\documentclass{gtart}

\newcommand{\pathtotrunk}{./}
\input{article_preamble.tex}

\title{The graph planar algebra embedding theorem}
\author{Scott Morrison}
\author{Kevin Walker}

\begin{abstract}
We prove the graph planar algebra embedding theorem, proposed by Vaughan Jones, which states that the representation theory of any finite index $II_1$ subfactor embeds in the graph planar algebra of its principal graph. In fact, we prove a generalization: for each collection of generators of an arbitrary semisimple pivotal $2$-category, we define an associated graph planar algebra and show that the $2$-category embeds. We work over an arbitrary field, and our proof is entirely diagrammatic. At the same time as this proof was discovered, Vaughan Jones and Dave Penneys gave an argument specific to the subfactor planar algebra case.
\end{abstract}

\keywords{Planar algebra, fusion category}

\begin{document}
\maketitle
\newcommand{\field}{\mathfrak{k}}

\textbf{Draft version (July 10, 2010), read with caution.}

\section{Introduction}
In this note we prove the following result

\begin{thm}
\label{thm:subfactor}
Every subfactor planar algebra $\cP$ embeds in the graph planar algebra of its principal graph.
\end{thm}

This theorem has been known to Vaughan Jones since he first defined the graph planar algebra (indeed, he's said ``if the theorem hadn't been true, I would have changed the definition of the graph planar algebra'') but a proof had not appeared in print until recently. Dave Penneys and Vaughan Jones published an alternative proof \cite{gpa2}, which was discovered simultaneously with the argument here.

We find it most convenient to prove a more general result. Associated to any set of generators $G$ for a semisimple pivotal $2$-category $\cC$ we define the principal graph $\Gamma(\cC, G)$ and an associated weight function $\mu$. Using only this data, we define a corresponding pivotal $2$-category $\cG(\Gamma, \mu)$ called the graph planar algebra. 
The most interesting new idea in this paper is an explicit diagrammatic functor $\cE: \cC \to \cG(\Gamma, \mu)$.
After having defined this functor, it's a relatively easy matter to show that it is faithful, and hence
Theorem \ref{thm:subfactor} becomes a special case of our main theorem
\begin{thm}
\label{thm:main}
Any semisimple evaluatable $G$-planar algebra over a field $\field$ embeds in the graph planar algebra over $\field$ of its $G$-principal graph.
\end{thm}
In fact, while Theorem \ref{thm:subfactor} was implicitly about linear categories over $\Complex$, our main theorem holds over any field.

The subfactor case of this result has already provided impetus for the construction of several subfactor planar algebras. 

The first author, with Stephen Bigelow, Emily Peters and Noah Snyder, constructed the extended Haagerup subfactor planar algebra  in \cite{0909.4099} by identifying it as a subalgebra of the appropriate graph planar.
In fact, this construction was inspired by Emily Peters' thesis work \cite{0902.1294} on the Haagerup subfactor planar algebra.

Soon, we expect obstructions to the existence of subfactor planar algebras on the basis of this theorem. Making certain assumptions about the principal graph we can show that there `isn't room' inside the graph planar algebra for a nontrivial subfactor planar algebra. We know of one such obstruction, but more powerful ones may be around the corner.

The proof presented here was first discovered by realising that the Turaev-Viro state sum invariant associated to a subfactor planar algebra naturally gave a map into the graph planar algebra of its principal graph. The formal properties (specifically, the gluing formulas) for the Turaev-Viro invariant ensure that this map is a map of graph planar algebras. However, we then realised that it is possible to describe the resulting map directly, and in fact easier to check that it is functorial straight from the definition. In \S \ref{sec:TV} we describe the connection between the map described here and the Turaev-Viro map.

\subsection{Acknowledgements}
The authors would like to thank Vaughan Jones and Dave Penneys for many interesting conversations on this subject. During this work, Scott Morrison was a Miller Fellow at UC Berkeley, and Kevin Walker was at Microsoft Station Q.

\section{Graph planar algebras}
\label{sec:gpa}
Fix a field $\field$.

We'll begin by defining a very general class of planar algebras, with an arbitrary set of strand labels and region labels. Each strand label has a pair of corresponding region labels. We encode all this data  as a bidirected graph.

\begin{defn}
A \emph{bidirected graph} is a finite directed graph together with an involution (called duality) of the edges that reverses sources and targets.
\end{defn}
In diagams we'll sometimes compress a pair of dual edges into a single undirected edge. In the following, all of our graphs are bidirected graphs.

\begin{defn}
Let $G$ be a bidirected graph. A $G$-planar tangle consists of a disc, with some number of embedded inner discs, all with a marked point on their boundaries, along with strands in the complementary region which are either embedded circles or embedded intervals meeting the boundaries of the discs, with a labelling of the regions between the strands by vertices of $G$, and a labelling of the strands by a pair consisting of an edge of $G$ between the vertices on either side and its reversal. 
\end{defn}

Given a $G$-planar tangle $T$ we will write $T_0$ for the cycle in $G$ appearing (always reading counterclockwise, from the marked point) around the outer boundary of $T$, and $T_i$ for the cycle appearing around the boundary of the $i$-th inner disc of $T$. 

\begin{defn}
Given a bidirected graph $G$, a \emph{$G$-planar algebra} $\cP$ consists of
\begin{itemize}
\item
a $\field$-vector space $\cP_g$ for each loop $g$ on $G$, and
\item
a multilinear map $\cP(T) : \tensor \cP_{T_i} \to \cP_{T_0}$ for each $G$-planar tangle $T$,
\end{itemize}
such that the maps only depend on the $G$-planar tangle up to isotopy, and the usual associativity constraints for planar algebras hold.
\end{defn}
%It is suggestive to think of this as associating a vector space to every endomorphism in the category freely generated by the edges of $G$.

%If the planar tangle \todo{draw these!} consisting of two rings labelled by involute edges $\{e, \bar{e}\}$ in $G$  acts by some scalar $D_e \in \field$, we say that these edges in $G$ have \emph{index} $D_e$. In particular, if $e$ an edge from $g_1$ to $g_2$ has a nonzero index, then the single ring labelled by $e$ is an isomorphism between the spaces $\cP_{g_1}$ and $\cP_{g_2}$.

If all the vector spaces $\cP_{g}$, for $g$ a length zero loop (that is, just a vertex), are $1$-dimensional, we say $\cP$ is \emph{evaluatable} (because every closed diagram can be `evaluated' as a multiple of the empty diagram). These $1$-dimensional vector spaces are canonically identified with $\field$, sending the empty diagram to $1$. %When $\cP$ is not evaluatable, it is often the case that the vector spaces $\cP_{g_0}$ are all isomorphic, and by enlarging the base field to the field of fractions of $\cP_{g_0}$ we can nevertheless think of $\cP$ as evaluatable.

\newcommand{\shadeddigraph}{\tikz[baseline=-0.5ex]{\draw[fill] (0,0) circle (2pt); \draw[] (0.5,0) circle (2pt); \draw[->] (0.1,0.05) .. controls (0.25,0.2) .. (0.4,0.05);  \draw[->] (0.4,-0.05) .. controls (0.25,-0.2) .. (0.1,-0.05);}}
\newcommand{\orienteddigraph}{\smash{\tikz[baseline=-0.5ex]{\draw[fill] (0,0) circle (2pt); \draw[->] (0,0) .. controls (-0.4,-0.4) and(-0.4,0.4) .. (-0.08,0.08); \draw[->] (0,0) .. controls (0.4,0.4) and(0.4,-0.4) .. (0.08,-0.08);}}}
\newcommand{\unorienteddigraph}{\smash{\tikz[baseline=-0.5ex]{\draw[fill] (0,0) circle (2pt); \draw[->] (0,0) .. controls (0.4,0.4) and(0.4,-0.4) .. (0.08,-0.08);}}}

\begin{example}
The most familiar example of a shaded planar algebra is in this language a $\shadeddigraph$-planar algebra (where the involution switches the edges). An unshaded oriented planar algebra is a $\orienteddigraph$-planar algebra (where the involution switches the edges). An unshaded unoriented planar algebra is a $\unorienteddigraph$-planar algebra (where the involution fixes the edge). 
\end{example}

\begin{example}[Kuperberg's spider for $\mathfrak{su}_3$]
Since Kuperberg's spider for $\mathfrak{su}_3$ has a single oriented strand type, it is naturally a $\orienteddigraph$-planar algebra. However, since the representation theory of $\mathfrak{su}_3$ is graded by $Z(SU(3)) = \Integer/3\Integer$, we can in fact $3$-color the regions, and think of it as a
$\tikz[baseline=1.5ex, shorten >=5pt,shorten <=5pt,x=20,y=20]{
\draw[fill] (0,0) circle (2pt);
\draw (0.5,1) circle (2pt);
\draw[fill=gray] (-0.5,1) circle (2pt);
\draw[<->] (0,0) -- (0.5,1);
\draw[<->] (0.5,1) -- (-0.5,1);
\draw[<->] (-0.5,1) -- (0,0);
}$-planar algebra. This has an obvious generalisation to the representation theory of $\mathfrak{su}_n$, which is $\Integer/n \Integer$ graded. It is a $C_n$-planar algebra when only the standard representation is allowed as a strand label, or a $K_n$-planar algebra when all fundamental representations are allowed.
\end{example}

\begin{example}[Intermediate subfactors]
Given a diagram $G$ of inclusions of $II_1$ subfactors, the intertwiners of tensor products of the corresponding bimodules are naturally a $G$-planar algebra.
\end{example}

If the loop labelled by a directed edge $e$ pointing from the outer region to the inner region is some multiple $\delta_{e}$ of the empty diagram we say $\cP$ has $e$-modulus $\delta_{e}$. In a shaded planar algebra the ``spherical'' axiom asks that the moduli are equal, while for a ``lopsided'' planar algebra we ask that one of the moduli is equal to $1$.

Given a bidirected graph homomorphism $\pi:\Gamma \to G$ we can take any $\Gamma$-planar algebra $\cP$ and `collapse along $\pi$' to form a $G$-planar algebra $\pi_*\cP$. This is defined by
$$(\pi_*\cP)_g = \DirectSum_{\gamma \in \pi^{-1}(g)} \cP^\Gamma_\gamma$$
and
$$(\pi_*\cP)(S) = \sum_{T \in \pi^{-1}(S)} \cP^{\Gamma}(T).$$
Here we implicitly think of $\pi$ as a map taking cycles in $\Gamma$ to cycles in $G$, and also as a map sending $\Gamma$-planar tangles to $G$-planar tangles by applying $\pi$ to all strand and region labels.

\begin{example}
Collapsing along the obvious homomorphism from $\shadeddigraph$ to $\orienteddigraph$ takes a shaded planar algebra and produces an oriented planar algebra, in which all spaces are zero except those where the boundary labels alternate incoming and outgoing, and the action of oriented planar tangles is determined by first filling in the checkerboard shading, and then acting.
\end{example}

\begin{example}
Collapsing along the homomorphism from $\orienteddigraph$ to $\unorienteddigraph$ takes an oriented planar algebra $\cP$ and produces an unoriented planar algebra $\cQ$, where the vector space $\cQ_k$ is the direct sum of the vector spaces for $\cP$ corresponding to the $2^k$ different loops of length $k$ on $\orienteddigraph$.
\end{example}

Fix a graph homomorphism $\pi :\Gamma \to G$ and a modulus $\delta_e \in \field$ for each edge of $G$. Dimension data $d$ for $(\pi,\delta)$ is an assignment $d(v) \in \field$ for each vertex of $\Gamma$ such that for each edge $a \to b$ of $G$ 
we have
$$d(A) \delta_{a \to b} = \sum_{(A \to B) \in \pi^{-1}(e)} d(B)$$
for each vertex $A \in \pi^{-1}(a)$ of $\Gamma$, and
$$d(B) \delta_{a \to b} = \sum_{(A \to B) \in \pi^{-1}(e)} d(A)$$
for each vertex $B \in \pi^{-1}(b)$.
If $d$ is positive on every vertex then we say that $d$ is \emph{Perron-Frobenius dimension data}  for $\pi$. 
If such $d$ exists, it is uniquely determined up to a normalization on each connected component. If $\Gamma$ has a base point in each connected component, we always choose the Perron-Frobenius dimension data with $d(\star) = 1$ for each base point $\star$.

\begin{example}
With $\tau$ the golden ratio,
$$\tikz[baseline=-0.5ex]{
	\draw[fill] (0,0) node [above] {$1$} node [below] {$\star$} circle (2pt);
	\draw[] (0.5,0) node [above] {$\tau$} circle (2pt);
	\draw[fill] (1,0) node [above] {$\tau$} circle (2pt);
	\draw[] (1.5,0) node [above] {$1$} circle (2pt);
	\draw[->] (0.1,0.05) .. controls (0.25,0.2) .. (0.4,0.05);
	\draw[->] (0.4,-0.05) .. controls (0.25,-0.2) .. (0.1,-0.05);
	\draw[->] (0.6,0.05) .. controls (0.75,0.2) .. (0.9,0.05);
	\draw[->] (0.9,-0.05) .. controls (0.75,-0.2) .. (0.6,-0.05);
	\draw[->] (1.1,0.05) .. controls (1.25,0.2) .. (1.4,0.05);
	\draw[->] (1.4,-0.05) .. controls (1.25,-0.2) .. (1.1,-0.05);
}
\to
\tikz[baseline=-0.5ex]{
	\draw[fill] (0,0) circle (2pt);
	\draw[] (0.5,0) circle (2pt);
	\node[label=above:$\tau$] at (0.25,0.025) {};
	\node[label=below:$\tau$] at (0.25,0.025) {};
	\draw[->] (0.1,0.05) .. controls (0.25,0.2) .. (0.4,0.05);
	\draw[->] (0.4,-0.05) .. controls (0.25,-0.2) .. (0.1,-0.05);
}
$$
is Perron-Frobenius dimension data for the $A_4$ digraph with spherical modulus, while 
$$\tikz[baseline=-0.5ex]{
	\draw[fill] (0,0) node [above] {$1$} node [below] {$\star$} circle (2pt);
	\draw[] (0.5,0) node [above] {$1$} circle (2pt);
	\draw[fill] (1,0) node [above] {$\tau$} circle (2pt);
	\draw[] (1.5,0) node [above] {$\tau^{\mbox{\tiny$-1$}}$} circle (2pt);
	\draw[->] (0.1,0.05) .. controls (0.25,0.2) .. (0.4,0.05);
	\draw[->] (0.4,-0.05) .. controls (0.25,-0.2) .. (0.1,-0.05);
	\draw[->] (0.6,0.05) .. controls (0.75,0.2) .. (0.9,0.05);
	\draw[->] (0.9,-0.05) .. controls (0.75,-0.2) .. (0.6,-0.05);
	\draw[->] (1.1,0.05) .. controls (1.25,0.2) .. (1.4,0.05);
	\draw[->] (1.4,-0.05) .. controls (1.25,-0.2) .. (1.1,-0.05);
}
\to
\tikz[baseline=-0.5ex]{
	\draw[fill] (0,0) circle (2pt);
	\draw[] (0.5,0) circle (2pt);
	\node[label=above:$1$] at (0.25,0.025) {};
	\node[label=below:$\tau^2$] at (0.25,0.025) {};
	\draw[->] (0.1,0.05) .. controls (0.25,0.2) .. (0.4,0.05);
	\draw[->] (0.4,-0.05) .. controls (0.25,-0.2) .. (0.1,-0.05);
}
$$
is Perron-Frobenius dimension data for lopsided modulus. In both cases, we've written the dimension data next to vertices on the initial graph, and the modulus of each edge on the final graph.
\end{example}

\begin{example}
\label{ex:su3-graph}
Perron-Frobenius dimension data for a particular graph homomorphism (see Example \ref{ex:su3-explanation} below for motivation). Here each arrow has a corresponding dual arrow, which we've omitted for clarity. There are two parallel arrows from the white vertex to the gray vertex.
$$
\tikz[baseline=2.5ex, shorten >=5pt,shorten <=5pt]{
\draw[fill] (0,0) node[below] {$1$} circle (2pt);
\draw (0.5,1)  node[right] {$2$} circle (2pt);
\draw[fill=gray] (-0.5,1)  node[left] {$2$} circle (2pt);
\draw[->] (0,0) -- (0.5,1);
\draw[->, double distance=1pt] (0.5,1) -- (-0.5,1);
\draw[->] (-0.5,1) -- (0,0);
\foreach \x in {0.25, 0, -0.25} {
	\draw[fill] (\x,2)  node[above] {$1$} circle (2pt);
	\draw[->] (\x,2) -- (0.5,1);
	\draw[<-] (\x,2) -- (-0.5,1);
}
}
\to
\tikz[baseline=2.5ex, shorten >=5pt,shorten <=5pt]{
\draw[fill] (0,0) circle (2pt);
\draw (0.5,1) circle (2pt);
\draw[fill=gray] (-0.5,1) circle (2pt);
\draw[->] (0,0) -- (0.5,1);
\draw[->] (0.5,1) -- (-0.5,1);
\draw[->] (-0.5,1) -- (0,0);
\node[label=above:$2$] at (0,1) {};
\node[label=right:$2$] at (0.25,0.5) {};
\node[label=left:$2$] at (-0.25,0.5) {};
}
$$
\end{example}

\begin{example}
Let $d=\sqrt{\frac{5+\sqrt{13}}{2}}$.
Lopsided dimension data for the Haagerup graph with moduluses $1$ and $d^2$ exists over the field $\Rational(\sqrt{13}) = \Rational(d^2)$,
$$
\tikz[shorten >=5pt,shorten <=5pt] {
\foreach \r/\c/\l in {3/black/$1$,2/white/$1$,1/black/$d^2-1$} {
	\foreach \q/\d in {60/right,180/above,300/right} {
		\draw[fill=\c] (\q:\r) node[\d] {\l} circle (2pt);
		\draw (\q:\r) -- (\q:\r-1);
	}
}
\draw[fill=white] (0,0) node[right] {$d^2-2$} circle (2pt);
}
$$
while for spherical dimension data with modulus $d$ we need to extend the field to include $d$:
$$
\tikz[shorten >=5pt,shorten <=5pt] {
\foreach \r/\c/\l in {3/black/$1$,2/white/$d$,1/black/$d^2-1$} {
	\foreach \q/\d in {60/right,180/above,300/right} {
		\draw[fill=\c] (\q:\r) node[\d] {\l} circle (2pt);
		\draw (\q:\r) -- (\q:\r-1);
	}
}
\draw[fill=white] (0,0) node[right] {$d^3-2d$} circle (2pt);
}
$$
\end{example}

\begin{defn}
The \emph{trivial planar algebra} $\cG(\Gamma, d)$ on a bidirected graph $\Gamma$ with dimension data $d$ is a  $\Gamma$-planar algebra. The vector spaces $\cG(\Gamma)_\gamma$ are all just $\field$. For any planar tangle $T$, choose an isotopy representative so all strands attach only along the bottom halves of the discs, with each disc's starred region on the right. Since all our vector spaces are $1$-dimensional, the multilinear map associated to a planar tangle is just a number, given by
$$\cG(\Gamma)(T) = \prod_{\substack{\text{critical}\\\text{points $c$}}} d(c^+)^{\text{sign}(c)}.$$
Here $c^+$ is the vertex of $\Gamma$ appearing above the critical point $c$.
\end{defn}
One checks that this satisfies the axioms of a planar algebra in the usual way, see e.g. \cite{MR1865703}. Note however that this definition is not the usual one, even in the ``shaded'' case: we've rearranged the critical point coefficients slightly. This rearrangement is essential, however, if we expect to work over an arbitrary field: the definition in \cite{MR1865703} takes square roots of dimensions.

\begin{defn}
The graph planar algebra $\cG(\pi)$ for a  homomorphism $\pi:\Gamma \to G$ of bidirected graphs with base points is the $G$-planar algebra obtained by collapsing the trivial planar algebra for $\Gamma$ with Perron-Frobenius dimension data along $\pi$: $$\cG(\pi) = \pi_*\cG(\Gamma, d_{\text{PF}}).$$
\end{defn}
Unravelling the definition, we have for $g$ a cycle in $G$
$$\cG(\pi)_g = \set{ \text{based directed cycles on $\Gamma$ over $g$} }^*$$  and
$$\cG(\pi)(T)\left(\Tensor f_i \right)= \sum_{\substack{\text{$\Gamma$-labels}\\\text{$\ell$ of $T$}}} \left( \prod_{\substack{\text{critical}\\\text{points $c$}}} d(c^+(\ell))^{\text{sign}(c)} \right) \left( \prod_i f_i(\ell) \right).$$
Here a $\Gamma$-label of $T$ is a lifting of the specified labelling of $T$ (recall, regions by vertices of $G$, strands by edges by $G$) to a labelling by $\Gamma$, $d(c^+(\ell))$ is the Perron-Frobenius dimension of the vertex of $\Gamma$ appearing above the critical point $c$ in the labelling $\ell$, and $f_i(\ell)$ is the functional $f_i$ evaluated on the cycle in $\Gamma$ given by reading the labelling $\ell$ around the boundary of the $i$-th inner disc. We've chosen to write the vector spaces as functionals on directed cycles, rather than formal linear combinations, for notational convenience below.

\section{Pivotal $2$-categories and planar algebras}
In this section we recall the now familiar translation between pivotal categories and unshaded planar algebras, and give the generalized translation between pivotal $2$-categories and $G$-planar algebras as described above. This section serves a double purpose: the embedding map will be easier to describe if we allow ourselves to move freely between the two descriptions, and we want to make clear that the embedding theorem is relevant for all (pseudo-unitary) pivotal $2$-categories.

We assume all our categories are linear, so all hom spaces are finite dimensional $\field$-vector spaces, and moreover that they are additive, so all direct sums of objects exist. (If a linear category is not additive, we can always embed it in the matrix category which has direct sums.) When discussing $2$-categories, linear means that the $2$-morphism spaces are vector spaces, and that direct sums of $1$-morphisms exist.

All our $2$-categories are strictly associative. We always write compositions of $1$-morphisms using $\tensor$, so tensor categories are simply $2$-categories with a single $0$-morphism. In attempt to avoid confusion translating between tensor categories and $2$-categories, we will refer to ``objects'' and ``homs'' of $1$-categories (including tensor categories), and to $0$-, $1$- and $2$-morphisms of $2$-categories. The axioms for pivotal categories from \cite{MR1686423} generalize immediately to give a definition of pivotal $2$-categories (we interpret pivotal categories as $2$-categories, then relax the requirement that there is only one $0$-morphism). Throughout, we assume all our pivotal categories are strictly pivotal (that is, the double dual functor is the identity, not just naturally isomorphic to the identity).

Fix a pivotal $2$-category $\cC$, and suppose $\cX$ is a finite set of dominant simple 1-morphisms (so every simple 1-morphism is a summand of some tensor product of elements of $\cX$) which is closed under duals. We can consider $\cX$ as the edges of a bidirected graph. Associated to $\cC$ and $\cX$ is a $\cX$-planar algebra.

\begin{defn}
 The $\cX$-planar algebra $\cP(\cC,\cX)$ has vector spaces 
$$\cP(\cC,\cX)_k = \Hom{\cC}{k}{\id}.$$
Here we interpret a directed cycle $k$ on $\cX$ as the corresponding tensor product of $1$-morphisms in $\cC$.
The multilinear maps for tangles are given by choosing an isotopy representative in which all strands attach only along the bottom halves of the discs, then interpreting the diagram as an element of the $2$-category by replacing critical points in strands with the evaluation and coevaluation maps in the usual manner.
\end{defn}

Going in the other direction, given a $\cX$-planar algebra $\cP$ we can define a pivotal $2$-category $\cC(\cP)$. We make an intermediate definition first.

\begin{defn}
The pivotal $2$-category $\check{\cC}(\cP)$ has
\begin{description}
\item[$0$-morphisms] vertices of $\cX$,
\item[$1$-morphisms] $\Hom{\check{\cC}(\cP)}{a}{b} = \set{\text{paths on $\cX$ from $a$ to $b$}}$,
\item[$2$-morphisms] $\Hom{\check{\cC}(\cP)}{\lambda}{\mu} = \cP_{\bar{\mu}\lambda}$, where $\bar{\mu}$ denotes the involute of $\mu$, and $\bar{\mu}\lambda$ is the concatenation of $\bar{\mu}$ and $\lambda$
\end{description}
where composition, tensor products and duality are all given by the obvious corresponding $\cX$-planar tangles.
\end{defn}
\begin{defn}
The pivotal $2$-category $\cC(\cP)$ is the idempotent completion of $\check{\cC}(\cP)$.
\end{defn}

%\begin{defn}
%The idempotent completion of a tensor category $\check{\cC}$ is a tensor category $\cC$ which contains $\check{\cC}$ as a full sub-category. The objects of $\cC$ are pairs $(o,p)$, where $o$ is an object of $\check{\cC}$ and $p:o \to o$ is an idempotent in $\check{\cC}$.  We define $$\Hom{\cC}{(o,p)}{(o',p')} = \setcl{ f \in \Hom{\check{\cC}}{o}{o'} }{ f p = f = p' f},$$ and inherit composition and tensor products.  (When we say $\check{\cC}$ is a full sub-category of $\cC$ we are implicitly identifying $o$ with $(o,\id_o)$.)
%\end{defn}
%A pivotal structure on $\check{\cC}$ induces a pivotal structure on the idempotent completion $\cC$.

If $P \in \cP_{\bar{\lambda} \lambda}$ and $Q \in \cP_{\bar{\mu} \mu}$ are idempotents (with respect to the multiplication tangles) in a $\cX$-planar algebra $\cP$, with $\lambda$ and $\mu$ paths in $\cX$ with the same start and end points, we have the notion of the $\operatorname{Hom}$-space between them, the subspace of $\cP_{\bar{\lambda} \mu}$ given by
$$\Hom{\cP}{P}{Q} = \setcr{f \in \cP_{\bar{\lambda} \mu}}{f P = f = Q f}.$$(In fact $P$ and $Q$ are $1$-morphisms in the associated $2$-category $\cC(\cP)$, and this just translates the $\operatorname{Hom}$-space of the idempotent completion back to the planar algebraic description.)


A semisimple pivotal $2$-category $\cC$ is pseudo-unitary if the Perron-Frobenius dimensions of objects and the quantum dimensions coincide. Note that if $\cC$ is unitary then it is automatically semisimple, non-degenerate and pseudo-unitary.

\begin{example}[$\Rep \mathfrak{g}$]
\label{ex:G}
Fix a complex semisimple Lie algebra $\mathfrak{g}$. As $\tensor$-categories, 
$\Rep \mathfrak{g}$, $\Rep U \mathfrak{g}$ and $\Rep U_q \mathfrak{q}$ are all equivalent. In each case, the irreducible representations with highest weight a fundamental weight $\tensor$-generate the category. Let $\chi$ be the graph with a single vertex and an edge for each fundamental representation of $\mathfrak{g}$ (and the representation theoretic duals). From $\Rep \mathfrak{g}$ we can construct a $\chi$-planar algebra, which we call $\FundRep \mathfrak{g}$.
\end{example}

\begin{example}[$\Rep \mathfrak{g}$ graded by $Z(G)$]
\label{ex:Z(G)}
Since the $\tensor$-product in $\Rep \mathfrak{g}$ is graded by the group $Z(G)$, the centre of the corresponding simply connected compact Lie group (equivalently, the group of weight vectors modulo root vectors), we can think of $\Rep \mathfrak{g}$ as a $2$-category with objects $Z(G)$. (Each $1$-morphism space is just a copy of $\Rep \mathfrak{g}$, and composition of $1$-morphisms is $\tensor$-product.) Now, we can take the graph $\chi_Z$ with vertices $Z(G)$, and an edge from $a$ to $b$ for each fundamental weight $\lambda$ such that $a+\lambda = b$ in $Z(G)$. Again, we can construct a $\chi_Z$-planar algebra, which we call $\FundRep^Z \mathfrak{g}$.
\end{example}

\begin{defn}
\label{defn:quantum-subgroup}
A quantum subgroup of $\mathfrak{g}$ is a $\chi$-planar algebra $\cP$ in the sense of Example \ref{ex:G}, along with a map $\FundRep U_q \mathfrak{g} \to \cP$.
\end{defn}

\begin{defn}
\label{defn:graded-quantum-subgroup}
A graded quantum subgroup of $\mathfrak{g}$ is a $\chi_Z$-planar algebra $\cP$ in the sense of  Example \ref{ex:Z(G)}, along with a map $\FundRep^Z U_q \mathfrak{g} \to \cP$.
\end{defn}


\begin{rem}
Collapsing along the graph homomorphism $\chi_Z \to \chi$ allows us to forget the grading on a quantum subgroup of $\mathfrak{g}$.
\end{rem}

\begin{example}
A graded quantum subgroup of $\mathfrak{su}_2$ is just a spherical shaded planar algebra with modulus $\qi{2}$. If $\qi{2} < 2$, then there is an ADE classification. A quantum subgroup is an oriented unshaded planar algebra with modulus $\qi{2}$, and if $\qi{2} < 2$ there is an ADET classification. See \cite{MR1936496} for details.
\end{example}

\begin{example}
\label{ex:su3-pa}
A graded quantum subgroup of $\mathfrak{su}_3$ is a
$\scalebox{0.75}{\tikz[baseline=2.5ex, shorten >=5pt,shorten <=5pt]{
\draw[fill] (0,0) circle (2pt);
\draw (0.5,1) circle (2pt);
\draw[fill=gray] (-0.5,1) circle (2pt);
\draw[->] (0,0) -- (0.5,1);
\draw[->] (0.5,1) -- (-0.5,1);
\draw[->] (-0.5,1) -- (0,0);
\node[label=above:$\qi{3}$] at (0,1) {};
\node[label=right:$\qi{3}$] at (0.25,0.5) {};
\node[label=left:$\qi{3}$] at (-0.25,0.5) {};
}}$-planar algebra which contains (a quotient of) Kuperberg's $U_q \mathfrak{su}_3$ spider.  See \cite{0906.4225} for the related notion of an $A_2$-planar algebra.
\end{example}

\section{The embedding map}
Given idempotents $\leftidx{_A}{P}{_B}, \leftidx{_A}{Q}{_C}$ and $\leftidx{_B}{X}{_C}$ in a $G$-planar algebra $\cP$ (here the subscripts $A, B$ and $C$ denote vertices of $G$, indicating the left and right shadings of the idempotents; equivalently, interpreting the idempotents as $1$-morphisms in the corresponding $2$-category, they are the sources and targets) there is a pairing $$(-,-)_{P,Q,X} : \Hom{\cP} {Q} {P \tensor X}\tensor \Hom{\cP}{P}{Q \tensor X^*} \to \operatorname{End}_{\cP}(Q)$$ given by
\begin{align}
(a,b) & = a \circ (b \tensor \id_{X}) \circ (\id_Q \tensor \operatorname{coev}_{X}) \notag \\
          & = d(Q) \times
          \begin{tikzpicture}[x=0.8cm,y=0.8cm,baseline=0.8cm]
          	\node(a)[draw, circle] at (0,0) {$a$};
		\node(Q) at (0,-1) {$Q$};
		\node(P) at (-1,1) {$P$};
		\node(X) at (1,1) {$X$};
		\draw (X) -- (a) -- (P) (a) -- (Q);
		\node(b)[draw,circle] at (-1,2) {$b$};
		\node(Q') at (-2,3) {$Q$};
		\node(X*) at (0,3) {$X^*$};
		\draw (P) -- (b) -- (Q') (b) -- (X*);
		\draw (X*) ++ (0,0.2) arc (180:0:0.5);
		\node(X') at (1,3) {$X$};
		\draw (X) -- (X');
		\draw (Q') -- +(0,0.8);
		\draw (Q) -- +(0,-0.6);
          \end{tikzpicture}. \label{eq:dual}
\end{align}
If $Q$ is simple, $\End{Q}$ is $1$-dimensional, canonically identified with $\field$. In this case the pairing $(-,-)_{P,Q,X}$ is non-degenerate as long as $\cP$ itself is non-degenerate.

\begin{defn}
\label{defn:principal-graph}
The \emph{principal graph} $\Gamma(\cP)$ of a nondegenerate $G$-planar algebra is a bidirected graph fibered over $G$, which has 
\begin{description}
\item[vertices]
a simple idempotent in $\cP$ from each isomorphism class,
\item[edges]
from $P$ to $Q$ over $X \in G$ corresponding to a $\field$-basis of $\Hom{\cC}{Q}{P \tensor X}$ and 
\item[involution]
sending a basis element of $\Hom{\cC}{Q}{P \tensor X}$ to the dual $\field$-basis element with respect to the pairing $(-,-)_{P,Q,X}$ in $\Hom{\cP}{P}{Q \tensor X^*}$.
\end{description}
\end{defn}

\todo{explain why it's always possible to choose bases so the involution works like this}

We're explicitly choosing bases for $\operatorname{Hom}$-spaces in this definition of the principal graph; a more usual definition would just have finite sets of edges with the same cardinality. We will make use of these extra choices in our construction of the embedding map. In the case that $\Gamma$ is simply-laced (that is $\Hom{\cC}{Q}{P \tensor X}$ is always $1$-dimensional) choosing a basis is just a choice of normalization: note that these choices nevertheless affect the embedding map below.

The principal graph splits into components according to the region label at the marked point of the idempotents. Each of these components is connected (if $Q$ is an idempotent in $\cP_{\bar{\lambda}\circ \lambda}$ then there is a path from the tensor identity to $Q$ in the principal graph that descends to $\lambda$), and \todo{fix this} moreover has a distinguished vertex given by the tensor identity, so $\Gamma(\cP)$ comes with Perron-Frobenius data.

\begin{example}
\label{ex:su3-explanation}
The representation theory of $U_q \mathfrak{su}_3$ at $q$ a $12$th root of unity is modular with the usual pivotal structure, but not modular with the unimodal pivotal structure. In particular, there are three invertible objects with twist factor $1$, those with highest weights $00$, $30$ and $03$. The modular quotient (equivalently, the M\"{u}ger deequivariantization by the subcategory of invertible objects) is a graded quantum subgroup of $\mathfrak{su}_3$ in the sense of Example \ref{ex:su3-pa}. The graph of Example \ref{ex:su3-graph} is the principal graph.
\end{example}

\begin{example}
The modular quotient of unimodal $U_q \mathfrak{su}_3$ at $q$ a $14$th root of unity can be realised as a quantum subgroup of $\mathfrak{su}_3$, but not as a graded quantum subgroup. This is because the subcategory of invertible objects is not concentrated in the trivial component of the $Z(G)$-grading.

The full classification of quantum subgroups of $\mathfrak{su}_3$ with $q$ a root unity has been announced by Ocneanu.
\end{example}

We now do a short calculation which will be used in the proof of Proposition \ref{prop:map}.
\begin{lem}
\label{lem:IH}
If $\cP$ is a semisimple $G$-planar algebra, $\gamma$ is an simple idempotent of $\cP$ and $X$ is an idempotent in $G$, then
\begin{align*}
\begin{tikzpicture}[baseline=1.75cm] 
\node(g) at (0,0) {$\gamma$};
\node(g') at (0,3.5) {} ;
\node(X) at (2,0) {$X$};
\node(X') at (2,3.5) {};
\draw (g)--(g') (X)--(X');
\end{tikzpicture} 
& = 
\sum_{\substack{\gamma \xrightarrow{\varepsilon} a \\ \varepsilon \mapsto X}} d(a)
\begin{tikzpicture}[baseline=1.75cm] 
\node(g) at (0,0) {$\gamma$};
\node(g') at (0,3.5) {$\gamma$};
\node(e')[draw,circle] at (1,1) {$\bar{\varepsilon}$};
\node(e)[draw,circle] at (1,2.5) {$\varepsilon$};
\node(X) at (2,0) {$X$};
\node(X') at (2,3.5) {$X$};
\draw (g)--(e') (X)--(e') (e')--(e) (e)--(g') (e)--(X');
\node(a) at (1.3,1.75) {$a$};
\end{tikzpicture} 
\end{align*}
Here the summation is over edges $\varepsilon$ from $\gamma$ to some other idempotent $a$, which descend to the edge $X$ of $G$ under the map $\Gamma(\cP) \to G$.
\end{lem}
\begin{proof}
Semisimplicity ensures that this identity holds with some coefficients. Squaring both sides, and using the fact that $\varepsilon$ and $\bar{\varepsilon}$ are dual in the sense of Definition \ref{defn:principal-graph} shows that the coefficients are just the dimensions of the idempotents $a$.
\end{proof}


Consider now a loop $\gamma$ in $\Gamma(\cP)$ for $\cP$ a $G$-planar algebra, with image $\chi$ in $G$ under $\Gamma(\cP) \to G$. This gives a sequence of idempotents and maps between them in $\cP$. We'll write $\gamma_i$ for the $i$-th vertex of $\gamma$, which is a minimal idempotent in $\cP$. Write $\chi_i$ for the edge of $\chi$ from the vertex $i$  to the vertex $i+1$ in $G$  (recall the edges of $G$ are the strand labels for $\cP$). (Notice that $\gamma_i$ is a vertex of $\Gamma(\cP)$ while $\chi_i$ is an edge of $G$.) For notational compactness in diagrams, we'll often introduce a second symbol to refer to the edges of $\gamma$. Thus when we say `let $\gamma$ be a loop with edges $a_i$' we mean that $a_i$ is the edge between the vertex $i$ and vertex $i+1$ in $\gamma$, which is a basis element for $\Hom{\cP}{P_{i+1}}{P_i \tensor \chi_i}$.

\begin{defn}
The graph planar algebra embedding map $\cE : \cP \into \cG(\Gamma(\cP))$ for a non-degenerate evaluatable $G$-planar algebra $\cP$ is given by
$$\cE(S)(\gamma) = \frac{
\begin{tikzpicture}[baseline=7ex]
\node(SSW) at (0,1.2) {};
\node(SNE) at (6,1.7) {};
\node(S)[draw, line width=2pt, rounded corners, fit=(SSW) (SNE)] {$S$};
\node(star)[right] at (S.east) {$\star$};
\foreach \index / \position in {0/0, 1/1.5, 2/3, n/6}
{
	\node(a\index)[draw, line width=2pt, circle] at (\position,0) {$a_\index$};
	\node[below=-3pt] at (a\index.south) {$\star$};
	\draw[->] (a\index) -- node[right] {$\chi_\index$} ($(S.south east)!(a\index)!(S.south west)$);
}
\draw[->] (a1) -- node[below] {$\gamma_1$} (a0);
\draw[->](a2)  -- node[below] {$\gamma_2$} (a1); 
\draw[->] ($(a2)+(0.6,0)$) -- (a2);
\draw[dashed] (a2) -- (an);
\draw (an) -- +(-0.6,0);
%\draw[dashed] (a2)+(0.6,0) -- (an)+(-0.6,0);
\draw[<-] (an.east) arc (-90:90:1.3) -- node[above] {$\gamma_0$} ($(a0.west)+(0,2.6)$) arc (90:270:1.3);
\end{tikzpicture}}
{\begin{tikzpicture}[baseline]\draw[<-] (0.4,0) arc (0:360:0.4); \node at (0.6,-0.4) {$\gamma_0$}; \end{tikzpicture}} 
,$$
where $S \in \cP_\chi$, with $\chi$ a loop in $G$ of length $n+1$ and $\gamma$ is a loop in $\Gamma(\cP)$ which descends to $\chi$ with edges $a_i$. The loop labelled by $\gamma_0$ appearing in the denominator is the quantum dimension of the idempotent $\gamma_0$.
\end{defn}
\begin{rem}
At this point, we only require that $\cP$ be non-degenerate so that $\qdim \gamma_0 \neq 0$. We'll need non-degeneracy for Propositions \ref{prop:map} and \ref{prop:injective}, however.
\end{rem}


\begin{prop}
\label{prop:map}
If $\cP$ is semisimple, then the map $\cE$ is a map of planar algebras.
\end{prop}
\begin{proof}
We first observe that every planar tangle can be written as composition of a collection of the four elementary tangles described below. Lemmas \ref{lem:emptiness} through \ref{lem:cap} then establish that $\cE$ commutes with each of these elementary tangles.

\begin{lem}
Every $G$-planar tangle can be expressed as a composition of tangles from the following four classes:
\begin{description}
\item[emptiness:] For any vertex $g$ of $G$, 
$$\eset_g = \begin{tikzpicture}[baseline]
\node[circle, draw, dashed, fill=black!5, minimum width=2cm](c) at (0,0) {};
\node[left] at (c.west) {$\star$};
\end{tikzpicture}: \field \to \cP_g$$
\item[rainbow:] For any edge $e$ of $G$ and loop $\lambda$ in $G$ with basepoint at the end of $e$ (i.e. such that $e \lambda \bar{e}$ is a loop),
$$R_{e,\lambda} =\begin{tikzpicture}[baseline=1.5cm]
\draw[dashed, fill=black!5, rounded corners] (-3,0) rectangle (3,3);
\draw[dashed, fill=white, rounded corners] (-2,1) rectangle (2,2);
\node at (-2.2,1.5) {$\star$};
\node at (-3.2,1.5) {$\star$};
\foreach \p / \l in {-1.5/$\lambda_1$, -0.5/$\lambda_2$, 0.5/$\cdots$, 1.5/$\lambda_l$} {
	\draw[->] (\p,0) -- node[right] {\l} (\p,1);
}
\draw[->] (-2.5,0) -- node[right] {$e$} (-2.5,1) -- (-2.5,1.2) to[bend left=90] (2.5,1.2) -- (2.5,0);
\end{tikzpicture}: \cP_\lambda \to \cP_{e \lambda \bar{e}}$$
\item[tensor product:] For any two loops $\lambda$ and $\mu$ in $G$ with the same basepoint (i.e. such that $\lambda \mu$ is a loop),
$$\tensor_{\lambda,\mu} = \begin{tikzpicture}[baseline=1.5cm]
\draw[dashed, fill=black!5, rounded corners] (-3.8,0) rectangle (3.8,3);
\draw[dashed, fill=white, rounded corners] (-3.4,1) rectangle (-0.2,2.5);
\draw[dashed, fill=white, rounded corners] (0.2,1) rectangle (3.4,2.5);
\node at (-4.0,1.75) {$\star$};
\node at (-3.6,1.75) {$\star$};
\node at (0,1.75) {$\star$};
\foreach \p / \l in {-3.0/$\lambda_1$, -2.2/$\lambda_2$, -1.4/$\cdots$, -0.6/$\lambda_l$} {
	\draw[->] (\p,0) -- node[right] {\l} (\p,1);
}
\foreach \p / \l in {0.6/$\mu_1$, 1.4/$\mu_2$, 2.2/$\cdots$, 3.0/$\mu_m$} {
	\draw[->] (\p,0) -- node[right] {\l} (\p,1);
}
\end{tikzpicture}: \cP_\lambda \tensor \cP_\mu \to \cP_{\lambda \mu}$$
\item[cap:] For any edge $e$ of $G$ and loop $\lambda$ in $G$ with basepoint the start of $e$ (i.e. such that $e \bar{e} \lambda$ is a loop),
$$C_{e, \lambda} = \begin{tikzpicture}[baseline=1.5cm]
\draw[dashed, fill=black!5, rounded corners] (-3.8,0) rectangle (1.8,3);
\draw[dashed, fill=white, rounded corners] (-3.4,1) rectangle (1.4,2.5);
\node at (-4.0,1.75) {$\star$};
\node at (-3.6,1.75) {$\star$};
\foreach \p / \l in {-1.4/$\lambda_1$, -0.6/$\lambda_2$, 0.2/$\cdots$, 1.0/$\lambda_l$} {
	\draw[->] (\p,0) -- node[right] {\l} (\p,1);
}
\draw[->] (-2.2,1) .. controls (-2.2,0.4) and (-3,0.4) .. node[below] {$e$}  (-3,1);
\end{tikzpicture}: \cP_{e \bar{e} \lambda} \to \cP_{\lambda}$$
\end{description}
\end{lem}
\begin{proof}
Rotation tangles can be built out of rainbows and caps. Conjugating the given cap tangle by a rotation allows a cap at an arbitrary pair of boundary points. Conjugating the rainbow tangle by a rotation allows inserting a cup between any pair of boundary points. Caps and cups together generate the category of annular planar tangles. Applying caps at the boundary of the tensor product tangle produces arbitrary quadratic compositions, and compositions of these and annular planar tangles produce all planar tangles.
\end{proof}

\begin{lem}
\label{lem:emptiness}
The map $\cE$ intertwines the emptiness tangle.
\end{lem}
\begin{proof}
For any vertex $g \in G$
\begin{align*}
\eset(\cE(1 \in \field))(g) & = \eset(1 \in \field)(g) \\ & = 1_G(g) \\ & = 1 \\
\intertext{while}
\cE(\eset(1))(g) & = \cE(\eset \in \cP_\eset)(g) \\ 
&= \frac{
	\begin{tikzpicture}[baseline]\draw[<-] (0.4,0) arc (0:360:0.4); \node at (0.6,-0.4) {$g$}; \end{tikzpicture}
}{
	\begin{tikzpicture}[baseline]\draw[<-] (0.4,0) arc (0:360:0.4); \node at (0.6,-0.4) {$g$}; \end{tikzpicture}
}  \\ & = 1
\end{align*}
and so $\eset \circ \cE = \cE \circ \eset$ on $\field = \cP_\eset$.
\end{proof}

\begin{lem}
\label{lem:rainbow}
The map $\cE$ intertwines the rainbow tangles.
\end{lem}
\begin{proof}
Consider $S \in \cP_\chi$ and $\gamma$ a loop in $\Gamma$ with edges $a_i$ descending to the loop $\chi$ in $G$, and $e$ an edge in $\Gamma$ from a vertex $\eta$ to $\gamma_0$ which descends to the edge $\varepsilon$ in $G$. We have


\begin{align*}
R(\cE(S))(e \gamma \bar{e}) & = d(\eta)^{-1} \cE(S)(\gamma) \\
	& = d(\eta)^{-1} d(\gamma_0)^{-1}
	\begin{tikzpicture}[baseline=7ex]
\node(SSW) at (0,1.2) {};
\node(SNE) at (6,1.7) {};
\node(S)[draw, line width=2pt, rounded corners, fit=(SSW) (SNE)] {$S$};
\node(star)[right] at (S.east) {$\star$};
\foreach \index / \position in {0/0, 1/1.5, 2/3, n/6}
{
	\node(a\index)[draw, line width=2pt, circle] at (\position,0) {$a_\index$};
	\node[below] at (a\index.south) {$\star$};
	\draw[->] (a\index) -- node[right] {$\chi_\index$} ($(S.south east)!(a\index)!(S.south west)$);
}
\draw[->] (a1) -- node[below] {$\gamma_1$} (a0);
\draw[->](a2)  -- node[below] {$\gamma_2$} (a1); 
\draw[->] ($(a2)+(0.6,0)$) -- (a2);
\draw[dashed] (a2) -- (an);
\draw (an) -- +(-0.6,0);
%\draw[dashed] (a2)+(0.6,0) -- (an)+(-0.6,0);
\draw[<-] (an.east) arc (-90:90:1.3) -- node[above] {$\gamma_0$} ($(a0.west)+(0,2.6)$) arc (90:270:1.3);
\end{tikzpicture}
\intertext{while}
\cE(R(S))(e \gamma \bar{e}) & = d(\eta)^{-1} 
\begin{tikzpicture}[baseline=7ex]
\node(SSW) at (0,1.2) {};
\node(SNE) at (6,1.7) {};
\node(S)[draw, line width=2pt, rounded corners, fit=(SSW) (SNE)] {$S$};
\node(star)[right] at (S.east) {$\star$};
\foreach \index / \position in {0/0, 1/1.5, 2/3, n/6}
{
	\node(a\index)[draw, line width=2pt, circle] at (\position,0) {$a_\index$};
	\node[below=-3pt] at (a\index.south) {$\star$};
	\draw[->] (a\index) -- node[right] {$\chi_\index$} ($(S.south east)!(a\index)!(S.south west)$);
}
\draw[->] (a1) -- node[below] {$\gamma_1$} (a0);
\draw[->](a2)  -- node[below] {$\gamma_2$} (a1); 
\draw[->] ($(a2)+(0.6,0)$) -- (a2);
\draw[dashed] (a2) -- (an);
\draw (an) -- +(-0.6,0);
\node(e)[draw, line width=2pt, circle] at (-1.5,0) {$e$};
\node(eb)[draw, line width=2pt, circle] at (7.5,0) {$\bar{e}$};
\draw[->] (a0) -- node[below] {$\gamma_0$} (e);
\draw[->] (eb) -- node[below] {$\gamma_0$}(an);
\draw[->] (e) -- +(0,1.25) .. controls +(0,3) and ($(eb)+(0,3)$) .. node[below] {$\varepsilon$} ($(eb)+(0,1.25)$) -- (eb);
\draw[<-] (eb.east) arc (-90:90:1.5) -- node[above] {$\eta$} ($(e.west)+(0,3)$) arc (90:270:1.5);
\end{tikzpicture}.
\end{align*}
These are equal, by the definition of the dual element $\bar{e}$ in Equation \eqref{eq:dual}.

Both $\cE(R(S))(g)$ and $R(\cE(S))(g)$ are zero for any other $g$ not of the form $e \gamma \bar{e}$, so we conclude that $\cE \circ R =  R \circ \cE$ on $\cP_\chi$.
\end{proof}

\newcommand{\boxandpath}[7][0]{
%parameters:
%x-offset, box-name, vertex-name, strand-name, node-name
	\node(SSW) at (#2,1.2) {};
	\node(SNE) at ($(#1,0)+(6,0)+(#2,1.7)$) {};
	\node(S)[draw, line width=2pt, rounded corners, fit=(SSW) (SNE)] {$#3$};
	\node(star)[right] at (S.east) {$\star$};
	\foreach \index / \name / \position in {0/#7-0/0, 1/#7-1/1.5, 2/#7-2/3, #6/#7-n/6}
	{
		\node(\name)[draw, line width=2pt, circle] at ($(#1,0)+(\position,-1.2)+(SSW)$) {$#4_\index$};
		\node[below=-3pt]  at (\name.south) {$\star$};
		\draw[->] (\name) -- node[right] {$#5_\index$} ($(S.south east)!(\name)!(S.south west)$);
	}
	\draw[->] (#7-1) --  (#7-0);
	\draw[->](#7-2)  --  (#7-1); 
	\draw[->] ($(#7-2)+(0.6,0)$) -- (#7-2);
	\draw[dashed] (#7-2) -- (#7-n);
	\draw (#7-n) -- +(-0.6,0);
}
%


\begin{lem}
\label{lem:tensor}
The map $\cE$ intertwines the tensor tangles.
\end{lem}
\begin{proof}
Take $S \in \cP_\lambda$,  $T \in \cP_\mu$ and $\alpha$ a loop with edges $a_i$ descending to $\lambda$ and $\beta$ a loop with edges $b_i$ descending to $\mu$, and suppose $\alpha_0 = \beta_0$.
We calculate 
\begin{align*}
\cE(S \tensor T)(\alpha \beta) & \\
 = d(\alpha_0)^{-1} & \scalebox{0.8}{\begin{tikzpicture}[baseline=7ex]
\boxandpath{0}{S}{a}{\lambda}{A}{a};
\boxandpath{8}{T}{b}{\mu}{B}{b};
\draw[->] (b-0) -- (a-n);
\draw[<-] (b-n.east) arc (-90:90:1.3) -- ($(a-0.west)+(0,2.6)$) arc (90:270:1.3);
\end{tikzpicture} }\displaybreak[1] \\
 & = 
d(\alpha_0)^{-2}  \scalebox{0.8}{\begin{tikzpicture}[baseline=7ex]
\boxandpath{0}{S}{a}{\lambda}{A}{x};
\draw[<-] (x-n.east) arc (-90:90:1.3) -- ($(x-0.west)+(0,2.6)$) arc (90:270:1.3);
\end{tikzpicture}} \quad \times \\
& \qquad \qquad \times \quad \scalebox{0.8}{\begin{tikzpicture}[baseline=7ex]
\boxandpath{0}{T}{b}{\mu}{B}{x};
\draw[<-] (x-n.east) arc (-90:90:1.3) -- ($(x-0.west)+(0,2.6)$) arc (90:270:1.3);
\end{tikzpicture}} \displaybreak[1] \\
& =  \cE(S)(\alpha) \cE(T)(\beta) \\
& = \tensor(\cE(S), \cE(T))(\alpha \circ \beta)
\end{align*}
Here we have used the fact that for endomorphisms $f$ and $g$ of a simple idempotent $\alpha_0$, $\tr{fg} = d(\alpha_0)^{-1} \tr{f} \tr{g}$.
\end{proof}

\begin{lem}
\label{lem:cap}
The map $\cE$ intertwines the cap tangles.
\end{lem}
\begin{proof}
Consider $S \in \cP_{e\bar{e} \lambda}$ and $\gamma$ a cycle in $\Gamma$ descending to $\lambda$, and check that
\begin{align*}
\cE(C(S))(\gamma)& = d(\gamma_0)^{-1}
\begin{tikzpicture}[baseline=7ex]
\boxandpath[3]{0}{S}{\alpha}{\lambda}{g}{a};
\draw[->] ($(S.south east)!(1.5,0)!(S.south west)$) to[bend left=90] node[below] {$e$} ($(S.south east)!(0,0)!(S.south west)$);
\draw[<-] (a-n.east) arc (-90:90:1.3) -- ($(a-0.west)+(-2,2.6)$) arc (90:270:1.3) -- (a-0.west);
\end{tikzpicture}
\displaybreak[1] \\
& = d(\gamma_0)^{-1} \sum_{\substack{\gamma_0 \xrightarrow{\varepsilon} a \\ \varepsilon \mapsto e}} d(a)
\begin{tikzpicture}[baseline=7ex]
\boxandpath[3]{0}{S}{\alpha}{\lambda}{g}{a};
\node(e)[draw, line width=2pt, circle] at (0,0) {$\varepsilon$};
\node(eb)[draw, line width=2pt, circle] at (1.5,0) {$\bar{\varepsilon}$};
\draw[<-] (a-n.east) arc (-90:90:1.3) -- ($(e.west)+(0,2.6)$) arc (90:270:1.3);
\draw[->] (a-0) -- (eb);
\draw[->] (eb) -- (e);
\draw[->] (e) -- node[right] {$e$} ($(S.south east)!(e)!(S.south west)$);
\draw[->] (eb) -- node[right] {$\bar{e}$} ($(S.south east)!(eb)!(S.south west)$);
\end{tikzpicture} \displaybreak[1] \\
	& = \sum_{\varepsilon} d(a) \cE(S)(\varepsilon \bar{\varepsilon} \gamma) \\
	& = C(\cE(S))(\gamma) 
\end{align*}
Passing from the first to the second line we have used Lemma \ref{lem:IH}, which requires semisimplicity.
\end{proof}

This concludes the proof of Proposition \ref{prop:map}.
\end{proof}

\begin{prop}
\label{prop:injective}
The graph planar algebra embedding map is injective.
\end{prop}
\begin{proof}
By the hypothesis of nondegeneracy, $\cP$ has no non-trivial planar ideals.  (See for example \cite[Proposition 3.5]{0909.4099}.) Since $\cE$ takes the empty diagram to something manifestly non-zero (the constant $1$ function on length $0$ cycles), the kernel must be trivial. 
\end{proof}

Together, Propositions \ref{prop:map} and \ref{prop:injective} give Theorem \ref{thm:main}.

\section{Turaev-Viro state sums}
\label{sec:TV}
Given a unitary pivotal category, we can construct a TQFT invariant of $k$-manifolds, for $k=0,1,2,3$ via the usual construction. The original Turaev-Viro construction of this invariant is via a state sum formula, and a suitable generalization of this formula reproduces the result above (at least in the case that our category $\cC$ is unitary).

Given a unitary pivotal $2$-category $\cC$, we can associate a vector space $A(\Sigma)$ to each closed $2$-manifold $\Sigma$ and a number $A(M)$ to each closed $3$-manifold $M$. Moreover, there are relative versions of these invariants. A $3$-manifold $M$ with boundary $\Sigma$ gives a vector in $A(\Sigma)$. A $2$-manifold $\Sigma$ with boundary ...

\todo{continue ...}


% ----------------------------------------------------------------
\hfuzz5pt 
\newcommand{\urlprefix}{}
\bibliographystyle{gtart}
\bibliography{../../bibliography/bibliography}
% ----------------------------------------------------------------

This paper is available online at \arxiv{?????}, and at
\url{http://tqft.net/gpa}.

% ----------------------------------------------------------------
\end{document}
% ----------------------------------------------------------------