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\author[Emily Peters, Noah Snyder, Stephen Bigelow]{
Emily Peters \\ \url{http://euclid.unh.edu/~eep} \\ 
Noah Snyder \\ \url{http://math.columbia.edu/~nsnyder} \\
Stephen Bigelow \\ \url{http://math.ucsb.edu/~bigelow}
\\ joint work with Scott Morrison}
\title{The extended Haagerup subfactor}
\date{Shanks Workshop on Subfactors and Tensor Categories \\ \url{http://euclid.uhn.edu/~eep/Shanks2010_1.pdf}}

\usepackage{multimedia}


\begin{document}

\frame{\titlepage}

\begin{frame}
       \frametitle{Outline}
       \tableofcontents
\end{frame}

\beamertemplatetransparentcovered 

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{\opaqueness<1->{60}} 
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\AtBeginSection[]
{
   \begin{frame}<beamer>
       \frametitle{Outline}
       \tableofcontents[currentsection]
   \end{frame}
}

\section{Classification of small-index subfactors}







	\begin{frame}{Subfactor planar algebras}
		A {\it subfactor planar algebra} has 
		\begin{itemize}
			\item $\dim{V_{0,+}} = \dim{V_{0,-}}=1$;
			\item spherical trace:  
			$\begin{tikzpicture}[baseline=.4cm]
				\fill[shaded] (-.7,-.7) rectangle (1.2,1.7);
				\filldraw[fill=white] (.5,1) arc (0:180:.5cm) -- (-.5,0) arc (-180:0:.5cm) arc (-90:90:.5cm);
				\draw[thick] (.5,.5) circle (.5);
				\node at (.5,.5) {$X$};
			\end{tikzpicture}
			=
			 \begin{tikzpicture}[baseline=.4cm]
				\node at (.5,.5) {$X$};
				\filldraw[shaded] (.5,1) arc (180:0:.5cm) -- (1.5,0) arc (-0:-180:.5cm) arc (-90:90:.5cm);
				\draw[thick] (.5,.5) circle (.5);
			\end{tikzpicture}
			$;
			\item an involution $^*$ on $V_{n,\pm}$, 
			such that $\left< x, y \right> = \tr{y^* x}$ is positive definite.
		\end{itemize}
		From these properties, it follows that closed circles count for a multiplicative constant $\delta$.
	\end{frame}

	\begin{frame}{Temperley-Lieb}
	$TL_{n,\pm}(\delta)$ is the span (over $\mathbb{C}$) of 
	non-crossing pairings of $2n$ points arranged around a circle, 
	with formal addition.  
	
	\begin{example} $TL_3=\operatorname{Span}_{\mathbb{C}}\{ 
		\begin{tikzpicture}[TLEG, scale=.8]
			\filldraw[shaded]  (30:1cm) arc (30:90:1cm) arc (30:-150:5mm) arc (150:210:1cm) -- cycle; 
			\filldraw[shaded]  (0,-1) arc (-90:-30:1cm) arc (30:210:5mm);
			\draw[thick] (0,0) circle (1cm);
			\node at (120:1.3cm) {$\star$};
		\end{tikzpicture},
		\begin{tikzpicture}[TLEG, scale=.8]
			\filldraw[shaded]  (0,1) arc (90:30:1cm) arc (90:270:5mm) arc (-30:-90:1cm) -- cycle; 
			\filldraw[shaded]  (-1,0) arc (180:210:1cm) arc (-90:90:5mm) arc (150:180:1cm);
			\draw[thick] (0,0) circle (1cm);
			\node at (120:1.3cm) {$\star$};
		\end{tikzpicture},
		\begin{tikzpicture}[TLEG, scale=.8, rotate=180]
			\filldraw[shaded]  (150:1cm) arc (150:90:1cm) arc (-210:-30:5mm) arc (30:-30:1cm)--cycle;
			\filldraw[shaded] (-90:1cm) arc (-90:-150:1cm) arc (150:-30:5mm);
			\draw[thick] (0,0) circle (1cm);
			\node at (-60:1.3cm) {$\star$};
		\end{tikzpicture},
		\begin{tikzpicture}[TLEG, scale=.8]
			\filldraw[shaded]  (0,-1) arc (-90:-30:1cm) arc (30:210:5mm);
			\filldraw[shaded]  (-1,0) arc (180:210:1cm) arc (-90:90:5mm) arc (150:180:1cm);
			\filldraw[shaded] (90:1cm) arc (90:30:1cm) arc (-30:-210:5mm);
			\draw[thick] (0,0) circle (1cm);
			\node at (120:1.3cm) {$\star$};
		\end{tikzpicture},
		\begin{tikzpicture}[TLEG, scale=.8]
			\filldraw[shaded]  (90:1cm) arc (30:-150:5mm) arc (150:210:1cm) arc (150:-30:5mm) arc (-90:-30:1cm) arc (-90:-270:5mm) arc (30:90:1cm);
			\draw[thick] (0,0) circle (1cm);
			\node at (120:1.3cm) {$\star$};
		\end{tikzpicture}
	\}$.
	\end{example}
	
	Planar tangles act on $TL$ by inserting diagrams into empty disks, smoothing strings,and throwing out closed loops at a cost of $\cdot \delta$.
	
	\begin{example}
		$\begin{tikzpicture}[TLEG, scale=.9]
			\filldraw[shaded] (90:2.3cm) -- (-90:2.3cm) arc (-90:90:2.3cm);
			\filldraw[unshaded] (0:1cm) circle (.5cm);
			\filldraw[shaded] (180:1cm) circle (.5cm);
			\node[Tbox,inner sep=3.5mm] at (0,0) {};
			\draw[thick] (0,0) circle (2.3cm);
			\node at (120:1.3cm) {$\star$};
			\node at (120:2.6cm) {$\star$};
		\end{tikzpicture}
		\left(
		\begin{tikzpicture}[TLEG, scale=.9]
			\filldraw[shaded]  (0,1) arc (90:30:1cm) arc (90:270:5mm) arc (-30:-90:1cm) -- cycle; 
			\filldraw[shaded]  (-1,0) arc (180:210:1cm) arc (-90:90:5mm) arc (150:180:1cm);
			\draw[thick] (0,0) circle (1cm);
			\node at (120:1.3cm) {$\star$};
		\end{tikzpicture}
		\right)
		=
		\begin{tikzpicture}[TLEG, scale=.9]
			\filldraw[shaded] (90:2.3cm) -- (-90:2.3cm) arc (-90:90:2.3cm);
			\filldraw[unshaded] (0:1cm) circle (.5cm);
			\filldraw[shaded] (180:1cm) circle (.5cm);
			\draw[thick] (0,0) circle (2.3cm);
			\node at (120:2.6cm) {$\star$};
		\end{tikzpicture}
		=\delta^2
		\begin{tikzpicture}[TLEG, scale=.9]
			\draw[thick]  (-2.3,0)arc (180:-180: 2.3cm);
			\filldraw[shaded]  (0,2.3)--(0,-2.3) arc (-90:90:2.3cm);
			\node at (120:2.6cm) {$\star$};
		\end{tikzpicture}
		$
	\end{example}
	\end{frame}





\begin{frame}{Invariants of subfactor planar algebras}

\begin{block}{Index}
The {\em index} of a  subfactor planar algebra $\delta^2$ ($\delta$ is the value of a loop).
\end{block}

\begin{block}{Principal graphs}
The {\em principal graphs} of a subfactor planar algebra encode the fusion ($\otimes$) rules of its irreducible idempotents. 
\end{block}

Note that the graph norm of the principal graphs is equal to $\delta$!

\begin{example}{The Haagerup principal graphs}
$$H=\begin{tikzpicture}[baseline,scale=.5]
\node at (0,0) {$\bullet$};
\node at (0,0) [above] {$*$};
\node at (1,0) {$\bullet$};
\node at (2,0) {$\bullet$};
\node at (3,0) {$\bullet$};
\draw (0,0)--(3,0);
\node at (4,.5) {$\bullet$};
\node at (5,.5) {$\bullet$};
\node at (6,.5) {$\bullet$};
\node at (4,-.5) {$\bullet$};
\node at (5,-.5) {$\bullet$};
\node at (6,-.5) {$\bullet$};
\draw (3,0)--(4,.5)--(6,.5);
\draw (3,0)--(4,-.5)--(6,-.5);
\end{tikzpicture}
\quad , \quad
H'=
\begin{tikzpicture}[baseline,scale=.5]
\node at (0,0) {$\bullet$};
\node at (0,0) [above] {$*$};
\node at (1,0) {$\bullet$};
\node at (2,0) {$\bullet$};
\node at (3,0) {$\bullet$};
\draw (0,0)--(5,0);
\node at (4,1) {$\bullet$};
\node at (4,0) {$\bullet$};
\node at (5,1) {$\bullet$};
\node at (5,0) {$\bullet$};
\draw (3,0) -- (4,1);
\draw (4,0)--(5,1);
\end{tikzpicture}
$$
\end{example}

\end{frame}






	\begin{frame}{Which graphs are principal graphs?}
		Subfactor planar algebras with index less than or equal to $4$ 
		have Dynkin diagrams or extended Dynkin diagrams
		as principal graphs.
				
		\begin{question}
		What are principal graphs 
		for (finite-depth) subfactors 
		with index slightly more than $4$?
		\end{question}
		
		Haagerup (1994) found two families of candidates and one additional candidate, 
		having index between $4$ and $3+\sqrt{3}$.
	\end{frame}
	
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\begin{frame}
\frametitle{Classification of small-index subfactors}
\begin{itemize}
\item<1-> Haagerup's possible principal graphs for subfactors with index less than $3+\sqrt{3}$:
\only<1|handout:0>{
\begin{itemize}
\item $\mathfig{0.15}{graphs/Haagerup}, \mathfig{0.225}{graphs/EH}, \mathfig{0.3}{graphs/EEH}, \ldots$,
\item $\mathfig{0.3}{graphs/HA}$,
\item \vspace{0.25cm} $\mathfig{0.15}{graphs/hexagon}, \mathfig{0.225}{graphs/Ehexagon}, \ldots.$
\end{itemize}}
\only<2|handout:0>{
\begin{itemize}
\item $\mathfig{0.15}{graphs/Haagerup-green}, \mathfig{0.225}{graphs/EH}, \mathfig{0.3}{graphs/EEH}, \ldots$,
\item $\mathfig{0.3}{graphs/HA-green}$,
\item \vspace{0.25cm} $\mathfig{0.15}{graphs/hexagon}, \mathfig{0.225}{graphs/Ehexagon}, \ldots.$
\end{itemize}}
\only<3|handout:0>{
\begin{itemize}
\item $\mathfig{0.15}{graphs/Haagerup-green}, \mathfig{0.225}{graphs/EH}\color{red}{, \mathfig{0.3}{graphs/EEH-red}, \ldots}$,
\item $\mathfig{0.3}{graphs/HA-green}$,
\item \vspace{0.25cm} \color{red}{$\mathfig{0.15}{graphs/hexagon-red}, \mathfig{0.225}{graphs/Ehexagon-red}, \ldots$}.
\end{itemize}}
\only<4>{
\begin{itemize}
\item $\mathfig{0.15}{graphs/Haagerup-green}, \mathfig{0.225}{graphs/EH-green}\color{red}{, \mathfig{0.3}{graphs/EEH-red}, \ldots}$,
\item $\mathfig{0.3}{graphs/HA-green}$,
\item \vspace{0.25cm} \color{red}{$\mathfig{0.15}{graphs/hexagon-red}, \mathfig{0.225}{graphs/Ehexagon-red}, \ldots$}.
\end{itemize}
}
\item<2-> Haagerup and \href{http://dx.doi.org/10.1007/s002200050574}{Asaeda \& Haagerup (1999)} constructed two of these possibilities.
\item<3-> \href{http://www.springerlink.com/content/c4885q02dflfttwm/}{Bisch (1998)} and \href{http://arxiv.org/abs/0711.4144}{Asaeda \& Yasuda (2007)} ruled out infinite families.
\item<4-> Today, we construct the missing example (`extended Haagerup'), and complete the classification.
\end{itemize}
\end{frame}




\section{The extended Haagerup planar algebra}

\begin{frame}{And now for something completely different ...}

The internal structure of the Haagerup and extended Haagerup planar algebras.

\begin{block}{Emily}
Generators and relations for the Haagerup family planar algebras
\end{block}

\begin{block}{Stephen}
Evaluation algorithm and basis for the Haagerup family planar algebras
\end{block}

\begin{block}{Noah}
Non-cyclotomicity of the Haagerup family planar algebras
\end{block}

\end{frame}

\end{document}
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