===== recycled on Wed Jun 27 20:37:50 EDT 2018 by drorbn on Ubuntu-1404 ====== 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}. ===== recycled on Wed Jun 27 20:38:02 EDT 2018 by drorbn on Ubuntu-1404 ====== 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}. ===== recycled on Wed Jun 27 20:38:37 EDT 2018 by drorbn on Ubuntu-1404 ====== 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. ===== recycled on Mon Aug 13 20:30:07 EDT 2018 by drorbn on Ubuntu-1404 ====== \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} }}