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%\parbox[b]{4in}{
%  {\tiny Dror Bar-Natan: MAT 1350 Knot Theory: \LARGE\bf Tentative Plan}
%  }
\parbox[b]{1.5in}{\tiny
  \href{http://www.math.toronto.edu/~drorbn}{Dror Bar-Natan}: \href{http://www.math.toronto.edu/~drorbn/classes}{Classes}:
  \newline \href{http://www.math.toronto.edu/~drorbn/classes/\#2122}{2021-22}: \href{http://drorbn.net/21-1350}{MAT 1350 Knot Theory}:
}~{\LARGE\bf IHOP Algebras, $R$-Elements, and $WG$}
\hfill\parbox[b]{1.4in}{\tiny
  \null\hfill\sheeturl
  \newline\null\hfill Source code: \href{http://drorbn.net/21-1350/nb/WG.pdf}{WG}
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\vskip -3mm
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\begin{multicols*}{2}

{\Large\bf IHOP Algebras and $R$-Elements}
\vskip 3mm

{\bf Definition.} An Involutive Hopf Algebra (IHOP, but only here) is a vector space $H$ that has
\begin{itemize}[leftmargin=4mm, labelwidth=4mm]
\item An algebra structure $(m\colon H\otimes H\to H, \eta\colon\bbQ\to H)$ satisfying the usual axioms of an algebra (set $1\coloneqq\eta(1)$).
\item A co-algebra structure $(\Delta\colon H\to H\otimes H, \varepsilon\colon H\to\bbQ)$ satisfying the ``dual'' axioms and compatible with the algebra structure in the sense that $\Delta$ and $\varepsilon$ are morphisms of algebras.
\item An ``antipode'' $S\colon H\to H$ which is an anti-homomorphism of both the algebra structure and the co-algebra structure, which is ``involutive'', $S^2=I$, and which is a ``convolution inverse'' of the identity map:
    \[ \Delta\act(I\otimes S)\act m = \varepsilon\act\eta = \Delta\act(S\otimes I)\act m. \]
\end{itemize}

Remember the interpretations!

$m$: Stitch strands. \hfill $\eta$: Insert an empty strand.
\newline $\Delta$: Double a strand. \hfill $\varepsilon$: Delete a strand.
\newline $S$: Reverse a strand. \hfill $R$ (below): A crossing.

\vskip 3mm
{\bf Definition.} An ``$R$-element'' for $H$ (related to ``a quasi-triangular structure'') is an invertible $R\in H\otimes H$ such that $\bar{R}\coloneqq R^{-1}$ inverts $R$ also in $H\otimes H^{op}$ and such that
\[
  (\Delta\otimes I)(R) = R_{23}R_{13}
  \quad\text{and}\quad (I\otimes\Delta)(R) = R_{12}R_{13},
\]
\[ (\eta\otimes I)(R) = 1 = (I\otimes\eta)(R), \]
\[ (S\otimes I)(R) = R^{-1} = (I\otimes S)(R). \]

\vskip 3mm {\Large\bf The $WG$ Example} \vskip 3mm

Let $G$ be a finite group with identity element $1$ and let $WG\coloneqq\bbQ\left\langle W(\alpha,\beta)\colon \alpha,\beta\in G\right\rangle$. Set
\begin{eqnarray*}
  W(\alpha,\beta)W(\gamma,\delta) &\coloneqq& \delta_{\alpha\beta,\beta\gamma}W(\alpha,\beta\delta), \\
  \eta(1) &\coloneqq& \sum_{\alpha}W(\alpha,1), \\
  \Delta W(\alpha,\beta) &\coloneqq& \sum_\gamma W(\gamma,\beta)\otimes W(\alpha\gamma^{-1},\beta), \\
  \varepsilon W(\alpha,\beta) &\coloneqq& \delta_{\alpha,1}, \\
  SW(\alpha,\beta) &\coloneqq& W(\beta^{-1}\alpha^{-1}\beta,\beta^{-1}), \\
  R &\coloneqq& \sum_{\alpha,\beta}W(\alpha,1)\otimes W(\beta,\alpha), \\
  \bar{R} &=& \sum_{\alpha,\beta}W(\alpha,1)\otimes W(\beta,\alpha^{-1}).
\end{eqnarray*}

{\bf Proposition. } $WG$ is an IHOP algebra and $R$ is an $R$-element for it.

{\bf Proof.} Think about homomorphisms from the fundamental group of the complement of a tangle to $G$.

\vskip 3mm {\Large\bf An Implementation of $WG$} \vskip 3mm

\input WG.tex

\end{multicols*}

\end{document}


