\section{$\sleps$, $CU$, and $QU$} \label{sec:U}

For a minimalistic reading of this paper it is enough to know the
definitions and some basic propeties of the Lie algebra $\sleps$ and its
associated associative algebras $CU$, and $QU$. Hence we start this section by
declaring these algebras by fiat and listing some of their properties,
postponing some of their proofs to Section~\ref{ssec:UProofs}. In
Section~\ref{ssec:UMotivation} we explain the motivation behind $\sleps$
and find that it extends to arbitrary semi-simple Lie algebras.

In anticipation of Section~\ref{sec:everything}, in which we show that
everything that matters around $\sleps$ is $\dpg$, we emphasize the first
occurrence of every object in this section that is later shown to be $\dpg$
with a lollipop symbol $\lollipop$. Within the context of the current section
the lollipops are purely motivational.

\subsection{Definitions and Basic Properties.}

Our ground ring throughout this section is $\bbQ[\eps]$, the ring of polynomials with rational coefficients
over a formal parameter $\eps$. Quantum algebra people should note that $\eps$ is distinct from $\hbar$.

\begin{definition} Let $\sleps$
be the Lie algebra $L\langle y,b,a,x\rangle$
with generators $\{y,b,a,x\}$ and with commutation relations
\begin{equation} \label{eq:slepsrelations}
  [a,x]=x,\quad
  [b,y]=-\epsilon y,\quad
  [a,b]=0,\quad
  [a,y]=-y,\quad
  [b,x]=\epsilon x,\quad
  [x,y]=b+\epsilon a.
\end{equation}
\end{definition}

\begin{remark} It is easy to verify that $t\coloneqq b-\eps a$ is central in $\sleps$, and that if $\eps$ is
invertible\footnote{E.g., if the ring of scalars is extended to $\bbQ(\eps)$ via
$\sleps\mapsto\bbQ(\eps)\otimes_{\bbQ[\eps]}\sleps$.}
then $\sleps$ splits as a direct sum: $\sleps\cong sl_2\oplus\langle t\rangle$, explaining its name.
(Though we will mostly care about the vicinity of $\eps=0$, and at $\eps=0$\footnote{Evaluation at
$\eps=\eps_0\in\bbQ$ makes sense via $\sleps\mapsto\left(\bbQ[\eps]/(\eps-\eps_0)\right)\otimes_{\bbQ[\eps]}\sleps$,
a Lie algebra over $\bbQ$.}
our algebra is not a direct sum).
\end{remark}

\begin{definition} Let $CU\coloneqq \calU(\sleps)$ be the universal
enveloping algebra of $\sleps$. Namely, $CU$ is the associative algebra
$A\langle y,b,a,x\rangle$ generated by the same $\{y,b,a,x\}$, subject to
the same relations as in~\eqref{eq:slepsrelations}. We denote the multiplication map of $CU$ with $\cm\colon
CU\otimes CU\to CU$ (or, with the language of Convention~\ref{conv:id} and Example~\ref{exa:m}, with
$\cm^{ij}_k\colon CU_i\otimes CU_j\to CU_k$)~\lollipop. $CU$ is a Hopf
algebra in the standard way; namely, with its given associative algebra
structure and with unit $\ceta\colon\bbQ\to CU$~\lollipop, counit $\ceps\colon
CU\to\bbQ$\footnotemark~\lollipop, antipode $\cS\colon CU\to CU$~\lollipop, and coproduct
$\cD\colon CU\to CU\otimes CU$~\lollipop\ given as follows:
\begin{equation} \label{eq:CUDef} \begin{aligned}
  \ceta(\lambda) &= \lambda\cdot 1, \\
  \ceps(1,y,b,a,x) &= (1,0,0,0,0), \\
  \cS(y,b,a,x) &= (-y,-b,-a,-x), \\
  \cD(y,b,a,x) &=
    (y\otimes 1+1\otimes y, b\otimes 1+1\otimes b, a\otimes 1+1\otimes a, x\otimes 1+1\otimes x).
\end{aligned} \end{equation}
\end{definition}

\footnotetext{We use \verb"\epsilon" ($\eps$) for a perturbation
parameter and \verb"\varepsilon" ($\varepsilon$) for counits. There's
rarely a reason for confusion.}

With the language of Convention~\ref{conv:id}, Equations~\eqref{eq:CUDef} become:
\begin{equation} \label{eq:CUDefId} \begin{aligned}
  & \ceta_i\colon\bbQ \to CU^{\otimes\{i\}}, & \ceta_i(\lambda) &= \lambda\cdot 1_i, \\
  & \ceps^i\colon CU^{\otimes\{i\}} \to \bbQ, & \ceps^i(1_i,y_i,b_i,a_i,x_i) &= (1,0,0,0,0), \\
  & \cS_i\coloneqq \cS^i_i\colon CU^{\otimes\{i\}} \to CU^{\otimes\{i\}},
    & \cS_i(y_i,b_i,a_i,x_i) &= (-y_i,-b_i,-a_i,-x_i), \\
  & \cD^i_{jk}\colon CU^{\otimes\{i\}} \to CU^{\otimes\{j,k\}}, & \cD^i_{jk}(y_i,b_i,a_i,x_i) &=
    (y_j+y_k, b_j+b_k, a_j+a_k, x_j+x_k).\hspace{-12mm}
\end{aligned} \end{equation}

\begin{definition} \label{def:QU}
Let $QU$, a ``quantization'' of $CU$, be the associative algebra $A\langle
y,b,a,x\rangle\llbracket\hbar\rrbracket$ over the ring $\bbQ\llbracket\hbar\rrbracket$ modulo to the relations
\[
  [a,x]=x,\quad
  [b,y]=-\epsilon y,\quad
  [a,b]=0,\quad
  [a,y]=-y,\quad
  [b,x]=\epsilon x,\quad
  xy-qyx=\frac{1-AB}{\hbar},
\]
where $q\coloneqq\bbe^{\hbar\epsilon}$, $A\coloneqq\bbe^{-\hbar\epsilon a}$, and $B\coloneqq\bbe^{-\hbar b}$.
We denote the multiplication map of $QU$ with $\qm\colon QU\otimes QU\to QU$~\lollipop. We also set
\begin{equation} \label{eq:QUDefId} \begin{aligned}
  \qeta_i(\lambda) &= \lambda\cdot 1_i & \lollipop, \\
  \qeps^i(1_i,y_i,b_i,a_i,x_i) &= (1,0,0,0,0) & \lollipop, \\
  \qS_i(y_i,b_i,a_i,x_i) &= (-B_i^{-1}y_i,-b_i,-a_i,-A_i^{-1}x_i) & \lollipop, \\
  \qD^i_{jk}(y_i,b_i,a_i,x_i) &= (y_j+B_jy_k, b_j+b_k, a_j+a_k, x_j+A_jx_k) & \lollipop.
\end{aligned} \end{equation}
\end{definition}

The following claim can be verified easily by explicit computations:

\begin{claim} With the above operations and relative to the $\hbar$-adic topology, $QU$ is a
complete topological\,\footnotemark\ Hopf algebra over the ring $\bbQ[\eps]\llbracket\hbar\rrbracket$. \qed
\end{claim}

\footnotetext{Most people can safely ignore the ``topological'' language:
it just means that everything can be a power series in $\hbar$, and only
reasonable things are done to such series.}

\begin{definition} \label{def:R}
Let $R$~\lollipop\ be the element of $QU\otimes QU$\footnote{Tensor
products are completed relative to the $\hbar$-adic topology with no
further mention.} given by the following formula:
\[ R=\sum_{m,n\geq 0}\frac{y^nb^m\otimes (\hbar a)^m(\hbar x)^n}{m![n]_q!},
  \quad\text{alternatively}\quad
  R_{ij}=\sum_{m,n\geq 0}\frac{y_i^nb_i^m(\hbar a_j)^m(\hbar x_j)^n}{m![n]_q!}
    \in \bbB_i\otimes\bbA_j,
\]
where $[n]_q!\coloneqq[1]_q[2]_q\cdots[n]_q$ and $[k]_q\coloneqq\frac{q^k-1}{q-1}=1+q+q^2+\ldots+q^{k-1}$
(recall that $q=\bbe^{\hbar\eps}$).
\end{definition}

\begin{proposition}[proof in Section~\ref{ssec:UProofs}] \label{prop:R}
$R$ is an $R$-matrix. Namely, it has the following properties: (This algebra section can be self
contained, yet when we can, we can't resist including
knot-theoretic interpretations, prefixed with ``KT''. Pure algebraists can ignore.)
\[ \begin{aligned}
  R_{13}\act \qD^1_{12} = (R_{14}R_{23})\act \qm^{34}_3 &
    & \qquad & \text{KT:} \raisebox{-4mm}{\input{figs/QT11.pdf_t}} \\
  R_{12}\act \qD^2_{23} = (R_{12}R_{43})\act \qm^{14}_1 &
    && \text{KT:} \raisebox{-4mm}{\input{figs/QT12.pdf_t}} \\
\end{aligned} \]
\[ 
  (\qD^1_{12}R_{34})\act(\qm^{13}_1\qm^{24}_2) = (R_{12}\,\qD^1_{34})\act(\qm^{14}_1\qm^{23}_2)
  \qquad \text{KT:} \quad \raisebox{-8mm}{\input{figs/QT2.pdf_t}}
\]
\[ 
  (R_{12}R_{63}R_{45}\act(\qm^{16}_1\qm^{24}_2\qm^{35}_3) = (R_{23}R_{14}R_{56})\act(\qm^{15}_1\qm^{26}_2\qm^{34}_3)
  \qquad \text{KT:} \quad \raisebox{-7mm}{\input{figs/QT3.pdf_t}}
\]
\end{proposition}

We have finished listing the atomic pieces we need for the purpose of knot theory. Yet these pieces in
themselves are assembled from even lower level pieces --- perhaps ``quarks'' --- and we need to introduce
those as they are necessary for both the proof of Proposition~\ref{prop:R} and for the proofs in
Section~\ref{sec:Everything} that all the lollipopped items above are indeed in \dpg. Here we go:

\begin{definition} \label{def:A}
Let $\fraka$ be the 2-dimensional Lie algebra $L\langle
a,x\rangle/[a,x]=x)$ and let $\bbA \coloneqq
\calU(\fraka)\llbracket\hbar\rrbracket$ be the $\hbar$-adic
completed universal enveloping algebra of the two dimensional Lie
algebra with generators $a$ and $x$ and with the same bracket as in
Definition~\ref{def:QU}. We turn $\bbA$ into a complete topological Hopf
algebra with the obvious definitions for $\am$, $\aeps$, and $\aeta$
(all~\lollipop), and with the
definitions for $\aS$ and $\aD$ (both \lollipop) induced from~\eqref{eq:QUDefId}. Namely,
\begin{equation} \label{eq:ADefId} \begin{aligned}
  \aS_i(a_i,x_i) &= (-a_i,-A_i^{-1}x_i), \\
  \aD^i_{jk}(a_i,x_i) &= (a_j+a_k, x_j+A_jx_k).
\end{aligned} \end{equation}
Let $\bbA'$ be the subalgebra of $\bbA$ generated by $\hbar a$ and by $\hbar x$\footnotemark. It is easy
to check that $\bbA'$ is a sub-Hopf-algebra of $\bbA$.
\end{definition}

\footnotetext{Elements of $\bbA$ are infinite series $\sum w_n\hbar^n$ where $w_n\in\calU(\fraka)$. Elements
of $\bbA'$ are such series in which each $w_n$ is a (non-commutative) polynomial in $a$ and $x$ of degree
at most $n$. So using language similar to the language of Section~\ref{sec:DoPeGDO}, $\bbA'$ is the
``docile'' subspace of $\bbA$.}

\begin{definition} \label{def:B}
Similarly let $\bbB \coloneqq \calU(L\langle y,b\rangle/[b,y]=-\eps
y)\llbracket\hbar\rrbracket$ be the $\hbar$-adic completed universal enveloping
algebra of the two dimensional Lie algebra with generators $y$ and $b$
and with the same bracket as in Definition~\ref{def:QU}. We turn $\bbB$ into a complete
topological Hopf algebra with the obvious definitions for $\bm$, $\beps$, and $\beeta$ (all 
\lollipop), with $\bS$ \lollipop\ taken to be the inverse of $\qS$ (but only on $y$ and $b$) and with
$\bD$ \lollipop\ taken to be the opposite of $\qD$ (but only on $y$ and $b$). Namely,
\begin{equation} \label{eq:BDefId} \begin{aligned}
  \bS_i(y_i,b_i) &= (-y_iB_i^{-1},-b_i), \\
  \bD^i_{jk}(y_i,b_i) &= (B_ky_j+y_k, b_j+b_k).
\end{aligned} \end{equation}
\end{definition}

Clearly, $R\in\bbB\otimes\bbA'$. We claim that it has an inverse, a pairing
$\Pi\in(\bbA')^\ast\otimes\bbB^\ast$ \lollipop:

\begin{proposition} There is a unique pairing $\Pi\in(\bbA')^\ast\otimes\bbB^\ast$ satisfying
\[ R_{ij}\act\Pi^{jk} = \sigma^k_i,\qquad \text{FD:} \qquad \raisebox{-8mm}{\input{figs/RP.pdf_t}} \]
where $\sigma^k_i\colon\bbB_k\to\bbB_i$ \lollipop\ is the identity map
(more preciely, the factor renaming map) and where ``FD'' stands
for ``Flow Diagram(s)'', a rather standard graphical language
for representing compositions of tensors (e.g.~\cite[Lecture
12]{EtingofSchiffman:QuantumGroups}) which nevertheless seems not to have
a standard name.
\end{proposition}

defined on the
generators by
\[
  \Pi\langle\hbar a,b\rangle = \Pi\langle\hbar x,y\rangle = 1,
  \qquad
  \Pi\langle\hbar a,y\rangle = \Pi\langle\hbar x,b\rangle = 0, 
\]

{\red MORE.}

{\red MORE.}

\subsection{Motivation for $\sleps$, $CU$, and $QU$} \label{ssec:UMotivation}

{\red MORE.}

\subsection{Proofs.} \label{ssec:UProofs}

{\red MORE.}
