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  \href{\myurl}{Dror Bar-Natan}:
  \href{\myurl/Talks}{Talks}:
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\begin{document} \latintext
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\parbox[b]{6in}{{\small \navigator} {\Large\bf\red Poly-Poly Extras}
 }
\hfill\parbox[b]{2in}{\tiny
  \null\hfill\webdef
  \newline\null\hfill Slides w/ no URL should be banned!
}
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\begin{multicols}{2} \raggedcolumns

{\red Warning.} Conventions on this page change randomly from line to line.

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{\red The Algebra.} $\calU_{\hbar;\alpha\beta}$ conventions: $q=\bbe^{\hbar\alpha\beta}$,
$H=\langle a,x\rangle/([a,x]=\alpha x)$ with
\[ A=\bbe^{-\hbar\beta a}, \quad xA=qAx, \quad S(a,A,x)=(-a, A^{-1}, -A^{-1}x), \]
\[ \Delta(a,A,x)=(a_1+a_2, A_1A_2, x_1+A_1x_2) \]
and dual $H^\ast=\langle b, y\rangle/([b,y]=-\beta y)$ with
\[ B=\bbe^{-\hbar\alpha b}, \quad By=qyB, \quad S(b,B,y)=(-b, B^{-1}, -yB^{-1}), \]
\[ \Delta(b,B,y)=(b_1+b_2, B_1B_2, y_1B_2+y_2). \]
Pairing by $(a,x)^\ast=\hbar(b,y)$ making $\langle y^lb^j,a^jx^k\rangle = \delta_{ij}\delta_{kl}i![k]_q!$.
Then $\calU=H^{\ast cop}\otimes H$
with $(\phi f)(\psi g) = \langle \psi_1S^{-1}f_3\rangle \langle \psi_3,f_1\rangle(\phi\psi_2)(f_2g)$.
With the central $t\coloneqq\beta a-\alpha b$, $T\coloneqq e^{\hbar t}=A^{-1}B$ get
\[ [a,y]=-\alpha y, \quad xy-qyx=(1-TA^2)/\hbar. \]
Benkart-Witherspoon, \href{http://drorbn.net/AcademicPensieve/2017-06/nb/BW.pdf}{2017-06/BW.nb}: At $\alpha\beta\hbar=\sigma-\rho$, represented by
$y\to\begin{pmatrix}0&0\\-\bbe^\rho&0\end{pmatrix}$,
$a\to\frac{\alpha}{\rho-\sigma}\begin{pmatrix}\rho&0\\0&\sigma\end{pmatrix}$,
$A\to\begin{pmatrix}\bbe^\rho&0\\0&\bbe^\sigma\end{pmatrix}$,
$x\to\frac{\bbe^\rho-\bbe^\sigma}{\hbar \bbe^{\rho+\sigma}}\begin{pmatrix}0&1\\0&0\end{pmatrix}$,
$t\to\frac{\rho+\sigma}{\hbar}\begin{pmatrix}1&0\\0&1\end{pmatrix}$,
$T\to \frac{1}{\bbe^{\rho+\sigma}}\begin{pmatrix}1&0\\0&1\end{pmatrix}$,
$b\to\frac{\beta}{\sigma-\rho}\begin{pmatrix}\sigma&0\\0&\rho\end{pmatrix}$,
$B\to\begin{pmatrix}\bbe^{-\sigma}&0\\0&\bbe^{-\rho}\end{pmatrix}$.

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{\red The $R$-Matrix.} With $[n]_q\coloneqq (q^n-1)/(q-1)$, $[n]_q! \coloneqq [1]q\cdot\ldots\cdot[n]_q$ and $\bbe_q^x \coloneqq \sum_{n\geq 0}\frac{x^n}{[n]_q!}$, we have the mysterious Quesne formula of \arXiv{math-ph/0305003}: $\bbe_q^x = \bbe^x\exp\left(\sum_{k\geq 2}\frac{(1-q)^kx^k}{k(1-q^k)}\right)$. Then $R_{ij} \coloneqq \bbO\left(yb\otimes ax\colon \bbe^{\hbar ba}\bbe_q^{\hbar yx}\right)$.

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{\red $xa$ Swaps.}

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{\red $ay$ Swaps.}

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{\red $xy$ Swaps.}

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{\red The Drinfel'd Element.}

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{\red Putting Everything Together.}

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\end{multicols}

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