%!TEX root = ../article.tex

\subsection{Background and Conventions}

The subject of quantum groups and quantum knot invariants suffers from a plethora of inconsistent conventions.  In this section we quickly recall important notions, specify our conventions, and give citations.  The reader is encouraged to skip this section and refer back to it when necessary.  In particular, most of Sections \ref{sec:invariants} and \ref{sec:identities} involve only diagram categories and do not require understanding quantum group constructions or their notation.  

\subsubsection{General conventions}
\begin{defn}
The $n^{th}$ quantum number $\qi{n}$ is defined as $$\frac{q^n-q^{-n}}{q-q^{-1}} = q^{n-1} + q^{n-3} + \cdots + q^{-n+1}.$$
 \end{defn}

Following \cite{MR2286123} the symbol $s$ will always denote a certain root of $q$ which will be specified as appropriate.

\subsubsection{Ribbon categories, diagrams, and knot invariants}

A ribbon category is a braided pivotal monoidal category satisfying a compatibility relation between the pivotal structure and the braiding.  See \cite{Selinger} for details (warning: that reference uses the word tortile in the place of ribbon). We use the optimistic convention where diagrams are read upward.

The key property of ribbon categories is that if $\cC$ is a ribbon category there is a functor $\cF$ from the category of ribbons labelled by objects of $\cC$ with coupons labelled by morphisms in $\cC$ to the category $\cC$ (see \cite{MR1036112, MR1292673, Selinger}).  In particular, if $V$ is an object in $\cC$ and $L_V$ denotes a framed oriented link $L$ labelled by $V$, then 
$$\tilde{\mathcal{J}}_{\cC,V}: L \mapsto \cF(L_V) \in \End{\id}$$ 
is an invariant of oriented framed links (due to Reshetikhin-Turaev \cite{MR1036112}). Whenever $V$ is a simple object, the invariant depends on the framing through a `twist factor'. That is, two links $L$ and $L'$ which are the same except that $w(L)=w(L')+1$, where $w$ denotes the writhe, have invariants satisfying $\tilde{\mathcal{J}}_{\cC,V}(L) = \theta_V \tilde{\mathcal{J}}_{\cC,V}(L')$ for some $\theta_V$ in the ground field (not depending on $L$). Thus $\tilde{\mathcal{J}}_{\cC,V}$ can be modified to give an invariant which does not depend on framing.  

\begin{thm}
Let $\J{\cC}{V}(L)=\theta_V^{-w(L)} \tilde{\mathcal{J}}_{\cC,V}(L)$.  
Then $\J{\cC}{V}(L)$ is an invariant of links.
\end{thm}

Given any pivotal tensor category $\cC$ (in particular any ribbon category) and a chosen object $X \in \cC$, one can consider the full subcategory whose objects are tensor products of $X$ and $X^*$.  This subcategory is more convenient from the diagrammatic perspective because one can drop the labeling of strands by objects and instead assume that all strands are labelled by $X$ (here $X^*$ appears as the downward oriented strand).  Thus this full subcategory becomes a spider \cite{MR1403861}, which is an oriented version of Jones's planar algebras \cite{math.QA/9909027}.  If $X$ is symmetrically self-dual then this full subcategory is an unoriented unshaded planar algebra in the sense of \cite{MR2559686}.

Often one only describes the full subcategory (via diagrams) but wishes to recover the whole category.  If the original category was semisimple and $X$ is a tensor generator, then this can be acheived via the idempotent completion.  This is explained in detail in \cite{MR2559686, MR1217386, MR1403861}.  The simple objects in the idempotent completion are the minimal projections in the full subcategory.

%We will abuse notation throughout and use the same notation to refer to the whole category and to the full subcategory.  We will also implicitly identify objects and projections.

\subsubsection{Conventions for knot polynomials and their diagram categories} \label{conventions}

In this subsection we give our conventions for the following knot polynomials: the Jones polynomial, the colored Jones polynomials, the HOMFLYPT polynomial, the Kauffman polynomial, and the Dubrovnik polynomial.  Each of these comes in a framed version as well as an unframed version.  The framed versions of these polynomials (other than the colored Jones polynomial) are given by simple skein relations.  These skein relations can be thought of as defining a ribbon category whose objects are collections of points (possibly with orientations) and whose morphisms are tangles modulo the skein relations and modulo all negligible morphisms (see \S \ref{subsec:semisimplification}).

We will often use the same name to refer to the knot polynomial and the category.  This is very convenient for keeping track of conventions.  The HOMFLYPT skein category and the Dubrovnik skein category are more commonly known as the Hecke category and the BMW category.

Contrary to historical practice, we normalize the polynomials so they are multiplicative for disjoint union. In particular, the invariant of the empty link is $1$, while the invariant of the unknot is typically nontrivial.

\paragraph{The Temperley-Lieb category}

We first fix our conventions for the Temperley-Lieb ribbon category $\TL(s)$.  Let $s$ be a complex number with $q=s^2$. 

The objects in Temperley-Lieb are natural numbers (thought of as disjoint unions of points).  The morphism space $\Hom{}{a}{b}$ consists of linear combinations of planar tangles with $a$ boundary points on the bottom and $b$ boundary points on the top, modulo the relation that each closed circle can be replaced by a multiplicative factor of $\qi{2} = q + q^{-1}$.  The endomorphism space of the object consisting of $k$ points will be called $\TL_{2k}$.

The braiding (which depends on the choice of $s = q^{\frac{1}{2}}$) is given by
\begin{equation*}
\inputtikz{ResolvedCrossing}.
\end{equation*}

We also use the following important diagrams.

\begin{itemize}
\item The Jones projections in $\TL_{2n}$:
$$e_i=\inputtikz{JonesProjection}, \, i \in \{ 1, \ldots, n-1\};$$
\item
The Jones-Wenzl projection  $\JW{n}$ in $\TL_{2n}$ \cite{MR873400} which is the unique projection
with the property
$$\JW{n} e_i = e_i \JW{n} =0, \, \text{for each $i \in \{1, \ldots, n-1 \}$}.$$
\end{itemize}

\paragraph{The Jones polynomial} 

The framed Jones polynomial $\widetilde{J}$ (or Kauffman bracket) is the invariant coming from the ribbon category $\TL(s)$.  In particular, it is defined for unoriented framed links by 
\begin{align}
\bigcirc & = q + q^{-1} \notag \\
\intertext{and}
\overcrossing & =is \identity - i s^{-1} \cupcap. \label{eq:jones-skein}
\end{align}

This implies that
\begin{equation*}
\positivetwist  = i s^3  \verticalstrand \qquad \text{and} \qquad \negativetwist  =  - i s^{-3} \verticalstrand
\end{equation*}
so the twist factor is $i s^{3}$. 

The framing-independent Jones polynomial is defined by $J(L)=(-is^{-3})^{\writhe(L)} \widetilde{J}(L)$.  It satisfies the following version of the Jones skein relation
\begin{align}
q^2 \Oovercrossing - q^{-2} \Oundercrossing & =-i s \overcrossing - i s^{-1} \undercrossing \notag \\
    & = q \identity - q^{-1} \identity  \notag \\
    & = (q - q^{-1}) \Oidentity. \label{eq:oriented-jones-skein} \notag
\end{align}

\paragraph{The colored Jones polynomial}

The framed colored Jones polynomial $\Jf{\SL{2}}{(k)}(K)$ is the invariant coming from the simple projection $\JW{k}$ in $\TL(s)$.  The twist factor is $i^{k^2} s^{k^2+2k}$.

\paragraph{The HOMFLYPT polynomial}

The framed HOMFLYPT polynomial is given by the following skein relation.

\begin{equation}
w \Oovercrossing - w^{-1} \Oundercrossing = z \Oidentity
\end{equation}
\begin{align}
\Opositivetwist & = w^{-1} a \Overticalstrand & \Onegativetwist & = a w^{-1} \Overticalstrand
\end{align}
%(This differs from that given in \cite{MR1710999} by replacing $a$ with $a^{-1}$, $v$ with $v^{-1}$, and $z$ with $s-s^{-1}$.)
The twist factor is just $w^{-1}a$.

Thus, the framing-independent HOMFLYPT polynomial is given by the following skein relation.

\begin{equation}
a \Oovercrossing - a^{-1} \Oundercrossing = z \Oidentity
\end{equation}


\paragraph{The Kauffman polynomial} The Kauffman polynomial comes in two closely related versions, known as the Kauffman and Dubrovnik normalizations. Both are invariants of unoriented framed links.
The framed Kauffman polynomial $\widetilde{\Kauffman}$ is defined by
\begin{equation} \label{eq:kauffman}
\overcrossing + \undercrossing = z\left(\identity + \cupcap\right)
\end{equation}
\begin{align}
\positivetwist & = a \verticalstrand & \negativetwist & = a^{-1} \verticalstrand.
\end{align}

Here the value of the unknot is $\frac{a+a^{-1}}{z} - 1$.

The framed Dubrovnik polynomial $\widetilde{\Dubrovnik}$ is defined by
\begin{equation} \label{eq:dubrovnik}
\overcrossing - \undercrossing = z\left(\identity - \cupcap\right)
\end{equation}
\begin{align}
\positivetwist & = a \verticalstrand & \negativetwist & = a^{-1} \verticalstrand.
\end{align}
Here the value of the unknot is $\frac{a-a^{-1}}{z} + 1$.

In both cases, the twist factor is $a$.  The unframed Kauffman and Dubrovnik polynomials do not satisfy any conveniently stated skein relations.  The Kauffman and Dubrovnik polynomials are closely related to each other by $$\widetilde{\Dubrovnik}(L)(a, z) =  i^{-w(L)}  (-1)^{\#L} \widetilde{\Kauffman}(L)(i a,-i z),$$ where $\#L$ is the number of components of the link and $w(L)$ is the writhe of any choice of orientation for $L$ (which turns out not to depend, modulo $4$, on the choice of orientation).  This is due to Lickorish \cite{MR966143}  \cite[p. 466]{MR958895}.

\paragraph{Kuperberg's $G_2$ Spider}
We recall Kuperberg's skein theoretic description of the quantum $G_2$ knot invariant \cite{MR1403861, MR1265145} (warning, there is a sign error in the former source).  Kuperberg's $q$ is our $q^2$ (which agrees with the usual quantum group conventions). The quantum $G_2$ invariant is defined by
$$
\overcrossing = \frac{1}{1+q^{2}} \mathfig{0.04}{G2/I} + \frac{1}{1+q^{-2}} \mathfig{0.08}{G2/H} + \frac{1}{q^{2}+q^{4}} \cupcap + \frac{1}{q^{-2}+q^{-4}} \identity
$$

where the trivalent vertex satisfies the following relations

\begin{align*}
\mathfig{0.04}{G2/loop} &= q^{10} + q^{8} + q^2 + 1 + q^{-2} + q^{-8} + q^{-10} \\
\mathfig{0.04}{G2/lollipop} &= 0 \\
\mathfig{0.04}{G2/bigon} &= -\left(q^6 + q^4 + q^2 + q^{-2} + q^{-4} + q^{-6}\right) \mathfig{0.0025}{G2/strand}\\
\mathfig{0.06}{G2/triangle} &=  \left(q^4 + 1 +q^{-4}\right)\mathfig{0.06}{G2/vertex}\\
\mathfig{0.06}{G2/square} &=  -\left(q^2+q^{-2}\right) \left(\mathfig{0.04}{G2/I} + \mathfig{0.08}{G2/H} \right)+ \left(q^2+1+q^{-2}\right) \left(\cupcap + \identity \right) \\
\mathfig{0.06}{G2/pentagon} &=  \mathfig{0.06}{G2/tree1} + \mathfig{0.06}{G2/tree2} + \mathfig{0.06}{G2/tree3} + \mathfig{0.06}{G2/tree4} + \mathfig{0.06}{G2/tree5} \\ 
& \qquad - \mathfig{0.06}{G2/forest1}-\mathfig{0.06}{G2/forest2}-\mathfig{0.06}{G2/forest3}-\mathfig{0.06}{G2/forest4}-\mathfig{0.06}{G2/forest5}.
\end{align*}

\subsubsection{Quantum groups}
\label{sec:quantumgroups}

Quantum groups are key sources of ribbon categories.  If $\mathfrak{g}$ is a complex semisimple Lie algebra, let $U_s(\mathfrak{g})$ denote the Drinfel'd-Jimbo quantum group, and let $\Rep{U_s(\mathfrak{g})}$ denote its category of representations.  This category is a ribbon category and hence given a quantum group and any representation the Reshetikhin-Turaev procedure gives a knot invariant.

We follow the conventions from \cite{MR2286123}.  See \cite[p. 2]{MR2286123} for a comprehensive summary of how his conventions line up with those in other sources. (In particular, our $q$ is the same as both Sawin's $q$ and Lusztig's $v$.) We make one significant change: we only require that the underlying Lie algebra $\mathfrak{g}$ be semi-simple rather than simple.  This does not cause any complications because $U_s(\mathfrak{g_1}\oplus \mathfrak{g_2}) \cong U_s(\mathfrak{g_1}) \boxtimes U_s(\mathfrak{g_2})$.

In particular, following  \cite{MR2286123}, we have variables $s$ and $q$ and the relation $s^L = q$ where $L$ is the smallest integer such that $L$ times any inner product of weights is an integer. The values of $L$ for each simple Lie algebra appear in Figure \ref{fig:quantum-group-data}. The quantum group itself and its representation theory only depend on $q$, while the braiding and the ribbon category depend on the additional choice of $s$.

For the quantum groups $U_s(\SO{n})$ we denote by $\vRep(U_q(\SO{n}))$ the collection of representations whose highest weight corresponds to a vector representation of $\SO{n}$ (that is, a representation of $\SO{n}$ which lifts to the non-simply connected Lie group $SO(n)$).  Note that the braiding on the vector representations does not depend on $s$ so we use $q$ as our subscript here instead.

Often the pivotal structure on a tensor category is not unique, and indeed for the representation theory of a quantum group the pivotal structures are a torsor over the group of maps from the weight lattice modulo the root lattice to $\pm 1$. In general there is no `standard' pivotal structure, but for the representation theory of a quantum group there is both the usual one defined by the Hopf algebra structure of the Drinfel'd-Jimbo quantum group, and Turaev's `unimodal' pivotal structure, $\uRep{U_s(\mathfrak{g})}$. Changing the pivotal structure by $\chi$, a map from the weight lattice modulo the root lattice to $\pm 1$, has two major effects: it changes both the dimension of an object and its twist factor by multiplying by  $\chi(V)$. 
%We will only consider the unimodal structure in situations where the weight lattice mod the root lattice is a cyclic group of even order, in which case the $\chi$ that relates the unimodal and standard pivotal structures is the only nontrivial map to $\pm 1$. 
The unimodal pivotal structure is characterized by the condition that every self-dual object is symmetrically self-dual.
One important particular case is that $\uRep{U_s(\SL{2})} \cong \TL(-is)$ \cite{0810.0084, 1002.0555}.

The twist factor for an irreducible representation $V$ is determined by the action of the ribbon element, giving (for the standard pivotal structure) $q^{\langle \lambda,\lambda+2\rho\rangle}$ where $\lambda$ is the highest weight of $V$.   Note that since $\langle \lambda,\lambda+2\rho\rangle \in \frac {1}{L} \mathbb{Z}$ (where $L$ is the exponent of the weight lattice mod the root lattice), the twist factor in general depends on a choice of $s=q^{\frac {1}{L}}$.

For $V=V_{(k)}$, the representation of $U_s(\SL{2})$ with highest weight $k$, the twist factor is $s^{k^2+2k}$ (notice this is the same as the $k$-colored Jones polynomial for $k$ even; for $k$ odd the twist factors differ by a sign as predicted by $\uRep{U_s(\SL{2})} \cong \TL(-is)$).  For $V=V^\natural$, the standard representation of $U_s(\SL{n})$, the twist factor is $s^{n^2-1}$. For the standard representations of $\SO{2n+1}$, $\SP{2n}$ and $\SO{2n}$ the twist factors are $q^{4n}$, $q^{2n+1}$ and $q^{2n-1}$ respectively.
 The twist factor for the representation $V_{ke_1}$ of $\SO{2n+1}$ is $q^{2k^2+(4n-2)k}$. 
Note the the representation $V_{ke_1}$ of $\SO{3}$ is the representation $V_{(2k)}$ of $\SL{2}$ and in this case the twist factor agrees with the first one given in this paragraph.
The twist factor for the representation $V_{ke_1}$ of $\SO{2n}$ is $q^{k^2+2(n-1)k}$. Later we will need the twist factors for the representations $V_{3e_1}$, $V_{e_{n-1}}$ and $V_{3e_{n-1}}$  of $\SO{2n}$. These are $q^{6n+3}$, $q^{\frac{1}{4}n(2n-1)}$ and $q^{\frac{3}{4}n(2n+1)}$. The twist factor for the $7$-dimensional representation of $G_2$ is $q^{12}$.

The invariants of the unknot are just the quantum dimensions. For the standard representations of $\SL{n}$, $\SO{2n+1}$, $\SP{2n}$ and $\SO{2n}$ these are $\qiq{n}{q}, \qiq{2n}{q^2} + 1,\qiq{2n+1}{q} - 1$ and $\qiq{2n-1}{q} + 1$ respectively.

The invariants of the standard representations are specializations of the HOMFLYPT or Dubrovnik polynomials. Written in terms of the framing-independent invariants, we have
\begin{align}
\label{eq:homflypt-A}%
\text{HOMFLYPT}(L)(q^n,q-q^{-1}) & = \J{\SL{n}}{V^\natural}(L)(q), \\
\label{eq:dubrovnik-B}%
\text{Dubrovnik}(L)(q^{4n}, q^2 - q^{-2}) & = \J{\SO{2n+1}}{V^\natural}(L)(q), \\
\label{eq:dubrovnik-C}%
\text{Dubrovnik}(L)(-q^{2n+1}, q-q^{-1}) & =  (-1)^{\sharp L} \J{\SP{2n}}{V^\natural}(L)(q), \\
\intertext{and}
\label{eq:dubrovnik-D}%
\text{Dubrovnik}(L)(q^{2n-1}, q - q^{-1}) & = \J{\SO{2n}}{V^\natural}(L)(q).
\end{align}

These identities are `well-known', but it's surprisingly hard to find precise statements in the literature, and we include these mostly for reference. The identities follow immediately from Theorem \ref{thm:eigenvalues} below, and the fact that the eigenvalues of the braiding on the natural representations of $\SL{n}, \SO{2n+1}, \SP{2n}$ and $\SO{2n}$ are 
$(-s^{-n-1}, s^{n-1}), (q^{-4n}, q^2, -q^{-2} ), (-q^{-2n-1}, q, -q^{-1})$ and $(q^{-2n+1}, q, -q^{-1})$ respectively. The sign in Equation \eqref{eq:dubrovnik-C} appears because Theorem \ref{thm:eigenvalues} does not apply immediately to the natural representation of $\SP{2n}$, which is antisymmetrically self-dual.  Changing to the unimodal pivotal structure fixes this, introduces the sign in the knot invariant, and explains the discrepancy between the value of $a$ in the specialization of the Dubrovnik polynomial and the twist factor for the natural representation of $\SP{2n}$.

We'll show using techniques inspired by \cite{MR1854694, MR1710999, MR2132671} that several of these identities between knot polynomials come from functors between the corresponding categories.

\subsubsection{Comparison with the \code{KnotTheory`} package}
The HOMFLYPT and Kauffman polynomials defined here agree with those available in the \code{Mathematica} package \code{KnotTheory`} (available at the Knot Atlas \cite{katlas}), except that in the package the invariants are normalized so that their value on the unknot is $1$. The Jones polynomial in the package uses `bad' conventions from the point of view of quantum groups. You'll need to substitute $q \mapsto q^{-2}$, and then multiply by $q+q^{-1}$ to get from the invariant implemented in \code{KnotTheory`} to the one described here. The $G_2$ spider invariant described here agrees with that calculated using the \code{QuantumKnotInvariant} function in the package. The function \code{QuantumKnotInvariant} in the package calculates the framing-independent invariants from quantum groups described here.

\subsubsection{Semisimplification} \label{subsec:semisimplification}

Suppose that $\cC$ is a spherical tensor category which is $\Complex$-linear and which is idempotent complete (every projection has a kernel and an image).  Let $N$ be the collection of negligible morphisms ($f$ is negligible if $\tr{f g} = 0$ for all $g$).  Call a collection of morphisms $I \in \cC$ an ideal if $I$ is closed under composition and tensor product with arbitrary morphisms in $\cC$. We recall the following facts:

\begin{itemize}
\item $N$ is an ideal.
\item Any ideal in $\cC$ is contained in $N$.
%\item $\cC^{ss} = \cC / N$ is a semisimple pivotal tensor category.
%\item Any pivotal functor from $\cC$ to a semisimple category factors through $\cC^{ss}$.
\item If $\cC$ semisimple then $N = 0$.
\item If $\cC$ is abelian, then $\cC/N = \cC^{ss}$ is semisimple.
\item If $\cD$ is psuedo-unitary (pivotal, and all quantum dimensions are positive, up to a fixed Galois conjugacy) and $\cF: \cC \rightarrow \cD$ is a functor of pivotal categories, then $\cF$ is trivial on $N$.
\end{itemize}

There are some technical issues which, while not immediately relevant to this paper, are important to keep in mind when dealing with semisimplifications. First, $\cC/N$ may not always be semisimple.  Furthermore, if $\cD$ is semisimple but not psuedo-unitary there may be a functor $\cF: \cC \rightarrow \cD$ which does not factor through $\cC/N$.

\begin{ex}
If $q$ is a root of unity, and $a$ is not an integer power of $q$, then the quotient of the Dubrovnik category at $(a, z=q-q^{-1})$ by negligibles is not semisimple \cite[Cor.~7.9]{MR2132671}.
\end{ex}

\begin{ex}
This example is adapted from \cite[Remark 8.26]{MR2183279}.  Let $$\cE = \vRep U_{q=\exp(\frac{2 \pi i}{10})}(\SO{3})$$ be the Yang-Lee category.  This fusion category has two objects, $1$ and $X$, satisfying $X\otimes X \cong X \oplus 1$. The object $X$ has dimension the golden ratio.  Let $\cE'$ be a Galois conjugate of $\cE$ where $X$ has dimension the conjugate of the golden ratio.  Let $\cD = \cE \boxtimes \cE'$; this is a non-pseudo-unitary semisimple category.  Note that $X \boxtimes X$ is a symmetrically self-dual object with dimension $-1$.  Hence there is a functor from $\cC= TL_{d=-1} \rightarrow \cD$ sending the single strand to $X\boxtimes X$ (see \S \ref{sec:ribbonfunctors} for more details).  The second Jones-Wenzl idempotent is negligible in $TL_{d=-1}$ but it is not killed by this functor.
\end{ex}

%In particular suppose we have a functor $\cF: \cC \rightarrow \cD$.  Then we get a functor $\cC \rightarrow \cD^{ss}$.  Since $\cD^{ss}$ is semisimple, this functor factors through $\cC^{ss}$ so we get a functor $\cC^{ss} \rightarrow \cD^{ss}.$  Furthermore this functor is automatically faithful because $\cC^{ss}$ has no ideals.

For further details see \cite{MR1686423} \cite{MR1217386} and \cite[Proposition 5.7]{MR2348906}.

\subsubsection{Quantum groups at roots of unity}

When $s$ is a root of unity, by $\Rep{U_s(\mathfrak{g})}$ we mean the semisimplified category of tilting modules of the Lusztig integral form.  We only ever consider cases where $q$ is a primitive $\ell$th root of unity with $\ell$ large enough in the sense of \cite[Theorem 2]{MR2286123}.  The key facts about this category are described in full generality in \cite{MR2286123} (based on earlier work by Andersen, Lusztig, and others):

\begin{itemize}
\item The isomorphism classes of simple objects correspond to weights in the fundamental alcove.  (Be careful, as when the Lie algebra is not simply laced the shape of the fundamental alcove depends on the factorization of the order of the root of unity \cite[Lemma 1]{MR2286123}.
\item The dimensions and twist factors for these simple objects are given by specializing the formulas for dimensions and twist factors from generic $q$.
\item The tensor product rule is given by the quantum Racah rule \cite[\S 5]{MR2286123}. 
\end{itemize}

\subsubsection{Modularization}
\label{sec:modularization}

We review the theory of modularization or deequivariantization developed by M\"uger \cite{MR1749250} and Brugui\`eres \cite{MR1741269}.  Suppose that $\cC$ is a ribbon category and that $\cG$ is a collection of invertible ($X \tensor X^* \iso \mathbf{1}$, or equivalently $\dim X = \pm 1$) simple objects in $\cC$ which satisfy four conditions:
\begin{itemize}
\item $\cG$ is closed under tensor product.
\item Every $V \in \cG$ is transparent (that is, the positive and negative braidings with any object $W \in \cC$ are equal).
\item $\dim V = 1$ for every $V \in \cG$. 
\item The twist factor $\theta_V$ is $1$ for every $V \in \cG$. 
\end{itemize}

Let $\cC // \cG$ denote the M\"uger-Brugui\`eres deequivariantization.  There is a faithful essentially surjective functor $\cC \rightarrow \cC // \cG$.  This functor is not full because in the deequivariantization there are more maps:  in $\cC // \cG$ every object in the image of $\cG$ becomes isomorphic to the trivial object. %In particular, if $\cG$ is a full subcategory equivalent to the representation theory of an abelian group $G$, with the symmetric braiding,  then we only need to check the second condition, transparency.  

We'll often write $\cC // X$ to denote the deequivariantization by the collection of tensor powers of some object $X$.

A ribbon functor between premodular (that is, ribbon and fusion) categories $\cF: \cC \rightarrow \cC'$ is called a modularization if it is dominant (every simple object in $\cC'$ appears as a summand of an object in the image of $\cF$) and if $\cC'$ is modular.

\begin{thm}
Suppose that $\cC$ is a premodular category whose global dimension is nonzero.  Any modularization is naturally isomorphic to $\cF: \cC \rightarrow \cC//\cG$ where $\cG$ is the set of all transparent objects.  A modularization exists if and only if every transparent object has dimension $1$ and twist factor $1$.
\end{thm}

In Section \ref{sec:KM}, we compute modularizations using the following lemma.

\begin{lem}
\label{lem:free-quotient}
Suppose $\cM$ is a modular $\tensor$-category, which is a full subcategory of a tensor category $\cC$. Denote by $\cI$ the subcategory of invertible objects in $\cC$, and assume they are all transparent, and that the group of objects $\cI$ acts (by tensor product) freely on $\cC$. 
Then the orbits of $\cI$ each contain exactly one object from $\cM$, and the modularization $\cC // \cI$ is equivalent to $\cM$.
\end{lem}

For further detail, see \cite[\S 1.3-1.4]{MR1854694}.  Related notions appear in the physics literature, for example \cite{0808.0627}.

