\documentclass[11pt]{amsart}

\newcommand{\cP}{\mathcal{P}}
\newcommand{\cV}{\mathcal{V}}
\newcommand{\Hom}{\operatorname{Hom}}
\newcommand{\tensor}{\otimes}

\theoremstyle{plain}
\newtheorem{prop}{Proposition}[section]
\newtheorem{thm}[prop]{Theorem}
\newtheorem{lem}[prop]{Lemma}
\newtheorem{cor}[prop]{Corollary}
\newtheorem*{cor*}{Corollary}
\newtheorem*{exc}{Exercise}
\newtheorem{defn}[prop]{Definition}         % numbered definition
\newtheorem{question}{Question}
%\newtheorem*{example}{Example}
\newenvironment{rem}{\noindent\textsl{Remark.}}{}  % perhaps looks better than rem above?
\numberwithin{equation}{section}
\newenvironment{example}{\noindent\textbf{Example.}}{\newline}  % perhaps looks better than rem above?

\newcommand{\su}{\mathfrak{su}(3)}

\begin{document}

Recall an oriented planar algebra $\cP$ has a vector space $\cP_\sigma$ for
each sign string $\sigma$, and these vector spaces form an algebra over the operad of
oriented planar tangles (the sign string determines the orientations
of strings meeting the boundary points of each disc).

\begin{defn}
The $\su$-spider is the oriented planar algebra $\cP$ with $\cP_\sigma
= \Hom_{U_q \su}(1, V^{\tensor \sigma})$, where $V^{\tensor \sigma}$
is the tensor product of a copy of $V$ for each $+$ in $\sigma$ and a
copy of $V^*$ for each $-$ in $\sigma$. The action of a planar tangle
is the usual representation theoretic one. Kuperberg gave generators
and relations for the $\su$-spider in \textbf{Greg Kuperberg} \emph{Spiders for rank {$2$} {L}ie algebras}, Comm.
  Math. Phys. 180 (1996) 109--151
\end{defn}

\begin{defn}
An $\su$-planar algebra $\cP$ is an oriented planar algebra along with
a map from Kuperberg's $\su$-spider.
\end{defn}
We allow either the generic case over $\mathbb{C}(q)$, or
specialisations with $q \in \mathbb{C}$.

An $\su$-planar algebra is $3$-colorable if the vector space
$\cP_\sigma$ is nonzero if and only if $\sigma$ sums to $0$ mod $3$.
Kuperberg's $\su$-spider is $3$-colourable since the representation
theory of $\su$ is graded by the centre of the corresponding Lie
group.

Given an $\su$-planar algebra $\cP$ we can extract a shaded planar
algebra $\cV$, by defining $\cV_{n,\pm} = \cP_{\sigma(n,\pm)}$, where
$\sigma(n,\pm)$ is the alternating sign string of length $n$ starting
with $\pm$.

We now note two examples of $\su$-planar algebras. \newline

\begin{example}
When $q$ is a root of unity of even order at least $6$, the
$\su$-spider has a nontrivial negligible ideal. We can take the
quotient by this ideal and produce a nondegenerate $\su$-planar algebra, that
is still $3$-colorable. The standard combinatorial rules for the
truncated tensor product at a root of unity determines the principal
graph of the associated shaded planar algebra: the vertices are the
simple objects appearing in $V^{\tensor \sigma(n,\pm)}$, and the edges
leaving a vertex corresponding to $X \subset V^{\tensor
\sigma(n,\pm)}$ connect to the simple summands of $X \tensor V^{\pm
(-1)^n}$.
\end{example}

\begin{example} (Deequivariantization, c.f. \textbf{Alain Brugui{\`e}res}, \emph{Cat\'egories pr\'emodulaires,
  modularisations et invariants des vari\'et\'es de dimension 3}, Math. Ann.
  316 (2000) 215--236)
The representation theory of $U_q \su$ at q a root of unity always
contains three transparent objects of dimension 1. We can apply
Brugui{\`e}res' deequivariantization procedure to produce another
$\su$-planar algebra. (That is, the dequivariantization is another
fusion category, which we can interpret as an oriented planar algebra,
with strand labelled by the image of the standard representation V.
The map from the $\su$-spider is just the deequivariantization map.)

Since every simple object is either in a free $Z/3Z$ orbit or is a fixed
point, it is easy to describe the combinatorial tensor product rule in
the deequivariantization. We'll write $[X]$ for the representative of a representation $X$. If $X$ is in a free orbit, then $[X]$  is simple, while if $X$ is a fixed point, then $[X]$ has $3$ simple summands.
If $X$ is in a free orbit,  then $[X] \tensor [V] = [X \tensor V]$, Similarly, if $X$ is a fixed point and $A$ is one of the simple summands of [X], $A \tensor [V] = 1/3 [X \tensor V]$ (each summand of $[X \tensor V]$ appears with multiplicity three). These rules
determine the principal graph of the corresponding shaded planar
algebra. We note that depending on the order of the root of unity, the
deequivariantization may or may not be $3$-colorable, depending on
whether the transparent objects are in $0$-grading or not.
\end{example}

\end{document}