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Ribbon Hopf algebras are immensely useful in low dimensional topology, as they lead to knot and
tangle invariants. See e.g.~\cite[Section~4.2]{Ohtsuki:QuantumInvariants} and our quick summary
in Aside~\ref{aside:URCMethod}.
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There is a standard ``quantum algebra'' methodology that associates
a framed knot invariant to certain triples $(U,R,C)$, where $U$ is
a unital algebra and $R\in U\otimes U$ and $C\in U$ are invertible
(see e.g.~\cite[Section~4.2]{Ohtsuki:QuantumInvariants}).
For convenience, we recall this methodology in Aside~\ref{aside:URCMethod}.
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We note that whenever $S$ is some finite set ``of strands'', the
methodology of Aside~\ref{aside:URCMethod} easily generalizes to the case
of ``$S$-component tangles'' --- knotted objects consisting of knotted
intervals labelled bijectively by the elements of $S$ (the precise
definition matters less than an example, so an example appears on the
right {\red MORE} while the precise definition is postponed to Section~\ref{sec:RVK})
--- except that in the $S$-component case $z$ takes values in $U^{\otimes
S}$, the $S$-fold tensor power of $U$, instead of merely in $U=U^{\otimes
1}$. Indeed, instead of multiplying all the $C$'s and $a_i$'s and $b_i$'s
of Aside~\ref{aside:URCMethod} as they appear along one interval, one
simply multiplies them as they appear along $S$ intervals, storing the
output in $U^{\otimes S}$ in the natural manner.
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\parpic[r]{\fbox{\begin{minipage}{0.5\linewidth}\sl
If $U$ is a vector space over $\bbQ$ (or another field) we set $U_n\coloneqq U^{\otimes n}$, so
$U_0=\bbQ$, $U_1=U$, $U_2=U\otimes U$, etc. We identify $U_1\cong U_1\otimes U_0\cong U_0\otimes
U_1$. With these conventions,
a {\em Hopf Algebra} is a vector space $U$ endowed with maps $m\colon U_2\to U$, $\Delta\colon U_1\to
U_2$, $\eta\colon U_0\to U_1$, $\epsilon\colon U_1\to U_0$, and an invertible
$S\colon U_1\to U_1$ such that:
\begin{itemize}[leftmargin=*,labelindent=0pt,itemsep=0pt,topsep=0pt]
\item $(m\otimes\Id)\act m = (\Id\otimes m)\act m$.
\item % In $\Hom((U_0\otimes U_1=U_1\otimes U_0=U_1)\to U_1)$,
$(\eta\otimes\Id)\act m = (\Id\otimes\eta)\act m = \Id$.
\item $\Delta\act(\Delta\otimes\Id) = \Delta\act(\Id\otimes\Delta$.
\item $\Delta\act(\epsilon\otimes\Id) = \Delta\act(\Id\otimes\epsilon) = \Id$.
\item $m\act\Delta = (\Delta\otimes\delta)\act(\Id\otimes\sigma\otimes\Id)\act(m\otimes m)$, where
$\sigma\colon U_2\to U_2$ is the transposition.
\item $\eta = (\eta\otimes\eta)\act m$.
\item $\epsilon = \Delta\act(\epsilon\otimes\epsilon)$.
\item $\Delta\act(S\otimes\Id)\act m = \epsilon\act\eta$.
\item $\Delta\act(\Id\otimes S)\act m = \epsilon\act\eta$.
\end{itemize}
\captionsetup{type=Aside}
\caption{Ordinary Hopf Algebras.} \label{aside:OrdinaryHopf}
\end{minipage} }}