text:
Q: Redirect any sub-domain to main domain by editing DNS Record Is it possible to redirect any sub-domain to the main domain by editing DNS Record in Windows Server? (ASP.NET WinForm)
example:
abc.domain.com >> www.domain.com
xyz.domain.com >> www.domain.com
What DNS Record should I add?
Can a CNAME record solve this problem?
Example of a CNAME record:
Name: *.domain.com
Type: CNAME
Data: www.domain.com

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Redirect 301 / http://www.domain.com

I don't know how this can work on IIS, I highly doubt it supports .htaccess files, but my guess is that you can do this with ISAPI filters.

redpajama_set_name: RedPajamaStackExchange


text:
\section{Introduction}
Two-dimensional conformal field theories (CFT)~\cite{BPZ1984b,DMS_yellow_CFT} form one of the most developed classes of quantum field theories.
The infinite-dimensional symmetry of a CFT allows one to identify the primary fields of the theory and the operator product expansion (OPE) of the primary fields in a closed form, sometimes leading to the exact solution.
Due to their integrability, CFTs have played important roles in string theory~\cite{GreenSchwarzWitten_superstring}, condensed matter physics~\cite{Ludwig_CFT_condensed_matter, Hansson_QHall_CFT}, and statistical physics~\cite{Cardy_lecture_2008,Mussardo-statistical_field_theory}.

Vertex operator algebras (VOA)~\cite{FLM-VOAs_Monster,Lepowsky_Li-VOA} and their representation theory give an algebraic counterpart of CFTs~\cite{Huang-CFT_and_VOA}.
According to the dictionary, a VOA gives the chiral algebra of a CFT, and the irreducible modules of the VOA correspond to the primary fields of the CFT.
Furthermore, the OPE of primary fields is translated into a tensor product of modules of the VOA.
Therefore, we could say that the study of a CFT partly comes down to the study of the category of modules of the corresponding VOA as a tensor category.

Quantum groups~\cite{Drinfeld1986} are algebras whose module categories are clearly tensor categories.
There are several known examples of equivalence between module categories of VOAs and those of quantum groups as tensor categories.
Prominent examples include the Kazhdan--Lusztig-type duality~\cite{Drinfeld1989,KL-tensor_structures_affine_Lie,Finkelberg1996,McRae-nonneg_integer_level_affine_Lie_algebra_tensor_cat}, and the duality between the triplet VOA and the small quantum $\mathfrak{sl}_{2}$~\cite{FeiginGainutdinovSemikhatovTipunin2006a, FeiginGainutdinovSemikhatovTipunin2006b,NagatomoTsuchiya2011,KondoSaito2011,TsuchiyaWood2013,GannonNegron-quantum_sltwo_and_log_VOAs,CreutzigLentnerRupert}.
The present work is to give another example, namely, the Virasoro VOA with a generic central charge and the quantum universal enveloping algebra of $\mathfrak{sl}_{2}$ with a generic quantization parameter denoted by $\UEA_{q}(\mathfrak{sl}_{2})$.

\subsubsection*{Relation to previous work}
This paper is a continuation of our previous work~\cite{KoshidaKytola2022}.
This work might have a different flavor compared to other works concerning categorical equivalence.
To be clear about our viewpoint and to motivate our method, it might be worthwhile to give a quick overview of the line of research.

Since the early stages of the research of CFT, hidden quantum group symmetry has been observed in several places~\cite{MR-comment_on_quantum_group_symmetry_in_CFT,PS-common_structures_between_finite_systems_and_CFTs,RRR-contour_picture_of_quantum_groups_in_CFT, FW-topological_representation_of_Uqsl2,SV-quantum_groups_and_homology_of_local_systems,Varchenko-multidimensional_hypergeometric_functions_and_representation_theory_of_Lie_algebras_and_quantum_groups,GRS-quantum_groups_in_2d_physics}.
In~\cite{KP-conformally_covariant_bdry_correlations}, the authors established one of the most concrete versions of the connection between a quantum group and CFT;
They gave maps from representations of $\UEA_{q}(\mathfrak{sl}_{2})$ to certain correlation functions of CFT.
Although their motivation was in application to the theory of Schramm--Loewner evolution~\cite{Schramm2000,BB-CFTs_of_SLEs}, their results were also of representation theoretical importance.
In fact, one consequence, among others, of their construction is that the asymptotic behavior of a correlation function when two points are close to each other is governed by the branching and associativity of the tensor product of representations of $\UEA_{q}(\mathfrak{sl}_{2})$.

In our previous work~\cite{KoshidaKytola2022}, we enhanced the results of \cite{KP-conformally_covariant_bdry_correlations} in the language of VOA.
Specifically, we identified the generic Virasoro VOA as the relevant VOA, determined the fusion rules among simple modules from the first row of the Kac table, and proved that the associativity of intertwining operators is governed by the quantum group $\UEA_{q}(\mathfrak{sl}_{2})$.

In the present work, we examine the category-theoretical implications of the results of the previous one~\cite{KoshidaKytola2022}.
As we shall see in the rest of the paper, the above-mentioned results allow us to establish the categorical equivalence between the category generated by the simple modules of the Virasoro VOA in the first row of the Kac table and that of finite-dimensional $\UEA_{q}(\mathfrak{sl}_{2})$-modules of type I (see Section~\ref{sect:quantum_group} for definition).
Also, we add discussion about the braiding in this paper.

\subsubsection*{Relation to other works}
The category of $C_{1}$-cofinite modules of the simple Virasoro VOA has been studied in \cite{CJORY-tensor_categories_arising_from_Virasoro_algebra} and has been shown to be a braided tensor category.
Our category of interest is a tensor subcategory of the category studied in \cite{CJORY-tensor_categories_arising_from_Virasoro_algebra}, therefore, it follows that our category is a braided tensor category. However, the relation to the quantum group is a new result.

In \cite{GannonNegron-quantum_sltwo_and_log_VOAs}, it has been established that the module category of the quantum $SL(2)$ at a root of unity and a certain module category of the Virasoro VOA in the logarithmic setting are equivalent.
Their approach employs the general theory of tensor categories, as well as the characterization of the module category of the quantum $SL(2)$.
Assuming that their method can be used in the case of generic central charges, we expect that it will give a more conceptual explanation to the question why the Virasoro VOA should be related to the quantum $SL(2)$.
Our approach is less conceptual in that regard, but is more direct and concrete, and we believe that it gives an alternative perspective.
We also mention the recent work \cite{McRaeYang2022} that studied the $\mathfrak{sl}_{2}$-type structure in the module category of the Virasoro VOA at central charge $25$.

\subsubsection*{Organization of the paper}
In the following Section~\ref{sect:quantum_group}, we give an account of the quantum group $\UEA_{q}(\mathfrak{sl}_{2})$
and fix several details of e.g. the Clebsch--Gordan coefficients and the universal $R$-matrix.
In Section~\ref{sect:module_cat_VOA}, we recall the necessary background information on the module category of a VOA.
In particular, we quickly look at the Huang--Lepowsky theory that equips the module category of a VOA with the structure of a ribbon tensor category.
We focus our attention on the generic Virasoro VOA in Section~\ref{sect:generic_VirVOA} and review the known results from our previous work.
Section~\ref{eq:cat_equiv} is the main part of this paper, where we define the first row module category of the generic Virasoro VOA and examine its structure in detail. Consequently, we will prove that it is equivalent to the category of type I modules of $\UEA_{q} (\mathfrak{sl}_{2})$.

\subsection*{Acknowledgements}
The author is grateful to Kalle Kyt{\"o}l{\"a}, Eveliina Peltola, and Ingo Runkel for fruitful discussions.
This work was supported by Academy of Finland (No.~340965).

\section{Quantum group}
\label{sect:quantum_group}
In this section, we give a brief overview of known facts about the quantum group $\UEA_{q} (\mathfrak{sl}_{2})$ and its representations.
We only consider the case where $q\in\mathbb{C}^{\times}$ is not a root of unity and simply write $\UEA_{q}$ for $\UEA_{q} (\mathfrak{sl}_{2})$.
Further details about quantum groups can be found in \cite{Lusztig1993, Kassel-quantum_groups,chari1995guide}.

\subsection{Quantum group \texorpdfstring{$\UEA_{q}$}{Uq}}
The algebra $\UEA_{q}$ is a unital associative $\mathbb{C}$-algebra generated by
$K, K^{-1}, E$, and $F$ subject to the relations
\begin{align*}
    &KK^{-1}=K^{-1}K=1, \\
    &KE=q^{2}EK,\quad KF=q^{-2}FK, \\
    &EF-FE=\frac{K-K^{-1}}{q-q^{-1}}.
\end{align*}
We equip $\UEA_{q}$ with the structure of a Hopf algebra in the following way.
The coproduct $\Delta\colon \UEA_{q}\to \UEA_{q}\otimes\UEA_{q}$ is given by
\begin{align*}
    \Delta (K) &:=K\otimes K, \\
    \Delta (E) &:=E\otimes 1+K\otimes E, \\
    \Delta (F) &:=F\otimes K^{-1} +1\otimes F.
\end{align*}
The counit $\varepsilon \colon \UEA_{q}\to\mathbb{C}$ and the antipode $S\colon \UEA_{q}\to\UEA_{q}$ are defined as
\begin{align*}
    &\varepsilon (E)=\varepsilon (F)=0,\quad \varepsilon (K)=1,\\
    &S (E)=-K^{-1}E,\quad S (F)=-FK,\quad S (K)=K^{-1}.
\end{align*}

\begin{rem}
The coproduct used in our previous work was the opposite: $\Delta^{\mathrm{op}}=\tau\circ \Delta$, where
\begin{equation*}
    \tau\colon \UEA_{q}\otimes\UEA_{q} \to \UEA_{q}\otimes \UEA_{q};\quad A\otimes B\to B\otimes A
\end{equation*}
is the permutation of tensor components.
As we shall see, the structure constants of associativity ($6j$-symbols; see below) manifestly observed are associated with the opposite coproduct $\Delta^{\mathrm{op}}$, but we will come back to the original $\Delta$ in the end.
\end{rem}

\subsection{Irreducible representations}
For each $\ell\in \mathbb{Z}_{\geq 0}$, $\QGirrep{\ell}$ is an $(\ell+1)$-dimensional complex vector space with a basis $(\QGirvec{\ell}{i})_{i=0,1,\dots, \ell}$.
We define a representation homomorphism
\begin{equation*}
    \QGirmap{\ell}\colon \UEA_{q}\to \mathrm{End} (\QGirrep{\ell})
\end{equation*}
by
\begin{align*}
    \QGirmap{\ell} (K) \QGirvec{\ell}{i} &=q^{\ell-2i}\QGirvec{\ell}{i}, \\
    \QGirmap{\ell}(E) \QGirvec{\ell}{i} &= \qint{i}\qint{\ell-i+1} \QGirvec{\ell}{i-1}, \\
    \QGirmap{\ell} (F) \QGirvec{\ell}{i} &= \QGirvec{\ell}{i+1}
\end{align*}
for $i=0,1,\dots,\ell$.
Here, we understand $\QGirvec{\ell}{i}=0$ when $i<0$ or $i>\ell$.
The $q$-integers are defined by
\begin{equation*}
    \qint{n}:=\frac{q^{n}-q^{-n}}{q-q^{-1}},\quad n\in \mathbb{Z}.
\end{equation*}

It is known that $(\QGirmap{\ell},\QGirrep{\ell})$, $\ell\in \mathbb{Z}_{\geq 0}$ are irreducible representations of $\UEA_{q}$.
On the other hand, a finite-dimensional irreducible representation of $\UEA_{q}$ is isomorphic to either $(\QGirmap{\ell},\QGirrep{\ell})$ or $(\QGirmap{\ell}\circ\chi,\QGirrep{\ell})$, where $\chi$ is the automorphism of $\UEA_{q}$ given by $\chi (K)=-K$, $\chi (E)=-E$, $\chi (F)=F$.
In the sequel, we always assume that the vector space $\QGirrep{\ell}$ is equipped with the representation homomorphism $\QGirmap{\ell}$ and will not specify it. These irreducible representations $\QGirrep{\ell}$, $\ell\in\mathbb{Z}_{\geq 0}$ are often referred to as type I.

Clearly, $\QGirrep{\ell}$ is generated by $\QGirvec{\ell}{0}$, which we call a highest weight vector of highest weight $\ell$.
Accordingly, we say that $\QGirrep{\ell}$ is a highest weight irreducible representation of highest weight $\ell$.

\subsection{Tensor product}
For $\ell_{1},\ell_{2}$, the tensor product $\QGirrep{\ell_{1}}\otimes \QGirrep{\ell_{2}}$ is equipped with the structure of a representation of $\UEA_{q}$ by the coproduct $\Delta$.
To emphasize the dependence on the coproduct, we write $\QGirrep{\ell}\otimes_{\Delta}\QGirrep{\ell_{2}}$ for the tensor product representation.
It decomposes into irreducible representations according to the Clebsch--Gordan rule:
\begin{equation}
\label{eq:QG_CGrule}
    \QGirrep{\ell_{1}}\otimes_{\Delta} \QGirrep{\ell_{2}}\simeq \bigoplus_{\ell\in\mathrm{Sel} (\ell_{1},\ell_{2})}\QGirrep{\ell},
\end{equation}
where we defined the set
\begin{equation*}
    \mathrm{Sel} (\ell_{1},\ell_{2})=\left\{\ell\in \mathbb{Z}_{\geq 0}: |\ell_{1}-\ell_{2}|\leq \ell \leq \ell_{1}+\ell_{2},\, \ell+\ell_{1}+\ell_{2}\equiv 0\, \mathrm{mod}\, 2\right\}
\end{equation*}
of the highest weights appearing in the tensor product.

For each triple $(\ell,\ell_{1},\ell_{2})$ such that $\ell_{1},\ell_{2}\in \mathbb{Z}_{\geq 0}$ and $\ell\in \mathrm{Sel} (\ell_{1},\ell_{2})$, we fix an embedding homomorphism $\CGemb{\ell}{\ell_{1}}{\ell_{2}}\colon \QGirrep{\ell}\to \QGirrep{\ell_{1}}\otimes_{\Delta} \QGirrep{\ell_{2}}$ so that
\begin{align}
    \CGemb{\ell}{\ell_{1}}{\ell_{2}}(\QGirvec{\ell}{0})&=\sum_{j=0}^{s}\CGcoeff{\ell}{\ell_{1}}{\ell_{2}}{j}\, \QGirvec{\ell_{1}}{j}\otimes \QGirvec{\ell_{2}}{s-j}, \notag \\
\label{eq:CG_coeffs}
    \CGcoeff{\ell}{\ell_{1}}{\ell_{2}}{j}&=(-1)^{j}\frac{\qint{\ell_{1}-j}!}{\qint{j}!\qint{s-j}!}\frac{\qint{\ell_{2}-s+j}!}{\qint{\ell_{1}}!\qint{\ell_{2}}!}\frac{q^{j(\ell_{1}-j+1)}}{(q-q^{-1})^{s}},\quad j=0,1,\dots, s,
\end{align}
where we set $s=(\ell_{1}+\ell_{2}-\ell)/2$ and the $q$-factorials are defined by
\begin{equation*}
    \qint{n}!:=
    \begin{cases}
    \qint{n}\qint{n-1}\cdots\qint{1},& n\in\mathbb{Z}_{> 0},\\
    1, &n=0.
    \end{cases}
\end{equation*}
Accordingly, the family of projections $\CGproj{\ell}{\ell_{1}}{\ell_{2}}\colon \QGirrep{\ell_{1}}\otimes_{\Delta} \QGirrep{\ell_{2}}\to \QGirrep{\ell}$, $\ell_{1},\ell_{2}\in\mathbb{Z}_{\geq 0}$, $\ell\in\mathrm{Sel} (\ell_{1},\ell_{2})$ is determined by the properties that
\begin{align*}
    \CGproj{\ell}{\ell_{1}}{\ell_{2}}\circ \CGemb{\ell'}{\ell_{1}}{\ell_{2}}&=
    \begin{cases}
    \mathrm{id}_{\QGirrep{\ell}}, & \ell=\ell', \\
    0, & \ell\neq\ell',
    \end{cases}\quad \ell,\ell'\in \mathrm{Sel} (\ell_{1},\ell_{2}), \\
    \sum_{\ell\in\mathrm{Sel}(\ell_{1},\ell_{2})}\CGemb{\ell}{\ell_{1}}{\ell_{2}}\circ \CGproj{\ell}{\ell_{1}}{\ell_{2}}&=\mathrm{id}_{\QGirrep{\ell_{1}}\otimes\QGirrep{\ell_{2}}},
\end{align*}
for each $\ell_{1},\ell_{2}\in\mathbb{Z}_{\geq 0}$.

Recall that there is another coproduct $\Delta^{\mathrm{op}}$ on $\UEA_{q}$, with which we can form another tensor product representation $\QGirrep{\ell_{1}}\otimes_{\Delta^{\mathrm{op}}}\QGirrep{\ell_{2}}$ for $\ell_{1},\ell_{2}\in\mathbb{Z}_{\geq 0}$.
The rule of decomposition into irreducible representations is the same as in (\ref{eq:QG_CGrule}).
For each $\ell\in \mathrm{Sel}(\ell_{1},\ell_{2})$, we obtain an injective homomorphism $\CGembop{\ell}{\ell_{1}}{\ell_{2}}\colon \QGirrep{\ell}\to \QGirrep{\ell_{1}}\otimes_{\Delta^{\mathrm{op}}}\QGirrep{\ell_{2}}$ by
\begin{equation*}
    \CGembop{\ell}{\ell_{1}}{\ell_{2}}:=P_{\QGirrep{\ell_{2}},\QGirrep{\ell_{1}}}\circ \CGemb{\ell}{\ell_{2}}{\ell_{1}}.
\end{equation*}
Here, we write $P_{\mathbf{U},\mathbf{V}}$ with vector spaces $\mathbf{U}$ and $\mathbf{V}$ for the permutation operator:
\begin{equation*}
    P_{\mathbf{U},\mathbf{V}}\colon \mathbf{U}\otimes\mathbf{V}\to \mathbf{V}\otimes\mathbf{U};\quad u\otimes v\mapsto v\otimes u.
\end{equation*}

\subsection{Representation category}
Here we describe the category of $\UEA_{q}$-modules as a ribbon tensor category.
The general theory of tensor categories can be found in \cite{BakalovKirillovJr2001,etingof2015tensor}.

\subsubsection{Abelian category}
We write $\mathsf{Mod} (\UEA_{q})$ for the category of finite-dimensional representations of $\UEA_{q}$, which is clearly an Abelian category.
As we have already noted, the simple objects of $\mathsf{Mod}(\UEA_{q})$ are exhausted by $\QGirrep{\ell}$, $\ell\in\mathbb{Z}_{\geq 0}$ and their twist by the automorphism $\chi$ up to isomorphism.
Under our assumption that $q$ is not a root of unity, it is also known that $\mathsf{Mod}(\UEA_{q})$ is semi-simple although it has infinitely many simple objects.

We define $\cC_{\mathrm{QG}}$ as the full subcategory of $\mathsf{Mod}(\UEA_{q})$ generated by $\QGirrep{\ell}$, $\ell\in\mathbb{Z}_{\geq 0}$.
In other words, each object of $\cC_{\mathrm{QG}}$ is isomorphic to a finite direct sum of $\QGirrep{\ell}$, $\ell\in\mathbb{Z}_{\geq 0}$.

\subsubsection{Tensor category}
We can equip the category $\mathsf{Mod} (\UEA_{q})$ with the tensor bifunctor $-\otimes_{\Delta}-\colon \mathsf{Mod}(\UEA_{q})\times\mathsf{Mod}(\UEA_{q})\to \mathsf{Mod}(\UEA_{q})$ defined by means of the coproduct $\Delta$.
The associativity isomorphisms are given by
\begin{equation*}
    \alpha_{\sfU,\sfV,\sfW}\colon (\sfU\otimes_{\Delta}\sfV)\otimes_{\Delta} \sfW\to \sfU\otimes_{\Delta} (\sfV\otimes_{\Delta}\sfW);\quad (u\otimes v)\otimes w\mapsto u\otimes (v\otimes w)
\end{equation*}
for $\sfU,\sfV,\sfW\in\mathsf{Mod} (\UEA_{q})$.
We can take $\QGirrep{0}$ as a unit object with respect to this tensor product and choose unit isomorphisms $\lambda_{\sfU}\colon \QGirrep{0}\otimes_{\Delta} \sfU\to\sfU$ and $\rho_{\sfU}\colon \sfU\otimes_{\Delta}\QGirrep{0}\to\sfU$.
In this way, $(\otimes_{\Delta},\alpha,\QGirrep{0},\lambda,\rho)$ defines a monoidal structure on the category $\mathsf{Mod} (\UEA_{q})$,
which turns out to make the category $\mathsf{Mod}(\UEA_{q})$ a tensor category.

Since the subcategory $\cC_{\mathrm{QG}}$ is closed under the tensor product, it is a tensor subcategory of $\mathsf{Mod} (\UEA_{q})$.
We suppose that the unit isomorphisms are chosen in such a way that $\lambda_{\QGirrep{\ell}}=\CGproj{\ell}{0}{\ell}$ and $\rho_{\QGirrep{\ell}}=\CGproj{\ell}{\ell}{0}$ for all $\ell\in\mathbb{Z}_{\geq 0}$.
Then, the monoidal structure on $\cC_{\mathrm{QG}}$ is uniquely determined.
To emphasize the dependence on the coproduct $\Delta$, we write $\cC_{\mathrm{QG}}^{\Delta}$ for the category $\cC_{\mathrm{QG}}$ equipped with the tensor structure determined by $\Delta$.

If we take the other coproduct $\Delta^{\mathrm{op}}$, the same underlying category $\cC_{\mathrm{QG}}$ is equipped with another monoidal structure $(\otimes_{\Delta^{\mathrm{op}}},\alpha^{\mathrm{op}},\QGirrep{0},\lambda^{\mathrm{op}},\rho^{\mathrm{op}})$.
This tensor category will be denoted by $\cC_{\mathrm{QG}}^{\Delta^{\mathrm{op}}}$.
Note that, however, as a linear map, each associativity isomorphism $\alpha^{\mathrm{op}}_{\sfU,\sfV,\sfW}$, $\sfU,\sfV,\sfW\in \cC_{\mathrm{QG}}$, can be taken as the same $\alpha_{\sfU,\sfV,\sfW}$.

At this point, we can discuss the so-called $6j$-symbols.
In our context, the $6j$-symbols of $\cC_{\mathrm{QG}}^{\Delta^{\mathrm{op}}}$, instead of those of $\cC_{\mathrm{QG}}^{\Delta}$, will be more manifest.
For an arbitrary choice of four $\ell_{1},\ell_{2},\ell_{3},\ell_{4}\in\mathbb{Z}_{\geq 0}$, we define the set
\begin{equation*}
    \Chan{\ell_{1}}{\ell_{2}}{\ell_{3}}{\ell_{4}}:=\mathrm{Sel} (\ell_{1},\ell_{2})\cap\mathrm{Sel} (\ell_{3},\ell_{4}).
\end{equation*}
Now, we fix $\ell_{1},\ell_{2},\ell_{3},\ell_{4}\in\mathbb{Z}_{\geq 0}$
and compare two spaces of homomorphisms:
one is $\mathrm{Hom}_{\UEA_{q}}(\QGirrep{\ell_{4}},(\QGirrep{\ell_{1}}\otimes_{\Delta^{\mathrm{op}}} \QGirrep{\ell_{2}})\otimes_{\Delta^{\mathrm{op}}} \QGirrep{\ell_{3}})$, which has a basis
\begin{equation*}
    (\CGembop{m}{\ell_{1}}{\ell_{2}}\otimes_{\Delta^{\mathrm{op}}} \mathrm{id}_{\QGirrep{\ell_{3}}})\circ \CGembop{\ell_{4}}{m}{\ell_{3}}
    =\hackcenter{
    \begin{tikzpicture}
    \draw[thick] (0,-1)--(0,0);
    \draw[thick] (0,0)--(-1,1);
    \draw[thick] (0,0)--(1,1);
    \draw[thick] (0,1)--(-.5,.5);
    \draw (0,-1)node[below]{$\ell_{4}$};
    \draw (-1,1)node[above]{$\ell_{1}$};
    \draw (0,1)node[above]{$\ell_{2}$};
    \draw (1,1)node[above]{$\ell_{3}$};
    \draw (-.25,.25)node[below left]{$m$};
    \end{tikzpicture}
    },\quad m\in \Chan{\ell_{1}}{\ell_{2}}{\ell_{3}}{\ell_{4}}
\end{equation*}
and the other is $\mathrm{Hom}_{\UEA_{q}}(\QGirrep{\ell_{4}},\QGirrep{\ell_{1}}\otimes_{\Delta^{\mathrm{op}}} (\QGirrep{\ell_{2}}\otimes_{\Delta^{\mathrm{op}}} \QGirrep{\ell_{3}}))$, which has a basis
\begin{equation*}
    (\mathrm{id}_{\QGirrep{\ell_{1}}}\otimes_{\Delta^{\mathrm{op}}} \CGembop{n}{\ell_{2}}{\ell_{3}})\circ \CGembop{\ell_{4}}{\ell_{1}}{n}
    =\hackcenter{
    \begin{tikzpicture}
    \draw[thick] (0,-1)--(0,0);
    \draw[thick] (0,0)--(-1,1);
    \draw[thick] (0,0)--(1,1);
    \draw[thick] (0,1)--(.5,.5);
    \draw (0,-1)node[below]{$\ell_{4}$};
    \draw (-1,1)node[above]{$\ell_{1}$};
    \draw (0,1)node[above]{$\ell_{2}$};
    \draw (1,1)node[above]{$\ell_{3}$};
    \draw (.25,.25)node[below right]{$n$};
    \end{tikzpicture}
    },\quad n\in \Chan{\ell_{2}}{\ell_{3}}{\ell_{1}}{\ell_{4}}.
\end{equation*}
Here we also drew diagrams representing the homomorphisms.
Each trivalent vertex depicts an injection and the composition is read from bottom to top.
The associativity isomorphism induces an isomorphism of these vector spaces of morphisms
\begin{align*}
    \mathrm{Hom}_{\UEA_{q}}(\QGirrep{\ell_{4}},(\QGirrep{\ell_{1}}\otimes_{\Delta^{\mathrm{op}}} \QGirrep{\ell_{2}})\otimes_{\Delta^{\mathrm{op}}} \QGirrep{\ell_{3}})&\to \mathrm{Hom}_{\UEA_{q}}(\QGirrep{\ell_{4}},\QGirrep{\ell_{1}}\otimes_{\Delta^{\mathrm{op}}} (\QGirrep{\ell_{2}}\otimes_{\Delta^{\mathrm{op}}} \QGirrep{\ell_{3}})) \\
    f &\mapsto \alpha^{\mathrm{op}}_{\QGirrep{\ell_{1}},\QGirrep{\ell_{2}},\QGirrep{\ell_{3}}}\circ f.
\end{align*}
Our $6j$-symbols $\sixjsymb{\ell_{1}}{\ell_{2}}{\ell_{3}}{\ell_{4}}{m}{n}$, $m\in \Chan{\ell_{1}}{\ell_{2}}{\ell_{3}}{\ell_{4}}$, $n\in \Chan{\ell_{2}}{\ell_{3}}{\ell_{1}}{\ell_{4}}$ are defined as the matrix elements of the {\it inverse} of this isomorphism in terms of the bases introduced above.
Explicitly, they are defined by
\begin{align}
\label{eq:QG_6j_definition}
    &(\alpha^{\mathrm{op}}_{\QGirrep{\ell_{1}},\QGirrep{\ell_{2}},\QGirrep{\ell_{3}}})^{-1}\circ (\mathrm{id}_{\QGirrep{\ell_{1}}}\otimes_{\Delta^{\mathrm{op}}} \CGembop{n}{\ell_{2}}{\ell_{3}})\circ \CGembop{\ell_{4}}{\ell_{1}}{n} \\
     =\, &
     \sum_{m\in \Chan{\ell_{1}}{\ell_{2}}{\ell_{3}}{\ell_{4}}}
     \sixjsymb{\ell_{1}}{\ell_{2}}{\ell_{3}}{\ell_{4}}{m}{n}
     \, (\CGembop{m}{\ell_{1}}{\ell_{2}}\otimes_{\Delta^{\mathrm{op}}} \mathrm{id}_{\QGirrep{\ell_{3}}})\circ \CGembop{\ell_{4}}{m}{\ell_{3}}\notag
\end{align}
for $n\in \Chan{\ell_{2}}{\ell_{3}}{\ell_{1}}{\ell_{4}}$, or diagramatically
\begin{equation*}
    (\alpha^{\mathrm{op}}_{\QGirrep{\ell_{1}},\QGirrep{\ell_{2}},\QGirrep{\ell_{3}}})^{-1}\circ -\colon\,
    \hackcenter{
    \begin{tikzpicture}
    \draw[thick] (0,-1)--(0,0);
    \draw[thick] (0,0)--(-1,1);
    \draw[thick] (0,0)--(1,1);
    \draw[thick] (0,1)--(.5,.5);
    \draw (0,-1)node[below]{$\ell_{4}$};
    \draw (-1,1)node[above]{$\ell_{1}$};
    \draw (0,1)node[above]{$\ell_{2}$};
    \draw (1,1)node[above]{$\ell_{3}$};
    \draw (.25,.25)node[below right]{$n$};
    \end{tikzpicture}
    }\mapsto
    \sum_{m\in \Chan{\ell_{1}}{\ell_{2}}{\ell_{3}}{\ell_{4}}}
     \sixjsymb{\ell_{1}}{\ell_{2}}{\ell_{3}}{\ell_{4}}{m}{n}\,
     \hackcenter{
    \begin{tikzpicture}
    \draw[thick] (0,-1)--(0,0);
    \draw[thick] (0,0)--(-1,1);
    \draw[thick] (0,0)--(1,1);
    \draw[thick] (0,1)--(-.5,.5);
    \draw (0,-1)node[below]{$\ell_{4}$};
    \draw (-1,1)node[above]{$\ell_{1}$};
    \draw (0,1)node[above]{$\ell_{2}$};
    \draw (1,1)node[above]{$\ell_{3}$};
    \draw (-.25,.25)node[below left]{$m$};
    \end{tikzpicture}
    }\, .
\end{equation*}

\subsubsection{Braided category}
Here we consider the quantum group $\UEA_{q}$ to be equipped with the coproduct $\Delta$ (but not $\Delta^{\mathrm{op}}$.) It admits the universal $R$-matrix given by the following formula:
\begin{equation*}
    \mathcal{R}=q^{\frac{1}{2}H\otimes H}\sum_{n=0}^{\infty}\frac{q^{n(n-1)/2}(q-q^{-1})^{n}}{\qint{n}!}\left(F^{n}\otimes E^{n}\right),
\end{equation*}
which makes sense in a certain completion of $\UEA_{q}\otimes\UEA_{q}$.
Here, $H$ is the symbol that is supposed to behave as $K=q^{H}$.
Its action on $\sfU\otimes_{\Delta}\sfV$ with $\sfU,\sfV\in \cC_{\mathrm{QG}}^{\Delta}$ is well defined.
Indeed, since these representations are finite dimensional, the infinite sum in the formula of $\cR$ truncates to a finite sum, and for $u\in\sfU$ and $v\in \sfV$ such that $K.u=q^{m}u$ and $K.v=q^{n}v$, we may understand
\begin{equation*}
    q^{\frac{1}{2}H\otimes H}.u\otimes v=q^{\frac{mn}{2}}u\otimes v.
\end{equation*}
We write $R_{\sfU,\sfV}$ for the action of $\cR$ on $\sfU\otimes_{\Delta}\sfV$
and set
\begin{equation*}
    c_{\sfU,\sfV}:=P_{\sfU,\sfV}R_{\sfU,\sfV}\colon \sfU\otimes_{\Delta}\sfV\to\sfV\otimes_{\Delta}\sfU.
\end{equation*}
The family of morphisms $c_{\sfU,\sfV}$, $\sfU,\sfV\in \cC_{\mathrm{QG}}^{\Delta}$ gives a braiding structure to $\cC_{\mathrm{QG}}^{\Delta}$.

\subsubsection{Ribbon structure and rigidity}
We make few comments on the ribbon structure and rigidity of $\cC_{\mathrm{QG}}^{\Delta}$.
It is a standard procedure to read a twist isomorphism out of the universal $R$-matrix.
For each $\sfU\in\cC_{\mathrm{QG}}^{\Delta}$, we can define the twist isomorphism $\theta_{\sfU}\colon \sfU\to \sfU$ by
\begin{equation}
\label{eq:QGtwist_inverse}
    \theta_{\sfU}^{-1}u=(-1)^{H}K\sum_{n=0}^{\infty}\frac{q^{n(n-1)/2}(q-q^{-1})^{n}}{\qint{n}!}(-K^{-1}E)^{n}q^{-\frac{1}{2}H^{2}}F^{n}u,\quad u\in\sfU.
\end{equation}

We can also equip $\cC_{\mathrm{QG}}^{\Delta}$ with the structure of duality by means of the andipode $S$ that recovers the twist morphisms together with the prescribed braiding.

\section{Category of modules for a VOA}
\label{sect:module_cat_VOA}
In this section, we review the construction of a tensor product and a braiding on the category of modules for a VOA developed in \cite{HuangLepowsky1992,HuangLepowsky1994,Huang--LepowskyI,Huang--LepowskyII,Huang--LepowskyIII,Huang1995,huang1999intertwining,Huang2005}.
For a concise account, \cite[Section 2]{Huang--Kirillov--Lepowsky} is also helpful.

\subsection{VOA and modules}
By a VOA, we mean a $\mathbb{Z}$-graded vector space $V=\bigoplus_{n\in\mathbb{Z}}V_{n}$ such that $\dimV_{n}<\infty$ for all $n\in\mathbb{Z}$, together with
a vacuum vector $\mathbf{1}\inV_{0}$,
a state-field correspondence map $Y(-,z)\colon V\to \mathrm{End}(V)[[z^{\pm 1}]]$ and a conformal vector $\omega\inV_{2}$.
They are subject to the set of axioms that can be found in, e.g.,~\cite{FLM-VOAs_Monster,Lepowsky_Li-VOA}. (See \cite{Kac-vertex_algebras, FrenkelBen-Zvi2004} for other sets of axioms.)

A $V$-module is a pair of a $\mathbb{Z}$-graded vector space $W=\bigoplus_{n\in\mathbb{Z}}W_{n}$ such that $\dimW_{n}<\infty$ for all $n\in\mathbb{Z}$ and a linear map $Y_{W}(-,z)\colon V\to \mathrm{End} (W) [[z^{\pm 1}]]$.
They satisfy properties similar to those required for $V$ except for those involving the vacuum vector.
For later use, we introduce the notation for the action of the Virasoro algebra on a $V$-module $W$:
\begin{equation*}
    Y_{W}(\omega,x)=\sum_{n\in\mathbb{Z}}L^{W}_{n}x^{-n-2}.
\end{equation*}
The Virasoro algebra $\mathfrak{vir}$ is defined by $\mathfrak{vir}=\bigoplus_{n\in\mathbb{Z}}\mathbb{C} L_{n}\oplus \mathbb{C} C$ with the relations
\begin{equation*}
    [L_{m},L_{n}]=(m-n)L_{m+n}+\frac{m^{3}-m}{12}\delta_{m+n,0}C,\quad m,n\in\mathbb{Z}.
\end{equation*}
There exists a unique representation of $\mathfrak{vir}$ on a $V$-module $W$ such that $L_{n}\mapsto L_{n}^{W}$, $n\in\mathbb{Z}$.

\subsection{Intertwining operators/maps}
Given three $V$-modules $W_{1}, W_{2}, W_{3}$,
an intertwining operator of type $\inttyp{W_{1}}{W_{2}}{W_{3}}$ is a linear map
\begin{equation*}
    \mathcal{Y} (-,x)\colon W_{1}\to\mathrm{Hom} (W_{2},W_{3})\{x\}
\end{equation*}
satisfying the Jacobi identity:
\begin{align*}
    &\, x_{0}^{-1}\delta \left(\frac{x_{1}-x_{2}}{x_{0}}\right)Y_{W_{3}}(v,x_{1})\mathcal{Y} (w_{1},x_{2})-x_{0}^{-1}\delta\left(\frac{x_{2}-x_{1}}{-x_{0}}\right)\mathcal{Y} (w_{1},x_{2})Y_{W_{2}}(v,x_{1}) \\
    =&\, x_{2}^{-1}\delta\left(\frac{x_{1}-x_{0}}{x_{2}}\right)\mathcal{Y}\left(Y_{W_{1}}(v,x_{0})w_{1},x_{2}\right)
\end{align*}
for all $v\in V$ and $w_{1}\inW_{1}$,
and the $L_{-1}$-derivation property:
\begin{equation*}
    \mathcal{Y}\left(L_{-1}^{W_{1}}w_{1},x\right)=\frac{d}{dx}\mathcal{Y} \left(w_{1},x\right)
\end{equation*}
for all $w_{1}\inW_{1}$.
The set of intertwining operators of type $\inttyp{W_{1}}{W_{2}}{W_{3}}$ is a vector space denoted by $\intsp{W_{1}}{W_{2}}{W_{3}}$, whose dimension is called the fusion rule of that type.

\begin{rem}
\label{rem:intertw_special_cases}
If all three modules are $V$ itself, $\intsp{V}{V}{V}$ contains the state-field correspondence map $Y(-,x)$.
If $W_{1}=V$ and $W_{2}=W_{3}=W$, then the space of intertwining operators $\intsp{V}{W}{W}$ contains the module map $Y_{W}(-,x)$.
In particular, the state-field correspondence map and the module map satisfy the Jacobi identity and the $L_{-1}$-derivation property, which are part of the axioms for a VOA and its modules that we have omitted.
\end{rem}

For $z\in\mathbb{C}^{\times}$, we define $\log (z)$ so that $\arg (z)\in [0,2\pi)$.
Then specializing the formal variable in an intertwining operator $\mathcal{Y}$ at $x=e^{\log (z)}$ makes sense, giving rise to the linear map
\begin{equation*}
    F_{\mathcal{Y},z}\colon W_{1}\otimesW_{2}\to\ol{W_{3}};\quad w_{1}\otimesw_{2}\to \mathcal{Y} (w_{1},x)w_{2}|_{x=e^{\log (z)}},
\end{equation*}
where we define the completion of $W_{3}$ by $\ol{W_{3}}=\prod_{n\in\mathbb{Z}}(W_{3})_{n}$.
In the sequel, we simply write
\begin{equation*}
    \mathcal{Y} (w_{1},z)w_{2}=\mathcal{Y} (w_{1},x)w_{2}|_{x=e^{\log (z)}}
\end{equation*}
for the evaluation of the formal variable as long as the branch is chosen as $\arg z\in [0,2\pi)$.
We call such a linear map obtained from an intertwining operator an {\it intertwining map} associated to $z$.
Since the correspondence $\mathcal{Y}\mapsto F_{\mathcal{Y},z}$ is one-to-one, we could say that they are the same notion, but intertwining maps are still more convenient when we define the tensor product of modules.

\subsection{\texorpdfstring{$P(z)$}{P(z)}-tensor product}
Let $W_{1}$ and $W_{2}$ be $V$-modules, and fix $z\in\mathbb{C}^{\times}$.
A $P(z)$-product of $W_{1}$ and $W_{2}$ is a $V$-module $W_{3}$ together with an intertwining map $F_{z}$ of type $\inttyp{W_{1}}{W_{2}}{W_{3}}$ associated to $z$.
The $P(z)$-{\it tensor} product is a universal object among $P(z)$-products.
To be precise, the $P(z)$-tensor product of $W_{1}$ and $W_{2}$ is the $P(z)$-product $(W_{1}\boxtimes_{P(z)}W_{2}, \boxtimes_{P(z)})$
such that for any $P(z)$-product $(W_{3},F_{z})$ of $W_{1}$ and $W_{2}$, there exists a unique $V$-module homomorphism $\eta\colon W_{1}\boxtimes_{P(z)}W_{2}\to W_{3}$ satisfying $F_{z}=\ol{\eta}\circ \boxtimes_{P(z)}$.
It is standard to show that the $P(z)$-tensor product is unique up to isomorphism if it exists.
We note that the $P(z)$-tensor product depends on the choice of a category of modules.

For the $P(z)$-tensor product $(W_{1}\boxtimes_{P(z)}W_{2}, \boxtimes_{P(z)})$,
there exists a unique intertwining operator $\mathcal{Y} (-,x)$ of type $\inttyp{W_{1}}{W_{2}}{W_{1}\boxtimes_{P(z)}W_{2}}$ such that
\begin{equation*}
    w_{1}\boxtimes_{P(z)}w_{2}=\mathcal{Y} (w_{1},z)w_{2},\quad w_{1}\inW_{1},\, w_{2}\inW_{2}.
\end{equation*}

The $P(z)$-tensor product defines a bifunctor on the category of modules of interest.
To see this, let $W_{1}, W_{2}, W_{3},W_{4}$ be $V$-modules and let $f\colon W_{1}\toW_{3}$ and $g\colonW_{2}\toW_{4}$ be morphisms.
Suppose that the $P(z)$-tensor products $(W_{1}\boxtimes_{P(z)}W_{2},\boxtimes_{P(z)}^{12})$ and $(W_{3}\boxtimes_{P(z)}W_{4},\boxtimes_{P(z)}^{34})$ exist.
Then the object $W_{3}\boxtimes_{P(z)}W_{4}$ together with $\boxtimes_{P(z)}^{34}\circ (f\otimes g)$ gives a $P(z)$-product of $W_{1}$ and $W_{2}$.
This means that there exists a unique morphism denoted by $f\boxtimes_{P(z)} g\colon W_{1}\boxtimes_{P(z)}W_{2}\to W_{3}\boxtimes_{P(z)}W_{4}$ such that
\begin{equation*}
    \boxtimes^{34}_{P(z)}\circ (f\otimes g)=\ol{f\boxtimes_{P(z)}g}\circ \boxtimes_{P(z)}^{12}.
\end{equation*}

\subsection{Composition and iteration}
Let $W_{1}$, $W_{2}$, $W_{3}$ be $V$-modules and $z_{1},z_{2}\in \mathbb{C}^{\times}$ be such that $|z_{1}|>|z_{2}|>|z_{1}-z_{2}|>0$.
In this setting, let us assume that the composition
\begin{equation*}
    W_{1}\boxtimes_{P(z_{1})}(W_{2}\boxtimes_{P(z_{2})}W_{3})
\end{equation*}
exists.
Then, associated with it, we have the composition of intertwining maps
\begin{equation*}
    \mathcal{Y}^{1}(-,z_{1})\mathcal{Y}^{2}(-,z_{2})-\colon W_{1}\otimes W_{2}\otimesW_{3}\to \ol{W_{1}\boxtimes_{P(z_{1})}(W_{2}\boxtimes_{P(z_{2})}W_{3})},
\end{equation*}
where
\begin{equation*}\mathcal{Y}^{1}(-,x)\in \intsp{W_{1}}{W_{2}\boxtimes_{P(z_{2})}W_{3}}{W_{1}\boxtimes_{P(z_{1})}(W_{2}\boxtimes_{P(z_{2})}W_{3})}\quad \mbox{and}\quad
\mathcal{Y}^{2}(-,x)\in \intsp{W_{2}}{W_{3}}{W_{2}\boxtimes_{P(z_{2})}W_{3}}
\end{equation*}
are the corresponding intertwining operators.
For $w_{i}\inW_{i}$, $i=1,2,3$, we can identify
\begin{equation*}
    w_{1}\boxtimes_{P(z_{1})}(w_{2}\boxtimes_{P(z_{2})}w_{3})=\mathcal{Y}^{1}(w_{1},z_{1})\mathcal{Y}^{2}(w_{2},z_{2})w_{3}.
\end{equation*}

Let us also assume that the iteration
\begin{equation*}
    (W_{1}\boxtimes_{P(z_{1}-z_{2})}W_{2})\boxtimes_{P(z_{2})}W_{3}
\end{equation*}
exists.
In this case, we get the iteration of the corresponding intertwining maps
\begin{equation*}
    \mathcal{Y}_{1}\left(\mathcal{Y}_{2}(-,z_{1}-z_{2})-,z_{2}\right)-\colon W_{1}\otimes W_{2}\otimes W_{3}\to \ol{(W_{1}\boxtimes_{P(z_{1}-z_{2})}W_{2})\boxtimes_{P(z_{2})}W_{3}},
\end{equation*}
where
\begin{equation*}
    \mathcal{Y}_{1}(-,x)\in\intsp{W_{1}\boxtimes_{P(z_{1}-z_{2})}W_{2}}{W_{3}}{(W_{1}\boxtimes_{P(z_{1}-z_{2})}W_{2})\boxtimes_{P(z_{2})}W_{3}}\quad \mbox{and}\quad
    \mathcal{Y}_{2}(-,x)\in \intsp{W_{1}}{W_{2}}{W_{1}\boxtimes_{P(z_{1}-z_{2})}W_{2}}.
\end{equation*}
We can make the identification
\begin{equation*}
    (w_{1}\boxtimes_{P(z_{1}-z_{2})}w_{2})\boxtimes_{P(z_{2})}w_{3}=\mathcal{Y}_{1}\left(\mathcal{Y}_{2}(w_{1},z_{1}-z_{2})w_{2},z_{2}\right)w_{3}
\end{equation*}
for $w_{i}\in W_{i}$, $i=1,2,3$.

The associativity isomorphism \begin{equation*}
    (\mathcal{A}_{P(z_{1}),P(z_{2})}^{P(z_{1}-z_{2}),P(z_{2})})_{W_{1},W_{2},W_{3}}\colon
    (W_{1}\boxtimes_{P(z_{1}-z_{2})}W_{2})\boxtimes_{P(z_{2})}W_{3}\to
    W_{1}\boxtimes_{P(z_{1})}(W_{2}\boxtimes_{P(z_{2})}W_{3}),
\end{equation*}
if it exists, is characterized by the property
\begin{equation*}
    \ol{(\mathcal{A}_{P(z_{1}),P(z_{2})}^{P(z_{1}-z_{2}),P(z_{2})})_{W_{1},W_{2},W_{3}}}\colon
    (w_{1}\boxtimes_{P(z_{1}-z_{2})}w_{2})\boxtimes_{P(z_{2})}w_{3} \mapsto
    w_{1}\boxtimes_{P(z_{1})}(w_{2}\boxtimes_{P(z_{2})}w_{3})
\end{equation*}
for $w_{i}\in W_{i}$, $i=1,2,3$.

\subsection{Parallel transport}
Let $W_{1}$, $W_{2}$ be $V$-modules,
and let us take $z_{1},z_{2}\in\mathbb{C}^{\times}$.
Then we may consider the $P(z_{1})$-tensor product $W_{1}\boxtimes_{P(z_{1})}W_{2}$ and the $P(z_{2})$-tensor product $W_{1}\boxtimes_{P(z_{2})}W_{2}$.
Depending on a path $\gamma$ in $\mathbb{C}^{\times}$ from $z_{1}$ to $z_{2}$,
the parallel transport isomorphism $\mathcal{T}_{\gamma}\colon W_{1}\boxtimes_{P(z_{1})}W_{2}\to W_{1}\boxtimes_{P(z_{2})}W_{2}$ is defined as follows.

As before, we fix a branch of $\log (z_{2})$ so that $\arg (z_{2})\in [0,2\pi)$.
Then we write $l_{\gamma}(z_{1})$ for the logarithm of $z_{1}$ determined by the analytic continuation along $\gamma$ from $\log (z_{2})$.
Let $\mathcal{Y} (-,x)$ be the corresponding intertwining operator to the $P(z_{2})$-tensor product $W_{1}\boxtimes_{P(z_{2})}W_{2}$.
Then, the parallel transport $\mathcal{T}_{\gamma}$ is characterized by the property
\begin{equation*}
    \ol{\mathcal{T}}_{\gamma}\left(w_{1}\boxtimes_{P(z_{1})}w_{2}\right)=\mathcal{Y}(w_{1},x)w_{2}|_{x=e^{l_{\gamma}(z_{1})}},\quad w_{1}\inW_{1},\, w_{2}\inW_{2}.
\end{equation*}

\subsection{Monoidal structure}
The $P(z)$-tensor product depends on $z\in \mathbb{C}^{\times}$, so varying $z$, we get a family of tensor products that are related by parallel transport.
Here, we fix a single monoidal structure at $z=1$.

First, we set $\boxtimes=\boxtimes_{P(1)}$ and take $V$ as a unit object.
For a $V$-module $W$, the unit isomorphisms $\lambda_{W}\colon V\boxtimesW\toW$ and $\rho_{W}\colonW\boxtimes V\to W$ are characterized by
\begin{equation*}
    \lambda_{W}\colon \mathbf{1}\boxtimes w\mapsto w,\quad \ol{\rho_{W}}\colon w\boxtimes\mathbf{1}\mapsto e^{L^{W}_{-1}}w,\quad w\inW.
\end{equation*}

To define the associativity isomorphism, we take $z_{1}$ and $z_{2}$ on the real axis so that $z_{1}>z_{2}>z_{1}-z_{2}>0$.
We also take several paths in $\mathbb{R}_{>0}$: $\gamma_{1}$ from $1$ to $z_{2}$, $\gamma_{2}$ from $1$ to $z_{1}-z_{2}$, $\gamma_{3}$ from $z_{1}$ to $1$, and $\gamma_{4}$ from $z_{2}$ to $1$.
Then, given three $V$-modules $W_{i}$, $i=1,2,3$, the isomorphism $\mathcal{A}_{W_{1},W_{2},W_{3}}\colon (W_{1}\boxtimes W_{2})\boxtimes W_{3}\to W_{1}\boxtimes (W_{2}\boxtimes W_{3})$ is the following compositions of isomorphisms:
\begin{align*}
    & (W_{1}\boxtimes W_{2})\boxtimes W_{3}
    \xrightarrow{\mathcal{T}_{\gamma_{1}}}
    (W_{1}\boxtimes W_{2})\boxtimes_{P(z_{2})} W_{3}
    \xrightarrow{\mathcal{T}_{\gamma_{2}}\boxtimes_{P(z_{2})}\mathrm{id}_{W_{3}}} \\
    & (W_{1}\boxtimes_{P(z_{1}-z_{2})} W_{2})\boxtimes_{P(z_{2})} W_{3}
    \xrightarrow{(\mathcal{A}_{P(z_{1}),P(z_{2})}^{P(z_{1}-z_{2}),P(z_{2})})_{W_{1},W_{2},W_{3}}}
    W_{1}\boxtimes_{P(z_{1})} (W_{2}\boxtimes_{P(z_{2})} W_{3}) \\
    &\xrightarrow{\mathrm{id}_{W_{1}}\boxtimes_{P(z_{1})}\mathcal{T}_{\gamma_{4}}}
    W_{1}\boxtimes_{P(z_{1})} (W_{2}\boxtimes W_{3})
    \xrightarrow{\mathcal{T}_{\gamma_{3}}}
    W_{1}\boxtimes (W_{2}\boxtimes W_{3}).
\end{align*}
Then, $(\boxtimes, \mathcal{A}, V, \lambda,\rho)$ gives a monoidal structure of the category of interest.

\subsection{Braiding and twist}
We can define a braiding on the category by means of the parallel transport.
Let $\gamma$ be a path from $-1$ to $1$ contained in the complex upper half plane except its end points.
Given two $V$-modules $W_{1}$ and $W_{2}$, the braiding isomorphism $c_{W_{1},W_{2}}\colon W_{1}\boxtimesW_{2}\to W_{2}\boxtimes W_{1}$ is characterized by
\begin{equation*}
    \ol{c_{W_{1},W_{2}}}\colon w_{1}\boxtimesw_{2}\mapsto e^{L^{W_{2}\boxtimesW_{1}}_{-1}}\ol{\mathcal{T}_{\gamma}}(w_{2}\boxtimes_{P(-1)}w_{1}),\quad w_{1}\inW_{1},\, w_{2}\in W_{2}.
\end{equation*}

For each $V$-module $W$, the twist $\theta_{W}\colon W\toW$ compatible to the braiding is given by $\theta_{W}=e^{2\pi\mathbbm{i} L_{0}^{W}}$.

\section{First-row modules of the generic Virasoro VOA}
\label{sect:generic_VirVOA}
In this section, we review the results of our previous work~\cite{KoshidaKytola2022} on the first-row modules of the generic Virsoro VOA.

\subsection{Generic Virasoro VOA and first-row modules}
For a fixed central charge $c\in \mathbb{C}$, the universal Virasoro VOA $\mathbb{V}_{c}$ is given by
\begin{equation*}
    \mathbb{V}_{c}=\cU (\mathfrak{vir})/\left(\cU (\mathfrak{vir})(C-c)+\textstyle{\sum_{n\geq -1}}\cU (\mathfrak{vir})L_{n}\right)
\end{equation*}
together with the vacuum vector $\mathbf{1}=[1]$ and the conformal vector $\omega=[L_{-2}]$.
The state field correspondence map $Y(-,x)$ is uniquely determined by
\begin{equation*}
    Y(\omega,x)=\sum_{n\in\mathbb{Z}}L_{n}x^{-n-2}.
\end{equation*}

We parameterize the central charge by another parameter $t$ as
\begin{equation}
\label{eq:vir_cc_in_t}
    c=c(t)=13-6(t+t^{-1}).
\end{equation}
It is known that, when $t\not\in\mathbb{Q}$, $\mathbb{V}_{c}$ is a simple VOA (see e.g.~\cite{IK-representation_theory_of_the_Virasoro_algebra}.)
In this case, we call $\mathbb{V}_{c}$ the generic Virsoro VOA of central charge $c$.

Let us consider modules of $\mathbb{V}_{c}$.
For a conformal weight $h\in\mathbb{C}$, the Verma module $\mathbb{M} (c,h)$ is given by
\begin{equation*}
    \mathbb{M} (c,h)= \cU (\mathfrak{vir})/\left(\cU (\mathfrak{vir})(C-c)+\cU(\mathfrak{vir})(L_{0}-h)+\textstyle{\sum_{n\geq 1}}\cU(\mathfrak{vir})L_{n}\right).
\end{equation*}
The Verma module is not only a representation of $\mathfrak{vir}$, but also a $\mathbb{V}_{c}$-module.
Under the parametrization (\ref{eq:vir_cc_in_t}), the conformal weights of the first row of the Kac table are given by
\begin{equation}
\label{eq:conf_weight_fr}
    h_{\ell}=h_{\ell}(t)=\frac{\ell(\ell+2)}{4}t-\frac{\ell}{2},\quad \ell\in\mathbb{Z}_{\geq 0}.
\end{equation}
For such a conformal weight, the Verma module $\mathbb{M} (c(t),h_{\ell}(t))$ is reducible, so we write its simple quotient as $\mathbb{W}_{\ell}$ and call it a first-row module of $\mathbb{V}_{c}$.
We remark that $\mathbb{W}_{0}$ is $\mathbb{V}_{c}$ itself because the maximal proper submodule of $\mathbb{M} (c(t),h_{0}(t))$ is generated by $[L_{-1}]$.

For later use, we fix the highest weight vector $\mathbf{w}_{\ell}$ of $\mathbb{W}_{\ell}$ as the image of $1\in\cU(\mathfrak{vir})$.

\subsection{Fusion rules}
The fusion rules among the first-row modules have been known for a long time~\cite{FrenkelZhu2012}, and we gave an alternative proof for them in our previous work~\cite{KoshidaKytola2022}.
Let us record the result here.

\begin{thm}[\cite{FrenkelZhu2012,KoshidaKytola2022}]
\label{thm:first-row_module_fusion}
For $\ell_{1},\ell_{2},\ell_{3}\in \mathbb{Z}_{\geq 0}$,
\begin{equation*}
    \dim \intsp{\mathbb{W}_{\ell_{1}}}{\mathbb{W}_{\ell_{2}}}{\mathbb{W}_{\ell_{3}}}=
    \begin{cases}
    1, & \ell_{3}\in \mathrm{Sel} (\ell_{1},\ell_{2}),\\
    0 & \mathrm{otherwise}.
    \end{cases}
\end{equation*}
Recall that $\mathrm{Sel}$ is the selection rule set of the Clebsch--Gordan rule.
\end{thm}

Suppose that $\ell_{1},\ell_{2},\ell_{3}\in\mathbb{Z}_{\geq 0}$ satisfy the selection rule $\ell_{3}\in\mathrm{Sel} (\ell_{1},\ell_{2})$.
Then there exists a unique intertwining operator of type $\inttyp{\mathbb{W}_{\ell_{1}}}{\mathbb{W}_{\ell_{2}}}{\mathbb{W}_{\ell_{3}}}$ up to constant.
We fix the normalization of the intertwining operator $\mathcal{Y}_{\ell_{1}\,\ell_{2}}^{\ell_{3}}(-,x)\in \intsp{\mathbb{W}_{\ell_{1}}}{\mathbb{W}_{\ell_{2}}}{\mathbb{W}_{\ell_{3}}}$ as
\begin{equation*}
    \mathcal{Y}_{\ell_{1}\,\ell_{2}}^{\ell_{3}}(\mathbf{w}_{\ell_{1}},x)\mathbf{w}_{\ell_{2}}\in \mathbf{B}_{\ell_{1}\,\ell_{2}}^{\ell_{3}}\mathbf{w}_{\ell_{3}}x^{h_{\ell_{3}}-h_{\ell_{1}}-h_{\ell_{2}}}+\mathbb{W}_{\ell_{3}}[[x]]x^{h_{\ell_{3}}-h_{\ell_{1}}-h_{\ell_{2}}+1},
\end{equation*}
where the constant $\mathbf{B}_{\ell_{1}\,\ell_{2}}^{\ell_{3}}$ is given by the formula
\begin{equation*}
    \mathbf{B}_{\ell_{1}\,\ell_{2}}^{\ell_{3}}=\frac{1}{s!}\prod_{j=1}^{s}\frac{\Gamma (1+tj) \Gamma (1-t(\ell_{1}+1-j)) \Gamma (1-t(\ell_{2}+1-j))}{\Gamma (1+t) \Gamma (2-t(2-p+\ell_{1}+\ell_{2}-s))},
\end{equation*}
where $s=(\ell_{1}+\ell_{2}-\ell_{3})/2$.

\begin{rem}
\label{rem:intertw_Virasoro_special_cases}
Let us remark on a few properties of the intertwining operator $\mathcal{Y}_{\ell_{1}\,\ell_{2}}^{\ell_{3}}(-,x)$ when $\ell_{1}=0$ or $\ell_{2}=0$.
When $\ell_{1}=0$, as we pointed out in Remark~\ref{rem:intertw_special_cases}, $\intsp{\mathbb{V}_{c}}{\mathbb{W}_{\ell}}{\mathbb{W}_{\ell}}$ contains $Y_{\mathbb{W}_{\ell}}(-,x)$, which now must span the space of intertwining operators.
Furthermore, we can identify $\mathcal{Y}_{0\,\ell}^{\ell}(-,x)=Y_{\mathbb{W}_{\ell}}(-,x)$ by observing
\begin{equation*}
    \mathcal{Y}_{0\,\ell}^{\ell}(\mathbf{1},x)=\mathrm{Id}_{\mathbb{W}_{\ell}}=Y_{\mathbb{W}_{\ell}}(\mathbf{1},x).
\end{equation*}
For the case where $\ell_{2}=0$, the normalization of $\mathcal{Y}_{\ell\, 0}^{\ell}(-,x)$ gives us
\begin{equation*}
    \mathcal{Y}_{\ell\, 0}^{\ell}(\mathbf{w}_{\ell},x)\mathbf{1}\in\mathbf{w}_{\ell}+\mathbb{W}_{\ell}[[x]]x.
\end{equation*}
By the Jacobi identity and the $L_{-1}$-derivation property, we can deduce that
\begin{equation*}
    \mathcal{Y}_{\ell\, 0}^{\ell}(\mathbf{w}_{\ell},x)\mathbf{1}=e^{L_{-1}^{\mathbb{W}_{\ell}}}\mathbf{w}_{\ell}.
\end{equation*}
\end{rem}

\subsection{Associativity of intertwining operators}
The general idea of associativity comes down to comparing the composition and iteration of intertwining operators,
but it is not even clear in general if the composition and iteration are possible.
The following theorem is a consequence of the general analysis by Huang~\cite{Huang2005},
or was proven in our previous work~\cite{KoshidaKytola2022}.

\begin{thm}[\cite{Huang2005,KoshidaKytola2022}]
Let $\ell_{1},\ell_{2},\ell_{3},\ell_{4}\in\mathbb{Z}_{\geq 0}$.
\begin{enumerate}
    \item For any $n\in \Chan{\ell_{2}}{\ell_{3}}{\ell_{1}}{\ell_{4}}$ and $w_{1}\in \mathbb{W}_{\ell_{1}}$, $w_{2}\in\mathbb{W}_{\ell_{2}}$, $w_{3}\in\mathbb{W}_{\ell_{3}}$, the formal series
    \begin{equation*}
        \mathcal{Y}_{\ell_{1}\, n}^{\ell_{4}}(w_{1},x_{1})\mathcal{Y}_{\ell_{2}\,\ell_{3}}^{n}(w_{2},x_{2})w_{3}
    \end{equation*}
    in $x_{1}$ and $x_{2}$ converges in $\ol{\mathbb{W}}_{\ell_{4}}$ at $x_{1}=e^{\log (z_{1})}$, $x_{2}=e^{\log (z_{2})}$ such that $|z_{1}|>|z_{2}|>0$.
    \item For any $m\in \Chan{\ell_{1}}{\ell_{2}}{\ell_{3}}{\ell_{4}}$ and $w_{1}\in \mathbb{W}_{\ell_{1}}$, $w_{2}\in\mathbb{W}_{\ell_{2}}$, $w_{3}\in\mathbb{W}_{\ell_{3}}$, the formal series
    \begin{equation*}
        \mathcal{Y}_{m\,\ell_{3}}^{\ell_{4}}\left(\mathcal{Y}_{\ell_{1}\,\ell_{2}}^{m}(w_{1},x_{0})w_{2},x_{2}\right)w_{3}
    \end{equation*}
    in $x_{0}$ and $x_{2}$ converges in $\ol{\mathbb{W}}_{\ell_{4}}$ at $x_{0}=e^{\log (z_{0})}$, $x_{2}=e^{\log (z_{2})}$ such that $|z_{2}|>|z_{0}|>0$.
\end{enumerate}
\end{thm}

From Theorem~\ref{thm:first-row_module_fusion}, we know that the fusion rules among first-row modules match those of finite dimensional irreducible representations of $\UEA_{q}$.
The Clebsch--Gordan rules are, however, independent of the parameter $q$.
The following theorem manifests the matching of the parameters for the generic Virasoro VOA and the quantum group $\UEA_{q}$.

\begin{thm}[\cite{KoshidaKytola2022}]
\label{thm:associativity_int_op_frmods}
Let $\ell_{1},\ell_{2},\ell_{3},\ell_{4}\in\mathbb{Z}_{\geq 0}$ and take $z_{1},z_{2}$ on the real axis so that $z_{1}>z_{2}>z_{1}-z_{2}>0$.
For any $n\in \Chan{\ell_{2}}{\ell_{3}}{\ell_{1}}{\ell_{4}}$ and $w_{1}\in\mathbb{W}_{\ell_{1}}$, $w_{2}\in\mathbb{W}_{\ell_{2}}$, $w_{3}\in \mathbb{W}_{\ell_{3}}$,
we get
\begin{align*}
    &\,\mathcal{Y}_{\ell_{1}\, n}^{\ell_{4}}(w_{1},z_{1})\mathcal{Y}_{\ell_{2}\,\ell_{3}}^{n}(w_{2},z_{2})w_{3} \\
    =&\,\sum_{m\in \Chan{\ell_{1}}{\ell_{2}}{\ell_{3}}{\ell_{4}}}\sixjsymb{\ell_{1}}{\ell_{2}}{\ell_{3}}{\ell_{4}}{m}{n}\mathcal{Y}_{k'\,\ell_{3}}^{\ell_{4}}\left(\mathcal{Y}_{\ell_{1}\,\ell_{2}}^{k'}(w_{1},z_{1}-z_{2})w_{2},z_{2}\right)w_{3}
\end{align*}
in $\ol{\mathbb{W}}_{\ell_{4}}$,
where the $6j$-symbols in the right hand side are those of $\UEA_{q}$ equipped with $\Delta^{\mathrm{op}}$ defined in (\ref{eq:QG_6j_definition}) at
\begin{equation*}
    q=e^{\pi \mathbbm{i} t},\quad t\not\in\mathbb{Q}.
\end{equation*}
\end{thm}

\section{Categorical equivalence}
\label{eq:cat_equiv}
In this last section, we introduce the first-row subcategory $\cC_{\mathfrak{vir}}^{+}$ of the module category of the generic Virasoro VOA $\mathbb{V}_{c}$, and establish the braided equivalence $\cC_{\mathfrak{vir}}^{+}\to \cC_{\mathrm{QG}}^{\Delta}$.

\subsection{Dual and opposite categories}
We first make tiny preliminaries on dual and opposite categories.
Let $\cC$ be a category.
The dual category $\cC^{\vee}$ is the category with the same objects as $\cC$ and the opposite morphisms.
If $\cC$ is equipped with a monoidal structure, then $\cC^{\vee}$ naturally becomes a monoidal category.

Next, suppose that $\cC$ is a monoidal category with the monoidal bifunctor denoted by $\otimes$.
The opposite category $\cC^{\mathrm{op}}$ is the monoidal category with the same underlying category as $\cC$ and the opposite monoidal structure $\otimes^{\mathrm{op}}$:
\begin{equation*}
    X\otimes^{\mathrm{op}}Y:=Y\otimes X,\quad X,Y\in\cC.
\end{equation*}

In the case where $\cC$ is rigid, the dual and opposite categories are equivalent under a functor such that $X\mapsto X^{*}$ \cite[Chapter 2]{etingof2015tensor}.

\begin{rem}
Sometimes the dual category is called opposite, and the opposite category above is called reversed.
However, we follow the terminology in~\cite{etingof2015tensor}.
\end{rem}

When we apply the above construction to $\cC=\cC_{\mathrm{QG}}^{\Delta^{\mathrm{op}}}$, which we have already noticed is rigid, we can say that $(\cC_{\mathrm{QG}}^{\Delta^{\mathrm{op}}})^{\vee}$ and $(\cC_{\mathrm{QG}}^{\Delta^{\mathrm{op}}})^{\mathrm{op}}$ are equivalent.
Let us also see that the category $(\cC_{\mathrm{QG}}^{\Delta^{\mathrm{op}}})^{\mathrm{op}}$ is equivalent to $\cC_{\mathrm{QG}}^{\Delta}$ as a tensor category.
Recall that $(\cC_{\mathrm{QG}}^{\Delta^{\mathrm{op}}})^{\mathrm{op}}$ and $\cC_{\mathrm{QG}}^{\Delta}$ have the same underlying abelian category $\cC_{\mathrm{QG}}$.
We take the natural isomorphism
\begin{equation*}
    J_{\sfU,\sfV}:=P_{\sfV,\sfU}\colon \sfV\otimes_{\Delta^{\mathrm{op}}}\sfU=\sfU\otimes_{\Delta^{\mathrm{op}}}^{\mathrm{op}}\sfV\to \sfU\otimes_{\Delta}\sfV
\end{equation*}
to be the permutation for each $\sfU,\sfV\in \cC_{\mathrm{QG}}$.
Indeed, $J_{\sfU,\sfV}$ lives in $\mathrm{Hom}_{\cC_{\mathrm{QG}}}(\sfV\otimes_{\Delta^{\mathrm{op}}}\sfU,\sfU\otimes_{\Delta}\sfV)$ as is checked as
\begin{align*}
    \Delta (a)P_{\sfV,\sfU}(v\otimes u)&=\sum_{(a)}a_{(1)}u\otimes a_{(2)}v \\
    &=P_{\sfV,\sfU}\Bigl(\sum_{(a)}a_{(2)}v\otimes a_{(1)}u\Bigr) \\
    &=P_{\sfV,\sfU}\left(\Delta^{\mathrm{op}}(a)(v\otimes u)\right),\quad v\in \sfV,\, u\in \sfU,\, a\in \UEA_{q}.
\end{align*}
Here, we wrote $\Delta (a)=\sum_{(a)}a_{(1)}\otimes a_{(2)}$ for $a\in \UEA_{q}$.
Therefore, the identity functor $\mathrm{Id}_{\cC_{\mathrm{QG}}}$ together with the natural isomorphisms $(J_{\sfU,\sfV})_{\sfU,\sfV\in \cC_{\mathrm{QG}}}$ defines an equivalence between $(\cC_{\mathrm{QG}}^{\Delta^{\mathrm{op}}})^{\mathrm{op}}$ and $\cC_{\mathrm{QG}}^{\Delta}$ as tensor categories.

In conclusion, we may identify $(\cC_{\mathrm{QG}}^{\Delta^{\mathrm{op}}})^{\vee}$, $(\cC_{\mathrm{QG}}^{\Delta^{\mathrm{op}}})^{\mathrm{op}}$, and $\cC_{\mathrm{QG}}^{\Delta}$ altogether as tensor categories.

\subsection{The first-row subcategory \texorpdfstring{$\cC_{\mathfrak{vir}}^{+}$}{catVir+}}
We define the category $\cC_{\mathfrak{vir}}^{+}$ as the full subcategory of the module category of the generic Virasoro VOA $\mathbb{V}_{c}$ generated by the first row modules $\mathbb{W}_{\ell}$, $\ell\in\mathbb{Z}_{\geq 0}$ as an additive category.
Therefore, any object of $\cC_{\mathfrak{vir}}^{+}$ is isomorphic to a finite direct sum of first row modules, and the morphism spaces are determined by
\begin{equation*}
    \mathrm{Hom}_{\cC_{\mathfrak{vir}}^{+}}(\mathbb{W}_{\ell_{1}},\mathbb{W}_{\ell_{2}})=
    \begin{cases}
    \mathbb{C}\, \mathrm{id}_{\mathbb{W}_{\ell_{1}}},& \ell_{1}=\ell_{2},\\
    0, &\mathrm{otherwise}
    \end{cases}
\end{equation*}
for $\ell_{1},\ell_{2}\in\mathbb{Z}_{\geq 0}$.

It must be already clear that $\cC_{\mathfrak{vir}}^{+}$ is equivalent to $\cC_{\mathrm{QG}}$ as an abelian category.
Nevertheless, we would like to make it explicit.
For each object $\mathbb{U}\in\cC_{\mathfrak{vir}}^{+}$, we fix an isomorphism $f_{\mathbb{U}}\colon \mathbb{U}\to\bigoplus_{\ell=0}^{\infty}\mathbb{W}_{\ell}^{\oplus m_{\ell}}$.
Then, we define a functor $F\colon \cC_{\mathfrak{vir}}^{+}\to\cC_{\mathrm{QG}}$ as follows.
At the object level, if $\mathbb{U}\in \cC_{\mathfrak{vir}}^{+}$ is isomorphic to $\bigoplus_{\ell=0}^{\infty}\mathbb{W}_{\ell}^{\oplus m_{\ell}}$, we send
\begin{equation*}
    F\colon \mathbb{U}\mapsto \bigoplus_{\ell=0}^{\ell}\QGirrep{\ell}^{\oplus m_{\ell}}.
\end{equation*}
At the morphism level, we first require that
\begin{equation*}
    F\colon \mathrm{Hom}_{\cC_{\mathfrak{vir}}^{+}}(\mathbb{W}_{\ell},\mathbb{W}_{\ell})\to \mathrm{Hom}_{\cC_{\mathrm{QG}}}(\QGirrep{\ell},\QGirrep{\ell});\quad \mathrm{id}_{\mathbb{W}_{\ell}}\mapsto \mathrm{id}_{\QGirrep{\ell}}
\end{equation*}
for each $\ell\in\mathbb{Z}_{\geq 0}$.
For a general $h\colon \mathbb{U}_{1}\to\mathbb{U}_{2}$,
there is a unique way to send $f_{\mathbb{U}_{2}}\circ h\circ f_{\mathbb{U}_{1}}^{-1}$ so that $F$ induces linear maps on morphism spaces.
Then, we can simply define $F(h)=F(f_{\mathbb{U}_{2}}\circ h\circ f_{\mathbb{U}_{1}}^{-1})$.

Similarly, on the side of $\cC_{\mathrm{QG}}$, we fix an isomorphism $g_{\sfU}\colon \sfU\to \bigoplus_{\ell=0}^{\infty}\QGirrep{\ell}^{\oplus n_{\ell}}$ for each $\sfU\in\cC_{\mathrm{QG}}$.
Then, we can define a functor $G\colon \cC_{\mathrm{QG}}\to\cC_{\mathfrak{vir}}^{+}$ in the exactly analogous way as defining $F$.

Now, let us observe that the composition $G\circ F\colon \cC_{\mathfrak{vir}}^{+}\to\cC_{\mathfrak{vir}}^{+}$ is isomorphic to $\mathrm{id}_{\cC_{\mathfrak{vir}}^{+}}$.
Indeed, the family $(f_{\mathbb{U}})_{\mathbb{U}\in\cC_{\mathfrak{vir}}^{+}}$ of the fixed isomorphisms gives a natural isomorphism $\mathrm{id}_{\cC_{\mathfrak{vir}}^{+}}\Rightarrow G\circ F $;
for any $\mathbb{U}_{1},\mathbb{U}_{2}\in \cC_{\mathfrak{vir}}^{+}$ and $h\colon \mathbb{U}_{1}\to\mathbb{U}_{2}$,
the commutativity of the diagram
\begin{equation*}
    \xymatrix{
    \mathbb{U}_{1} \ar[r]^{f_{\mathbb{U}_{1}}} \ar[d]_{h} & G\circ F(\mathbb{U}_{1}) \ar[d]^{f_{\mathbb{U}_{2}}\circ h\circ f_{\mathbb{U}_{1}}^{-1}} \\
    \mathbb{U}_{2} \ar[r]_{f_{\mathbb{U}_{2}}} & G\circ F(\mathbb{U}_{2})
    }
\end{equation*}
follows from the definitions of $F$ and $G$.
Similarly, $F\circ G\colon \cC_{\mathrm{QG}}\to \cC_{\mathrm{QG}}$ is shown to be isomorphic to $\mathrm{id}_{\cC_{\mathrm{QG}}}$.

At this point, we state our main result.
\begin{thm}
The first-row subcategory $\cC_{\mathfrak{vir}}^{+}$ is equipped with the structure of a braided tensor category and is equivalent to the category $\cC_{\mathrm{QG}}^{\Delta}$ under the matching of the parameters $q=e^{\pi\mathbbm{i} t}$.
\end{thm}

In the rest of the paper, we will define a monoidal structure and braiding on $\cC_{\mathfrak{vir}}^{+}$ and, at each step, we compare the structure with that of $\cC_{\mathrm{QG}}^{\Delta}$, which consequently proves the theorem.

\subsection{Tensor product}
First, we show that the category $\cC_{\mathfrak{vir}}^{+}$ is closed under the $P(z)$-tensor product for any $z\in \mathbb{C}^{\times}$.
The following formula (\ref{eq:Pz-tensor_frmods}) has been recorded in \cite[Theorem 5.2.2]{CJORY-tensor_categories_arising_from_Virasoro_algebra}, but we give a proof of it to keep the text elementary.

\begin{thm}
For any $z\in \mathbb{C}^{\times}$, the category $\cC_{\mathfrak{vir}}^{+}$ is closed under the $P(z)$-tensor product.
Furthermore, for $\ell_{1},\ell_{2}\in\mathbb{Z}_{\geq 0}$, the $P(z)$-tensor product $\mathbb{W}_{\ell_{1}}\boxtimes_{P(z)}\mathbb{W}_{\ell_{2}}$ is given by
\begin{equation}
\label{eq:Pz-tensor_frmods}
    \mathbb{W}_{\ell_{1}}\boxtimes_{P(z)}\mathbb{W}_{\ell_{2}}=\bigoplus_{\ell\in\mathrm{Sel} (\ell_{1},\ell_{2})}\mathbb{W}_{\ell}
\end{equation}
together with the $P(z)$-intertwining map
\begin{equation}
\label{eq:Pz-intmap_frmods}
    \mathcal{Y}_{\mathbb{W}_{\ell_{1}}\boxtimes_{P(z)}\mathbb{W}_{\ell_{2}}}(-,z)=\sum_{\ell\in\mathrm{Sel} (\ell_{1},\ell_{2})}\mathcal{Y}_{\ell_{1}\,\ell_{2}}^{\ell}(-,z)-.
\end{equation}
\end{thm}
\begin{proof}
Since the category $\cC_{\mathfrak{vir}}^{+}$ is semi-simple, it suffices to show that the $P(z)$-tensor product of simple objects exists and is given by the formulas (\ref{eq:Pz-tensor_frmods}) and (\ref{eq:Pz-intmap_frmods}).
Let $\mathbb{U}\in \cC_{\mathfrak{vir}}^{+}$ together with $\mathcal{Y}_{\mathbb{U}} (-,z)-\colon \mathbb{W}_{\ell_{1}}\otimes\mathbb{W}_{\ell_{2}}\to\ol{\mathbb{U}}$ be a $P(z)$-product of $\mathbb{W}_{\ell_{1}}$ and $\mathbb{W}_{\ell_{2}}$.
The object $\mathbb{U}$ can be decomposed into a direct sum of simple objects
\begin{equation}
\label{eq:virmod_general_branching}
    \mathbb{U}\simeq \bigoplus_{i=1}^{m}\mathbb{W}_{k_{i}},
\end{equation}
where $k_{i}\in\mathbb{Z}_{\geq 0}$, $i=1,\dots, m$.
In other words, we can find a family of injections $\iota_{i}\colon \mathbb{W}_{k_{i}}\to \mathbb{U}$ and projections $p_{i}\colon \mathbb{U}\to\mathbb{W}_{k_{i}}$, $i=1,\dots, m$ such that
\begin{equation}
\label{eq:virmod_direct_sum_structure}
    p_{i}\circ \iota_{j}=\delta_{i,j}\mathrm{id}_{\mathbb{W}_{k_{i}}},\quad i,j=1,\dots,m,\quad \sum_{i=1}^{m}\iota_{i}\circ p_{i}=\mathrm{id}_{\mathbb{U}}.
\end{equation}
Then, for each $i=1,\dots, m$, the composition
$\ol{p}_{i}\circ \mathcal{Y}_{\mathbb{U}}(-,z)-$
is an intertwining map of type $\inttyp{\mathbb{W}_{\ell_{1}}}{\mathbb{W}_{\ell_{2}}}{\mathbb{W}_{k_{i}}}$.
Hence, there exists a unique homomorphism \begin{equation*}
    f_{i}\colon\bigoplus_{\ell\in\mathrm{Sel}(\ell_{1},\ell_{2})}\mathbb{W}_{\ell}\to\mathbb{W}_{k_{i}}
\end{equation*}
such that
\begin{equation*}
    \ol{p}_{i}\circ \mathcal{Y}_{\mathbb{U}}(-,z)-=\ol{f}_{i}\circ \mathcal{Y}_{\mathbb{W}_{\ell_{1}}\boxtimes_{P(z)}\mathbb{W}_{\ell_{2}}}(-,z)-.
\end{equation*}
Therefore, the sum $f=\sum_{i=1}^{m}\iota_{i}\circ f_{i}$ is the unique homomorphism such that
\begin{equation}
\label{eq:Pz-tensor_universality_hom}
    \mathcal{Y}_{\mathbb{U}}(-,z)-=\sum_{i=1}^{m}\ol{\iota_{i}\circ p_{i}}\circ \mathcal{Y}_{\mathbb{U}}(-,z)-=\ol{f}\circ \mathcal{Y}_{\mathbb{W}_{\ell_{1}}\boxtimes_{P(z)}\mathbb{W}_{\ell_{2}}}(-,z)-.
\end{equation}

It remains to show that the homomorphism $f$ is even independent of an irreducible decomposition.
We can still assume the branching (\ref{eq:virmod_general_branching}) and suppose that there are other injections $\iota'_{i}\colon \mathbb{W}_{k_{i}}\to\mathbb{U}$ and projections $p'_{i}\colon\mathbb{U}\to\mathbb{W}_{k_{i}}$, $i=1,\dots, m$ that satisfy the same equations as (\ref{eq:virmod_direct_sum_structure}) for $\iota_{i}$ and $p_{i}$, $i=1,\dots, m$.
Then, we get $f'=\sum_{i=1}^{m}\iota'_{i}\circ f'_{i}$, where each $f'_{i}$ is determined by
\begin{equation*}
    \ol{p}'_{i}\circ \mathcal{Y}_{\mathbb{U}}(-,z)-=\ol{f}'_{i}\circ \mathcal{Y}_{\mathbb{W}_{\ell_{1}}\boxtimes_{P(z)}\mathbb{W}_{\ell_{2}}}(-,z)-.
\end{equation*}
From (\ref{eq:Pz-tensor_universality_hom}) and the uniqueness of $f'_{i}$, we can relate $\{f_{i}\}_{i=1}^{m}$ and $\{f'_{i}\}_{i=1}^{m}$ as
\begin{equation*}
    f'_{i}=\sum_{j=1}^{m}p'_{i}\circ \iota_{j}\circ f_{j},\quad i=1,\dots, m,
\end{equation*}
from which we get
\begin{equation*}
    f'=\sum_{i=1}^{m}\iota'_{i}\circ f'_{i}=\sum_{j=1}^{m}\iota_{j}\circ f_{j}=f.
\end{equation*}
Therefore, the desired universal property has been proved.
\end{proof}

We write $\pzproj{z}{\ell_{1}}{\ell_{2}}{\ell}$ for the canonical projection from $\mathbb{W}_{\ell_{1}}\boxtimes_{P(z)}\mathbb{W}_{\ell_{2}}$ to $\mathbb{W}_{\ell}$ for $\ell\in \mathrm{Sel} (\ell_{1},\ell_{2})$ according to the realization (\ref{eq:Pz-tensor_frmods}).
Then, it is characterized by the property
\begin{equation*}
    \ol{\pzproj{z}{\ell_{1}}{\ell_{2}}{\ell}} (w_{1}\boxtimes_{P(z)}w_{2})=\mathcal{Y}_{\ell_{1}\,\ell_{2}}^{\ell}(w_{1},z)w_{2},\quad w_{1}\in\mathbb{W}_{\ell_{1}},\,w_{2}\in\mathbb{W}_{\ell_{2}}.
\end{equation*}

Given two points $z_{1},z_{2}\in \mathbb{C}^{\times}$ and a path $\gamma$ from $z_{1}$ to $z_{2}$, we can define the parallel transport isomorphism $\mathcal{T}_{\gamma}\colon \boxtimes_{P(z_{1})}\Rightarrow \boxtimes_{P(z_{2})}$ as we have explained in Section~\ref{sect:module_cat_VOA}.
Although the parallel transport depends on (the homotopy class of) the path $\gamma$, its action can be described in a simple way, especially in the case when $\gamma$ does not change the branch.

\begin{prop}
\label{prop:triviality_parallel_transport}
Let $z_{1},z_{2}\in \mathbb{C}^{\times}$ and $\gamma$ be a path from $z_{1}$ to $z_{2}$, along which $e^{l_{\gamma}(z_{1})}$ has the same angle as $z_{1}$ in $[0,2\pi)$.
Suppose that, for $\ell_{1},\ell_{2}\in \mathbb{Z}_{\geq 0}$, the $P(z)$-tensor product $\mathbb{W}_{\ell_{1}}\boxtimes_{P(z)}\mathbb{W}_{\ell_{2}}$ is realized by the formulas (\ref{eq:Pz-tensor_frmods}) and (\ref{eq:Pz-intmap_frmods}).
Then, the parallel transport $\mathcal{T}_{\gamma}$ acts as $\mathrm{id}_{\mathbb{W}_{\ell}}$ on each component $\mathbb{W}_{\ell}$, $\ell\in \mathrm{Sel}(\ell_{1},\ell_{2})$ appearing in (\ref{eq:Pz-tensor_frmods}).
\end{prop}
\begin{proof}
In the realization in (\ref{eq:Pz-tensor_frmods}) and (\ref{eq:Pz-intmap_frmods}), the $P(z)$-tensor product does not depend on $z$ at the object level, so we can think that $\mathbb{W}_{\ell_{1}}\boxtimes_{P(z_{1})}\mathbb{W}_{\ell_{2}}=\mathbb{W}_{\ell_{1}}\boxtimes_{P(z_{2})}\mathbb{W}_{\ell_{2}}$ as objects.
Recall that $\mathcal{T}_{\gamma}$ is characterized by the property
\begin{equation*}
    \ol{\mathcal{T}}_{\gamma}\colon \sum_{\ell\in\mathrm{Sel}(\ell_{1},\ell_{2})}\mathcal{Y}_{\ell_{1}\,\ell_{2}}^{\ell}(w_{1},z_{1})w_{2}\mapsto \sum_{\ell\in\mathrm{Sel}(\ell_{1},\ell_{2})}\mathcal{Y}_{\ell_{1}\,\ell_{2}}^{\ell}(w_{1},e^{l_{\gamma}(z_{1})})w_{2}
\end{equation*}
for $w_{1}\in\mathbb{W}_{\ell_{1}}, w_{2}\in\mathbb{W}_{\ell_{2}}$,
but under the assumption on $\gamma$, we have
\begin{equation*}
    \mathcal{Y}_{\ell_{1}\,\ell_{2}}^{\ell}(w_{1},e^{l_{\gamma}(z_{1})})w_{2}=\mathcal{Y}_{\ell_{1}\,\ell_{2}}^{\ell}(w_{1},z_{1})w_{2},\quad \ell\in \mathrm{Sel} (\ell_{1},\ell_{2}).
\end{equation*}
Therefore, $\mathcal{T}_{\gamma}$ acts as $\mathrm{id}_{\mathbb{W}_{\ell}}$ on each $\mathbb{W}_{\ell}$, $\ell\in\mathrm{Sel} (\ell_{1},\ell_{2})$.
\end{proof}

Let us look into the associativity isomorphism on $\cC_{\mathfrak{vir}}^{+}$.
First, we clarify the structure of the composition and iteration of the tensor product in more detail, focusing our attention on simple objects.
We take $z_{1},z_{2}$ on the real axis so that $z_{1}>z_{2}>z_{1}-z_{2}>0$ and the paths $\gamma_{1},\gamma_{2},\gamma_{3},\gamma_{4}$ as before.
Let us fix $\ell_{1},\ell_{2},\ell_{3}\in\mathbb{Z}_{\geq 0}$.
As we have already noticed, according to the realization of the $P(z)$-tensor product as in (\ref{eq:Pz-tensor_frmods}) and (\ref{eq:Pz-intmap_frmods}), we can identify
\begin{equation}
\label{eq:composition_frmods}
    \mathbb{W}_{\ell_{1}}\boxtimes_{P(z_{1})} (\mathbb{W}_{\ell_{2}}\boxtimes_{P(z_{2})} \mathbb{W}_{\ell_{3}})
    =\mathbb{W}_{\ell_{1}}\boxtimes (\mathbb{W}_{\ell_{2}}\boxtimes \mathbb{W}_{\ell_{3}})=\bigoplus_{\ell_{\infty}\in\mathbb{Z}_{\geq 0}}\bigoplus_{n\in \Chan{\ell_{2}}{\ell_{3}}{\ell_{1}}{\ell_{\infty}}}(\mathbb{W}_{\ell_{\infty}})^{(n)},
\end{equation}
where $(\mathbb{W}_{\ell_{\infty}})^{(n)}$ are copies of $\mathbb{W}_{\ell_{\infty}}$ and the canonical projection to each $(\mathbb{W}_{\ell_{\infty}})^{(n)}$ is given by
\begin{equation*}
    \pzproj{z_{1}}{\ell_{1}}{n}{\ell_{\infty}}\circ \left(\mathrm{id}_{\mathbb{W}_{\ell_{1}}}\boxtimes_{P(z_{1})}\pzproj{z_{2}}{\ell_{2}}{\ell_{3}}{n}\right)
\end{equation*}
From the definition, it is clear that
\begin{align*}
    \ol{\pzproj{z_{1}}{\ell_{1}}{n}{\ell_{\infty}}\circ \left(\mathrm{id}_{\mathbb{W}_{\ell_{1}}}\boxtimes_{P(z_{1})}\pzproj{z_{2}}{\ell_{2}}{\ell_{3}}{n}\right)}\colon &w_{1}\boxtimes_{P(z_{1})}(w_{2}\boxtimes_{P(z_{2})}w_{3})\\
    \mapsto &\mathcal{Y}_{\ell_{1}\,n}^{\ell_{\infty}}(w_{1},z_{1})\mathcal{Y}_{\ell_{2}\,\ell_{3}}^{n}(w_{2},z_{2})w_{3}
\end{align*}
for $w_{i}\in\mathbb{W}_{\ell_{i}}$, $i=1,2,3$.
Furthermore, from Proposition~\ref{prop:triviality_parallel_transport}, the composition of parallel transports $\mathcal{T}_{\gamma_{1}}\circ (\mathrm{id}_{\mathbb{W}_{\ell_{1}}}\boxtimes_{P(z_{1})} \mathcal{T}_{\gamma_{2}})$ acts as $\mathrm{id}_{(\mathbb{W}_{\ell_{\infty}})^{(n)}}$ on each component $(\mathbb{W}_{\ell_{\infty}})^{(n)}$, $\ell_{\infty}\in\mathbb{Z}_{\geq 0}$, $n\in \Chan{\ell_{2}}{\ell_{3}}{\ell_{1}}{\ell_{\infty}}$.

Similarly, for the iteration, we have the identification
\begin{equation}
\label{eq:iteration_frmods}
    (\mathbb{W}_{\ell_{1}}\boxtimes_{P(z_{1}-z_{2})}\mathbb{W}_{\ell_{2}})\boxtimes_{P(z_{2})}\mathbb{W}_{\ell_{3}}
    =(\mathbb{W}_{\ell_{1}}\boxtimes \mathbb{W}_{\ell_{2}})\boxtimes\mathbb{W}_{\ell_{3}}
    =\bigoplus_{\ell_{\infty}\in\mathbb{Z}_{\geq 0}}\bigoplus_{m\in I^{\ell_{1}\,\ell_{2}}_{\ell_{3}\,\ell_{\infty}}}(\mathbb{W}_{\ell_{\infty}})_{(m)},
\end{equation}
where each copy $(\mathbb{W}_{\ell_{\infty}})_{(m)}$ is the image of the projection
\begin{equation*}
    \pzproj{z_{2}}{m}{\ell_{3}}{\ell_{\infty}}\circ \left(\pzproj{z_{1}-z_{2}}{\ell_{1}}{\ell_{2}}{m}\boxtimes_{P(z_{2})}\mathrm{id}_{\mathbb{W}_{\ell_{3}}}\right).
\end{equation*}
Again, from the definition, this projection is characterized by the property
\begin{align*}
    \ol{\pzproj{z_{2}}{m}{\ell_{3}}{\ell_{\infty}}\circ \left(\pzproj{z_{1}-z_{2}}{\ell_{1}}{\ell_{2}}{m}\boxtimes_{P(z_{2})}\mathrm{id}_{\mathbb{W}_{\ell_{3}}}\right)}\colon & (w_{1}\boxtimes_{P(z_{1}-z_{2})}w_{2})\boxtimes_{P(z_{2})}w_{3} \\
    \mapsto & \mathcal{Y}_{m\,\ell_{3}}^{\ell_{\infty}}\left(\mathcal{Y}_{\ell_{1}\,\ell_{2}}^{m}(w_{1},z_{1}-z_{2})w_{2},z_{2}\right)w_{3}
\end{align*}
for $w_{i}\in\mathbb{W}_{\ell_{i}}$, $i=1,2,3$.
By Proposition~\ref{prop:triviality_parallel_transport}, the composition of parallel transports $(\mathcal{T}_{\gamma_{4}}\boxtimes_{P(z_{2})}\mathrm{id}_{\mathbb{W}_{\ell_{3}}})\circ \mathcal{T}_{\gamma_{3}}$ acts as $\mathrm{id}_{(\mathbb{W}_{\ell_{\infty}})_{(m)}}$ on each component $(\mathbb{W}_{\ell_{\infty}})_{(m)}$, $\ell_{\infty}\in \mathbb{Z}_{\geq 0}$, $m\in \Chan{\ell_{1}}{\ell_{2}}{\ell_{3}}{\ell_{\infty}}$.

The above observations allow us to conclude that the associativity isomorphism $\mathcal{A}_{\mathbb{W}_{\ell_{1}},\mathbb{W}_{\ell_{2}},\mathbb{W}_{\ell_{3}}}$ coincides with the resolved version $(\mathcal{A}_{P(z_{1}),P(z_{2})}^{P(z_{1}-z_{2}),P(z_{2})})_{\mathbb{W}_{\ell_{1}},\mathbb{W}_{\ell_{2}},\mathbb{W}_{\ell_{3}}}$ under the identifications (\ref{eq:composition_frmods}) and (\ref{eq:iteration_frmods}) and amounts to a homomorphism of the form
\begin{equation*}
    \mathcal{A}_{\mathbb{W}_{\ell_{1}},\mathbb{W}_{\ell_{2}},\mathbb{W}_{\ell_{3}}}=\sum_{\ell_{\infty}\in\mathbb{Z}_{\geq 0}}\sum_{m\in \Chan{\ell_{1}}{\ell_{2}}{\ell_{3}}{\ell_{\infty}}}\sum_{n\in \Chan{\ell_{2}}{\ell_{3}}{\ell_{1}}{\ell_{\infty}}}\ascmtx{\ell_{1}}{\ell_{2}}{\ell_{3}}{\ell_{\infty}}{m}{n},\quad \ascmtx{\ell_{1}}{\ell_{2}}{\ell_{3}}{\ell_{\infty}}{m}{n}\colon (\mathbb{W}_{\ell_{\infty}})_{(m)}\to (\mathbb{W}_{\ell_{\infty}})^{(n)}.
\end{equation*}

\begin{thm}
The associativity isomorphism $\mathcal{A}_{\mathbb{W}_{\ell_{1}},\mathbb{W}_{\ell_{2}},\mathbb{W}_{\ell_{3}}}$ is given by
\begin{equation*}
    \ascmtx{\ell_{1}}{\ell_{2}}{\ell_{3}}{\ell_{\infty}}{m}{n}=\sixjsymb{\ell_{1}}{\ell_{2}}{\ell_{3}}{\ell_{\infty}}{m}{n}\mathrm{id}_{\mathbb{W}_{\ell_{\infty}}},\quad \ell_{\infty}\in\mathbb{Z}_{\geq 0},\, m\in \Chan{\ell_{1}}{\ell_{2}}{\ell_{3}}{\ell_{\infty}},\, n\in \Chan{\ell_{2}}{\ell_{3}}{\ell_{1}}{\ell_{\infty}}.
\end{equation*}
Here, the 6j-symbols are those of $\UEA_{q}$ at $q=e^{\pi\mathbbm{i} t}$ equipped with the opposite coproduct $\Delta^{\mathrm{op}}$.
\end{thm}
\begin{proof}
It suffices to show that the isomorphism $\mathcal{A}_{\mathbb{W}_{\ell_{1}},\mathbb{W}_{\ell_{2}},\mathbb{W}_{\ell_{3}}}$ defined in such a way behaves as
\begin{equation*}
    \mathcal{A}_{\mathbb{W}_{\ell_{1}},\mathbb{W}_{\ell_{2}},\mathbb{W}_{\ell_{3}}}\colon (w_{1}\boxtimes_{P(z_{1}-z_{2})}w_{2})\boxtimes_{P(z_{2})}w_{3}\mapsto w_{1}\boxtimes_{P(z_{1})}(w_{2}\boxtimes_{P(z_{2})}w_{3})
\end{equation*}
for $w_{i}\in\mathbb{W}_{\ell_{i}}$, $i=1,2,3$.
Recall that
\begin{equation*}
    (w_{1}\boxtimes_{P(z_{1}-z_{2})}w_{2})\boxtimes_{P(z_{2})}w_{3}=\sum_{\ell_{\infty}\in\mathbb{Z}_{\geq 0}}\sum_{m\in \Chan{\ell_{1}}{\ell_{2}}{\ell_{3}}{\ell_{\infty}}}\mathcal{Y}_{m\,\ell_{3}}^{\ell_{\infty}}\left(\mathcal{Y}_{\ell_{1}\,\ell_{2}}^{m}(w_{1},z_{1}-z_{2})w_{2},z_{2}\right)w_{3},
\end{equation*}
which is sent by $\mathcal{A}_{\mathbb{W}_{\ell_{1}},\mathbb{W}_{\ell_{2}},\mathbb{W}_{\ell_{3}}}$ to
\begin{align}
\label{eq:associativity_computation_intermediate}
    \sum_{\ell_{\infty}\in\mathbb{Z}_{\geq 0}}\sum_{n\in \Chan{\ell_{2}}{\ell_{3}}{\ell_{1}}{\ell_{\infty}}}\left(\sum_{m\in \Chan{\ell_{1}}{\ell_{2}}{\ell_{3}}{\ell_{\infty}}}\sixjsymb{\ell_{1}}{\ell_{2}}{\ell_{3}}{\ell_{\infty}}{m}{n}\mathcal{Y}_{m\,\ell_{3}}^{\ell_{\infty}}\left(\mathcal{Y}_{\ell_{1}\,\ell_{2}}^{m}(w_{1},z_{1}-z_{2})w_{2},z_{2}\right)w_{3}\right),
\end{align}
where each summand
\begin{equation*}
    \sum_{m\in \Chan{\ell_{1}}{\ell_{2}}{\ell_{3}}{\ell_{\infty}}}\sixjsymb{\ell_{1}}{\ell_{2}}{\ell_{3}}{\ell_{\infty}}{m}{n}\mathcal{Y}_{m\,\ell_{3}}^{\ell_{\infty}}\left(\mathcal{Y}_{\ell_{1}\,\ell_{2}}^{m}(w_{1},z_{1}-z_{2})w_{2},z_{2}\right)w_{3}
\end{equation*}
lives in $(\mathbb{W}_{\ell_{\infty}})^{(n)}$.
Here, we use Theorem~\ref{thm:associativity_int_op_frmods} to conclude that (\ref{eq:associativity_computation_intermediate}) coincides with
\begin{align*}
    \sum_{\ell_{\infty}\in\mathbb{Z}_{\geq 0}}\sum_{n\in \Chan{\ell_{2}}{\ell_{3}}{\ell_{1}}{\ell_{\infty}}}\mathcal{Y}_{\ell_{1}\,n}^{\ell_{\infty}}(w_{1},z_{1})\mathcal{Y}_{\ell_{2}\,\ell_{3}}^{n}(w_{2},z_{2})w_{3}=w_{1}\boxtimes_{P(z_{1})}(w_{2}\boxtimes_{P(z_{2})}w_{3}).
\end{align*}
Therefore, the isomorphism $\mathcal{A}_{\mathbb{W}_{\ell_{1}},\mathbb{W}_{\ell_{2}},\mathbb{W}_{\ell_{3}}}$ satisfies the desired property.
\end{proof}

We can define unit isomorphisms $\lambda_{\mathbb{U}}\colon \mathbb{V}_{c}\boxtimes\mathbb{U}\to\mathbb{U}$ and $\rho_{\mathbb{U}}\colon \mathbb{U}\boxtimes\mathbb{V}_{c}\to \mathbb{U}$ for each $\mathbb{U}\in\cC_{\mathfrak{vir}}^{+}$ according to the general theory.
If $\mathbb{U}=\mathbb{W}_{\ell}$, $\ell\in\mathbb{Z}_{\geq 0}$ is a simple object, we think that $\mathbb{V}_{c}\boxtimes\mathbb{W}_{\ell}=\mathbb{W}_{\ell}=\mathbb{W}_{\ell}\boxtimes\mathbb{V}_{c}$ under the realization in (\ref{eq:Pz-tensor_frmods}) and (\ref{eq:Pz-intmap_frmods}).
\begin{thm}
\label{thm:unit_isom_catvir}
Let $\ell\in\mathbb{Z}_{\geq 0}$. Under the realization in (\ref{eq:Pz-tensor_frmods}) and (\ref{eq:Pz-intmap_frmods}), both $\lambda_{\mathbb{W}_{\ell}}\colon \mathbb{V}_{c}\boxtimes \mathbb{W}_{\ell}=\mathbb{W}_{\ell}\to\mathbb{W}_{\ell}$ and $\rho_{\mathbb{W}_{\ell}}\colon \mathbb{W}_{\ell}\boxtimes\mathbb{V}_{c}=\mathbb{W}_{\ell}\to\mathbb{W}_{\ell}$ are $\mathrm{id}_{\mathbb{W}_{\ell}}$.
\end{thm}
\begin{proof}
Recall that the left unit isomorphism $\lambda_{\mathbb{W}_{\ell}}$ is characterized by $\lambda_{\mathbb{W}_{\ell}}(\mathbf{1}\boxtimes w)=w$, $w\in\mathbb{W}_{\ell}$.
From (\ref{eq:Pz-intmap_frmods}), we have
\begin{equation*}
    \mathbf{1}\boxtimes w=\mathcal{Y}_{0\,\ell}^{\ell}(\mathbf{1},1)w=w.
\end{equation*}
Here we used $\mathcal{Y}_{0\,\ell}^{\ell}(\mathbf{1},x)=\mathrm{id}$, which we have noticed in Remark~\ref{rem:intertw_Virasoro_special_cases}.
Hence, we observe $\lambda_{\mathbb{W}_{\ell}}=\mathrm{id}_{\mathbb{W}_{\ell}}$.
As for the right unit isomorphism, it is characterized by $\ol{\rho_{\mathbb{W}_{\ell}}}(w\boxtimes \mathbf{1})=e^{L^{\mathbb{W}_{\ell}}_{-1}}w$, $w\in\mathbb{W}_{\ell}$.
Again, from (\ref{eq:Pz-intmap_frmods}) and the property of $\mathcal{Y}_{\ell\,0}^{\ell}(-,x)$ pointed out in Remark~\ref{rem:intertw_Virasoro_special_cases}, we see that
\begin{equation*}
    w\boxtimes \mathbf{1} = \mathcal{Y}_{\ell\,0}^{\ell}(w,1)=e^{L^{\mathbb{W}_{\ell}}_{-1}}w,
\end{equation*}
which verifies that $\rho_{\mathbb{W}_{\ell}}=\mathrm{id}_{\mathbb{W}_{\ell}}$.
\end{proof}

Now we can compare the structure of $\cC_{\mathfrak{vir}}^{+}$ as a tensor category to that of $\cC_{\mathrm{QG}}^{\Delta}$.
\begin{thm}
The functor $F\colon \cC_{\mathfrak{vir}}^{+}\to \cC_{\mathrm{QG}}^{\Delta}$ gives an equivalence of tensor categories under the parameter matching $q=e^{\pi\mathbbm{i} t}$.
\end{thm}
\begin{proof}
From Theorem~\ref{thm:unit_isom_catvir}, it is clear that $F(\lambda_{\mathbb{W}_{\ell}})=F(\rho_{\mathbb{W}_{\ell}})=\mathrm{id}_{\QGirrep{\ell}}$ for each $\ell\in\mathbb{Z}_{\geq 0}$.
Therefore, it remains to show the coincidence of associativity isomorphisms.

Recall the equivalence $\cC_{\mathrm{QG}}^{\Delta}\simeq (\cC_{\mathrm{QG}}^{\Delta^{\mathrm{op}}})^{\vee}$ of tensor categories.
We compare the monoidal structure of $\cC_{\mathfrak{vir}}^{+}$ with that of $(\cC_{\mathrm{QG}}^{\Delta^{\mathrm{op}}})^{\vee}$ instead of $\cC_{\mathrm{QG}}^{\Delta}$.
The associativity isomorphism $\mathcal{A}_{\mathbb{W}_{\ell_{1}},\mathbb{W}_{\ell_{2}},\mathbb{W}_{\ell_{3}}}$ of $\cC_{\mathfrak{vir}}^{+}$ is characterized by the property
\begin{align*}
    &\pzproj{z_{1}}{\ell_{1}}{n}{\ell_{\infty}}\circ \left(\mathrm{id}_{\mathbb{W}_{\ell_{1}}}\boxtimes_{P(z_{1})}\pzproj{z_{2}}{\ell_{2}}{\ell_{3}}{n}\right)\circ \mathcal{A}_{\mathbb{W}_{\ell_{1}},\mathbb{W}_{\ell_{2}},\mathbb{W}_{\ell_{3}}} \\
    =& \sum_{m\in \Chan{\ell_{1}}{\ell_{2}}{\ell_{3}}{\ell_{\infty}}}\sixjsymb{\ell_{1}}{\ell_{2}}{\ell_{3}}{\ell_{\infty}}{m}{n}\pzproj{z_{2}}{m}{\ell_{3}}{\ell_{\infty}}\circ \left(\pzproj{z_{1}-z_{2}}{\ell_{1}}{\ell_{2}}{m}\boxtimes_{P(z_{2})}\mathrm{id}_{\mathbb{W}_{\ell_{3}}}\right).
\end{align*}
for $n\in \Chan{\ell_{2}}{\ell_{3}}{\ell_{1}}{\ell_{\infty}}$.
We compare this with (\ref{eq:QG_6j_definition}) to conclude that
\begin{align*}
    F(\mathcal{A}_{\mathbb{W}_{\ell_{1}},\mathbb{W}_{\ell_{2}},\mathbb{W}_{\ell_{3}}})&= (\alpha_{\QGirrep{\ell_{1}},\QGirrep{\ell_{2}},\QGirrep{\ell_{3}}}^{\mathrm{op}})^{-1}\\
    &\in  \mathrm{Hom}_{\cC_{\mathrm{QG}}^{\Delta^{\mathrm{op}}}}(\QGirrep{\ell_{1}}\otimes_{\Delta^{\mathrm{op}}} (\QGirrep{\ell_{2}}\otimes_{\Delta^{\mathrm{op}}}\QGirrep{\ell_{3}}),(\QGirrep{\ell_{1}}\otimes_{\Delta^{\mathrm{op}}}\QGirrep{\ell_{2}})\otimes_{\Delta^{\mathrm{op}}}\QGirrep{\ell_{3}}) \\
    &=  \mathrm{Hom}_{(\cC_{\mathrm{QG}}^{\Delta^{\mathrm{op}}})^{\vee}}((\QGirrep{\ell_{1}}\otimes_{\Delta^{\mathrm{op}}}\QGirrep{\ell_{2}})\otimes_{\Delta^{\mathrm{op}}}\QGirrep{\ell_{3}},\QGirrep{\ell_{1}}\otimes_{\Delta^{\mathrm{op}}} (\QGirrep{\ell_{2}}\otimes_{\Delta^{\mathrm{op}}}\QGirrep{\ell_{3}})).
\end{align*}
This proves the desired result.
\end{proof}

\begin{rem}
We make a comment on the reason why we compared the associativity of $\cC_{\mathfrak{vir}}^{+}$ with that of $(\cC_{\mathrm{QG}}^{\Delta^{\mathrm{op}}})^{\vee}$, but not $\cC_{\mathrm{QG}}^{\Delta}$.
Recall that our $6j$-symbols are defined as the matrix elements in terms of injections.
On the VOA side, the same $6j$-symbols naturally appear as matrix elements in projections.
Therefore, to match injections to projections, one needs to take the dual of either category.
This also explains why we defined the $6j$ symbols associated with the opposite coproduct $\Delta^{\mathrm{op}}$.
Since we have decided to take the dual category on the quantum group side, we can eventually return to the original coproduct $\Delta$ under the equivalence $(\cC_{\mathrm{QG}}^{\Delta^{\mathrm{op}}})^{\vee}\simeq \cC_{\mathrm{QG}}^{\Delta}$.
\end{rem}

\subsection{Braiding}
We move on to comparing the structures of braiding on $\cC_{\mathfrak{vir}}^{+}$ and $\cC_{\mathrm{QG}}^{\Delta}$.
We first calculate the braiding of $\cC_{\mathfrak{vir}}^{+}$ following the definition given in Section~\ref{sect:module_cat_VOA}.
Under the realization (\ref{eq:Pz-tensor_frmods}), we can consider $\mathbb{W}_{\ell_{1}}\boxtimes \mathbb{W}_{\ell_{2}}$ and $\mathbb{W}_{\ell_{2}}\boxtimes\mathbb{W}_{\ell_{1}}$ to be identical as objects.
\begin{prop}
Let $\ell_{1},\ell_{2}\in \mathbb{Z}_{\geq 0}$.
Under the realization of $\mathbb{W}_{\ell_{1}}\boxtimes \mathbb{W}_{\ell_{2}}$ and $\mathbb{W}_{\ell_{2}}\boxtimes\mathbb{W}_{\ell_{1}}$ given by the formulas (\ref{eq:Pz-tensor_frmods}) and (\ref{eq:Pz-intmap_frmods}), the braiding isomorphism $c_{\mathbb{W}_{\ell_{1}},\mathbb{W}_{\ell_{2}}}$ is given by
\begin{equation*}
    c_{\mathbb{W}_{\ell_{1}},\mathbb{W}_{\ell_{2}}}=\sum_{\ell\in \mathrm{Sel} (\ell_{1},\ell_{2})}e^{\pi\mathbbm{i} (h_{\ell}-h_{\ell_{1}}-h_{\ell_{2}})}\mathrm{id}_{\mathbb{W}_{\ell}}.
\end{equation*}
\end{prop}
\begin{proof}
We compare $w_{1}\boxtimesw_{2}$ and $e^{L^{\mathbb{W}_{\ell_{2}}\boxtimes\mathbb{W}_{\ell_{1}}}_{-1}}\ol{\mathcal{T}_{\gamma}}(w_{2}\boxtimes_{P(-1)}w_{1})$ for $w_{1}\in\mathbb{W}_{\ell_{1}}$, $w_{2}\in \mathbb{W}_{\ell_{2}}$.
From (\ref{eq:Pz-intmap_frmods}) we have
\begin{equation*}
    w_{1}\boxtimesw_{2}=\sum_{\ell\in\mathrm{Sel} (\ell_{1},\ell_{2})}\mathcal{Y}_{\ell_{1}\ell_{2}}^{\ell}(w_{1},1)w_{2}
\end{equation*}
and on the other hand, we can see that
\begin{equation*}
    e^{L^{\mathbb{W}_{\ell_{2}}\boxtimes\mathbb{W}_{\ell_{1}}}_{-1}}\ol{\mathcal{T}_{\gamma}}\left(w_{2}\boxtimes_{P(-1)}w_{1}\right)=\sum_{\ell\in\mathrm{Sel} (\ell_{1},\ell_{2})}e^{xL^{\mathbb{W}_{\ell}}_{-1}}\mathcal{Y}_{\ell_{2}\ell_{1}}^{\ell}(w_{2},e^{\pi\mathbbm{i}}x)w_{1}\Big|_{x=1}.
\end{equation*}
For each $\ell\in\mathbb{Z}_{\geq 0}$, we define
\begin{equation*}
    (\Omega\mathcal{Y}_{\ell_{2}\ell_{1}}^{\ell})(-,x)\colon \mathbb{W}_{\ell_{1}}\to\mathrm{Hom} (\mathbb{W}_{\ell_{2}},\mathbb{W}_{\ell})\{x\}
\end{equation*}
by
\begin{equation*}
    (\Omega\mathcal{Y}_{\ell_{2}\ell_{1}}^{\ell})(w_{1},x)w_{2}:=e^{xL^{\mathbb{W}_{\ell}}_{-1}}\mathcal{Y}_{\ell_{2}\ell_{1}}^{\ell}(w_{2},e^{\pi\mathbbm{i}}x)w_{1},\quad w_{1}\in\mathbb{W}_{\ell_{1}},\, w_{2}\in\mathbb{W}_{\ell_{2}}.
\end{equation*}
It is known \cite[Proposition~7.1]{Huang--LepowskyII} that
\begin{equation*}
(\Omega\mathcal{Y}_{\ell_{2}\ell_{1}}^{\ell})(-,x)\in\intsp{\mathbb{W}_{\ell_{1}}}{\mathbb{W}_{\ell_{2}}}{\mathbb{W}_{\ell}}.
\end{equation*}
In particular, it must be proportional to $\mathcal{Y}_{\ell_{1}\ell_{2}}^{\ell}(-,x)$ since the corresponding space of intertwining operators is one-dimensional.
We can fix the constant of proportionality by looking at the image of the highest weight vectors:
\begin{equation*}
    (\Omega\mathcal{Y}_{\ell_{2}\ell_{1}}^{\ell})(\mathbf{w}_{\ell_{1}},x)\mathbf{w}_{\ell_{2}}\in
    \mathbf{B}_{\ell_{2}\,\ell_{1}}^{\ell}e^{\pi\mathbbm{i} (h_{\ell}-h_{\ell_{1}}-h_{\ell_{2}})}\mathbf{w}_{\ell}x^{h_{\ell}-h_{\ell_{1}}-h_{\ell_{2}}}+\mathbb{W}_{\ell}[[x]]x^{h_{\ell}-h_{\ell_{1}}-h_{\ell_{2}}+1}.
\end{equation*}
Since $\mathbf{B}_{\ell_{1}\,\ell_{2}}^{\ell}=\mathbf{B}_{\ell_{2}\,\ell_{1}}^{\ell}$, we can see that
\begin{equation*}
    (\Omega\mathcal{Y}_{\ell_{2}\ell_{1}}^{\ell})(-,x)=e^{\pi\mathbbm{i} (h_{\ell}-h_{\ell_{1}}-h_{\ell_{2}})}\mathcal{Y}_{\ell_{1}\ell_{2}}^{\ell}(-,x).
\end{equation*}
Therefore, we obtain the formula
\begin{equation*}
     e^{L^{\mathbb{W}_{\ell_{2}}\boxtimes\mathbb{W}_{\ell_{1}}}_{-1}}\ol{\mathcal{T}_{\gamma}}\left(w_{2}\boxtimes_{P(-1)}w_{1}\right)=\sum_{\ell\in\mathrm{Sel} (\ell_{1},\ell_{2})}e^{\pi\mathbbm{i} (h_{\ell}-h_{\ell_{1}}-h_{\ell_{2}})}\mathcal{Y}_{\ell_{1}\ell_{2}}^{\ell}(w_{1},1)w_{2},
\end{equation*}
which concludes the desired result.
\end{proof}

We now see that the braiding in the above theorem matches that on $\cC_{\mathrm{QG}}^{\Delta}$.

\begin{thm}
Along the functor $F\colon \cC_{\mathfrak{vir}}^{+}\to\cC_{\mathrm{QG}}^{\Delta}$,
\begin{equation*}
    F(c_{\mathbb{W}_{\ell_{1}},\mathbb{W}_{\ell_{2}}})=c_{\QGirrep{\ell_{1}},\QGirrep{\ell_{2}}}
\end{equation*}
for each $\ell_{1},\ell_{2}\in\mathbb{Z}_{\geq 0}$.
\end{thm}
\begin{proof}
It suffices to show that
\begin{equation*}
    \CGproj{\ell}{\ell_{2}}{\ell_{1}}\circ c_{\QGirrep{\ell_{1}},\QGirrep{\ell_{2}}}\circ \CGemb{\ell}{\ell_{1}}{\ell_{2}}=e^{\pi \mathbbm{i} (h_{\ell}-h_{\ell_{1}}-h_{\ell_{2}})}\mathrm{id}_{\mathbb{W}_{\ell}}
\end{equation*}
for each $\ell\in\mathrm{Sel}(\ell_{1},\ell_{2})$.
Let us apply $c_{\QGirrep{\ell_{1}},\QGirrep{\ell_{2}}}$ on $\CGemb{\ell}{\ell_{1}}{\ell_{2}}(\QGirvec{\ell}{0})$ to find
\begin{equation*}
    c_{\QGirrep{\ell_{1}},\QGirrep{\ell_{2}}}\circ \CGemb{\ell}{\ell_{1}}{\ell_{2}}(\QGirvec{\ell}{0})
    \in \CGcoeff{\ell}{\ell_{1}}{\ell_{2}}{0}q^{\frac{1}{2}\ell_{1}(\ell_{2}-2s)}\QGirvec{\ell_{2}}{s}\otimes\QGirvec{\ell_{1}}{0}+\sum_{j=1}^{s}\mathbb{C} \QGirvec{\ell_{2}}{s-j}\otimes\QGirvec{\ell_{1}}{j}.
\end{equation*}
Here we set $s=(\ell_{1}+\ell_{2}-\ell)/2$ as before.
Since the composition $\CGproj{\ell}{\ell_{2}}{\ell_{1}}\circ c_{\QGirrep{\ell_{1}},\QGirrep{\ell_{2}}}\circ \CGemb{\ell}{\ell_{1}}{\ell_{2}}$ must be proportional to the identity,
we have
\begin{align*}
    \CGproj{\ell}{\ell_{2}}{\ell_{1}}\circ c_{\QGirrep{\ell_{1}},\QGirrep{\ell_{2}}}\circ \CGemb{\ell}{\ell_{1}}{\ell_{2}}=\frac{\CGcoeff{\ell}{\ell_{1}}{\ell_{2}}{0}}{\CGcoeff{\ell}{\ell_{2}}{\ell_{1}}{s}}q^{\frac{1}{2}\ell_{1}(\ell_{2}-2s)}\mathrm{id}_{\QGirrep{\ell}}.
\end{align*}
Substituting the explicit formula for the Clebsch--Gordan coefficients (\ref{eq:CG_coeffs}), the constant becomes
\begin{equation*}
    \frac{\CGcoeff{\ell}{\ell_{1}}{\ell_{2}}{0}}{\CGcoeff{\ell}{\ell_{2}}{\ell_{1}}{s}}q^{\frac{1}{2}\ell_{1}(\ell_{2}-2s)}
    =(-1)^{s}q^{\frac{1}{2}\ell_{1}\ell_{2}-s(\ell_{1}+\ell_{2})+s^{2}-s}.
\end{equation*}
When we recall that $q$ is related to $t$ by $q=e^{\pi\mathbbm{i} t}$ and use the formula (\ref{eq:conf_weight_fr}), we can check that this constant coincides with $e^{\pi\mathbbm{i} (h_{\ell}-h_{\ell_{1}}-h_{\ell_{2}})}$ to complete the proof.
\end{proof}

\subsection{Ribbon structure}
Finally, we compare the ribbon structures.
Let us state the result immediately.
\begin{thm}
Along the functor $F\colon \cC_{\mathfrak{vir}}^{+}\to\cC_{\mathrm{QG}}^{\Delta}$, we have
\begin{equation*}
    F(\theta_{\mathbb{W}_{\ell}})=\theta_{\QGirrep{\ell}}
\end{equation*}
for each $\ell\in\mathbb{Z}_{\geq 0}$.
\end{thm}
\begin{proof}
It is already clear that $\theta_{\mathbb{W}_{\ell}}=e^{2\pi\mathbbm{i} h_{\ell}}\mathrm{id}_{\mathbb{W}_{\ell}}$.
When we apply the formula for $\theta_{\QGirrep{\ell}}^{-1}$ in (\ref{eq:QGtwist_inverse}) to the lowest weight vector $\QGirvec{\ell}{\ell}$, we immediately get
\begin{equation*}
    \theta_{\QGirrep{\ell}}^{-1} \QGirvec{\ell}{\ell}=e^{-2\pi\mathbbm{i} h_{\ell}}\QGirvec{\ell}{\ell}.
\end{equation*}
Therefore, we get $\theta_{\QGirrep{\ell}}=e^{2\pi\mathbbm{i} h_{\ell}}\mathrm{id}_{\QGirrep{\ell}}$ concluding the desired result.
\end{proof}

\bibliographystyle{alpha}

redpajama_set_name: RedPajamaArXiv


text:
Bedard: Patriots should do the right thing and adjust Cam Newton's contract … eventually
By Greg A. Bedard
The Patriots potentially have one of the greatest all-time bargains if Cam Newton leads the team to the playoffs.
The Patriots could have the best QB in the division, and a top-10 talent in the NFL, and are paying him $1.05 million in base salary, of which just $550,000 is guaranteed. His cap number of $1.137 million ranks behind such illustrious Patriots names as Jermaine Eluemunor, Justin Bethel, Terrence Brooks, Cody Davis, Shilique Calhoun, Brandon King and the team's last two second-round picks, who have accomplished nothing in their short careers: Joejuan Williams and Kyle Dugger.
It's ridiculous.
It's also a little embarrassing now that the team has about $35 million in cap space after magically settling their grievances with Antonio Brown and the estate of Aaron Hernandez days after Newton signed his contract, and the Covid-19 opt-outs.
Yes, there are incentives in the deal that give Newton's contract a maximum value of $7.5 million. But, realistically, Newton will make $6 million if he's the starter, leads the team to the playoffs and is named to the Pro Bowl (only if there is one).
That's not a truly terrible deal, but it's a bit insulting when you consider that all of the first-round rookies this year average more per season, and Mitch Trubisky, Daniel Jones, Marus Mariota and Taysom Hill are among those that do as well.
That's not right.
Yes, the Patriots should drive a hard bargain because business is business, but everyone knows that without the hurdles associated with Covid-19, Newton would have been signed as a starter by another team. More importantly, other players know this. And the good thing is Bill Belichick knows that players know this. There is perhaps no GM in the game who is more in tune with the locker room dynamics when it comes to finances.
Last year, I said the same thing about Julian Edelman's contract ... and later on, Edelman did get a new deal. That laid the groundwork for the Patriots to add Antonio Brown and pay him that mind-boggling signing bonus. Without the Edelman extension, the Brown deal (or maybe another one that actually would have benefitted the team and kept the offense from having the potency of a Pop Warner double-wing offense) couldn't have happened.
Belichick knows this better than anyone. It's not rocket science when you're dealing with professional athletes who, believe it or not, are not always "just about winning" or "playing for the fans."
I would expect Belichick to tweak Newton's contract at some point, because Belichick knows that can benefit the team in many different ways:
BSJ Live Q&A: Bedard on Patriots/NFL, 2 p.m. Friday
Bedard: Patriots fans should be rooting for these teams to make bold QB moves; McDaniels stays
Bedard: Now that the CBA tweaks are done, Patriots & other teams know ground rules for 2020 season
Video: Cam Newton speaks to the media for first time with the Patriots

redpajama_set_name: RedPajamaCommonCrawl


The top stress inducers are work overload, lack of recognition, insufficient salary and negative work environment. Look at your life and your habits, your mood and social interactions. Do you have digestive problems, eating disorders, muscle tension, irritability or depression, forgetfulness and poor concentration? These are common signs of stress, and if you have at least some of them for a long time, look closely at your life and try changing something, until some more serious health issues appear.
Try to avoid too much caffeine and cut out all the junk foods and sugar. If you are smoking, quit as soon as you can. These changes may be tough at the beginning, but soon your body will thank you.
No one says you should head to Olympic Games – a little cardio would change the situation for better. Swimming or jogging, yoga and whatever you like better would be great.
Don't try to do everything at once – delegate your tasks to others if you can do so, and don't judge yourself if something goes wrong. We are our worst critics, so get out of your own head and take it easy. Share work problems with colleagues – this will not only help your communication and reduce your level of stress. But avoid gossips and negative thinking.
And the main thing – always stay positive, no matter what comes.