\label{eq:EshEta}
\newline\hspace{16pt}
$\ds (\lambda;\,\omega)_s \act \glosm{steta}{t\eta}^u
\coloneqq (\lambda\act(u\to 0) ;\, \omega\act(u\to 0))_s
$\hfill\inlineeq\label{eq:EstEta}
\item
$\ds
(\lambda;\,\omega)_s \act \glosm{shA}{hA}^x
\coloneqq ((\lambda\remove x)\sqcup(x\to-\lambda_x) ;\, \omega)_s
$\hfill\inlineeq\label{eq:EshA}
\newline\hspace{16pt}
$\ds \glosm{stA}{tA}^u \coloneqq I $\hfill\inlineeq\label{eq:EstA}
\item
$\ds \glosm{shS}{hS}^x\coloneqq hA^x,$\hfill\inlineeq\label{eq:EshS}
\newline\hspace{16pt}
$\ds
(\lambda;\,\omega)_s \act \glosm{stS}{tS}^u
\coloneqq (\lambda\act(u\to -u) ;\, \omega\act(u\to -u))_s
$\hfill\inlineeq\label{eq:EstS}
\item
$\ds
(\lambda;\,\omega)_s \act \glosm{shm}{hm}^{xy}_z
\coloneqq ((\lambda\remove\{x,y\})\sqcup(z\to\BCH(\lambda_x, \lambda_y));\,\omega)_s
$\hfill\inlineeq\label{eq:Eshm}
\newline\hspace{16pt}$\ds
(\lambda;\,\omega)_s \act \glosm{stm}{tm}^{uv}_w
\coloneqq (\lambda\act(u,v\to w) ;\, \omega\act(u,v\to w))_s
$\hfill\inlineeq\label{eq:Estm}
\item
$\ds
(\lambda;\,\omega)_s \act \glosm{shDelta}{h\Delta}^x_{yz}
\coloneqq ((\lambda\remove x)\sqcup(y\to\lambda_x, z\to\lambda_x);\,\omega)_s
$\hfill\inlineeq\label{eq:EshDelta}
\newline\hspace{16pt}$\ds
(\lambda;\,\omega)_s \act \glosm{stDelta}{t\Delta}^u_{vw}
\coloneqq (\lambda\act(u\to v+w) ;\, \omega\act(u\to v+w))_s
$\hfill\inlineeq\label{eq:EstDelta}
\item
$\ds
(\lambda;\,\omega)_s \act \glosm{shsigma}{h\sigma}^x_y
\coloneqq ((\lambda\remove x)\sqcup(y\to\lambda_x);\,\omega)_s
$\hfill\inlineeq\label{eq:EshSigma}
\newline\hspace{16pt}$\ds
(\lambda;\,\omega)_s \act \glosm{stsigma}{t\sigma}^u_v
\coloneqq (\lambda\act(u\to v) ;\, \omega\act(u\to v))_s
$\hfill\inlineeq\label{eq:EstSigma}
\item $\ds
(\lambda;\,\omega)_s \act \glosm{stha}{tha}^{ux}
\coloneqq \left(
\lambda \act RC_u^{\lambda_x} ;\,
(\omega+J_u(\lambda_x))\act RC_u^{\lambda_x}
\right)_s.
$\hfill\inlineeq\label{eq:Estha}
\end{enumerate}
\end{defprop}
\noindent{\em Proof.} The first 8
assertions (14 operations) are very easy. The main challenge to the reader
should be to gather her concentration for the 14-times repetitive task
of unwrapping definitions. If you are ready to cut corners, only go over
\eqref{eq:EsCup}, \eqref{eq:Eshm}, \eqref{eq:Estm}, \eqref{eq:EshDelta},
and \eqref{eq:EstDelta}. Let us turn to the proof of the last assertion,
Equation~\eqref{eq:Estha}. That proof is in fact in~\cite{KBH}, or
at least can be assembled from pieces already in~\cite{KBH}. Yet the
assembly would be a bit delicate, and hence a proof is reproduced below
which refers back to~\cite{KBH} only at one technical point.
By inspecting the definition of $tha^{ux}$, it is clear that there
is {\em some} assignment $\gamma\mapsto R_u^\gamma$ that assigns an
operator $R_u^\gamma\colon\FL(T)\to\FL(T)$ to every $\gamma\in\FL(T)$
and that there is {\em some} functional $K_u\colon\FL(T)\to\CW(T)$,
for which a version of Equation~\eqref{eq:Estha} holds:
\begin{equation} \label{eq:Esthap}
E_s(\lambda;\,\omega)_s \act tha^{ux}
= E_s\left(
\lambda \act R_u^{\lambda_x} ;\,
(\omega+K_u(\lambda_x))\act R_u^{\lambda_x}
\right)_s
\end{equation}
Indeed, $tha^{ux}$ acts on $E_s(\lambda;\,\omega)_s$ by placing a copy
of $\exp(\lambda_x)$ at the top of the tail strand $u$, and then
re-writing the result without having any heads on strand $u$ so as to
invert $E_s$ back again. The re-writing is done by sliding the heads of
$\exp(\lambda_x)$ down to the bottom of strand $u$, where they cancel
by $CP$. Every time a head slides past a tail we get a contribution from
$\aSTU_2$. Sometimes a head of a $\lambda_x$ will slide against a tail of
another $\lambda_x$, whose head will have to slide down too, leading to
a rather complicated iterative process. Nevertheless, these contributions
are the same for every tail on strand $u$, namely for every occurrence of
the variable $u$ in $\FL(T)^H$ and/or in $\CW(T)$. This explains the terms
$\lambda \act R_u^{\lambda_x}$ and $\omega \act R_u^{\lambda_x}$ in
Equation~\eqref{eq:Esthap}. We note that the degree $0$ part of the
operator $R_u^{\lambda_x}$ is the identity, and hence it is invertible.
But yet another type of term arises in the process --- sometimes a head of
some tree will slide against a tail of its own, and then the contribution
arising from $\aSTU_2$ will be a wheel. Hence there is an additional
contribution to the output, some $L_u(\lambda_x)$ which clearly can depend
only on $u$ and $\lambda_x$. Using the invertibility of $R_u^{\lambda_x}$
to write $L_u(\lambda_x)=K_u(\lambda_x)\act R_u^{\lambda_x}$ we completely
reproduce Equation~\eqref{eq:Esthap}.
We now need to show that $R_u^\gamma$ and $K_u(\gamma)$ are $RC_u^\gamma$
and $J_u(\gamma)$ of Definitions~\ref{def:CRC} and~\ref{def:J}. Tracing
again through the discussion in the previous two paragraphs, we see that at
any fixed degree, $R_u^\gamma$ and $K_u(\gamma)$ depend polynomially on the
coefficients of $\gamma$, and hence it is legitimate to study their
variation with
respect to $\gamma$. It is also easy to verify that $R_u^0=RC_u^0=I$ and
that $K_u(0)=J_u(0)=0$, and hence it is enough to show that, with an
indeterminate scalar $\tau$,
\begin{equation} \label{eq:DerEqns}
\frac{d}{d\tau}R_u^{\tau\gamma}=\frac{d}{d\tau}RC_u^{\tau\gamma}
\qquad\text{and}\qquad
\frac{d}{d\tau}K_u(\tau\gamma)=\frac{d}{d\tau}J_u(\tau\gamma).
\end{equation}
Let us compute the left-hand-sides of the above equations. If $\tau$ is an
infinitesimal (so $\tau^2=0$), or more precisely, computing the above
left-hand-sides at $\tau=0$, we can re-trace the process described in the
two paragraphs following Equation~\eqref{eq:Esthap} keeping in mind that
with $\lambda_x=\tau\gamma$ the $\aSTU_2$ relation can only by applied
once (or else terms proportional to $\tau^2$ will arise). The result is
\begin{equation} \label{eq:DersAtZero}
\left.\frac{d}{d\tau}R_u^{\tau\gamma}\right|_{\tau=0}
= \ad_u^\gamma
\qquad\text{and}\qquad
\left.\frac{d}{d\tau}K_u(\tau\gamma)\right|_{\tau=0}
= \atdiv_u(\gamma),
\end{equation}
where $\glosm{adugamma}{\ad_u^\gamma}\colon\FL(T)\to\FL(T)$
is the derivation which maps the generator $u$ of $\FL(T)$ to
$[\gamma,u]$ and annihilates all other generators of $\FL(T)$
(compare~\cite[Definition~\ref{KBH-def:adu}]{KBH}) and where
$\atdiv_u(\gamma)$ is the same as in Definition~\ref{def:J}.
Moving on to general $\tau$, we
note that the operations $hm$ and $tha$ satisfy\footnoteC{None
should believe without a verification:
\shortdialoginclude{haction}
}
\begin{equation} \label{eq:haction}
hm^{xy}_z\act tha^{uz}=tha^{ux}\act tha^{uy}\act hm^{xy}_z
\end{equation}
(stitching strands $x$ and $y$ and then stitching a copy
of the result to $u$ is the same as stitching a copy of $x$
to $u$, then a copy of $y$, and then stitching $x$ to $y$;
compare~\cite[Equation~\eqref{KBH-eq:haction}]{KBH}). Applying the
operators on the two sides of Equation~\eqref{eq:haction} to
$E_s(\lambda;\,\omega)$ (assuming $H$ and $T$ are such that it makes
sense), then expanding using~\eqref{eq:Eshm} and~\eqref{eq:Esthap}, and
then ignoring the wheels in the resulting
equality, we find that $R_u$ satisfies
\begin{equation} \label{eq:Rh}
R_u^{\BCH(\lambda_x,\lambda_y)}
= R_u^{\lambda_x}\act R_u^{\lambda_y\act R_u^{\lambda_x}}
\end{equation}
(compare~\cite[Equation~\eqref{KBH-eq:RCh}]{KBH}). Similarly, looking only
at the wheel part of~\eqref{eq:haction} we get
\[
K_u(\BCH(\lambda_x,\lambda_y))\act R_u^{\BCH(\lambda_x,\lambda_y)}
= K_u(\lambda_x)\act
R_u^{\lambda_x}\act R_u^{\lambda_y\act R_u^{\lambda_x}}
+ K_u(\lambda_y\act R_u^{\lambda_x})\act
R_u^{\lambda_y\act R_u^{\lambda_x}},
\]
which, composing on the right with $R_u^{\BCH(\lambda_x,\lambda_y)}$
and using~\eqref{eq:Rh}, is equivalent to
\begin{equation} \label{eq:Kh}
K_u(\BCH(\lambda_x,\lambda_y))
= K_u(\lambda_x)\act R_u^{\lambda_x}
+ K_u(\lambda_y\act R_u^{\lambda_x})\act C_u^{-\lambda_x}
\end{equation}
(compare~\cite[Equation~\eqref{KBH-eq:JhProperty}]{KBH}).
Equations~\eqref{eq:Rh} and~\eqref{eq:Kh} hold for any $\lambda$,
and hence for any $\lambda_x$ and $\lambda_y$. Specializing
to $\lambda_x=\tau\gamma$ and $\lambda_y=\epsilon\gamma$, where
$\epsilon$ is some new indeterminate scalar, and using the fact that
$\BCH(\tau\gamma,\epsilon\gamma) = (\tau+\epsilon)\gamma$,
Equations~\eqref{eq:Rh} and~\eqref{eq:Kh} become
\[
R_u^{(\tau+\epsilon)\gamma}
= R_u^{\tau\gamma}\act R_u^{\epsilon\gamma\act R_u^{\tau\gamma}}
\qquad\text{and}\qquad
K_u((\tau+\epsilon)\gamma)
= K_u(\tau\gamma)\act R_u^{\tau\gamma}
+ K_u(\epsilon\gamma\act R_u^{\tau\gamma})\act C_u^{-\tau\gamma}.
\]
Now differentiating with respect to $\epsilon$ at $\epsilon=0$ and using
Equation~\eqref{eq:DersAtZero} with $\tau$ replaced with $\epsilon$, we get
\[
\frac{d}{d\tau}R_u^{\tau\gamma}
= R_u^{\tau\gamma}\act\ad_u^{\gamma\act R_u^{\tau\gamma}}
\qquad\text{and}\qquad
\frac{d}{d\tau}K_u(\tau\gamma)
= \atdiv_u(\gamma\act R_u^{\tau\gamma})\act C_u^{-\tau\gamma}.
\]
The first of these equations is the same equation that is satisfied by
$RC_u$ (see \cite[Lemma \ref{KBH-lem:dC}]{KBH}, with $\delta\gamma$
proportional to $\gamma$), and hence $R_u=RC_u$. By a simple change
of variables, $J_u(\tau\gamma)=\int_0^\tau dt\,\atdiv_u\!\left(
\gamma \sslash RC_u^{t\gamma} \right) \sslash C_u^{-t\gamma}$,
and hence $\frac{d}{d\tau}J_u(\tau\gamma) = \atdiv_u(\gamma\act
RC_u^{\tau\gamma})\act C_u^{-\tau\gamma}$ (compare with the
formula for the full differential of $J$, \cite[Proposition
\ref{KBH-prop:dJ}]{KBH}). Comparing with the above formula for the
derivative of $K_u$, we find that $K_u=J_u$. \qed
\subsubsection{The inclusion $\{\calA^w(S)\} \hookrightarrow
\{\calA^w(H;T)\}$.} \label{sssec:Inclusion}
The following definition and proposition imply that there is no
loss in studying the spaces $\calA^w(H;T)$ rather than the spaces
$\calA^w(S)$.
\begin{definition} Let
$\glosm{delta}{\delta}\colon\calA^w(S)\to\calA^w(S;S)$
be the composition of the ``double every strand'' map
$\prod_{a\in S}\Delta^a_{ha,ta} \colon \calA^w(S)
\to \calA^w({hS\sqcup tS})$ with the projection
$\calA^w({hS\sqcup tS})\to\calA^w(S;S)$ (as an exception to
the rule of Footnote~\ref{foot:BruteDisjoint} we temporarily highlight
the distinction between head and tail labels by affixing them with the
prefixes $h$ and $t$).
\end{definition}
\Needspace{1.5in}
{
\makeatletter\def\thm@space@setup{%
\thm@preskip=0cm plus 0cm minus 0cm %\thm@postskip=\thm@preskip
}\makeatother
\parpic[r]{\input{figs/delta.pstex_t}}
\begin{comment} \label{com:delta} If $D\in\calA^w(S)$ is sorted
``heads below tails'' as in Comment~\ref{com:SortedForm}, then $\delta
D$ is $D$ with its arrow heads placed on the head strands and its arrow
tails placed on the tail strands, as shown on the right.
\end{comment}
}
\begin{proposition} $\delta$ is a (non-multiplicative) vector space
isomorphism. The inverse of $\delta$ on $D\in\calA^w(S;S)$ is given by
the process
\begin{enumerate}
\item Write $D$ with only arrow heads on the head strands and only arrow
tails on the tail strands. By Comment~\ref{com:PureForm} this produces a
well-defined element $D'$ of $\calA^w({hS\sqcup tS})$.
\item Stitch all the head-tail pairs of strands in
$D'$ by putting each head ahead of its corresponding tail: $\delta^{-1}D =
D'\act\prod_a dm^{ha,ta}_a$.
\end{enumerate}
\end{proposition}
\begin{proof} $\delta^{-1}\act\delta=I$ by inspection, and
$\delta\act\delta^{-1}$ is clearly the identity on diagrams sorted to have
heads ahead of tails as in Comment~\ref{com:SortedForm}. \qed
\end{proof}
\TopLie In topology, $\delta$ agrees with the
$\delta$ of~\cite[Section~\ref{KBH-subsec:delta}]{KBH}. In Lie theory,
it agrees with the linear (non-multiplicative) isomorphism
$\calU(I\frakg)\simeq\calU(\frakg)\otimes\calS(\frakg^\ast)$ and with
similar isomorphisms considered by Etingof and Kazhdan within their work
on the quantization of Lie bialgebras~\cite{EtingofKazhdan:BialgebrasI}
(albeit only when the Lie bialgebras in question are cocommutative).
\begin{definition} The product $\#$ of $\calA^w(S;S)$ induces a new
product, also denoted $\glosm{wjail}{\#}$, on $\calA^w(S)$. If
$D_1$ and $D_2$ are in $\calA^w(S)$, set
\begin{equation} \label{eq:FProduct}
D_1\#D_2\coloneqq(\delta(D_1)\#\delta(D_2))\act\delta^{-1}.
\end{equation}
\end{definition}
\Needspace{1in}
{
\makeatletter\def\thm@space@setup{%
\thm@preskip=0cm plus 0cm minus 0cm %\thm@postskip=\thm@preskip
}\makeatother
\parpic[r]{\input{figs/jail.pstex_t}}
\begin{comment}
With Comment~\ref{com:delta} in mind, we see that if $D_1$ and $D_2$ are
sorted as in Comment~\ref{com:SortedForm}, then $D_1\#D_2$ is ``heads of
$D_1$, then of $D_2$, then tails of $D_1$, then of $D_2$''
(with the last two parts interchangeable, by $TC$). The picture is nicer
when rotated, as on the right.
\end{comment}
}
\TopLieSmall See the comments following Discussion~\ref{disc:WhyTwo}.\newline
The next proposition shows how the operations of defined on the
$\calA^w(S)$-spaces in Definition~\ref{def:Operations}
can be written in terms of the ``head and tail'' operations of
Definition~\ref{def:AHTOperations}, thus completing the description of
the $E_s$ presentation.
\begin{proposition} \label{prop:dinht}
\begin{enumerate}[leftmargin=*,labelindent=0pt]
\item \label{it:HTsqcup} If $S_1$ and $S_2$ are disjoint and
$D_1\in\calA^w({S_1})$ and $D_2\in\calA^w({S_2})$,
then $\delta(D_1\sqcup D_2)=\delta(D_1)\sqcup\delta(D_2)$.
\item \label{it:HTStacking}
Let $D_1,D_2\in\calA^w(S)$. Then $\delta(D_1D_2)$ can be
written in terms of $\delta(D_1)$ and $\delta(D_2)$ using its description
in terms of $\sqcup$, $d\sigma$, and $dm$ in Equation~\eqref{eq:multiplem} and
using the formulas for $\sqcup$, $d\sigma$, and $dm$ that appear in
parts (\ref{it:HTsqcup}), (\ref{it:HTsigma}), and (\ref{it:HTdm}) of this
proposition.\footnotemarkC
\addtocounter{footnoteC}{-1}
\begin{multicols}{2}
\item $d\eta^a\act\delta = \delta\act h\eta^a\act t\eta^a$.
\item \label{it:HTdA}
$\dA^a \act \delta = \delta \act hA^a \act tA^a\act tha^{aa}$.
\item \label{it:HTdS}
$\dS^a \act \delta = \delta \act hS^a \act tS^a\act tha^{aa}$.
\item \label{it:HTdm}
$dm^{ab}_c \act \delta = \delta \act tha^{ab} \act hm^{ab}_c \act
tm^{ab}_c$.\footnotemarkC
\item $d\Delta^a_{bc} \act \delta =
\delta \act h\Delta^a_{bc} \act t\Delta^a_{bc}$.
\item \label{it:HTsigma} $d\sigma^a_b \act \delta = \delta \act
h\sigma^a_b \act t\sigma^a_b$.
\end{multicols}
\end{enumerate}
\end{proposition}
\footnotetextC{As a sample for the whole proposition, we quote
the implementation of $dm$ and verify its meta-associativity
$dm^{ab}_a \act dm^{ac}_a = dm^{bc}_b \act dm^{ab}_a$
(compare~\cite[Equation~\eqref{KBH-eq:assoc}]{KBH}). We then include our
implementation of the stacking product (item {\it(\ref{it:HTStacking})}
above) without further explanations:
\ACQuote{Esdm}
\shortdialoginclude{metaassoc}
\ACQuote{EsNCM}
}
\begin{proof} The only difficulty is with items
{\it(\ref{it:HTdA})}--{\it(\ref{it:HTdm})}. Item~{\it(\ref{it:HTdA})}
is easier to understand in the form $\delta^{-1}\act \dA^a= hA^a \act
tA^a\act tha^{aa}\act\delta^{-1}$. Indeed, $\delta^{-1}$ plants heads
ahead of tails on strand $a$. Applying $\dA^a$ reverses that strand
(and adds some signs). This reversal can be achieved by reversing the
head part (with signs), then the tail part (with signs), and then
by swapping the two parts across each other. The first reversal
is $hA^a$, the second is $tA^a$, and the swap is $tha^{aa}$
followed by $\delta^{-1}$. Item~{\it(\ref{it:HTdS})} is proven
in exactly the same way, and item~{\it(\ref{it:HTdm})} is proven
in a similar way, where the right hand side traces the schematics
$(ha\,ta\,hb\,tb)\xrightarrow{tha}(ha\,hb\,ta\,tb)\xrightarrow{hm\act
tm}((ha\,hb)(ta\,tb))$. \qed
\end{proof}
\begin{discussion} \label{disc:coalg} It is easy to verify
that $\delta\colon \calA^w(S) \to \calA^w(S;\,S)$
is a co-algebra morphism,
and hence it restricts to an isomorphism $\delta\colon
{\calA^w_{\exp}(S)} \to \calA^w_{\exp}(S;\,S)$. Therefore
$E_s\act\delta^{-1}$ is a bijection between $\TW_s(S)\coloneqq\TW_s(S;S)$ and
$\calA^w_{\exp}(S)$. Proposition~\ref{prop:dinht} now tells us
how to write all the ``$d$'' operations of Definition~\ref{def:Operations}
as compositions of ``$h$'' and ``$t$'' operations, and
Definition-Proposition~\ref{dp:EsOps} tells us how to write these as
operations on $\TW_s(H;T)$ (the $H$ and $T$ label sets that occur here
are always $S$ with one or two labels
added or removed). Hence overall $E_s\act\delta^{-1}$, acting on $\TW_s(S)$,
is a complete presentation of $\calA^w_{\exp}(S)$.
\end{discussion}
\Needspace{5cm}
\vskip 1mm
\makeatletter\def\thm@space@setup{%
\thm@preskip=0cm plus 0cm minus 0cm %\thm@postskip=\thm@preskip
}\makeatother
\parpic[r]{\parbox{1.75in}{
\centering{\input{figs/Ef.pstex_t}}
\begin{figcap} \label{fig:Ef} $E_f(\lambda;\,\omega)_s$. \end{figcap}
}}
\begin{definition} The ``factored'' presentation $\glosm{Ef}{E_f}$
of $\calA^w_{\exp}$ is the composition ${E_f\coloneqq
E_s\act\delta^{-1}}$. Namely, for a set $S$ of strands, we define
$
E_f\colon\TW_s(S)
\overset{\sim}{\longrightarrow} \calA^w_{\exp}(S)
$
by
$(\lambda;\,\omega)_s\mapsto
E_s(\lambda;\,\omega)_s\act\delta^{-1}
= \exp_\#\left(l\lambda + \iota\omega\right)
$. See the illustration on the right.
\end{definition}
\draftcut
\subsection{Converting between the $E_l$ and the $E_f$ presentations.}
\label{subsec:Conversion}
We now have two presentations for elements of $\calA^w_{\exp}(S)$,
and we wish to be able to convert between the two. This turns out to
involve the maps $\Gamma$ and $\Lambda$ of Propositions~\ref{prop:Gamma}
and~\ref{prop:Lambda}.
\begin{definition} Define a pair of inverse maps
$\glosm{TWGamma}{\Gamma}\colon\TW_l(S)\to\TW_s(S)$ and
$\glosm{TWLambda}{\Lambda}\colon\TW_s(S)\to\TW_l(S)$ by
\[ \Gamma\colon(\lambda;\,\omega)_l \mapsto (\Gamma(\lambda);\,\omega)_s
\quad\text{and}\quad
\Lambda\colon(\lambda;\,\omega)_s \mapsto (\Lambda(\lambda);\,\omega)_l.
\]
\end{definition}
\begin{theorem} \label{thm:GammaLambda}
The left-most triangle in Figure~\ref{fig:diagram} commutes. Namely,
\begin{equation} \label{eq:convertion}
E_l = \Gamma \act E_f
\quad\text{and}\quad
E_f = \Lambda \act E_l.
\end{equation}
(All other parts of Figure~\ref{fig:diagram} commute by definition).
\end{theorem}
Before we can prove this theorem we need a few preliminaries. For an
element $D\in{\calA^w_{\exp}(S)}$, we can define three associated
quantities:
\begin{myitemize}
\item The projection of $D$ to the degree 1 part of $\calA^w(S)$, and
especially, the projection $\glosm{piA}{\pi_A}(D)$ of the degree $1$
part to its ``framing'' part $A_S$ (consisting of self-arrows,
that begin and end on the same strand and point, say, up).
\item A conjugation automorphism $C_D$ of $\FL(S)$, defined as follows.
First, embed $\FL(S)$ into $\calA^w({S\sqcup\{\infty\}})$
by mapping any generator $a\in S$ to a degree 1 diagram in
${\calA^w({S\sqcup\{\infty\}})}$, the arrow whose tail is on
strand $a$ and whose head is on the new ``$\infty$'' strand and extending
in a bracket-preserving way, using the commutator of the stacking product
as the bracket on $\calA^w({S\sqcup\{\infty\}})$. Then note that
$\FL(S)\subset\calA^w({S\sqcup\{\infty\}})$ is invariant under
conjugation by $D$ and let $C_D$ denote this conjugation action.
\vskip 1mm
\Topology This is a direct analog of the Artin action of the pure braid
groups $\PuB_n$ /
$\PwB_n$ on the free group $\FG(n)$.
\item $\glosm{pidowncap}{\pi_{\!\downcap}}(D)$ is the result of
adding a bullet at the bottom of every strand of $D$, in the
same sense as in Section~\ref{sssec:Family}. Equivalently,
$\pi_{\!\downcap}=\delta\act\prod_{a\in S}h\eta^a$ is the composition
of $\delta$ with ``delete all head strands''. The target space of
$\pi_{\!\downcap}$ is $\calA^w(\emptyset;S)$, which is the symmetric algebra
$\calS(\CW(S))$ generated by wheels.
\end{myitemize}
\begin{proposition} $D$ is determined by the above three quantities
$\pi_A(D)$, $C_D$, and $\pi_{\!\downcap}(D)$.
\end{proposition}
\begin{proof} As in Section~\ref{subsec:AT}, every
$D\in\calA^w_{\exp}(S)$ can be written uniquely in the
form $D=e^{l\lambda} e^{\iota\omega}$, where $\lambda\in\FL(S)^S$
and $\omega\in\CW(S)$. One may easily verify that $\pi_{\!\downcap}(D)$ is
$\omega$, that $C_D$ is the exponential of the derivation in $\tder_S$
corresponding to $\lambda$, and that $\pi_A(D)$ determines the part of
$\lambda$ lost by the projection $\FL(S)^S\to\tder_S$.\qed
\end{proof}
\noindent{\em Proof of Theorem~\ref{thm:GammaLambda}.} For $\lambda\in\FL(S)^S$ let $\lambda'=\Gamma(\lambda)$.
Comparing Figures~\ref{fig:El} and~\ref{fig:Ef}, we find that the $\omega$
parts drop out and we need to prove, schematically, that in
$\calA^w_{\exp}(S)$,
\[ \input{figs/ElEf.pstex_t} \]
A simple degree 1 calculation shows that $\pi_A(A)=\pi_A(B)=0$.
The CP relation of Section~\ref{sssec:Family} shows that
$\pi_{\!\downcap}(A)=\pi_{\!\downcap}(B)=0$. Finally, it is easy to verify that
$C_A=e^{-\partial_\lambda}$ while $C_B=C^{\lambda'}$, and hence $C_A=C_B$
follows from Proposition~\ref{prop:Gamma}.\qed
\input comp.tex
\draftcut\eject\input glossary.tex
\draftcut\input refs.tex
\if\draft y
\newpage
Everything below is to be blanked out before the completion of this paper.
\input recycling.tex
\input ToDo.tex
\fi
\end{document}
\endinput