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\author[Noah Snyder]{Noah Snyder \\ \texttt{http://math.columbia.edu/$\sim$nsnyder} \\ joint work with Vaughan Jones, Scott Morrison, and Emily Peters}
\institute{Columbia University}
\title{Small fusion categories and subfactors}
\date{Fusion categories and Applications, Baylor, October 18 2009 \\ \url{http://tqft.net/waco1}}

\begin{document}

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       \frametitle{Outline}
       \tableofcontents
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\section{Small fusion categories}

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

\begin{frame}{What does small mean?}
\begin{block}{}
We want to find all ``small" tensor categories.  But what definition of small?
\end{block}

\begin{block}{Some notions of small}
\begin{itemize}
\item Small global dimension.
\item Small rank.
\item $X^{\otimes n}$ not too complicated for small $n$.
\item One object of small dimension.
\end{itemize}
\end{block}

\begin{block}{}
If the tensor category is not unitary then you need to think harder about what ``small" means for dimensions, so we'll concentrate on unitary tensor categories.
\end{block}

\end{frame}

\begin{frame}{Current progress}
\begin{block}{Global dimension}
Lots of progress in the case where the global dimension is an integer (Etingof, Gelaki, Jordan, Larsen, Nikshych, Ostrik, etc.).  But essentially no progress for general global dimension.
\end{block}
\begin{block}{Rank}
Ostrik solved rank 2.  Assuming a braiding or modular structure additional progress made by Ostrik, Rowell-Strong-Wang, Hong-Rowell.  Big open question: ``are there finitely many?"
\end{block}
\begin{block}{$X^{\otimes n}$ not too complicated for small $n$}
Kazhdan and Wenzl did $X \otimes X \cong A \oplus B$ and $X^{\otimes 3}$ not too bad.  Assuming a braiding Wenzl and Tuba did the case of $X \otimes X \cong 1 \oplus A \oplus B$, and Snyder did $X \otimes X \cong 1 \oplus X \oplus A \oplus B$.
\end{block}
\end{frame}

\begin{frame}
\begin{block}{One small object}
See this talk! 
\end{block}

\begin{block}{Motivating question}
What is the smallest possible dimension strictly larger than $2$ among all objects in all unitary fusion categories?
\end{block}

\begin{block}{Why should you care?}
\begin{itemize}
\item Source of new examples.
\item The analogous question for finite groups, namely finite subgroups of small Lie groups, has lead to very interesting math like the McKay correspondence.
\item Small global dimension means all individual representations are small.
\item Subfactor results make it easy to make progress.
\end{itemize}
\end{block}
\end{frame}

\section{Connection to subfactors}

\AtBeginSection[]
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       \frametitle{Outline}
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\begin{frame}{Relationship between unitary tensor categories and subfactors}
\begin{block}{What is a subfactor?}
A factor is a von Neumann algebra with trivial center.  A subfactor is an inclusion $A \subset B$ of factors.  For nice applications to category theory we want this inclusion to be finite index, extremal, and we probably want $A$ and $B$ to be type $II_1$ factors
\end{block}

\begin{block}{But I hate analysis!}
Don't worry, the whole theory of subfactors can be translated into algebra or into pictures.  The only vestige of analysis is assuming that a certain inner product is positive definite.
\end{block}
\end{frame}

\begin{frame}{Can you say that in algebra?}
\begin{block}{Yes!}
Notice that $B$ is an algebra object in the category of $A-A$ bimodules.  This suggests several algebraic versions of subfactor theory.
\begin{itemize}
\item A Frobenius algebra object in a unitary tensor category.
\item A categorical Morita equivalence between two unitary tensor categories.
\item A $2$-category with exactly $2$ objects together with a choice of generating $1$-morphism.
\end{itemize}
\end{block}
\end{frame}

\begin{frame}{Can you say that in pictures?}
\begin{block}{Yes!}
\begin{tabular}{c|c}
spherical tensor categories & subfactor planar algebras \\
\hline\hline
$\mathfig{0.2}{category-tangle}$ & $\mathfig{0.2}{subfactor-tangle}$ \\
oriented edges, one shading & unoriented edges, two shadings \\
an edge label for each object & edges only labelled by $X$ \\
many $\operatorname{Hom}$-spaces & just $\mathcal{P}_{k,\pm}$
\end{tabular}
\end{block}
\end{frame}

\begin{frame}{Going from a unitary tensor category to a subfactor}
Given a unitary tensor category and a preferred object $X$ we can get a subfactor by taking the ``alternating part."

\begin{block}{In terms of diagrams}
Restrict your attention to diagrams with edges labelled with only with $X$ and which alternate in and out.  Alternating diagrams can always be checkerboard shaded.
\end{block}

\begin{block}{In terms of algebra}
Inside the tensor category is a Frobenius algebra object, $X \otimes X^*$.  The algebra structure is given by contraction.  The Frobenius structure comes from the pivotal structure, namely $X \otimes X^* \rightarrow X^{**} \otimes X^* \rightarrow \mathbf{1}$.
\end{block}
\end{frame}


\begin{frame}{What about fusion categories?}
\begin{block}{}
If you start with a unitary fusion category then you get a finite depth subfactor.
\end{block}

\begin{block}{Warning!}
\begin{itemize}
\item Finite depth-ness of the subfactor does not imply fusion-ness for the tensor category.
\item Not every finite depth subfactor comes from a fusion category.
\item By looking at the even parts (A-A or B-B) you can recover two fusion categories from any subfactor, but this is not inverse to the ``alternating part" construction.
\end{itemize}
\end{block}
\end{frame}

\section{Small subfactors}

\AtBeginSection[]
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       \frametitle{Outline}
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\begin{frame}{Small subfactors}
\begin{block}{Global dimension}
In subfactor theory this is called the ``global index."  There's been very little work.
\end{block}

\begin{block}{Rank}
This has not been studied much in subfactor theory.  But see depth.
\end{block}

\begin{block}{$X^{\otimes n}$ not too complicated for small $n$}
There has been some work in this direction, most notably papers of Bisch and Jones.  Also D. Thurston's work on dominoes.
\end{block}
\end{frame}

\begin{frame}{Small subfactors}
\begin{block}{One object of small dimension}
The square of the dimension of the chosen object $X$ is called the index.  This is where there's been the most work in subfactor theory.  See Scott's talk for further details!
\end{block}

\begin{block}{Depth}
The depth is the smallest $n$ such that $(X \otimes X^*)^{\otimes \frac {n}{2}}$ contains all simples.  Finite rank is equivalent to finite depth.  Depth $2$ always comes from Hopf algebras (Ocneanu-Szymanski).
\end{block}

%There is one more measure of size that is very important in subfactor theory called the depth.  This measures how high a power of $X$ you need to generate everything.
\end{frame}

\begin{frame}{Summary}
\begin{tikzpicture}
\path (0,4) node (fusion) [shape = rectangle, draw] {\begin{tabular}{c}Unitary fusion category\\ Fixed ``small" object $X$ \end{tabular}};
\path (6.5,2) node (subfactor) [shape = rectangle, draw]{\begin{tabular}{c}Finite depth subfactor \\ ``Small" index $(\dim X)^2$\end{tabular}};
\path (0,0) node (even) [shape = rectangle, draw]{\begin{tabular}{c} Two unitary fusion categories \\ Objects of size $(\dim X)^2 -1$\end{tabular}};

\draw [->, very thick] (fusion.east) to [bend left , auto] node {Alternating part}  (subfactor.north);
\draw [->, very thick] (subfactor.south) to [bend left , auto] node {Even part}  (even.east);
\draw [->, very thick, dashed] (subfactor.west) to [bend left, auto] node {Not invertible!} (fusion.330);
\draw [dashed] (even.140) to [bend left, auto] node {Different!} (fusion.220);
\end{tikzpicture}
\end{frame}

\begin{frame}{Fusion graphs}
\begin{block}{Fusion graphs give a visual depiction of the fusion rules.}
\begin{itemize}
\item Vertices correspond to simple objects.
\item Edge(V,W) correspond to $\dim \Hom{}{X \otimes V}{W}$
\end{itemize}

\end{block}

\begin{block}{Example: $U_q(\mathfrak{su}_3)$}
$$
\mathfig{0.3}{su3-principal-graphs/weyl-chamber} 
\mathfig{0.3}{su3-principal-graphs/principal-graph} 
\mathfig{0.3}{su3-principal-graphs/dual-principal-graph}
$$
\end{block}


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