# Quantum affine vertex algebras associated to untwisted quantum affinization
algebras

Fei Kong  Key Laboratory of Computing and Stochastic Mathematics (Ministry of
Education), School of Mathematics and Statistics, Hunan Normal University,
Changsha, China 410081<EMAIL_ADDRESS>
###### Abstract.

Let ${\mathcal{U}}_{\hbar}(\hat{\mathfrak{g}})$ be the untwisted affinization
of a symmetrizable quantum Kac-Moody algebra
${\mathcal{U}}_{\hbar}({\mathfrak{g}})$. For $\ell\in{\mathbb{C}}$, we
construct an $\hbar$-adic quantum vertex algebra
$V_{\hat{\mathfrak{g}},\hbar}(\ell,0)$, and establish a one-to-one
correspondence between $\phi$-coordinated
$V_{\hat{\mathfrak{g}},\hbar}(\ell,0)$-modules and restricted
${\mathcal{U}}_{\hbar}(\hat{\mathfrak{g}})$-modules of level $\ell$. Suppose
that $\ell$ is a positive integer. We construct a quotient $\hbar$-adic
quantum vertex algebra $L_{\hat{\mathfrak{g}},\hbar}(\ell,0)$ of
$V_{\hat{\mathfrak{g}},\hbar}(\ell,0)$, and establish a one-to-one
correspondence between certain $\phi$-coordinated
$L_{\hat{\mathfrak{g}},\hbar}(\ell,0)$-modules and restricted integrable
${\mathcal{U}}_{\hbar}(\hat{\mathfrak{g}})$-modules of level $\ell$. Suppose
further that ${\mathfrak{g}}$ is of finite type. We prove that
$L_{\hat{\mathfrak{g}},\hbar}(\ell,0)/\hbar
L_{\hat{\mathfrak{g}},\hbar}(\ell,0)$ is isomorphic to the simple affine
vertex algebra $L_{\hat{\mathfrak{g}}}(\ell,0)$.

###### Key words and phrases:

Quantum vertex algebra, affine vertex algebra, $\phi$-coordinated module,
quantum affine algebra, quantum affinization

###### 2020 Mathematics Subject Classification:

17B69, 17B37

##  1\. Introduction

Let ${\mathfrak{g}}$ be a finite dimensional simple Lie algebra, and let
$\hat{\mathfrak{g}}={\mathfrak{g}}\otimes{\mathbb{C}}[t,t^{-1}]\oplus{\mathbb{C}}c$
be the corresponding affine Kac-Moody Lie algebra. For any complex number
$\ell\in{\mathbb{C}}$, we define the induced module

$\displaystyle
V_{\hat{\mathfrak{g}}}(\ell,0)={\mathcal{U}}(\hat{\mathfrak{g}})\otimes_{{\mathcal{U}}(\hat{\mathfrak{g}}_{+})}{\mathbb{C}}_{\ell},\quad\mbox{where}\,\,\hat{\mathfrak{g}}_{+}={\mathfrak{g}}\otimes{\mathbb{C}}[t]\oplus{\mathbb{C}}c$

and ${\mathbb{C}}_{\ell}={\mathbb{C}}$ is a $\hat{\mathfrak{g}}_{+}$-module
with ${\mathfrak{g}}\otimes{\mathbb{C}}[t]$ acting trivially and $c$ acting as
scalar $\ell$. The induced module $V_{\hat{\mathfrak{g}}}(\ell,0)$ carries a
vertex algebra structure, and the category of
$V_{\hat{\mathfrak{g}}}(\ell,0)$-modules is naturally isomorphic to the
category of restricted $\hat{\mathfrak{g}}$-modules of level $\ell$ [FZ, Li1,
LL]. If $\ell$ is a positive integer, then the unique simple quotient
$\hat{\mathfrak{g}}$-module $L_{\hat{\mathfrak{g}}}(\ell,0)$ of
$V_{\hat{\mathfrak{g}}}(\ell,0)$ is integrable. Moreover,
$L_{\hat{\mathfrak{g}}}(\ell,0)$ carries a simple quotient vertex algebra
structure, and the category of $L_{\hat{\mathfrak{g}}}(\ell,0)$-modules is
naturally isomorphic to the category of restricted integrable
$\hat{\mathfrak{g}}$-modules of level $\ell$ [FZ, DL, MP1, MP2, DLM1, LL].

Denote by ${\mathcal{U}}_{\hbar}(\hat{\mathfrak{g}})$ the (untwisted) quantum
affine algebra associated to $\hat{\mathfrak{g}}$ ([Dr1] and [Jim]). Like the
affinization realization of affine Kac-Moody Lie algebras, Drinfeld provided a
quantum affinization realization of
${\mathcal{U}}_{\hbar}(\hat{\mathfrak{g}})$ in [Dr2]. Based on the Drinfeld
presentation, Frenkel and Jing constructed vertex representations for simply-
laced untwisted quantum affine algebras in [FJ]. In the very paper, they
formulated a fundamental problem of developing certain “quantum vertex algebra
theory” associated to quantum affine algebras, in parallel with the connection
between affine Kac-Moody Lie algebras and vertex algebras.

As one of the fundamental works, Etingof and Kazhdan ([EK]) developed a theory
of quantum vertex operator algebras in the sense of formal deformations of
vertex algebras. The most visible difference between these and vertex algebras
is that the usual locality is replaced by the $S$-locality. Such $S$-locality
is controlled by a rational quantum Yang-Baxter operator. Partly motivated by
the work of Etingof and Kazhdan, H. Li conducted a series of studies. While
vertex algebras are analogues of commutative associative algebras, H. Li
introduced the notion of nonlocal vertex algebras [Li4], which are analogues
of noncommutative associative algebras. A nonlocal vertex algebra is a weak
quantum vertex algebra [Li4] if it satisfies the $S$-locality. In addition, it
becomes a quantum vertex algebra [Li4] if the $S$-locality is controlled by a
rational quantum Yang-Baxter operator. Mainly in order to associate quantum
vertex algebras to quantum affine algebras, a theory of $\phi$-coordinated
quasi modules was developed in [Li7, Li8]. The $\hbar$-adic counterparts of
these notions were introduced in [Li6]. In this framework, a quantum vertex
operator algebra in sense of Etingof-Kazhdan is an $\hbar$-adic quantum vertex
algebra whose classical limit is a vertex algebra.

In the pioneer work [EK], Etingof and Kazhdan constructed quantum vertex
operator algebras as formal deformations of
$V_{\hat{\mathfrak{g}l}_{n}}(\ell,0)$ and
$V_{\hat{\mathfrak{s}l}_{n}}(\ell,0)$, by using the $R$-matrix type relations
given in [RSTS]. Recently, Butorac, Jing and Kožić ([BJK]) extended Etingof-
Kazhdan’s construction to type $B$, $C$ and $D$ rational $R$-matrices. The
modules of these quantum vertex operator algebras are in one-to-one
correspondence with restricted modules for the corresponding Yangian doubles
(see [Ko1]). Based on the $R$-matrix presentation of quantum affine algebras
(see [DF, JLM1, JLM2]), Kožić constructed the quantum vertex operator algebras
associated with trigonometric $R$-matrices of types $A$, $B$, $C$ and $D$
([Ko2, Ko1]), and established a one-to-one correspondence between
$\phi$-coordinated modules and restricted modules for quantum affine algebras.

In [JKLT2], N. Jing, H. Li, S. Tan and I developed a method to construct
desired quantum vertex algebras. Let $V$ be a nonlocal vertex algebra, and let
$(H,\rho,\tau)$ be a triple consisting of a cocommutative nonlocal vertex
bialgebra $H$, a right $H$-comodule nonlocal vertex algebra structure $\rho$
on $V$, and a “compatible” $H$-module nonlocal vertex algebra structure $\tau$
on $V$. Then a new nonlocal vertex algebra $\mathfrak{D}_{\tau}^{\rho}(V)$
with $V$ as the underlying space was obtained. It was also proved that under
certain conditions, $\mathfrak{D}_{\tau}^{\rho}(V)$ is a quantum vertex
algebra. On the representation side, it was proved that for any
$\phi$-coordinated $V$-module $W$ with a “compatible” $\phi$-coordinated
$H$-module structure, there exists a $\phi$-coordinated
$\mathfrak{D}_{\tau}^{\rho}(V)$-module structure on $W$. Later in [JKLT3], we
introduced an $\hbar$-adic version of this construction, and constructed a
family of $\hbar$-adic quantum vertex algebras $V_{L}[[\hbar]]^{\eta}$ as
formal deformations of the lattice vertex algebras $V_{L}$. When $L$ is a root
lattice of $A$, $D$, $E$ type, we established a natural connection between
twisted quantum affine algebras and equivariant $\phi$-coordinated quasi
modules for $V_{L}[[\hbar]]^{\eta}$ with suitable $\eta$.

It is remarkable that Drinfeld’s quantum affinization process can be extended
to general symmetrizable quantum Kac-Moody algebras ([GKV, Jin, N, CJKT]). We
denote by ${\mathcal{U}}_{\hbar}(\hat{\mathfrak{g}})$ the untwisted quantum
affinization of a symmetrizable quantum Kac-Moody algebra
${\mathcal{U}}_{\hbar}({\mathfrak{g}})$. In this paper, we use the language of
Drinfeld presentations to construct the $\hbar$-adic quantum vertex algebra
$V_{\hat{\mathfrak{g}},\hbar}(\ell,0)$ and establish a one-to-one
correspondence between restricted
${\mathcal{U}}_{\hbar}({\mathfrak{g}})$-modules of level $\ell$ and
$\phi$-coordinated $V_{\hat{\mathfrak{g}},\hbar}(\ell,0)$-modules. Assume
further that $\ell$ is a positive integer. We will study a quotient
$\hbar$-adic quantum vertex algebra $L_{\hat{\mathfrak{g}},\hbar}(\ell,0)$ of
$V_{\hat{\mathfrak{g}},\hbar}(\ell,0)$, and establish a one-to-one
correspondence between restricted integrable
${\mathcal{U}}_{\hbar}(\hat{\mathfrak{g}})$-modules of level $\ell$ and
$\phi$-coordinated weight modules of $L_{\hat{\mathfrak{g}},\hbar}(\ell,0)$.
Finally, when ${\mathfrak{g}}$ is of finite type, we show that
$L_{\hat{\mathfrak{g}},\hbar}(\ell,0)/\hbar
L_{\hat{\mathfrak{g}},\hbar}(\ell,0)\cong L_{\hat{\mathfrak{g}}}(\ell,0)$ as
vertex algebras.

Now we provide some detailed information. Let $A=(a_{ij})_{i,j\in I}$ be a
symmetrizable generalized Cartan matrix (GCM). Then there are unique
relatively prime positive integers $r_{i}$ ($i\in I$) such that $DA$ is
symmetric with $D=\operatorname{diag}\\{r_{i}\\}_{i\in I}$. Let
${\mathfrak{g}}=[{\mathfrak{g}}(A),{\mathfrak{g}}(A)]$ be the derived
subalgebra of the Kac-Moody Lie algebra ${\mathfrak{g}}(A)$, and let
$\hat{\mathfrak{g}}$ be an “affinization Lie algebra” associated to
${\mathfrak{g}}$ (see Definition 2.4). Recall that the (untwisted) quantum
affinization algebra (see [CJKT]) ${\mathcal{U}}_{\hbar}(\hat{\mathfrak{g}})$
is the ${\mathbb{C}}[[\hbar]]$-algebra topologically generated by

(1.1) $\displaystyle{\left.\left\\{{h_{i,q}(n),\ x^{\pm}_{i,q}(n),\
c}\,\right|\,{i\in I,n\in{\mathbb{Z}}}\right\\}},$

and subject to the relations in terms of the generating functions

$\displaystyle\phi_{i,q}^{\pm}(z)=q^{\pm
h_{i,q}(0)}\,\mbox{exp}\left(\pm(q-q^{-1})\sum\limits_{\pm
m>0}h_{i,q}(m)z^{-m}\right),$ $\displaystyle
x^{\pm}_{i,q}(z)=\sum\limits_{m\in{\mathbb{Z}}}x^{\pm}_{i,q}(m)z^{-m},$

The relations are ($i,j\in I$):

(Q1)
$\displaystyle[c,\phi_{i,q}^{\pm}(z)]=0=[\phi_{i,q}^{\pm}(z),\phi_{j,q}^{\pm}(w)]=[c,x_{i,q}^{\pm}(z)],$
(Q2)
$\displaystyle\phi^{+}_{i,q}(z_{1})\phi^{-}_{j,q}(z_{2})=\phi^{-}_{j,q}(z_{2})\phi^{+}_{i,q}(z_{1})\iota_{z_{1},z_{2}}g_{ij,q}(q^{rc}z_{2}/z_{1})^{-1}g_{ij,q}(q^{-rc}z_{2}/z_{1}),$
(Q3)
$\displaystyle\phi^{+}_{i,q}(z_{1})x^{\pm}_{j,q}(z_{2})=x^{\pm}_{j,q}(z_{2})\phi^{+}_{i,q}(z_{1})\iota_{z_{1},z_{2}}g_{ij,q}(q^{\mp{\frac{1}{2}}rc}z_{2}/z_{1})^{\pm
1},$ (Q4)
$\displaystyle\phi^{-}_{i,q}(z_{1})x^{\pm}_{j,q}(z_{2})=x^{\pm}_{j,q}(z_{2})\phi^{-}_{i,q}(z_{1})\iota_{z_{2},z_{1}}g_{ji,q}(q^{\mp{\frac{1}{2}}rc}z_{1}/z_{2})^{\mp
1},$ (Q5) $\displaystyle[x_{i,q}^{+}(z_{1}),x_{j,q}^{-}(z_{2})]$
$\displaystyle=\frac{\delta_{ij}}{q_{i}-q_{i}^{-1}}\Bigg{(}\phi_{i,q}^{+}(z_{1}q^{-{\frac{1}{2}}rc})\delta\left(\frac{z_{2}q^{rc}}{z_{1}}\right)-\phi_{i,q}^{-}(z_{1}q^{{\frac{1}{2}}rc})\delta\left(\frac{z_{2}q^{-rc}}{z_{1}}\right)\Bigg{)},$
(Q6) $\displaystyle
F^{\pm}_{ij,q}(z_{1},z_{2})x^{\pm}_{i,q}(z_{1})x^{\pm}_{j,q}(z_{2})=G^{\pm}_{ij,q}(z_{1},z_{2})x^{\pm}_{j,q}(z_{2})x^{\pm}_{i,q}(z_{1}),$
(Q7) $\displaystyle\sum_{\sigma\in
S_{{m}_{ij}}}\sum_{k=0}^{{m}_{ij}}(-1)^{k}\binom{{m}_{ij}}{k}_{q_{i}}x_{i,q}^{\pm}(z_{\sigma(1)})\cdots
x_{i,q}^{\pm}(z_{\sigma(k)})x_{j,q}^{\pm}(w)$ $\displaystyle\qquad\times
x_{i,q}^{\pm}(z_{\sigma(k+1)})\cdots x_{i,q}^{\pm}(z_{\sigma({m}_{ij})})\
=0,\quad\mbox{if}\ \ {a}_{ij}<0,$

where $q=\exp\hbar\in{\mathbb{C}}[[\hbar]]$, $q_{i}=q^{r_{i}}$, $r$ is the
least common multiple of $\\{r_{i}\\}_{i\in I}$ and

$\displaystyle F^{\pm}_{ij,q}(z_{1},z_{2})=z_{1}-q_{i}^{\pm a_{ij}}z_{2},\quad
G^{\pm}_{ij,q}(z_{1},z_{2})=q_{i}^{\pm a_{ij}}z_{1}-z_{2},$ $\displaystyle
g_{ij,q}(z)=\frac{G_{ij,q}^{+}(1,z)}{F_{ij,q}^{+}(1,z)}$

and the map $\iota_{z_{1},z_{2}}$ is defined as in [FHL, §3.1]. The classical
limit ${\mathcal{U}}_{\hbar}(\hat{\mathfrak{g}})/\hbar
U_{\hbar}(\hat{\mathfrak{g}})$ is isomorphic to the universal enveloping
algebra of $\hat{\mathfrak{g}}$.

Let $V_{\hat{\mathfrak{g}}}(\ell,0)$ be the vertex algebra associated to
$\hat{\mathfrak{g}}$ (see Definition 2.6). Different from the construction of
$V_{L}[[\hbar]]^{\eta}$ in [JKLT3], we are not able to find a suitable
“deforming triple” for $V_{\hat{\mathfrak{g}}}(\ell,0)$, such that the
corresponding $\hbar$-adic quantum vertex algebra can be associated to
${\mathcal{U}}_{\hbar}(\hat{\mathfrak{g}})$. Let
${\mathcal{U}}_{\hbar}^{f}(\hat{\mathfrak{g}})$ be the unital associative
algebra over ${\mathbb{C}}[[\hbar]]$ topologically generated by the elements
(1.1) subject to relations (Q1)-(Q4), (Q6) and

(Q5.5)
$\displaystyle(1-q^{r\ell}z_{2}/z_{1})^{\delta_{ij}}(1-q^{-r\ell}z_{2}/z_{1})^{\delta_{ij}}[x_{i,q}^{+}(z_{1}),x_{j,q}^{-}(z_{2})]=0,\quad
i,j\in I.$

It is easy to see that ${\mathcal{U}}_{\hbar}(\hat{\mathfrak{g}})$ is a
quotient algebra of ${\mathcal{U}}_{\hbar}^{f}(\hat{\mathfrak{g}})$. Using the
theory of free vertex algebras developed by Roitman ([R]), we construct a
vertex algebra $F(A,\ell)$ (see Definition 2.6). Then by using the
construction given in [JKLT2, JKLT3], we get a family of $\hbar$-adic quantum
vertex algebras $F_{\tau}(A,\ell)$ as formal deformations of $F(A,\ell)$ (see
Theorem 5.13). The category of restricted
${\mathcal{U}}_{\hbar}^{f}(\hat{\mathfrak{g}})$-modules is isomorphic to the
category of $\phi$-coordinated $F_{\tau}(A,\ell)$-modules with suitable
$\tau$. For this suitable $\tau$, we construct the quotient $\hbar$-adic
quantum vertex algebras $V_{\hat{\mathfrak{g}},\hbar}(\ell,0)$ and
$L_{\hat{\mathfrak{g}},\hbar}(\ell,0)$ of $F_{\tau}(A,\ell)$. Finally, based
on the “normal ordered” type quantum Serre relations given in [CJKT] and the
integrable conditions developed in [DM], we establish a natural connection
between restricted (resp. restricted integrable)
${\mathcal{U}}_{\hbar}(\hat{\mathfrak{g}})$-modules of level $\ell$ and
$\phi$-coordinated modules (resp. $\phi$-coordinated weighted modules) for
$V_{\hat{\mathfrak{g}},\hbar}(\ell,0)$ (resp.
$L_{\hat{\mathfrak{g}},\hbar}(\ell,0)$).

The paper is organized as follows. In Section 2, we recall the basics about
vertex algebras and free vertex algebras. Define the vertex algebra
$F(A,\ell)$ associated to a symmetrizable generalized Cartan matrix $A$ and a
complex number $\ell$. Then we realize $V_{\hat{\mathfrak{g}}}(\ell,0)$ and
$L_{\hat{\mathfrak{g}}}(\ell,0)$ as quotient vertex algebras of $F(A,\ell)$.
In Section 3, we recall the notion of $\hbar$-adic quantum vertex algebras,
and the theory of constructing $\hbar$-adic nonlocal vertex algebras
introduced in [Li6]. In Section 5, we construct an $\hbar$-adic nonlocal
vertex algebra $F_{\tau}(A,\ell)$. By using the deformation method introduced
in [JKLT2, JKLT3] (see Section 4), we prove that $F_{\tau}(A,\ell)$ is an
$\hbar$-adic quantum vertex algebra and $F_{\tau}(A,\ell)/\hbar
F_{\tau}(A,\ell)\cong F(A,\ell)$ as vertex algebras. In Section 6, we
construct two quotient $\hbar$-adic quantum vertex algebras
$V_{\hat{\mathfrak{g}},\hbar}(\ell,0)$ and
$L_{\hat{\mathfrak{g}},\hbar}(\ell,0)$ of $F_{\tau}(A,\ell)$. In Section 7, we
recall the notion of $\phi$-coordinated modules of $\hbar$-adic nonlocal
vertex algebras, and study the category of $\phi$-coordinated modules of
$V_{\hat{\mathfrak{g}},\hbar}(\ell,0)$. In Section 8, we establish the natural
connections between certain
${\mathcal{U}}_{\hbar}(\hat{\mathfrak{g}})$-modules and $\phi$-coordinated
modules for the two $\hbar$-adic quantum vertex algebras
$V_{\hat{\mathfrak{g}},\hbar}(\ell,0)$ and
$L_{\hat{\mathfrak{g}},\hbar}(\ell,0)$.

Throughout this paper, we denote the set of nonnegative integer and positive
integers by ${\mathbb{N}}$ and ${\mathbb{Z}}_{+}$, respectively. For any ring
$R$, we denote the set of invertible elements by $R^{\times}$.

##  2\. Vertex algebras and affine vertex algebras

In this section, we recall the notion of vertex algebras and define some
vertex algebras associated to the symmetrizable GCM $A=(a_{ij})_{i,j\in I}$.

A _vertex algebra_ is a vector space $V$ equipped with a linear map

(2.1) $\displaystyle Y(\cdot,z):$ $\displaystyle
V\to{\mathcal{E}}(V):=\operatorname{Hom}(V,V((z)));\quad v\mapsto
Y(v,z)=\sum_{n\in{\mathbb{Z}}}v_{n}z^{-n-1}$

and equipped with a distinguished vector ${{\bf 1}}$ of $V$, called the
_vacuum vector_ , such that

(2.2) $\displaystyle Y({{\bf 1}},z)v=v,\quad Y(v,z){{\bf 1}}\in
V[[z]],\quad\lim_{z\to 0}Y(v,z){{\bf 1}}=v\quad\mbox{for }v\in V$

and such that for $u,v,w\in V$, the following _Jacobi identity_ holds true

$\displaystyle z_{0}^{-1}\delta\left(\frac{z_{1}-z_{2}}{z_{0}}\right)$
$\displaystyle
Y(u,z_{1})Y(v,z_{2})w-z_{0}^{-1}\delta\left(\frac{z_{2}-z_{1}}{-z_{0}}\right)Y(v,z_{2})Y(u,z_{1})w$
(2.3)
$\displaystyle=z_{1}^{-1}\delta\left(\frac{z_{2}+z_{0}}{z_{1}}\right)Y(Y(u,z_{0})v,z_{2})w.$

A _conformal Lie algebra_ $(C,Y^{+},T)$ ([Kac]), also known as a _vertex Lie
algebra_ ([DLM2, P]), is a vector space $C$ equipped with a linear operator
$T$ and a linear map

(2.4) $\displaystyle
Y^{+}(\cdot,z):\,\,C\to\mathrm{Hom}(C,z^{-1}C[z^{-1}]);\quad u\mapsto
Y^{+}(u,z)=\sum_{n\geq 0}u_{n}z^{-n-1}$

such that for any $u,v\in C$,

$\displaystyle[T,Y^{+}(u,z)]=Y^{+}(Tu,z)=\frac{d}{dz}Y^{+}(u,z),$
$\displaystyle
Y^{+}(u,z)v=\operatorname{Sing}_{z}\left(e^{zT}Y^{+}(v,-z)u\right),$
$\displaystyle[Y^{+}(u,z_{1}),Y^{+}(v,z_{2})]=\operatorname{Sing}_{z_{1}}\operatorname{Sing}_{z_{2}}(Y^{+}(Y^{+}(u,z_{1}-z_{2})v,z_{2})),$

where $\operatorname{Sing}$ stands for the singular part. Let $(C,Y^{+},T)$,
$(C_{1},Y_{1}^{+},T_{1})$ be conformal Lie algebras. A linear map $f:C\to
C_{1}$ is called conformal Lie algebra homomorphism if
$f\big{(}Y^{+}(u,z)v\big{)}=Y^{+}_{1}(f(u),z)f(v)$ and
$f\big{(}Tu\big{)}=T_{1}f(u)$ for $u,v\in C$.

Let $C$ be an arbitrary conformal Lie algebra. Recall the _coefficient
algebra_ $\widehat{C}$ ([R]), also known as the _local vertex Lie algebra_
([DLM2]) is the Lie algebra with the underlying vector space
$\big{(}C\otimes{\mathbb{C}}[t,t^{-1}]\big{)}/\mbox{Im}\big{(}T\otimes
1+1\otimes d/dt\big{)}$, and the Lie bracket

(2.5) $\displaystyle[u(m),v(n)]=\sum_{k\geq
0}\binom{m}{k}\big{(}u_{k}v\big{)}(m+n-k),$

where $u,v\in C$, $m,n\in{\mathbb{Z}}$ and $u(m)$ is the image of $u\otimes
t^{m}$ in $\widehat{C}$. Set
$\widehat{C}^{-}=\operatorname{Span}_{\mathbb{C}}{\left.\left\\{{u(m)}\,\right|\,{u\in
C,m<0}\right\\}}$ and
$\widehat{C}^{+}=\operatorname{Span}_{\mathbb{C}}{\left.\left\\{{u(m)}\,\right|\,{u\in
C,m\geq 0}\right\\}}$, and define

(2.6) $\displaystyle
V_{C}={\mathcal{U}}(\widehat{C})\otimes_{{\mathcal{U}}(\widehat{C}^{+})}{\mathbb{C}},$

where ${\mathbb{C}}$ is a trivial $\widehat{C}^{+}$-module. It is proved in
[R, Proposition 1.3] (see also [DLM2, Proposition 3.4]) that
$\widehat{C}^{\pm}$ are Lie subalgebras of $\widehat{C}$ and
$\widehat{C}=\widehat{C}^{-}\oplus\widehat{C}^{+}$. Then
$V_{C}\cong{\mathcal{U}}(\widehat{C}^{-})$ as vector spaces. For convenience,
we identify $u$ with $u(-1)\otimes 1\in V_{C}$ for $u\in C$. A
$\widehat{C}$-module $W$ is called a _restricted_ module if

$\displaystyle u(z)v=\sum_{m\in{\mathbb{Z}}}u(m)vz^{-m-1}\in
W((z)),\quad\mbox{for }u\in C,\,\,v\in W.$

It is proved in [DLM2] that

######  Theorem 2.1 ([DLM2, Theorem 4.8]).

There exists a vertex algebra structure on $V_{C}$ such that ${{\bf
1}}=1\otimes 1\in V_{C}$ is the vacuum vector and

$\displaystyle Y(u,z)=\sum_{m\in{\mathbb{Z}}}u(m)z^{-m-1}\quad\mbox{for }u\in
C.$

For any restricted $\widehat{C}$-module $W$, there exists a $V_{C}$-module
structure $Y_{W}$ on $W$ such that

$\displaystyle Y_{W}(u,z)=u(z)=\sum_{m\in{\mathbb{Z}}}u(m)z^{-m-1},\quad u\in
C.$

On the other hand, for any $V_{C}$-module $W$, there exists a restricted
$\widehat{C}$-module structure on $W$ such that

$\displaystyle u(z).w=Y_{W}(u,z)w$

where $u\in C$ and $w\in W$.

Let $\mathcal{B}$ be a set, and let
$N:\mathcal{B}\times\mathcal{B}\to{\mathbb{N}}$ be a symmetric function called
a _locality function_ ([R]). Denote by $\mathfrak{Conf}(\mathcal{B},N)$ the
category whose objects are conformal Lie algebras $C$ which contain
$\mathcal{B}$ as a generating subset and

$\displaystyle a_{n}b=0\quad\mbox{for all }a,b\in\mathcal{B},\,n\geq N(a,b).$

The following result is given in [R, Proposition 3.1]

######  Proposition 2.2.

There exists a conformal Lie algebra
$C(\mathcal{B},N)\in\mathfrak{Conf}(\mathcal{B},N)$ such that for any
$C\in\mathfrak{Conf}(\mathcal{B},N)$, there is a unique conformal Lie algebra
homomorphism $f$ from $C(\mathcal{B},N)$ to $C$ such that

$\displaystyle f(a)=a\in\mathcal{B}\subset C\quad\mbox{for
}a\in\mathcal{B}\subset C(\mathcal{B},N).$

Moreover, the coefficient algebra $\widehat{C}(\mathcal{B},N)$ of
$C(\mathcal{B},N)$ is isomorphic to the Lie algebra generated by

(2.7)
$\displaystyle{\left.\left\\{{b(n)}\,\right|\,{b\in\mathcal{B},\,n\in{\mathbb{Z}}}\right\\}}$

and subject to the relations

(2.8) $\displaystyle\sum_{i\geq
0}(-1)^{i}\binom{N(a,b)}{i}[a(n-i),b(m+i)]=0\quad\mbox{for
}a,b\in\mathcal{B},\,m,n\in{\mathbb{Z}}.$

The conformal Lie algebra $C(\mathcal{B},N)$ is called the _free conformal Lie
algebra_ in [R]. From Theorem 2.1, we get a vertex algebra $V(\mathcal{B},N)$
corresponding to $C(\mathcal{B},N)$, which is called the _free vertex algebra_
in [R].

We note that the relation (2.8) is equivalent to the following relation

(2.9) $\displaystyle(z_{1}-z_{2})^{N(a,b)}[a(z_{1}),b(z_{2})]=0\quad\mbox{for
}a,b\in\mathcal{B}.$

######  Proposition 2.3.

Let $V$ be a vertex algebra, and let $f:\mathcal{B}\to V$ such that

$\displaystyle(z_{1}-z_{2})^{N(a,b)}[Y(f(a),z_{1}),Y(f(b),z_{2})]=0\quad\mbox{for
}a,b\in\mathcal{B}.$

Then there exists a vertex algebra homomorphism
$\widehat{f}:V(\mathcal{B},N)\to V$ such that $\widehat{f}|_{\mathcal{B}}=f$.

We introduce the following Lie algebra.

######  Definition 2.4.

Let $\hat{\mathfrak{g}}$ be the Lie algebra generated by

${\left.\left\\{{h_{i}(m),\,x_{i}^{\pm}(m)}\,\right|\,{i\in
I,\,m\in{\mathbb{Z}}}\right\\}}$

and a central element $c$, subject to the relations written in terms of
generating functions in $z$:

$\displaystyle h_{i}(z)=\sum_{m\in{\mathbb{Z}}}h_{i}(m)z^{-m-1},\quad
x_{i}^{\pm}(z)=\sum_{m\in{\mathbb{Z}}}x_{i}^{\pm}(m)z^{-m-1},\quad i\in I.$

The relations are ($i,j\in I$)

(L1)
$\displaystyle[h_{i}(z_{1}),h_{j}(z_{2})]=r_{i}a_{ij}rc\frac{\partial}{\partial{z_{2}}}z_{1}^{-1}\delta\left(\frac{z_{2}}{z_{1}}\right),$
(L2) $\displaystyle[h_{i}(z_{1}),x_{j}^{\pm}(z_{2})]=\pm
r_{i}a_{ij}x_{j}^{\pm}(z_{2})z_{1}^{-1}\delta\left(\frac{z_{2}}{z_{1}}\right),$
(L3)
$\displaystyle[x_{i}^{+}(z_{1}),x_{j}^{-}(z_{2})]=\frac{\delta_{ij}}{r_{i}}\left(h_{i}(z_{2})z_{1}^{-1}\delta\left(\frac{z_{2}}{z_{1}}\right)+rc\frac{\partial}{\partial{z_{2}}}z_{1}^{-1}\delta\left(\frac{z_{2}}{z_{1}}\right)\right),$
(L4)
$\displaystyle(z_{1}-z_{2})^{n_{ij}}[x_{i}^{\pm}(z_{1}),x_{j}^{\pm}(z_{2})]=0,$
(S)
$\displaystyle[x_{i}^{\pm}(z_{1}),[x_{i}^{\pm}(z_{2}),\dots,[x_{i}^{\pm}(z_{m_{ij}}),x_{j}^{\pm}(z_{0})]\cdots]]=0,\quad\mbox{if
}a_{ij}<0,$

where $n_{ij}=1-\delta_{ij}$ for $i,j\in I$ and $m_{ij}=1-a_{ij}$ for $i,j\in
I$ with $a_{ij}<0$.

######  Remark 2.5.

Suppose that the GCM $A$ is of finite or affine type. Recall that
${\mathfrak{g}}=[{\mathfrak{g}}(A),{\mathfrak{g}}(A)]$ is the derived
subalgebra of the Kac-Moody Lie algebra associated to the GCM $A$. If $A$ is
not of type $A_{1}^{(1)}$, then $\hat{\mathfrak{g}}$ is the universal central
extension of ${\mathfrak{g}}\otimes{\mathbb{C}}[t,t^{-1}]$ (see [Gar] when $A$
is of finite type and [MERY] when $A$ is of affine type). If $A$ is of type
$A_{1}^{(1)}$, then $\hat{\mathfrak{g}}$ is the quotient algebra of the
universal central extension of ${\mathfrak{g}}\otimes{\mathbb{C}}[t,t^{-1}]$
modulo the ideal generated by the following relations

$\displaystyle(z_{1}-z_{2})[x_{i}^{\pm}(z_{1}),x_{j}^{\pm}(z_{2})]=0,\quad\mbox{for
}i\neq j.$

Introduce a set
$\mathcal{B}={\left.\left\\{{h_{i},\,x_{i}^{\pm}}\,\right|\,{i\in
I}\right\\}}$, and define a function
$N:\mathcal{B}\times\mathcal{B}\to{\mathbb{N}}$ by

$\displaystyle N(h_{i},h_{j})=2,\quad N(h_{i},x_{j}^{\pm})=1,\quad
N(x_{i}^{\pm},x_{j}^{\pm})=n_{ij},\quad N(x_{i}^{+},x_{j}^{-})=\delta_{ij}2.$

######  Definition 2.6.

For $\ell\in{\mathbb{C}}$, we let $F(A,\ell)$ be the quotient vertex algebra
of $V(\mathcal{B},N)$ modulo the ideal generated by

$\displaystyle(h_{i})_{0}(h_{j}),\quad(h_{i})_{1}(h_{j})-r_{i}a_{ij}r\ell{{\bf
1}},\quad(h_{i})_{0}(x_{j}^{\pm})\mp r_{i}a_{ij}x_{j}^{\pm}\quad\mbox{for
}i,j\in I.$

Furthermore, let $V_{\hat{\mathfrak{g}}}(\ell,0)$ be the quotient vertex
algebra of $F(A,\ell)$ modulo the ideal generated by

$\displaystyle(x_{i}^{+})_{0}(x_{j}^{-})-\frac{\delta_{ij}}{r_{i}}h_{i},\quad(x_{i}^{+})_{1}(x_{j}^{-})-\frac{\delta_{ij}}{r_{i}}r\ell{{\bf
1}}\quad\mbox{for }i,j\in I,$
$\displaystyle\left((x_{i}^{\pm})_{0}\right)^{m_{ij}}(x_{j}^{\pm})\quad\mbox{for
}i,j\in I\,\mbox{with }a_{ij}<0.$

If $\ell\in{\mathbb{Z}}_{+}$, we define $L_{\hat{\mathfrak{g}}}(\ell,0)$ to be
the quotient vertex algebra of $V_{\hat{\mathfrak{g}}}(\ell,0)$ modulo the
ideal generated by

$\displaystyle\left((x_{i}^{\pm})_{-1}\right)^{\ell
r/r_{i}}x_{i}^{\pm}\quad\mbox{for }i\in I.$

######  Remark 2.7.

Set
$\hat{\mathfrak{g}}_{+}={\mathfrak{g}}\otimes{\mathbb{C}}[t]\oplus{\mathbb{C}}c$.
For $\ell\in{\mathbb{C}}$, we let ${\mathbb{C}}_{\ell}:={\mathbb{C}}$ be a
$\hat{\mathfrak{g}}_{+}$-module, with
${\mathfrak{g}}\otimes{\mathbb{C}}[t].{\mathbb{C}}_{\ell}=0$ and $c=\ell$.
Then there are $\hat{\mathfrak{g}}$-module structures on both
$V_{\hat{\mathfrak{g}}}(\ell,0)$ and $L_{\hat{\mathfrak{g}}}(\ell,0)$, such
that

$\displaystyle h_{i}(z)=Y(h_{i},z),\quad x_{i}^{\pm}(z)=Y(x_{i}^{\pm},z),\quad
i\in I.$

In addition, as $\hat{\mathfrak{g}}$-modules, we have that

$\displaystyle
V_{\hat{\mathfrak{g}}}(\ell,0)\cong{\mathcal{U}}(\hat{\mathfrak{g}})\otimes_{{\mathcal{U}}(\hat{\mathfrak{g}}_{+})}{\mathbb{C}}_{\ell}$

Suppose that the GCM $A$ is of finite type. Then ${\mathfrak{g}}$ is a finite
dimensional simple Lie algebra. And $L_{\hat{\mathfrak{g}}}(\ell,0)$ is the
unique simple quotient $\hat{\mathfrak{g}}$-module of
$V_{\hat{\mathfrak{g}}}(\ell,0)$, when $\ell\in{\mathbb{Z}}_{+}$.

##  3\. Quantum vertex algebras

A ${\mathbb{C}}[[\hbar]]$-module $V$ is said to be _torsion-free_ if $\hbar
v\neq 0$ for every $0\neq v\in V$, and said to be _separated_ if $\cap_{n\geq
1}\hbar^{n}V=0$. For a ${\mathbb{C}}[[\hbar]]$-module $V$, using subsets
$v+\hbar^{n}V$ for $v\in V$, $n\geq 1$ as the basis of open subsets one
obtains a topology on $V$, which is called the _$\hbar$ -adic topology_. A
${\mathbb{C}}[[\hbar]]$-module $V$ is said to be _$\hbar$ -adically complete_
if every Cauchy sequence in $V$ with respect to this $\hbar$-adic topology has
a limit in $V$. A ${\mathbb{C}}[[\hbar]]$-module $V$ is _topologically free_
if $V=V_{0}[[\hbar]]$ for some vector space $V_{0}$ over ${\mathbb{C}}$. It is
known that a ${\mathbb{C}}[[\hbar]]$-module is topologically free if and only
if it is torsion-free, separated, and $\hbar$-adically complete ([Kas], see
[Li6]). For another topologically free ${\mathbb{C}}[[\hbar]]$-module
$U=U_{0}[[\hbar]]$, we recall the complete tensor

$\displaystyle U\widehat{\otimes}V=(U_{0}\otimes V_{0})[[\hbar]].$

We view a vector space as a ${\mathbb{C}}[[\hbar]]$-module by letting
$\hbar=0$. Fix a ${\mathbb{C}}[[\hbar]]$-module $W$. For
$k\in{\mathbb{Z}}_{+}$, and some formal variables $z_{1},\dots,z_{k}$, we
define

(3.1)
$\displaystyle{\mathcal{E}}^{(k)}(W;z_{1},\dots,z_{k})=\operatorname{Hom}_{{\mathbb{C}}[[\hbar]]}(W,W((z_{1},\dots,z_{k}))).$

Recall from [Li2, Definition 5.3] that an ordered sequence
$(a_{1}(z),\dots,a_{k}(z))$ in ${\mathcal{E}}^{(1)}(W)$ is said to be
_compatible_ if there exists an $m\in{\mathbb{Z}}_{+}$, such that

$\displaystyle\left(\prod_{1\leq i<j\leq
k}(z_{i}-z_{j})^{m}\right)a_{1}(z_{1})\cdots
a_{k}(z_{k})\in{\mathcal{E}}^{(k)}(W;z_{1},\dots,z_{k}).$

Now, we assume $W=W_{0}[[\hbar]]$ for some vector space $W_{0}$. Then $W$ is
topologically free. Define

(3.2)
$\displaystyle{\mathcal{E}}_{\hbar}^{(k)}(W;z_{1},\dots,z_{k})=\operatorname{Hom}_{{\mathbb{C}}[[\hbar]]}(W,W_{0}((z_{1},\dots,z_{k}))[[\hbar]]).$

Note that
${\mathcal{E}}_{\hbar}^{(k)}(W;z_{1},\dots,z_{k})={\mathcal{E}}^{(k)}(W;z_{1},\dots,z_{k})[[\hbar]]$
is topologically free. For convenience, we will also write
${\mathcal{E}}_{\hbar}^{(k)}(W)={\mathcal{E}}_{\hbar}^{(k)}(W;z_{1},\dots,z_{k})$
and write ${\mathcal{E}}_{\hbar}(W)={\mathcal{E}}_{\hbar}^{(1)}(W)$. For
$n,k\in{\mathbb{Z}}_{+}$, the quotient map from $W$ to $W/\hbar^{n}W$ induces
the following ${\mathbb{C}}[[\hbar]]$-module map

$\displaystyle\widetilde{\pi}_{n}^{(k)}:(\operatorname{End}_{{\mathbb{C}}[[\hbar]]}(W))[[z_{1}^{\pm
1},\dots,z_{k}^{\pm
1}]]\to(\operatorname{End}_{{\mathbb{C}}[[\hbar]]}(W/\hbar^{n}W))[[z_{1}^{\pm
1},\dots,z_{k}^{\pm 1}]].$

For
$A(z_{1},z_{2}),B(z_{1},z_{2})\in\operatorname{Hom}_{{\mathbb{C}}[[\hbar]]}(W,W_{0}((z_{1}))((z_{2}))[[\hbar]])$,
we write $A(z_{1},z_{2})\sim B(z_{2},z_{1})$ if for each
$n\in{\mathbb{Z}}_{+}$ there is $k\in{\mathbb{N}}$ such that

$\displaystyle(z_{1}-z_{2})^{k}\widetilde{\pi}_{n}^{(2)}(A(z_{1},z_{2}))=(z_{1}-z_{2})^{k}\widetilde{\pi}_{n}^{(2)}(B(z_{2},z_{1})).$

Let
$Z(z_{1},z_{2}):{\mathcal{E}}_{\hbar}(W)\widehat{\otimes}{\mathcal{E}}_{\hbar}(W)\widehat{\otimes}{\mathbb{C}}((z))[[\hbar]]\to\operatorname{End}_{{\mathbb{C}}[[\hbar]]}(W)[[z_{1}^{\pm
1},z_{2}^{\pm 1}]]$ be defined by

$\displaystyle Z(z_{1},z_{2})\left(a(z)\otimes b(z)\otimes
f(z)\right)=\iota_{z_{1},z_{2}}f(z_{1}-z_{2})a(z_{1})b(z_{2}).$

A subset $U$ of ${\mathcal{E}}_{\hbar}(W)$ is said to be _$\hbar$ -adically
$S$-local_ if for any $a(z),b(z)\in U$, there exists
$A(z)\in\left({\mathbb{C}}U\otimes{\mathbb{C}}U\otimes{\mathbb{C}}((z))\right)[[\hbar]]$
such that

$\displaystyle a(z_{1})b(z_{2})\sim Z(z_{2},z_{1})\left(A(z)\right),$

where ${\mathbb{C}}U$ denotes the subspace spanned by $U$.

Recall from [Li6, Remark 4.7] that $\widetilde{\pi}_{n}^{(k)}$ induces a
${\mathbb{C}}[[\hbar]]$-module map
$\pi_{n}^{(k)}:{\mathcal{E}}_{\hbar}^{(k)}(W)\to{\mathcal{E}}^{(k)}(W/\hbar^{n}W)$
with kernel $\hbar^{n}{\mathcal{E}}_{\hbar}^{(k)}(W)$. And
${\mathcal{E}}_{\hbar}^{(k)}(W)$ is isomorphic to the inverse limit of the
following inverse system

If $k=1$, we will also write $\pi_{n}=\pi_{n}^{(1)}$. Then an ordered sequence
$(a_{1}(z)\dots,a_{r}(z))$ in ${\mathcal{E}}_{\hbar}(W)$ is said to be
_$\hbar$ -adically compatible_ if for every $n\in{\mathbb{Z}}_{+}$, the
sequence $(\pi_{n}(a_{1}(z)),\dots,\pi_{n}(a_{r}(z)))$ in
${\mathcal{E}}(W/\hbar^{n}W)$ is compatible. A subset $U$ of
${\mathcal{E}}_{\hbar}(W)$ is said to be _$\hbar$ -adically compatible_ if
every finite sequence in $U$ is $\hbar$-adically compatible.

Let $(a(z),b(z))$ in ${\mathcal{E}}_{\hbar}(W)$ be $\hbar$-adically
compatible. That is, for any $n\in{\mathbb{Z}}_{+}$, we have that

$\displaystyle(z_{1}-z_{2})^{k_{n}}\pi_{n}\left(a(z_{1})\right)\pi_{n}\left(b(z_{2})\right)\in{\mathcal{E}}^{(2)}(W/\hbar^{n}W)\quad\mbox{for
some }k_{n}\in{\mathbb{Z}}_{+}.$

We recall the following vertex operator introduced in [Li6, Definition 4.11]:

(3.3) $\displaystyle
Y_{\mathcal{E}}\left(a(z),z_{0}\right)b(x)=\sum_{n\in{\mathbb{Z}}}a(z)_{n}b(z)z_{0}^{-n-1}$
$\displaystyle=$
$\displaystyle\varprojlim_{n>0}z_{0}^{-k_{n}}\left((z_{1}-z)^{k_{n}}\pi_{n}(a(z_{1}))\pi_{n}(b(z))\right)|_{z_{1}=z+z_{0}}.$

An _$\hbar$ -adic nonlocal vertex algebra_ [Li6, Definition 2.9] is a
topologically free ${\mathbb{C}}[[\hbar]]$-module $V$ equipped with a
${\mathbb{C}}[[\hbar]]$-module map $Y(\cdot,z):V\to{\mathcal{E}}_{\hbar}(V)$
and a distinguished vacuum vector ${{\bf 1}}$ such that the vacuum property
(2.2) holds, and for $u,v\in V$, $(Y(u,z),Y(v,z))$ is an $\hbar$-adic
compatible pair with

(3.4) $\displaystyle
Y_{\mathcal{E}}\left(Y(u,z),z_{0}\right)Y(v,z)=Y\left(Y(u,z_{0})v,z\right)$

Denote by $\partial$ the canonical derivation on $V$:

(3.5) $\displaystyle u\mapsto\partial u=\lim_{z\to 0}\frac{d}{dz}Y(u,z){{\bf
1}}.$

An _$\hbar$ -adic weak quantum vertex algebra_ ([Li6, Definition 2.9]) is an
$\hbar$-adic nonlocal vertex algebra $V$, such that
${\left.\left\\{{Y(u,z)}\,\right|\,{u\in V}\right\\}}$ is $\hbar$-adically
$S$-local. And an _$\hbar$ -adic quantum vertex algebra_ ([Li6, Definition
2.20]) is an $\hbar$-adic weak quantum vertex algebra $V$ equipped with a
${\mathbb{C}}[[\hbar]]$-module map (called a _quantum Yang-Baxter operator_)

(3.6) $\displaystyle S(z):V\widehat{\otimes}V\to
V\widehat{\otimes}V\widehat{\otimes}{\mathbb{C}}((z))[[\hbar]],$

which satisfies the _shift condition_ :

(3.7) $\displaystyle[\partial\otimes 1,S(z)]=-\frac{d}{dz}S(z),$

the _quantum Yang-Baxter equation_ :

(3.8) $\displaystyle
S^{12}(z_{1})S^{13}(z_{1}+z_{2})S^{23}(z_{2})=S^{23}(z_{2})S^{13}(z_{1}+z_{2})S^{12}(z_{1}),$

and the _unitarity condition_ :

(3.9) $\displaystyle S^{21}(z)S(-z)=1,$

satisfying the following conditions:

(1) The _vacuum property_ :

(3.10) $\displaystyle S(z)({{\bf 1}}\otimes v)={{\bf 1}}\otimes
v,\quad\mbox{for }v\in V.$

(2) The _$S$ -locality_: For any $u,v\in V$, one has

(3.11) $\displaystyle Y(u,z_{1})Y(v,z_{2})\sim Y(z_{2})(1\otimes
Y(z_{1}))S(z_{2}-z_{1})(v\otimes u).$

(3) The _hexagon identity_ :

(3.12) $\displaystyle S(z_{1})(Y(z_{2})\otimes 1)=(Y(z_{2})\otimes
1)S^{23}(z_{1})S^{13}(z_{1}+z_{2}).$

We list some technical results that will be used later on.

######  Lemma 3.1.

Let $(V,S(z))$ be an $\hbar$-adic quantum vertex algebra. Then

(3.13) $\displaystyle S(z)(v\otimes{{\bf 1}})=v\otimes{{\bf 1}}\quad\mbox{for
}v\in V,$ (3.14) $\displaystyle[1\otimes\partial,S(z)]=\frac{d}{dz}S(z),$
(3.15) $\displaystyle S(z_{1})(1\otimes Y(z_{2}))=(1\otimes
Y(z_{2}))S^{12}(z_{1}-z_{2})S^{13}(z_{1}),$ (3.16) $\displaystyle
S(z)f(\partial\otimes 1)=f\left(\partial\otimes
1+\frac{\partial}{\partial{z}}\right)S(z)\quad\mbox{for
}f(z)\in{\mathbb{C}}[z][[\hbar]],$ (3.17) $\displaystyle
S(z)f(1\otimes\partial)=f\left(1\otimes\partial-\frac{\partial}{\partial{z}}\right)S(z)\quad\mbox{for
}f(z)\in{\mathbb{C}}[z][[\hbar]].$

######  Lemma 3.2.

Let $(V,S(z))$ be an $\hbar$-adic quantum vertex algebra, and let $u,v\in V$,
$f(z)\in{\mathbb{C}}((z))[[\hbar]]$.

  * (1)

If $S(z)(v\otimes u)=v\otimes u+{{\bf 1}}\otimes{{\bf 1}}\otimes f(z)$, then

$\displaystyle S(z)((v_{-1})^{n}{{\bf 1}}\otimes u)$ $\displaystyle=$
$\displaystyle(v_{-1})^{n}{{\bf 1}}\otimes u+n(v_{-1})^{n-1}{{\bf
1}}\otimes{{\bf 1}}\otimes f(z),$ $\displaystyle
S(z)(v\otimes(u_{-1})^{n}{{\bf 1}})$ $\displaystyle=$ $\displaystyle
v\otimes(u_{-1})^{n}{{\bf 1}}+n{{\bf 1}}\otimes(u_{-1})^{n-1}{{\bf 1}}\otimes
f(z),$ $\displaystyle S(z)\left(\exp(v_{-1}){{\bf 1}}\otimes u\right)$
$\displaystyle=$ $\displaystyle\exp(v_{-1}){{\bf 1}}\otimes
u+\exp(v_{-1}){{\bf 1}}\otimes{{\bf 1}}\otimes f(z),\quad\mbox{if }v\in\hbar
V,$ $\displaystyle S(z)\left(v\otimes\exp(u_{-1}){{\bf 1}}\right)$
$\displaystyle=$ $\displaystyle v\otimes\exp(u_{-1}){{\bf 1}}+{{\bf
1}}\otimes\exp(u_{-1}){{\bf 1}}\otimes f(z),\quad\mbox{if }u\in\hbar V.$

  * (2)

If $S(z)(v\otimes u)=v\otimes u+{{\bf 1}}\otimes u\otimes f(z)$, then

$\displaystyle S(z)((v_{-1})^{n}{{\bf 1}}\otimes
u)=\sum_{i=0}^{n}\binom{n}{i}(v_{-1})^{i}{{\bf 1}}\otimes u\otimes
f(z)^{n-i},$ $\displaystyle S(z)\left(\exp(v_{-1}){{\bf 1}}\otimes
u\right)=\exp(v_{-1}){{\bf 1}}\otimes u\otimes\exp\left(f(z)\right),$
$\displaystyle\quad\mbox{if }v\in\hbar
V,f(z)\in\hbar{\mathbb{C}}((z))[[\hbar]].$

  * (3)

If $S(z)(v\otimes u)=v\otimes u+v\otimes{{\bf 1}}\otimes f(z)$, then

$\displaystyle S(z)(v\otimes(u_{-1})^{n}{{\bf
1}})=\sum_{i=0}^{n}\binom{n}{i}v\otimes(u_{-1})^{i}{{\bf 1}}\otimes
f(z)^{n-i},$ $\displaystyle S(z)\left(v\otimes\exp(u_{-1}){{\bf
1}}\right)=v\otimes\exp\left(u_{-1}\right){{\bf 1}}\otimes\exp(f(z)),$
$\displaystyle\quad\mbox{if }u\in\hbar
V,f(z)\in\hbar{\mathbb{C}}((z))[[\hbar]].$

For a topologically free ${\mathbb{C}}[[\hbar]]$-module $V$ and a submodule
$M$, we recall the notation given in [Li6, Definition 3.4]:

$\displaystyle[M]={\left.\left\\{{v\in V}\,\right|\,{\hbar^{n}V\in
M\,\,\mbox{for some }n\in{\mathbb{N}}}\right\\}}.$

It is straightforward to verify that

######  Lemma 3.3.

Let $(V,S(z))$ be an $\hbar$-adic quantum vertex algebra, and let $U\subset V$
be a generating subset of $V$. Let $M\subset V$ be a closed ideal of $V$ such
that $[M]=M$ and

$\displaystyle S(z)(M\otimes U),\,S(z)(U\otimes M)\subset
M\widehat{\otimes}V\widehat{\otimes}{\mathbb{C}}((z))[[\hbar]]+V\widehat{\otimes}M\widehat{\otimes}{\mathbb{C}}((z))[[\hbar]].$

Then $V/M$ is an $\hbar$-adic quantum vertex algebra with quantum Yang-Baxter
operator induced from $S(z)$.

In the rest of this section, we recall the construction of $\hbar$-adic
nonlocal vertex algebras and their modules introduced in [Li6]. Here, a
_module_ ([Li6, Definition 2.33]) for an $\hbar$-adic nonlocal vertex algebra
$V$ is a topologically free ${\mathbb{C}}[[\hbar]]$-module $W$, equipped with
a ${\mathbb{C}}[[\hbar]]$-module map
$Y_{W}(\cdot,z):V\to{\mathcal{E}}_{\hbar}(W)$, satisfying the conditions that
$Y_{W}({{\bf 1}},z)=1_{W}$ and that for $u,v\in V$, $(Y_{W}(u,z),Y_{W}(v,z))$
is $\hbar$-adically compatible and

$\displaystyle
Y_{\mathcal{E}}\left(Y_{W}(u,z),z_{0}\right)Y_{W}(v,z)=Y_{W}\left(Y(u,z_{0})v,z\right).$

The following result is given in [Li6]:

######  Theorem 3.4.

Let $W$ be a topologically free ${\mathbb{C}}[[\hbar]]$-module, and let
$U\subset{\mathcal{E}}_{\hbar}(W)$ be an $\hbar$-adically compatible subset.
Then there is a minimal $\hbar$-adically $S$-local subset
${\langle}U{\rangle}\subset{\mathcal{E}}_{\hbar}(W)$ containing $1_{W}$ and
$U$, such that

  * (1)

${\langle}U{\rangle}$ is topologically free and
$[{\langle}U{\rangle}]={\langle}U{\rangle}$.

  * (2)

${\langle}U{\rangle}$ is $Y_{\mathcal{E}}$ closed in the sense that for any
$a(z),b(z)\in U$ and any $n\in{\mathbb{Z}}$, we have
$a(z)_{n}b(z)\in{\langle}U{\rangle}$.

Then $({\langle}U{\rangle},Y_{\mathcal{E}},1_{W})$ carries the structure of an
$\hbar$-adic nonlocal vertex algebra and $W$ is a faithful
${\langle}U{\rangle}$-module with the module map $Y_{W}$ defined by
$Y_{W}(a(z),z_{0})=a(z_{0})$ for $a(z)\in{\langle}U{\rangle}$.

######  Remark 3.5.

If $U$ is $\hbar$-adically $S$-local, then ${\langle}U{\rangle}$ is an
$\hbar$-adic weak quantum vertex algebra.

We recall the following result from [Li6, Theorem 4.24] for later use.

######  Theorem 3.6.

Let $V$ be a topologically free ${\mathbb{C}}[[\hbar]]$-module, $U\subset V$,
${{\bf 1}}\in V$, and $Y^{0}$ a map

$\displaystyle Y^{0}:U\to{\mathcal{E}}_{\hbar}(V);\quad u\mapsto
Y^{0}(u,z)=u(z)=\sum_{n\in{\mathbb{Z}}}u_{n}z^{-n-1}.$

Assume that all the following conditions hold:

$\displaystyle Y^{0}(u,z){{\bf 1}}\in V[[z]]\quad\mbox{and}\quad\lim_{z\to
0}Y^{0}(u,z){{\bf 1}}=u\quad\mbox{for }u\in U,$

$U(z)={\left.\left\\{{u(z)}\,\right|\,{u\in U}\right\\}}$ is $\hbar$-adically
$S$-local, and $V$ is $\hbar$-adically spanned by vectors

(3.18) $\displaystyle u_{m_{1}}^{(1)}\cdots u_{m_{r}}^{(r)}{{\bf 1}}$

for $r\in{\mathbb{N}}$, $u^{(i)}\in U$, $m_{i}\in{\mathbb{Z}}$. In addition we
assume that there exists a ${\mathbb{C}}[[\hbar]]$-module map $\psi$ from $V$
to ${\langle}U(z){\rangle}\subset{\mathcal{E}}_{\hbar}(V)$ such that
$\psi({{\bf 1}})=1_{V}$ and

(3.19)
$\displaystyle\psi\left(Y(u,z_{0})v\right)=Y_{\mathcal{E}}\left(u(z),z_{0}\right)\psi(v)\quad\mbox{for
}u\in U,\,v\in V.$

Then the map $Y^{0}$ extends uniquely to a ${\mathbb{C}}[[\hbar]]$-module map
$Y$ from $V$ to ${\mathcal{E}}_{\hbar}(V)$ such that $(V,Y,{{\bf 1}})$ carries
the structure of an $\hbar$-adic weak quantum vertex algebra.

##  4\. Deformation by vertex bialgebras

In this section, we recall the deformation of $\hbar$-adic nonlocal vertex
algebras by using vertex bialgebras introduced in [JKLT2, JKLT3].

We start by recalling the notion of vertex bialgebras and smash products given
in [Li5] (see also [JKLT3]). An _$\hbar$ -adic (nonlocal) vertex bialgebra_ is
an $\hbar$-adic (nonlocal) vertex algebra $V$ equipped with a classical
coalgebra structure $(\Delta,\varepsilon)$ such that (the coproduct)
$\Delta:V\to V\widehat{\otimes}V$ and (the counit)
$\varepsilon:V\to{\mathbb{C}}[[\hbar]]$ are homomorphisms of $\hbar$-adic
nonlocal vertex algebras.

######  Remark 4.1.

Let $(H,\Delta,\varepsilon)$ be a bialgebra over ${\mathbb{C}}[[\hbar]]$
equipped with a derivation $\partial$. Suppose that $H$ is topologically free.
Then $H$ is an $\hbar$-adic nonlocal vertex bialgebra with vacuum $1$ and
vertex operator defined by

$\displaystyle Y(a,z)b=\left(e^{z\partial}a\right)b\quad\mbox{for }a,b\in H.$

We denote this $\hbar$-adic nonlocal vertex bialgebra by
$(H,\partial,\Delta,\varepsilon)$.

Let $(H,\Delta,\varepsilon)$ be an $\hbar$-adic nonlocal vertex bialgebra. A
(left) _$H$ -module (nonlocal) vertex algebra_ ([Li5]) is an $\hbar$-adic
nonlocal vertex algebra $V$ equipped with a module structure $\tau$ on $V$ for
$H$ viewed as an $\hbar$-adic nonlocal vertex algebra such that

(4.1) $\displaystyle\tau(h,z)v\in
V\widehat{\otimes}{\mathbb{C}}((z))[[\hbar]],\qquad\tau(h,z){{\bf
1}}_{V}=\varepsilon(h){{\bf 1}}_{V},$ (4.2)
$\displaystyle\tau(h,z_{1})Y(u,z_{2})v=\sum
Y(\tau(h_{(1)},z_{1}-z_{2})u,z_{2})\tau(h_{(2)},z_{1})v$

for $h\in H$, $u,v\in V$, where ${{\bf 1}}_{V}$ denotes the vacuum vector of
$V$ and $\Delta(h)=\sum h_{(1)}\otimes h_{(2)}$ is the coproduct in the
Sweedler notation.

A _(right) $H$-comodule nonlocal vertex algebra_ ([JKLT2]) is a nonlocal
vertex algebra $V$ equipped with a homomorphism $\rho:V\to
V\widehat{\otimes}H$ of $\hbar$-adic nonlocal vertex algebras such that

(4.3) $\displaystyle(\rho\otimes
1)\rho=(1\otimes\Delta)\rho,\quad(1\otimes\varepsilon)\rho=\mbox{Id}_{V}.$

$\rho$ is _compatible_ with an $H$-module nonlocal vertex algebra structure
$\tau$ (see [JKLT2]) if

(4.4) $\displaystyle\rho(\tau(h,z)v)=(\tau(h,z)\otimes 1)\rho(v)\quad\mbox{for
}h\in H,\,v\in V.$

We introduce the following notion.

######  Definition 4.2.

Let $V$ be an $\hbar$-adic nonlocal vertex algebra. A _deforming triple_ for
$V$ is a triple $(H,\rho,\tau)$, where $H$ is a cocommutative $\hbar$-adic
nonlocal vertex bialgebra, $(V,\rho)$ is a right $H$-comodule nonlocal vertex
algebra and $(V,\tau)$ is an $H$-module nonlocal vertex algebra, such that
$\rho$ and $\tau$ are compatible.

The following result is given in [JKLT2].

######  Theorem 4.3.

Let $V$ be an $\hbar$-adic nonlocal vertex algebra, and let $(H,\rho,\tau)$ be
a deforming triple. Set

(4.5) $\displaystyle\mathfrak{D}_{\tau}^{\rho}(Y)(a,z)=\sum
Y(a_{(1)},z)\tau(a_{(2)},z)\quad\mbox{for }a\in V,$

where $\rho(a)=\sum a_{(1)}\otimes a_{(2)}\in V\otimes H$. Then
$(V,\mathfrak{D}_{\tau}^{\rho}(Y),{{\bf 1}})$ carries the structure of an
$\hbar$-adic nonlocal vertex algebra. Denote this $\hbar$-adic nonlocal vertex
algebra by $\mathfrak{D}_{\tau}^{\rho}(V)$. Moreover,
$(\mathfrak{D}_{\tau}^{\rho}(V),\rho)$ is also a right $H$-comodule nonlocal
vertex algebra.

######  Remark 4.4.

Let $H$ be a cocommutative $\hbar$-adic nonlocal vertex bialgebra, and let
$(V,\rho)$ be an $H$-comodule nonlocal vertex algebra. We view the counit
$\varepsilon$ as an $H$-module nonlocal vertex algebra structure on $V$ as
follows

(4.6) $\displaystyle\varepsilon(h,z)v=\varepsilon(h)v\quad\mbox{for }h\in
H,\,v\in V.$

Then it is easy to verify that $\varepsilon$ and $\rho$ are compatible, and
$\mathfrak{D}_{\varepsilon}^{\rho}(V)=V$.

######  Remark 4.5.

Let $V_{1},V_{2}$ be two $\hbar$-adic nonlocal vertex algebras. Suppose that
$(H,\rho_{1},\tau_{1})$ and $(H,\rho_{2},\tau_{2})$ are deforming triples of
$V_{1}$ and $V_{2}$ respectively. Suppose that $f:V_{1}\to V_{2}$ is an
$\hbar$-adic nonlocal vertex algebra homomorphism such that

$\displaystyle\rho_{2}\circ f=(f\otimes 1)\circ\rho_{1},\quad\mbox{and}\quad
f(\tau_{1}(h,z)v)=\tau_{2}(h,z)f(v)\quad\mbox{for }h\in H,\,v\in V_{1}.$

Then $f$ is an $\hbar$-adic nonlocal vertex algebra homomorphism from
$\mathfrak{D}_{\tau_{1}}^{\rho_{1}}(V_{1})$ to
$\mathfrak{D}_{\tau_{2}}^{\rho_{2}}(V_{2})$.

Now, we fix a cocommutative $\hbar$-adic nonlocal vertex bialgebra $H$, and an
$H$-comodule nonlocal vertex algebra $(V,\rho)$. Note that

$\operatorname{Hom}\left(H,\operatorname{Hom}(V,V\widehat{\otimes}{\mathbb{C}}((x))[[\hbar]])\right)$

is a unital associative algebra, where the multiplication is defined by

$\displaystyle(f\ast g)(h,z)=\sum f(h_{(1)},z)g(h_{(2)},z)$

for
$f,g\in\operatorname{Hom}\left(H,\operatorname{Hom}(V,V\widehat{\otimes}{\mathbb{C}}((z))[[\hbar]])\right)$,
and the unit $\varepsilon$. For
$f,g\in\operatorname{Hom}\left(H,\operatorname{Hom}(V,V\widehat{\otimes}{\mathbb{C}}((z))[[\hbar]])\right)$,
we say $f$ and $g$ commute if

$\displaystyle[f(h,z_{1}),g(k,z_{2})]=0\quad\mbox{for }h,k\in H.$

Let $\mathfrak{L}_{H}^{\rho}(V)$ be the set of $H$-module nonlocal vertex
algebra structures on $V$ that are compatible with $\rho$. The following is an
immediate $\hbar$-adic analogue of [JKLT2, Proposition 3.3 and Proposition
3.4].

######  Proposition 4.6.

Let $\tau$ and $\tau^{\prime}$ are commuting elements in
$\mathfrak{L}_{H}^{\rho}(V)$. Then
$\tau\ast\tau^{\prime}\in\mathfrak{L}_{H}^{\rho}(V)$ and
$\tau\ast\tau^{\prime}=\tau^{\prime}\ast\tau$. Moreover,
$\tau\in\mathfrak{L}_{H}^{\rho}\left(\mathfrak{D}_{\tau^{\prime}}^{\rho}(V)\right)$
and

(4.7)
$\displaystyle\mathfrak{D}_{\tau}^{\rho}\left(\mathfrak{D}_{\tau^{\prime}}^{\rho}(V)\right)=\mathfrak{D}_{\tau\ast\tau^{\prime}}^{\rho}(V).$

Recall that an element $\tau\in\mathfrak{L}_{H}^{\rho}(V)$ is said to be
_invertible_ if there exists $\tau^{-1}\in\mathfrak{L}_{H}^{\rho}(V)$, such
that $\tau$ and $\tau^{-1}$ are commute and $\tau\ast\tau^{-1}=\varepsilon$.
We have the immediate $\hbar$-adic analogue of [JKLT2, Theorem 3.6].

######  Theorem 4.7.

Let $V_{0}$ be a vertex algebra, and let $V=V_{0}[[\hbar]]$ be the
corresponding $\hbar$-adic vertex algebra. Suppose that $(H,\rho,\tau)$ is a
deforming triple of $V$, such that $\tau$ is invertible in
$\mathfrak{L}_{H}^{\rho}(V)$. Then $\mathfrak{D}_{\tau}^{\rho}(V)$ is an
$\hbar$-adic quantum vertex algebra with quantum Yang-Baxter operator $S(z)$
defined by

$\displaystyle S(z)(v\otimes
u)=\sum\tau(u_{(2)},-z)v_{(1)}\otimes\tau^{-1}(v_{(2)},z)u_{(1)}\quad\mbox{for
}u,v\in V,$

where $\rho(u)=\sum u_{(1)}\otimes u_{(2)}$ and $\rho(v)=\sum v_{(1)}\otimes
v_{(2)}$.

##  5\. Construction of $F_{\tau}(A,\ell)$

In this section, we fix a GCM $A=(a_{ij})_{i,j\in I}$ and a complex number
$\ell$. Let $\mathfrak{T}$ be the set of tuples

$\displaystyle\tau=\big{(}\tau_{ij}(z),\tau_{ij}^{1,\pm}(z),\tau_{ij}^{2,\pm}(z),\tau_{ij}^{\epsilon_{1},\epsilon_{2}}(z)\big{)}_{i,j\in
I}^{\epsilon_{1},\epsilon_{2}=\pm},$

where
$\tau_{ij}(z),\,\tau_{ij}^{1,\pm}(z),\,\tau_{ij}^{2,\pm}(z),\,\tau_{ij}^{\epsilon_{1},\epsilon_{2}}(z)\in{\mathbb{C}}((z))[[\hbar]]$,
such that

$\displaystyle\lim_{\hbar\to 0}\tau_{ij}(z)=\lim_{\hbar\to
0}\tau_{ji}(-z),\quad\lim_{\hbar\to 0}\tau_{ij}^{1,\pm}(z)=-\lim_{\hbar\to
0}\tau_{ji}^{2,\pm}(-z),$ $\displaystyle\lim_{\hbar\to
0}\tau_{ij}^{\epsilon_{1},\epsilon_{2}}(z)=\lim_{\hbar\to
0}\tau_{ji}^{\epsilon_{2},\epsilon_{1}}(-z)\in{\mathbb{C}}[[z]]^{\times}.$

######  Definition 5.1.

Let $\tau\in\mathfrak{T}$. Define $\mathcal{M}_{\tau}$ to be the category
consisting of topologically free ${\mathbb{C}}[[\hbar]]$-modules $W$, equipped
with fields $h_{i,\hbar}(z),x_{i,\hbar}^{\pm}(z)\in{\mathcal{E}}_{\hbar}(W)$
($i\in I$) satisfying the following conditions

(5.1) $\displaystyle[h_{i,\hbar}(z_{1}),h_{j,\hbar}(z_{2})]$
$\displaystyle\quad=r_{i}a_{ij}r\ell\frac{\partial}{\partial{z_{2}}}z_{1}^{-1}\delta\left(\frac{z_{2}}{z_{1}}\right)+\tau_{ij}(z_{1}-z_{2})-\tau_{ji}(z_{2}-z_{1}),$
(5.2) $\displaystyle[h_{i,\hbar}(z_{1}),x_{j,\hbar}^{\pm}(z_{2})]$
$\displaystyle\quad=\pm
x_{j,\hbar}^{\pm}(z_{2})\left(r_{i}a_{ij}z_{1}^{-1}\delta\left(\frac{z_{2}}{z_{1}}\right)+\tau_{ij}^{1,\pm}(z_{1}-z_{2})+\tau_{ji}^{2,\pm}(z_{2}-z_{1})\right),$
(5.3)
$\displaystyle\tau_{ij}^{\pm,\pm}(z_{1}-z_{2})(z_{1}-z_{2})^{n_{ij}}x_{i,\hbar}^{\pm}(z_{1})x_{j,\hbar}^{\pm}(z_{2})$
$\displaystyle\quad=\tau_{ji}^{\pm,\pm}(z_{2}-z_{1})(z_{1}-z_{2})^{n_{ij}}x_{j,\hbar}^{\pm}(z_{2})x_{i,\hbar}^{\pm}(z_{1}),$
(5.4)
$\displaystyle\tau_{ij}^{\pm,\mp}(z_{1}-z_{2})(z_{1}-z_{2})^{2\delta_{ij}}x_{i,\hbar}^{\pm}(z_{1})x_{j,\hbar}^{\mp}(z_{2})$
$\displaystyle\quad=\tau_{ji}^{\mp,\pm}(z_{2}-z_{1})(z_{1}-z_{2})^{2\delta_{ij}}x_{j,\hbar}^{\mp}(z_{2})x_{i,\hbar}^{\pm}(z_{1}).$

Let $\mathcal{F}_{\tau}$ be the forgetful functor from $\mathcal{M}_{\tau}$ to
the category of topologically free ${\mathbb{C}}[[\hbar]]$-modules. Define
$\operatorname{End}_{{\mathbb{C}}[[\hbar]]}(\mathcal{F}_{\tau})$ to be the
algebra of endomorphisms of the functor $\mathcal{F}_{\tau}$. For each
$W\in\mathcal{M}_{\tau}$, $\operatorname{End}_{{\mathbb{C}}[[\hbar]]}(W)$ is a
topological algebra over ${\mathbb{C}}[[\hbar]]$ such that

$\displaystyle{\left.\left\\{{(K:\hbar^{n}W)}\,\right|\,{K\subset
W,\,|K|<\infty,\,n\in{\mathbb{Z}}_{+}}\right\\}}$

forms a local basis at $0$, where

$\displaystyle(K:\hbar^{n}W)={\left.\left\\{{\varphi\in\operatorname{End}_{{\mathbb{C}}[[\hbar]]}(W)}\,\right|\,{\varphi(K)\subset\hbar^{n}W}\right\\}}.$

Equip $\operatorname{End}_{{\mathbb{C}}[[\hbar]]}(\mathcal{F}_{\tau})$ with
the coarsest topology such that for any $W\in\mathcal{M}_{\tau}$, the
canonical ${\mathbb{C}}[[\hbar]]$-algebra epimorphism from $\mathcal{A}$ to
$\operatorname{End}_{{\mathbb{C}}[[\hbar]]}(W)$ is continuous. It is easy to
verify that both $\operatorname{End}_{{\mathbb{C}}[[\hbar]]}(W)$ and
$\operatorname{End}_{{\mathbb{C}}[[\hbar]]}(\mathcal{F}_{\tau})$ are
topologically free.

For each $i\in I$, we define endomorphisms $h_{i,\hbar}(n)$ and
$x_{i,\hbar}^{\pm}(n)$ ($n\in{\mathbb{Z}}$) of $\mathcal{F}_{\tau}$ as
follows:

$\displaystyle\sum_{n\in{\mathbb{Z}}}h_{i,\hbar}(n).vz^{-n-1}=h_{i,\hbar}(z)v,\quad\sum_{n\in{\mathbb{Z}}}x_{i,\hbar}^{\pm}(n).vz^{-n-1}=x_{i,\hbar}^{\pm}(z)v.$

where $v\in W$ and $W\in\mathcal{M}_{\tau}$. We denote by $\mathcal{A}$ the
closed subalgebra of
$\operatorname{End}_{{\mathbb{C}}[[\hbar]]}(\mathcal{F}_{\tau})$ generated by
${\left.\left\\{{h_{i,\hbar}(n),x_{i,\hbar}^{\pm}(n)}\,\right|\,{i\in
I,n\in{\mathbb{Z}}}\right\\}}$. Then $\mathcal{A}$ is topologically free. Let
$\mathcal{A}_{+}$ be the minimal closed left ideal of $\mathcal{A}$ containing
$h_{i,\hbar}(n)$ and $x_{i,\hbar}^{\pm}(n)$ ($i\in I$, $n\geq 0$), such that
$[\mathcal{A}_{+}]=\mathcal{A}_{+}$. Define

(5.5) $\displaystyle F_{\tau}(A,\ell)=\mathcal{A}/\mathcal{A}_{+}.$

Set ${{\bf 1}}=1+\mathcal{A}_{+}\in F_{\tau}(A,\ell)$. And for each $i\in I$,
we identify $h_{i,\hbar}$ and $x_{i,\hbar}^{\pm}$ with $h_{i,\hbar}(-1){{\bf
1}}$ and $x_{i,\hbar}^{\pm}(-1){{\bf 1}}$ in $F_{\tau}(A,\ell)$, respectively.
It is straightforward to verify the following result.

######  Lemma 5.2.

The topologically free ${\mathbb{C}}[[\hbar]]$-module $F_{\tau}(A,\ell)$
equipped with fields $h_{i,\hbar}(z),x_{i,\hbar}^{\pm}(z)$ ($i\in I$) is an
object in $\mathcal{M}_{\tau}$. Moreover, let $W$ be an object in
$\mathcal{M}_{\tau}$, and let $v_{+}\in W$ be such that
$h_{i,\hbar}(n).v_{+}=0=x_{i,\hbar}^{\pm}(n).v_{+}$ for $i\in I$ and $n\geq
0$. Then there exists a unique $\mathcal{A}$-module homomorphism
$\varphi:F_{\tau}(A,\ell)\to W$ such that $\varphi({{\bf 1}})=v_{+}$.

######  Proposition 5.3.

There exists an $\hbar$-adic nonlocal vertex algebra structure on
$F_{\tau}(A,\ell)$ such that ${{\bf 1}}$ is the vacuum vector and the vertex
operator map $Y_{\tau}$ is determined by the following conditions

(5.6) $\displaystyle Y_{\tau}(h_{i,\hbar},z)=h_{i,\hbar}(z),\quad
Y_{\tau}(x_{i,\hbar}^{\pm},z)=x_{i,\hbar}^{\pm}(z),\quad i\in I.$

Moreover, there exists a natural isomorphism between the category of
$F_{\tau}(A,\ell)$-modules and $\mathcal{M}_{\tau}$.

###### Proof.

Let $(W,h_{i,\hbar}(z),x_{i,\hbar}^{\pm}(z))$ be an object in the category
$\mathcal{M}_{\tau}$, and let

$\displaystyle
U_{W}={\left.\left\\{{h_{i,\hbar}(z),\,x_{i,\hbar}^{\pm}(z),\,c}\,\right|\,{i\in
I}\right\\}}\subset{\mathcal{E}}_{\hbar}(W).$

It follows from the relations (5.1), (5.2), (5.3) and (5.4) that $U_{W}$ is an
$\hbar$-adically $S$-local subset. From Theorem 3.4, we get an $\hbar$-adic
weak quantum vertex algebra
${\langle}U_{W}{\rangle}\subset{\mathcal{E}}_{\hbar}(W)$ containing
$U_{W},1_{W}$, such that $1_{W}$ is the vacuum vector and $Y_{\mathcal{E}}$ is
the vertex operator (see (3.3)). Moreover, $W$ is a faithful
${\langle}U_{W}{\rangle}$-module with $Y_{W}(a(z),z_{0})=a(z_{0})$ for
$a(z)\in{\langle}U_{W}{\rangle}$.

Using [Li6, Lemma 4.21], we get that
$\left({\langle}U_{W}{\rangle},Y_{\mathcal{E}}(h_{i,\hbar}(z_{1}),z),Y_{\mathcal{E}}(x_{i,\hbar}^{\pm}(z_{1}),z)\right)$
is an object in $\mathcal{M}_{\tau}$. From the vacuum property (2.2), we have
that

$\displaystyle
h_{i,\hbar}(z)1_{W}=Y_{\mathcal{E}}(h_{i,\hbar}(z_{1}),z)1_{W},$
$\displaystyle
x_{i,\hbar}^{\pm}(z)1_{W}=Y_{\mathcal{E}}(x_{i,\hbar}^{\pm}(z_{1}),z)1_{W}\in{\langle}U_{W}{\rangle}[[z]]\quad\mbox{for
}i\in I.$

Combining this with Lemma 5.2, we get an $\mathcal{A}$-module map
$\varphi_{W}:F_{\tau}(A,\ell)\to{\langle}U_{W}{\rangle}$, such that
$\varphi_{W}({{\bf 1}})=1_{W}$.

Note that $F_{\tau}(A,\ell)$ is an object in $\mathcal{M}_{\tau}$ (see Lemma
5.2). Since $\varphi_{F_{\tau}(A,\ell)}$ is an $\mathcal{A}$-module map, we
have that

$\displaystyle\varphi_{F_{\tau}(A,\ell)}(a(z_{0})v)=a(z_{0})\varphi_{F_{\tau}(A,\ell)}(v)=Y_{\mathcal{E}}(a(z),z_{0})\varphi_{F_{\tau}(A,\ell)}(v)$
$\displaystyle\quad\mbox{for
}a=h_{i,\hbar}\,\,\mbox{or}\,\,x_{i,\hbar}^{\pm},\,\,v\in F_{\tau}(A,\ell).$

From the definition of $F_{\tau}(A,\ell)$ we have that $a(z){{\bf 1}}\in
F_{\tau}(A,\ell)[[z]]$ for $a=h_{i,\hbar},\,x_{i,\hbar}^{\pm}$. Then by using
Theorem 3.6, we get a unique ${\mathbb{C}}[[\hbar]]$-module map $Y_{\tau}$
such that

$\displaystyle Y_{\tau}(h_{i,\hbar},z)=h_{i,\hbar}(z),\quad
Y_{\tau}(x_{i,\hbar}^{\pm},z)=x_{i,\hbar}^{\pm}(z),\quad i\in I,$

and $(F_{\tau}(A,\ell),Y_{\tau},{{\bf 1}})$ carries the structure of an
$\hbar$-adic weak quantum vertex algebra.

An immediate $\hbar$-adic analogue of [Li4, Proposition 6.7] implies that
every $F_{\tau}(A,\ell)$-module is an object in $\mathcal{M}_{\tau}$. On the
other hand, let $(W,h_{i,\hbar}(z),x_{i,\hbar}^{\pm}(z))$ be an object in the
category $\mathcal{M}_{\tau}$. Since
$\varphi_{W}:F_{\tau}(A,\ell)\to{\langle}U_{W}{\rangle}$ is an
$\mathcal{A}$-module map, we have that

$\displaystyle\varphi_{W}(Y_{\tau}(a,z_{0})v)=\varphi_{W}(a(z_{0})v)=a(z_{0})\varphi_{W}(v)=Y_{\mathcal{E}}(a(z),z_{0})\varphi_{W}(v),$

for any $a=h_{i,\hbar},\,x_{i,\hbar}^{\pm}$ and $v\in F_{\tau}(A,\ell)$.
Hence, $\varphi_{W}$ is an $\hbar$-adic weak quantum vertex algebra
homomorphism. Since $W$ is a faithful ${\langle}U_{W}{\rangle}$-module, we get
that $W$ is a $F_{\tau}(A,\ell)$-module. ∎

The proof of Proposition 5.3 implies that $F_{\tau}(A,\ell)$ admits the
following universal property.

######  Proposition 5.4.

Let $V$ be an $\hbar$-adic nonlocal vertex algebra, and let $Y$ be the vertex
operator map of $V$. Suppose there exists
$\bar{h}_{i,\hbar},\bar{x}_{i,\hbar}^{\pm}\in V$, such that
$(V,Y(\bar{h}_{i,\hbar},z),Y(\bar{x}_{i,\hbar}^{\pm},z))$ becomes an object of
$\mathcal{M}_{\tau}$. Then there exists a unique $\hbar$-adic nonlocal vertex
algebra homomorphism $\varphi:F_{\tau}(A,\ell)\to V$, such that
$\varphi(h_{i,\hbar})=\bar{h}_{i,\hbar}$ and
$\varphi(x_{i,\hbar}^{\pm})=\bar{x}_{i,\hbar}^{\pm}$ for $i\in I$.

######  Corollary 5.5.

Let $V$ be an $\hbar$-adic commutative vertex algebra, and let
$\alpha_{i},\,e_{i}^{\pm}\in V$ ($i\in I$). Then there exists a unique
$\hbar$-adic nonlocal vertex algebra homomorphism $\rho:F_{\tau}(A,\ell)\to
F_{\tau}(A,\ell)\widehat{\otimes}V$, such that

$\displaystyle\rho(h_{i,\hbar})=h_{i,\hbar}\otimes 1+{{\bf
1}}\otimes\alpha_{i},\quad\rho(x_{i,\hbar}^{\pm})=x_{i,\hbar}^{\pm}\otimes
e_{i}^{\pm},\quad i\in I.$

Let $H_{0}$ be the symmetric algebra of the following vector space, and let
$H=H_{0}[[\hbar]]$:

$\displaystyle\bigoplus_{i\in
I}\left({\mathbb{C}}[\partial]\left({\mathbb{C}}\alpha_{i}\oplus{\mathbb{C}}e_{i}^{+}\oplus{\mathbb{C}}e_{i}^{-}\right)\right).$

Then $H$ is a commutative and cocommutative bialgebra with $\Delta$ and
$\varepsilon$ uniquely determined by ($i\in I$, $n\in{\mathbb{N}}$):

$\displaystyle\Delta(\partial^{n}\alpha_{i})=\partial^{n}\alpha_{i}\otimes
1+1\otimes\partial^{n}\alpha_{i},\quad\Delta(\partial^{n}e_{i}^{\pm})=\sum_{k=0}^{n}\binom{n}{k}\partial^{k}e_{i}^{\pm}\otimes\partial^{n-k}e_{i}^{\pm},$
$\displaystyle\varepsilon(\partial^{n}\alpha_{i})=0,\quad\varepsilon(\partial^{n}e_{i}^{\pm})=\delta_{n,0}.$

Let $\partial$ be the derivation on $H$ such that

$\displaystyle\partial(\partial^{n}\alpha_{i})=\partial^{n+1}\alpha_{i},\quad\partial(\partial^{n}e_{i}^{\pm})=\partial^{n+1}e_{i}^{\pm}\quad\mbox{for
}i\in I,\,n\in{\mathbb{N}}.$

It is straightforward to see that $\Delta\circ\partial=(\partial\otimes
1+1\otimes\partial)\circ\Delta$ and $\varepsilon\circ\partial=0$. From Remark
4.1 we have that $(H,\partial,\Delta,\varepsilon)$ carries an $\hbar$-adic
vertex bialgebra structure. It is immediate from Corollary 5.5 that

######  Lemma 5.6.

There is a unique $\hbar$-adic nonlocal vertex algebra map
$\rho:F_{\tau}(A,\ell)\to F_{\tau}(A,\ell)\widehat{\otimes}H$, such that

(5.7) $\displaystyle\rho(h_{i,\hbar})=h_{i,\hbar}\otimes 1+{{\bf
1}}\otimes\alpha_{i},\quad\rho(x_{i,\hbar}^{\pm})=x_{i,\hbar}^{\pm}\otimes
e_{i}^{\pm}\quad\mbox{for }i\in I.$

Moreover, $(F_{\tau}(A,\ell),\rho)$ is an $H$-comodule nonlocal vertex
algebra.

Recall from [Li5] that a _pseudo-endomorphism_ of an $\hbar$-adic nonlocal
vertex algebra $V$ is a ${\mathbb{C}}[[\hbar]]$-module map $A(z):V\to
V\widehat{\otimes}{\mathbb{C}}((z))[[\hbar]]$, such that

$\displaystyle A(z){{\bf 1}}={{\bf 1}}\otimes 1,\quad
A(z_{1})Y(u,z_{2})v=Y(A(z_{1}-z_{2})u,z_{2})v\quad\mbox{for }u,v\in V.$

And a _pseudo-derivation_ of $V$ is a ${\mathbb{C}}[[\hbar]]$-module map
$D(z):V\to V\widehat{\otimes}{\mathbb{C}}((z))[[\hbar]]$, such that

$\displaystyle[D(z_{1}),Y(u,z_{2})]=Y(D(z_{1}-z_{2})u,z_{2})\quad\mbox{for
}u\in V.$

As a straightforward $\hbar$-adic analogue of [Li3, Proposition 2.11], we have
that

######  Proposition 5.7.

Let $V$ be an $\hbar$-adic nonlocal vertex algebra, and view
${\mathbb{C}}((z))[[\hbar]]$ as an $\hbar$-adic vertex algebra with the vertex
operator map

$Y(f(z),z_{0})g(z)=f(z-z_{0})g(z).$

Suppose that $A(z)$ is an $\hbar$-adic nonlocal vertex algebra homomorphism
from $V$ to the tensor product $\hbar$-adic nonlocal vertex algebra
$V\widehat{\otimes}{\mathbb{C}}((z))[[\hbar]]$. Then $A(z)$ is a pseudo-
endomorphism of $V$. Moreover, let

$D(z):V\to V\widehat{\otimes}{\mathbb{C}}((z))[[\hbar]]$

be a ${\mathbb{C}}[[\hbar]]$-module map. We view
$V[\delta]/\delta^{2}V[\delta]$ as an $\hbar$-adic nonlocal vertex algebra. If
$1+\delta D(z)$ is a ${\mathbb{C}}[\delta]$-linear pseudo-endomorphism of
$V[\delta]/\delta^{2}V[\delta]$, then $D(z)$ is a pseudo-derivation of $V$.

Using Corollary 5.5 and Proposition 5.7, we have:

######  Lemma 5.8.

Let $\sigma\in\mathfrak{T}$. Then for each $i\in I$ there exists a pseudo-
derivation $\sigma_{i}(z)$ on $F_{\tau}(A,\ell)$ such that

(5.8) $\displaystyle\sigma_{i}(z)h_{j,\hbar}={{\bf
1}}\otimes\sigma_{ij}(z),\quad\sigma_{i}(z)x_{j,\hbar}^{\pm}=\pm
x_{j,\hbar}^{\pm}\otimes\sigma_{ij}^{1,\pm}(z).$

And there exists a pseudo-endomorphism $\sigma_{i}^{\pm}(x)$ on
$F_{\tau}(A,\ell)$ such that

(5.9) $\displaystyle\sigma_{i}^{\pm}(z)h_{j,\hbar}=h_{j,\hbar}\otimes
1\mp{{\bf
1}}\otimes\sigma_{ij}^{2,\pm}(z),\quad\sigma_{i}^{\pm}(z)x_{j,\hbar}^{\epsilon}=x_{j,\hbar}^{\epsilon}\otimes\sigma_{ij}^{\pm,\epsilon}(z)^{-1},$

where $\epsilon=\pm$ and $i,j\in I$.

Let $V$ be an $\hbar$-adic nonlocal vertex algebra. A subset $U$ of

$\operatorname{Hom}(V,V\widehat{\otimes}{\mathbb{C}}((z))[[\hbar]])$

is said to be _$\Delta$ -closed_ (see [Li5]) if for any $a(z)\in U$, there
exist two sequences $\\{a_{(1n)}(z)\\}_{n\geq 1},\\{a_{(2n)}(z)\\}_{n\geq
1}\subset
U\subset\operatorname{Hom}(V,V\widehat{\otimes}{\mathbb{C}}((z))[[\hbar]])$
that converge to $0$ under the $\hbar$-adic topology and

$\displaystyle a(z_{1})Y(v,z_{2})=\sum_{n\geq
1}Y(a_{(1n)}(z_{1}-z_{2})v,z_{2})a_{(2n)}(z_{1})\quad\mbox{for }v\in V.$

Let $B(V)$ be the sum of all $\Delta$-closed subspaces $U$ of
$\operatorname{Hom}(V,V\widehat{\otimes}{\mathbb{C}}((z))[[\hbar]])$ such that

(5.10) $\displaystyle a(z){{\bf 1}}\in{\mathbb{C}}[[\hbar]]{{\bf
1}}\quad\mbox{for }a(z)\in U.$

Note that
$\operatorname{Hom}(V,V\widehat{\otimes}{\mathbb{C}}((z))[[\hbar]])\cong\mbox{End}_{{\mathbb{C}}((z))[[\hbar]]}(V\widehat{\otimes}{\mathbb{C}}((z))[[\hbar]])$
is an associative algebra.

The following is the immediate $\hbar$-adic analogue of [Li5, Proposition
3.4].

######  Proposition 5.9.

For any $\hbar$-adic nonlocal vertex algebra $V$, $B(V)$ is a $\Delta$-closed
associative subalgebra of
$\operatorname{Hom}(V,V\widehat{\otimes}{\mathbb{C}}((z))[[\hbar]])$ and it is
closed under the derivation $\partial=\frac{d}{dz}$. Moreover, $B(V)$ is an
$\hbar$-adic nonlocal vertex algebra with the vertex operator defined by

$\displaystyle Y(a(z),z_{0})b(z)=a(z+z_{0})b(z)\quad\mbox{for }a(z),\,b(z)\in
B(V).$

Furthermore, $V$ is a $B(V)$-module with the module action
$Y_{V}(a(z),z_{0})=a(z_{0})$ for $a(z)\in B(V)$.

We have that

######  Proposition 5.10.

For each $\sigma\in\mathfrak{T}$, there exists an $H$-module structure
$\sigma(\cdot,z)$ on $F_{\tau}(A,\ell)$ such that

(5.11)
$\displaystyle\sigma(\alpha_{i},z)=\sigma_{i}(z),\quad\sigma(e_{i}^{\pm},z)=\sigma_{i}^{\pm}(z),\quad
i\in I.$

Moreover, $(H,\rho,\sigma)$ is a deforming triple for $F_{\tau}(A,\ell)$.

###### Proof.

Note that there is a ${\mathbb{C}}[[\hbar]]$-algebra homomorphism
$\varphi:H\to B(F_{\tau}(A,\ell))$ defined by

$\displaystyle\varphi(\partial^{n_{1}}\alpha_{i_{1}}\partial^{n_{2}}\alpha_{i_{2}}\cdots\partial^{n_{r}}\alpha_{i_{r}})=\frac{d^{n_{1}}\sigma_{i_{1}}(z)}{dz^{n_{1}}}\frac{d^{n_{2}}\sigma_{i_{2}}(z)}{dz^{n_{2}}}\cdots\frac{d^{n_{r}}\sigma_{i_{r}}(z)}{dz^{n_{r}}},$
$\displaystyle\varphi(\partial^{n_{1}}e_{i_{1}}^{\epsilon_{1}}\partial^{n_{2}}e_{i_{2}}^{\epsilon_{2}}\cdots\partial^{n_{r}}e_{i_{r}}^{\epsilon_{r}})=\frac{d^{n_{1}}\sigma_{i_{1}}^{\epsilon_{1}}(z)}{dz^{n_{1}}}\frac{d^{n_{2}}\sigma_{i_{2}}^{\epsilon_{2}}(z)}{dz^{n_{2}}}\cdots\frac{d^{n_{r}}\sigma_{i_{r}}^{\epsilon_{r}}(z)}{dz^{n_{r}}},$

where $i_{1},i_{2},\dots,i_{r}\in I$,
$\epsilon_{1},\epsilon_{2},\dots,\epsilon_{r}=\pm$ and $0\leq
n_{1},\dots,n_{r}\in{\mathbb{Z}}$. It is easy to see that

$\displaystyle\varphi(\partial h)=\frac{d}{dz}\varphi(h)\quad\mbox{for }h\in
H.$

Then $\varphi$ is an $\hbar$-adic vertex algebra homomorphism. Hence,
$F_{\tau}(A,\ell)$ is an $H$-module, and we denote this module action by
$\sigma(\cdot,z)$.

Let $H^{\prime}$ be the maximal ${\mathbb{C}}[[\hbar]]$-submodule of $H$ such
that for any $h\in H^{\prime}$, one has $\sigma(h,z){{\bf
1}}=\varepsilon(h){{\bf 1}}$ and

$\displaystyle\sigma(h,z_{1})Y_{\tau}(u,z_{2})v=\sum
Y_{\tau}(\sigma(h_{(1)},z_{1}-z_{2})u,z_{2})\sigma(h_{(2)},z_{1})v,$

for all $u,v\in F_{\tau}(A,\ell)$, where $\Delta(h)=\sum h_{(1)}\otimes
h_{(2)}$. It is straightforward to check that $H^{\prime}$ is an $\hbar$-adic
vertex subalgebra of $H$. Since
${\left.\left\\{{\alpha_{i},\,e_{i}^{\pm}}\,\right|\,{i\in I}\right\\}}\subset
H^{\prime}$ and $H$ is generated by
${\left.\left\\{{\alpha_{i},\,e_{i}^{\pm}}\,\right|\,{i\in I}\right\\}}$, we
get that $H^{\prime}=H$. Hence, $(F_{\tau}(A,\ell),\sigma)$ is an $H$-module
nonlocal vertex algebra.

Similarly, one can easily checks that

(5.12) $\displaystyle\rho(\sigma(h,z)v)=\left(\sigma(h,z)\otimes
1\right)\circ\rho(v)\quad\mbox{for }h\in H,\,v\in F_{\tau}(A,\ell).$

This proves that $(H,\rho,\sigma)$ is a deforming triple for
$F_{\tau}(A,\ell)$. ∎

Let $\tau,\sigma\in\mathfrak{T}$. Combining Theorem 4.3 and Proposition 5.10,
we have the $\hbar$-adic nonlocal vertex algebra

(5.13) $\displaystyle\mathfrak{D}_{\sigma}^{\rho}(F_{\tau}(A,\ell)).$

Define $\tau\ast\sigma$ to be

$\displaystyle(\tau_{ij}(z)+\sigma_{ij}(z),\tau_{ij}^{1,\pm}(z)+\sigma_{ij}^{1,\pm}(z),\tau_{ij}^{2,\pm}(z)+\sigma_{ij}^{2,\pm}(z),\tau_{ij}^{\epsilon_{1},\epsilon_{2}}(z)\sigma_{ij}^{\epsilon_{1},\epsilon_{2}}(z))_{i,j\in
I}^{\epsilon_{1},\epsilon_{2}=\pm}.$

Then $\mathfrak{T}$ is an abelian group under $\ast$ with identity
$\varepsilon$ defined by

(5.14)
$\displaystyle\varepsilon_{ij}(z)=0=\varepsilon_{ij}^{1,\pm}(z)=\varepsilon_{ij}^{2,\pm}(z),\quad\varepsilon_{ij}^{\epsilon_{1},\epsilon_{2}}(z)=1,$

where $i,j\in I$, $\epsilon_{1},\epsilon_{2}=\pm$.

######  Lemma 5.11.

Let $\tau,\tau^{\prime},\tau^{\prime\prime}\in\mathfrak{T}$. Then the
following relations hold true on $F_{\tau}(A,\ell)$

$\displaystyle[\tau^{\prime}(h,z_{1}),\tau^{\prime\prime}(h^{\prime},z_{2})]=0\quad\mbox{for
}h,h^{\prime}\in H.$

###### Proof.

From (5.11), (5.8) and (5.9), we have that

$\displaystyle[\tau^{\prime}(a,z_{1}),\tau^{\prime\prime}(b,z_{2})]v=0.$

for $a,b\in{\left.\left\\{{\alpha_{i},e_{i}^{\pm}}\,\right|\,{i\in
I}\right\\}}$, $v\in\\{h_{i,\hbar},x_{i,\hbar}^{\pm}\,\mid\,i\in I\\}$. Let
$V$ be the ${\mathbb{C}}[[\hbar]]$-submodule of $F_{\tau}(A,\ell)$ consisting
of elements $v$ such that

$\displaystyle[\tau^{\prime}(a,z_{1}),\tau^{\prime\prime}(b,z_{2})]v=0\quad\mbox{for
}a,b=\alpha_{i},e_{i}^{\pm},\,i\in I.$

It is easy to verify that $V$ is a subalgebra of $F_{\tau}(A,\ell)$. Since
$F_{\tau}(A,\ell)$ is generated by
$\\{h_{i,\hbar},\,x_{i,\hbar}^{\pm}\,\mid\,i\in I\\}$, we get
$V=F_{\tau}(A,\ell)$. Since $H$ is a commutative $\hbar$-adic vertex algebra
generated by ${\left.\left\\{{\alpha_{i},e_{i}^{\pm}}\,\right|\,{i\in
I}\right\\}}$, we have that

$\displaystyle[\tau^{\prime}(h,z_{1}),\tau^{\prime\prime}(h^{\prime},z_{2})]v=0\quad\mbox{for
}h,h^{\prime}\in H,\,v\in F_{\tau}(A,\ell).$

Therefore, we complete the proof. ∎

It follows from Lemma 5.11 and Proposition 4.6 that

(5.15)
$\displaystyle\mathfrak{D}_{\tau^{\prime}}^{\rho}\left(\mathfrak{D}_{\tau^{\prime\prime}}^{\rho}(F_{\tau}(A,\ell))\right)=\mathfrak{D}_{\tau^{\prime}\ast\tau^{\prime\prime}}^{\rho}(F_{\tau}(A,\ell))\quad\mbox{for
}\tau,\tau^{\prime},\tau^{\prime\prime}\in\mathfrak{T}.$

Moreover, we have that

######  Proposition 5.12.

Let $\tau,\sigma\in\mathfrak{T}$. Then

$\displaystyle\mathfrak{D}_{\sigma}^{\rho}(F_{\tau}(A,\ell))=F_{\tau\ast\sigma}(A,\ell).$

Moreover, $F_{\tau}(A,\ell)/\hbar F_{\tau}(A,\ell)\cong F(A,\ell)$.

###### Proof.

From Proposition 4.6, we have that

$\displaystyle\mathfrak{D}_{\sigma}^{\rho}(Y_{\tau})(h_{i,\hbar},z)=Y_{\tau}(h_{i,\hbar},z)+\sigma_{i}(z),\quad\mathfrak{D}_{\sigma}^{\rho}(Y_{\tau})(x_{i,\hbar}^{\pm},z)=Y_{\tau}(x_{i,\hbar}^{\pm},z)\sigma_{i}^{\pm}(z).$

Using Lemma 5.8, one can easily check that

$(\mathfrak{D}_{\sigma}^{\rho}(F_{\tau}(A,\ell)),\mathfrak{D}_{\sigma}^{\rho}(Y_{\tau})(h_{i,\hbar},z),\mathfrak{D}_{\sigma}^{\rho}(Y_{\tau})(x_{i,\hbar}^{\pm},z))$

is an object of $\mathcal{M}_{\tau\ast\sigma}$. Using Proposition 5.4, we get
an $\hbar$-adic nonlocal vertex algebra homomorphism $f_{\tau,\sigma}$ from
$F_{\tau\ast\sigma}(A,\ell)$ to
$\mathfrak{D}_{\sigma}^{\rho}(F_{\tau}(A,\ell))$. From Remark 4.5 we see that
$f_{\tau,\sigma}$ is also an $\hbar$-adic nonlocal vertex algebra homomorphism
from $\mathfrak{D}_{\sigma^{-1}}^{\rho}(F_{\tau\ast\sigma}(A,\ell))$ to
$\mathfrak{D}_{\sigma^{-1}}^{\rho}\left(\mathfrak{D}_{\sigma}^{\rho}(F_{\tau}(A,\ell))\right)$.
It follows from Remark 4.4 and (5.15) that
$\mathfrak{D}_{\sigma^{-1}}^{\rho}\left(\mathfrak{D}_{\sigma}^{\rho}(F_{\tau}(A,\ell))\right)=F_{\tau}(A,\ell)$.
Then $f_{\tau,\sigma}$ becomes an $\hbar$-adic nonlocal vertex algebra
homomorphism from
$\mathfrak{D}_{\sigma^{-1}}^{\rho}(F_{\tau\ast\sigma}(A,\ell))$ to
$F_{\tau}(A,\ell)$. Replacing $\tau$ with $\tau\ast\sigma$ and replacing
$\sigma$ with $\sigma^{-1}$, we get an $\hbar$-adic nonlocal vertex algebra
homomorphism

$\displaystyle
f_{\tau\ast\sigma,\sigma^{-1}}:F_{\tau}(A,\ell)\to\mathfrak{D}_{\sigma^{-1}}^{\rho}(F_{\tau\ast\sigma}(A,\ell)).$

Hence, $f_{\tau,\sigma}\circ f_{\tau\ast\sigma,\sigma^{-1}}$ is an
$\hbar$-adic nonlocal vertex algebra endomorphism on $F_{\tau}(A,\ell)$ which
maps $a$ to $a$ for all
$a\in{\left.\left\\{{h_{i,\hbar},x_{i,\hbar}^{\pm}}\,\right|\,{i\in
I}\right\\}}$. From Proposition 5.4, we get that

(5.16)
${F_{\tau}(A,\ell)}$${\mathfrak{D}_{\sigma^{-1}}^{\rho}(F_{\tau\ast\sigma}(A,\ell))}$${F_{\tau}(A,\ell)}$$\scriptstyle{f_{\tau\ast\sigma,\sigma^{-1}}}$$\scriptstyle{f_{\tau,\sigma}}$

By replacing $\tau$ and $\sigma$ with $\tau\ast\sigma$ and $\sigma^{-1}$ in
(5.16), we get that

(5.17)
${F_{\tau\ast\sigma}(A,\ell)}$${\mathfrak{D}_{\sigma}^{\rho}(F_{\tau}(A,\ell))}$${F_{\tau\ast\sigma}(A,\ell)}$$\scriptstyle{f_{\tau,\sigma}}$$\scriptstyle{f_{\tau\ast\sigma,\sigma^{-1}}}$

Notice that

$\displaystyle\mathfrak{D}_{\sigma^{-1}}^{\rho}(F_{\tau\ast\sigma}(A,\ell))/\hbar\mathfrak{D}_{\sigma^{-1}}^{\rho}(F_{\tau\ast\sigma}(A,\ell))=F_{\tau\ast\sigma}(A,\ell)/\hbar
F_{\tau\ast\sigma}(A,\ell),$
$\displaystyle\mathfrak{D}_{\sigma}^{\rho}(F_{\tau}(A,\ell))/\hbar\mathfrak{D}_{\sigma}^{\rho}(F_{\tau}(A,\ell))=F_{\tau}(A,\ell)/\hbar
F_{\tau}(A,\ell).$

Then $f_{\tau,\sigma}$ and $f_{\tau\ast\sigma,\sigma^{-1}}$ induce the
following ${\mathbb{C}}$-linear isomorphisms

(5.18) ${F_{\tau\ast\sigma}(A,\ell)/\hbar
F_{\tau\ast\sigma}(A,\ell)}$${F_{\tau}(A,\ell)/\hbar
F_{\tau}(A,\ell)}$${\mathfrak{D}_{\sigma}^{\rho}(F_{\tau}(A,\ell))/\hbar\mathfrak{D}_{\sigma}^{\rho}(F_{\tau}(A,\ell)).}$

Since both $F_{\tau\ast\sigma}(A,\ell)$ and
$\mathfrak{D}_{\sigma}^{\rho}(F_{\tau}(A,\ell))$ are topologically free, we
get that
$f_{\tau,\sigma}:F_{\tau\ast\sigma}(A,\ell)\to\mathfrak{D}_{\sigma}^{\rho}(F_{\tau}(A,\ell))$
is an $\hbar$-adic nonlocal vertex algebra isomorphism.

From (5.14), we see that $F_{\varepsilon}(A,\ell)=F(A,\ell)[[\hbar]]$ as
$\hbar$-adic vertex algebras. Recall that $\varepsilon$ is the identity of
$\mathcal{T}$. Then by replacing $\tau$ and $\sigma$ with $\varepsilon$ and
$\tau$ in (5.18), we get that $F_{\tau}(A,\ell)/\hbar F_{\tau}(A,\ell)\cong
F(A,\ell)$. ∎

For $\tau\in\mathfrak{T}$, we note that

$\displaystyle\tau^{-1}=(-\tau_{ij}(z),-\tau_{ij}^{1,\pm}(z),-\tau_{ij}^{2,\pm}(z),\tau_{ij}^{\epsilon_{1},\epsilon_{2}}(z)^{-1})_{i,j\in
I}^{\epsilon_{1},\epsilon_{2}=\pm}.$

It is immediate from Theorem 4.7 that

######  Theorem 5.13.

For any $\tau\in\mathfrak{T}$, $F_{\tau}(A,\ell)$ is an $\hbar$-adic quantum
vertex algebra with the quantum Yang-Baxter operator $S_{\tau}(z)$ defined by

$\displaystyle S_{\tau}(z)(v\otimes
u)=\sum\tau(u_{(2)},-z)v_{(1)}\otimes\tau^{-1}(v_{(2)},z)u_{(1)}\quad\mbox{for
}u,v\in F_{\tau}(A,\ell).$

Moreover, for any $i,j\in I$ and $\epsilon_{1},\epsilon_{2}=\pm$, we have that

(5.19) $\displaystyle S_{\tau}(z)(h_{j,\hbar}\otimes
h_{i,\hbar})=h_{j,\hbar}\otimes h_{i,\hbar}+{{\bf 1}}\otimes{{\bf
1}}\otimes\left(\tau_{ij}(-z)-\tau_{ji}(z)\right),$ (5.20) $\displaystyle
S_{\tau}(z)(x_{j,\hbar}^{\pm}\otimes h_{i,\hbar})=x_{j,\hbar}^{\pm}\otimes
h_{i,\hbar}\pm x_{j,\hbar}^{\pm}\otimes{{\bf
1}}\otimes\big{(}\tau_{ij}^{1,\pm}(-z)+\tau_{ji}^{2,\pm}(z)\big{)},$ (5.21)
$\displaystyle S_{\tau}(z)(h_{j,\hbar}\otimes
x_{i,\hbar}^{\pm})=h_{j,\hbar}\otimes x_{i,\hbar}^{\pm}\mp{{\bf 1}}\otimes
x_{i,\hbar}^{\pm}\otimes\big{(}\tau_{ij}^{2,\pm}(-z)+\tau_{ji}^{1,\pm}(z)\big{)},$
(5.22) $\displaystyle S_{\tau}(z)(x_{j,\hbar}^{\epsilon_{1}}\otimes
x_{i,\hbar}^{\epsilon_{2}})=x_{j,\hbar}^{\epsilon_{1}}\otimes
x_{i,\hbar}^{\epsilon_{2}}\otimes\tau_{ji}^{\epsilon_{1},\epsilon_{2}}(z)\tau_{ij}^{\epsilon_{2},\epsilon_{1}}(-z)^{-1}.$

##  6\. Construction of quantum affine vertex algebras

Let $\ell\in{\mathbb{C}}$. In the rest of this paper, we fix a special

$\tau=(\tau_{ij}(z),\tau_{ij}^{1,\pm}(z),\tau_{ij}^{2,\pm}(z),\tau_{ij}^{\epsilon_{1},\epsilon}(z))_{i,j\in
I}^{\epsilon_{1},\epsilon_{2}=\pm}\in\mathfrak{T}$

defined as follows

(6.1)
$\displaystyle\tau_{ij}(z)=[r_{i}a_{ij}]_{q^{\frac{\partial}{\partial{z}}}}[r\ell]_{q^{\frac{\partial}{\partial{z}}}}q^{r\ell\frac{\partial}{\partial{z}}}\frac{e^{-z}}{(1-e^{-z})^{2}}-r_{i}a_{ij}r\ell
z^{-2},$ (6.2)
$\displaystyle\tau_{ij}^{1,\pm}(z)=\tau_{ij}^{2,\pm}(z)=[r_{i}a_{ij}]_{q^{\frac{\partial}{\partial{z}}}}q^{r\ell\frac{\partial}{\partial{z}}}\frac{1+e^{-z}}{2-2e^{-z}}-r_{i}a_{ij}z^{-1},$
(6.3)
$\displaystyle\tau_{ij}^{\pm,\pm}(z)=\begin{cases}\left(e^{z/2}-e^{-z/2}\right)^{-1}\left(q_{i}^{-1}e^{z/2}-q_{i}e^{-z/2}\right),&\mbox{if
}a_{ij}>0,\\\
z^{-1}\left(q_{i}^{-a_{ij}/2}e^{z/2}-q_{i}^{a_{ij}/2}e^{-z/2}\right),&\mbox{if
}a_{ij}\leq 0,\end{cases}$ (6.4)
$\displaystyle\tau_{ij}^{+,-}(z)=z^{-\delta_{ij}}(z+2r\ell\hbar)^{\delta_{ij}},$
(6.5)
$\displaystyle\tau_{ij}^{-,+}(z)=z^{-\delta_{ij}}(z-2r\ell\hbar)^{\delta_{ij}}g_{ij,\hbar}(z)^{-1},$

where $g_{ij,\hbar}(z)=(1-q_{i}^{a_{ij}}e^{-z})/(q_{i}^{a_{ij}}-e^{-z})$. Then
the equations (5.1), (5.2), (5.3), (5.4) become:

(6.6) $\displaystyle[h_{i,\hbar}(z_{1}),h_{j,\hbar}(z_{2})]$
$\displaystyle\quad=[r_{i}a_{ij}]_{q^{\frac{\partial}{\partial{z_{2}}}}}[r\ell]_{q^{\frac{\partial}{\partial{z_{2}}}}}\left(\iota_{z_{1},z_{2}}q^{-r\ell\frac{\partial}{\partial{z_{2}}}}-\iota_{z_{2},z_{1}}q^{r\ell\frac{\partial}{\partial{z_{2}}}}\right)\frac{e^{-z_{1}+z_{2}}}{(1-e^{-z_{1}+z_{2}})^{2}},$
(6.7) $\displaystyle[h_{i,\hbar}(z_{1}),x_{j,\hbar}^{\pm}(z_{2})]$
$\displaystyle\quad=\pm
x_{j,\hbar}^{\pm}(z_{2})[r_{i}a_{ij}]_{q^{\frac{\partial}{\partial{z_{2}}}}}\left(\iota_{z_{1},z_{2}}q^{-r\ell\frac{\partial}{\partial{z_{2}}}}-\iota_{z_{2},z_{1}}q^{r\ell\frac{\partial}{\partial{z_{2}}}}\right)\frac{1+e^{-z_{1}+z_{2}}}{2-2e^{-z_{1}+z_{2}}},$
(6.8)
$\displaystyle\iota_{z_{1},z_{2}}\frac{1-q_{i}^{a_{ij}}e^{-z_{1}+z_{2}}}{(1-e^{-z_{1}+z_{2}})^{\delta_{ij}}}x_{i,\hbar}^{\pm}(z_{1})x_{j,\hbar}^{\pm}(z_{2})$
$\displaystyle\quad=\iota_{z_{2},z_{1}}\frac{q_{i}^{a_{ij}}-e^{-z_{1}+z_{2}}}{(1-e^{-z_{1}+z_{2}})^{\delta_{ij}}}x_{j,\hbar}^{\pm}(z_{2})x_{i,\hbar}^{\pm}(z_{1}),$
(6.9)
$\displaystyle(z_{1}-z_{2})^{\delta_{ij}}(z_{1}-z_{2}+2r\ell\hbar)^{\delta_{ij}}$
$\displaystyle\quad\times\left(x_{i,\hbar}^{+}(z_{1})x_{j,\hbar}^{-}(z_{2})-\iota_{z_{2},z_{1}}g_{ij,\hbar}(z_{1}-z_{2})x_{j,\hbar}^{-}(z_{2})x_{i,\hbar}^{+}(z_{1})\right)=0.$

For $i,j\in I$, we set

(6.10) $\displaystyle
f_{ij,\hbar}(z)=z^{n_{ij}}\tau_{ij}^{+,+}(z)=\frac{(q_{i}^{-a_{ij}/2}e^{z/2}-q_{i}^{a_{ij}/2}e^{-z/2})}{(e^{z/2}-e^{-z/2})^{\delta_{ij}}}.$

Then the relation (6.8) is equivalent to the following equation:

(6.11)
$\displaystyle\iota_{z_{1},z_{2}}f_{ij}(z_{1}-z_{2})x_{i,\hbar}^{\pm}(z_{1})x_{j,\hbar}^{\pm}(z_{2})$
$\displaystyle=$
$\displaystyle-(-1)^{\delta_{ij}}\iota_{z_{2},z_{1}}f_{ji}(z_{2}-z_{1})x_{j,\hbar}^{\pm}(z_{2})x_{i,\hbar}^{\pm}(z_{1}).$

It is straightforward to verify that

######  Lemma 6.1.

Let $i,j\in I$. Then

(6.12)
$\displaystyle\tau_{ij}(z)-\tau_{ji}(-z)=[r_{i}a_{ij}]_{q^{\frac{\partial}{\partial{z}}}}[r\ell]_{q^{\frac{\partial}{\partial{z}}}}\left(q^{r\ell\frac{\partial}{\partial{z}}}-q^{-r\ell\frac{\partial}{\partial{z}}}\right)\frac{e^{-z}}{(1-e^{-z})^{2}},$
(6.13)
$\displaystyle\tau_{ij}^{1,+}(z)+\tau_{ji}^{2,+}(-z)=[r_{i}a_{ij}]_{q^{\frac{\partial}{\partial{z}}}}\left(q^{r\ell\frac{\partial}{\partial{z}}}-q^{-r\ell\frac{\partial}{\partial{z}}}\right)\frac{1+e^{-z}}{2-2e^{-z}}$
$\displaystyle\quad=\tau_{ij}^{1,-}(z)+\tau_{ji}^{2,-}(-z),$ (6.14)
$\displaystyle\tau_{ij}^{\pm,\pm}(z)\tau_{ji}^{\pm,\pm}(-z)^{-1}=g_{ij,\hbar}(z)=\tau_{ij}^{\pm,\mp}(z)^{-1}\tau_{ji}^{\mp,\pm}(-z).$

Moreover, we have that

######  Lemma 6.2.

Let

(6.15) $\displaystyle
F(z)=(q^{z}-q^{-z})/z\in\hbar{\mathbb{C}}[z^{2}][[\hbar]].$

Then for any $i,j\in I$, we have that

(6.16) $\displaystyle
F\left(\frac{\partial}{\partial{z}}\right)\left(\tau_{ij}(z)-\tau_{ji}(-z)\right)$
$\displaystyle\quad=\left(q^{-r\ell\frac{\partial}{\partial{z}}}-q^{r\ell\frac{\partial}{\partial{z}}}\right)\left(\tau_{ij}^{1,+}(z)+\tau_{ji}^{2,+}(-z)\right),$
(6.17) $\displaystyle
F\left(\frac{\partial}{\partial{z}}\right)\left(\tau_{ij}^{1,\pm}(z)+\tau_{ji}^{2,\pm}(-z)\right)=\mp\left(q^{r\ell\frac{\partial}{\partial{z}}}-q^{-r\ell\frac{\partial}{\partial{z}}}\right)\log\frac{\tau_{ij}^{+,\pm}(z)}{\tau_{ji}^{\pm,+}(-z)}.$

###### Proof.

From (6.13), we have that

$\displaystyle\left(q^{-r\ell\frac{\partial}{\partial{z}}}-q^{r\ell\frac{\partial}{\partial{z}}}\right)(\tau_{ij}^{1,+}(z)+\tau_{ji}^{2,+}(-z))$
$\displaystyle=$
$\displaystyle\left(q^{-\frac{\partial}{\partial{z}}}-q^{\frac{\partial}{\partial{z}}}\right)[r\ell]_{q^{\frac{\partial}{\partial{z}}}}(\tau_{ij}^{1,+}(z)+\tau_{ji}^{2,+}(-z))$
$\displaystyle=$ $\displaystyle
F\left(\frac{\partial}{\partial{z}}\right)\frac{\partial}{\partial{z}}[r\ell]_{q^{\frac{\partial}{\partial{z}}}}(\tau_{ij}^{1,+}(z)+\tau_{ji}^{2,+}(-z))$
$\displaystyle=$ $\displaystyle
F\left(\frac{\partial}{\partial{z}}\right)[r_{i}a_{ij}]_{q^{\frac{\partial}{\partial{z}}}}[r\ell]_{q^{\frac{\partial}{\partial{z}}}}\left(q^{r\ell\frac{\partial}{\partial{z}}}-q^{-r\ell\frac{\partial}{\partial{z}}}\right)\frac{e^{-z}}{(1-e^{-z})^{2}}.$

Combining this with (6.12), we prove the equation (6.16). The proof of the
equation (6.17) is similar. ∎

######  Definition 6.3.

For $\ell\in{\mathbb{C}}$, we let $R_{1}^{\ell}$ be the minimal closed ideal
of $F_{\tau}(A,\ell)$ such that $[R_{1}^{\ell}]=R_{1}^{\ell}$ and contains the
following elements

(6.18)
$\displaystyle\left(x_{i,\hbar}^{+}\right)_{0}x_{i,\hbar}^{-}-(q_{i}-q_{i}^{-1})^{-1}\left({{\bf
1}}-E(h_{i,\hbar})\right)\quad\mbox{for }i\in I,$ (6.19)
$\displaystyle\left(x_{i,\hbar}^{+}\right)_{1}x_{i,\hbar}^{-}+2r\ell\hbar(q_{i}-q_{i}^{-1})^{-1}E(h_{i,\hbar})\quad\mbox{for
}i\in I,$ (6.20)
$\displaystyle\left(x_{i,\hbar}^{\pm}\right)_{0}^{m_{ij}}x_{j,\hbar}^{\pm}\quad\mbox{for
}i,j\in I\,\mbox{with }a_{ij}<0,$

where

(6.21) $\displaystyle
E(h_{i,\hbar})=\left(\frac{F(r_{i}+r\ell)}{F(r_{i}-r\ell)}\right)^{\frac{1}{2}}\exp\left(\left(-q^{-r\ell\partial}F\left(\partial\right)h_{i,\hbar}\right)_{-1}\right){{\bf
1}}.$

Define

(6.22) $\displaystyle
V_{\hat{\mathfrak{g}},\hbar}(\ell,0)=F_{\tau}(A,\ell)/R_{1}^{\ell}.$

######  Definition 6.4.

Let $\ell\in{\mathbb{Z}}_{+}$. We let $R_{2}^{\ell}$ be the minimal closed
ideal of $V_{\hat{\mathfrak{g}},\hbar}(\ell,0)$ such that
$[R_{2}^{\ell}]=R_{2}^{\ell}$ and contains the following elements

(6.23)
$\displaystyle\left(x_{i,\hbar}^{\pm}\right)_{-1}^{r\ell/r_{i}}x_{i,\hbar}^{\pm}\quad\mbox{for
}i\in I.$

Define

(6.24) $\displaystyle
L_{\hat{\mathfrak{g}},\hbar}(\ell,0)=V_{\hat{\mathfrak{g}},\hbar}(\ell,0)/R_{2}^{\ell}.$

It is immediate from Propositions 2.3, 5.12 and Definitions 2.6, 6.3, 6.4 that

######  Proposition 6.5.

For $\ell\in{\mathbb{C}}$, there is a surjective vertex algebra homomorphism
from $V_{\hat{\mathfrak{g}}}(\ell,0)$ to
$V_{\hat{\mathfrak{g}},\hbar}(\ell,0)/\hbar
V_{\hat{\mathfrak{g}},\hbar}(\ell,0)$ such that

(6.25) $\displaystyle h_{i}\mapsto h_{i,\hbar},\quad x_{i}^{\pm}\mapsto
x_{i,\hbar}^{\pm}\quad\mbox{for }i\in I.$

If $\ell\in{\mathbb{Z}}_{+}$, then (6.25) defines a surjective vertex algebra
homomorphism from $L_{\hat{\mathfrak{g}}}(\ell,0)$ to
$L_{\hat{\mathfrak{g}},\hbar}(\ell,0)/\hbar
L_{\hat{\mathfrak{g}},\hbar}(\ell,0)$.

The main purpose of this section is to prove the following results.

######  Theorem 6.6.

Let $\ell\in{\mathbb{C}}$. Then $V_{\hat{\mathfrak{g}},\hbar}(\ell,0)$ is an
$\hbar$-adic quantum vertex algebra. Moreover, if $\ell\in{\mathbb{Z}}_{+}$,
then $L_{\hat{\mathfrak{g}},\hbar}(\ell,0)$ is also an $\hbar$-adic quantum
vertex algebra. Furthermore, the quantum Yang-Baxter operators of both
$V_{\hat{\mathfrak{g}},\hbar}(\ell,0)$ and
$L_{\hat{\mathfrak{g}},\hbar}(\ell,0)$ satisfy the relations (5.19), (5.20),
(5.21) and (5.22).

We start with some technical results.

######  Lemma 6.7.

Let $W$ be a topologically free ${\mathbb{C}}[[\hbar]]$-module,

$\beta(z)=\sum_{m\in{\mathbb{Z}}}\beta_{m}z^{-m-1}\in W((z))$

and $f(z)=\sum_{i=0}^{n}a_{i}z^{i}\in{\mathbb{C}}[z]$, such that $a_{n}=1$.
Suppose that

$\displaystyle\operatorname{Sing}_{z}z^{k}\hbar^{n}f(z/\hbar)\beta(z)=0\quad\mbox{for
some }k\in{\mathbb{Z}}.$

Then
$\beta_{m}\in\operatorname{Span}_{{\mathbb{C}}[[\hbar]]}{\left.\left\\{{\beta_{i}}\,\right|\,{k\leq
i<k+n}\right\\}}$ for all $m\geq k$. Moreover, if $\beta_{i}=0$ for all $k\leq
i<k+n$, then $\operatorname{Sing}_{z}z^{k}\beta(z)=0$.

###### Proof.

Notice that

$\displaystyle
0=\operatorname{Sing}_{z}z^{k}\hbar^{n}f(z/\hbar)\beta(z)=\sum_{m\geq
0}z^{-m-1}\sum_{i=0}^{n}a_{i}\hbar^{n-i}\beta_{m+k+i}.$

It follows that

$\displaystyle\beta_{m+n}=-\sum_{i=0}^{n-1}a_{i}\hbar^{n-i}\beta_{m+i}\quad\mbox{for
}m\geq k.$

By using induction on $m$, we complete the proof. ∎

######  Lemma 6.8.

Let $V$ be an $\hbar$-adic nonlocal vertex algebra, and let $u,v\in V$,
$f(z)=(\sum_{i=0}^{n}a_{i}z^{i})/(\sum_{j=0}^{m}b_{j}z^{j})\in{\mathbb{C}}(z)$.
Suppose

(6.26)
$\displaystyle\iota_{z_{1},z_{2},\hbar}\hbar^{n-m}f\big{(}(z_{1}-z_{2})/\hbar\big{)}Y(u,z_{1})Y(v,z_{2})\in{\mathcal{E}}_{\hbar}^{(2)}(V).$

Then we have that

$\displaystyle\operatorname{Sing}_{z}\iota_{z,\hbar}\hbar^{n-m}f(z/\hbar)Y(u,z)v=0.$

###### Proof.

Let $U={\left.\left\\{{Y(u,z)}\,\right|\,{u\in
V}\right\\}}\subset{\mathcal{E}}_{\hbar}(V)$. From (3.4), we get an
$\hbar$-adic nonlocal vertex algebra homomorphism $V\to{\langle}U{\rangle}$
defined by $u\mapsto Y(u,z)$. Then from (6.26), we have that

$\displaystyle\hbar^{n-m}f(z/\hbar)Y(Y(u,z)v,z_{2})$ $\displaystyle=$
$\displaystyle\hbar^{n-m}f\big{(}(z_{1}-z_{2})/\hbar\big{)}Y_{\mathcal{E}}\left(Y(u,z_{2}),z\right)Y(v,z_{2})$
$\displaystyle=$
$\displaystyle\left(\hbar^{n-m}f\big{(}(z_{1}-z_{2})/\hbar\big{)}Y(u,z_{1})Y(v,z_{2})\right)|_{z_{1}=z_{2}+z}.$

Notice that

$\displaystyle\left(\hbar^{n-m}f\big{(}(z_{1}-z_{2})/\hbar\big{)}Y(u,z_{1})Y(v,z_{2})\right)|_{z_{1}=z_{2}+z}$
$\displaystyle\quad\in\operatorname{Hom}(V,V((z_{2}))[[z]])+\hbar^{n}\operatorname{End}V[[z_{2}^{\pm
1},z]]\quad\mbox{for all }n\geq 0,$

and that $f(z,\hbar)Y(Y(u,z)v,z_{2}){{\bf 1}}\in V[[z,z^{-1},z_{2}]]$. It
follows that

$\displaystyle\hbar^{n-m}f(z/\hbar)Y(Y(u,z)v,z_{2}){{\bf 1}}$ $\displaystyle=$
$\displaystyle\left(\hbar^{n-m}f\big{(}(z_{1}-z_{2})/\hbar\big{)}Y(u,z_{1})Y(v,z_{2}){{\bf
1}}\right)|_{z_{1}=z_{2}+z}\in V[[z_{2},z]].$

Setting $z_{2}=0$, we get that

$\displaystyle\hbar^{n-m}f(z/\hbar)Y(u,z)v=\left(\hbar^{n-m}f(z/\hbar)Y(Y(u,z)v,z_{2}){{\bf
1}}\right)|_{z_{2}=0}\in V[[z]].$

Therefore,
$\operatorname{Sing}_{z}\iota_{z,\hbar}\hbar^{n-m}f(z/\hbar)Y(u,z)v=0$. ∎

Combining Lemmas 6.8 and equations (6.11), (6.9), we have that

######  Lemma 6.9.

In the $\hbar$-adic quantum vertex algebra $F_{\tau}(A,\ell)$, we have that

(6.27)
$\displaystyle\operatorname{Sing}_{z}(z-r_{i}a_{ij}\hbar)Y_{\tau}(x_{i,\hbar}^{\pm},z)x_{j,\hbar}^{\pm}=0,\quad\mbox{if
}a_{ij}<0,$ (6.28)
$\displaystyle\operatorname{Sing}_{z}z^{-1}(z-r_{i}a_{ii}\hbar)Y_{\tau}(x_{i,\hbar}^{\pm},z)x_{i,\hbar}^{\pm}=0,$
(6.29)
$\displaystyle\operatorname{Sing}_{z}Y_{\tau}(x_{i,\hbar}^{+},z)x_{j,\hbar}^{-}=0,\quad\mbox{if
}i\neq j,$ (6.30)
$\displaystyle\operatorname{Sing}_{z}z(z+2r\ell\hbar)Y_{\tau}(x_{i,\hbar}^{+},z)x_{i,\hbar}^{-}=0.$

We have that

######  Proposition 6.10.

For $i\in I$, we define $A_{i}(z)$ to be

$\displaystyle\operatorname{Sing}_{z}Y_{\tau}(x_{i,\hbar}^{+},z)x_{i,\hbar}^{-}-(q_{i}-q_{i}^{-1})^{-1}\left({{\bf
1}}z^{-1}-E(h_{i,\hbar})(z+2r\ell\hbar)^{-1}\right).$

Then $A_{i}(z)=0$ in $V_{\hat{\mathfrak{g}},\hbar}(\ell,0)$. Moreover, we have
that

$\displaystyle
Y_{\tau}(x_{i,\hbar}^{+},z_{1})Y_{\tau}(x_{j,\hbar}^{-},z_{2})-\iota_{z_{2},z_{1}}g_{ij,\hbar}(z_{1}-z_{2})Y_{\tau}(x_{j,\hbar}^{-},z_{2})Y_{\tau}(x_{i,\hbar}^{+},z_{1})$
(6.31) $\displaystyle=$
$\displaystyle\frac{\delta_{ij}}{q_{i}-q_{i}^{-1}}\left(z_{1}^{-1}\delta\left(\frac{z_{2}}{z_{1}}\right)-Y_{\tau}\left(E(h_{i,\hbar}),z_{2}\right)z_{1}^{-1}\delta\left(\frac{z_{2}-2r\ell\hbar}{z_{1}}\right)\right).$

###### Proof.

Let

$\displaystyle\bar{A}_{i}(z)=Y_{\tau}(x_{i,\hbar}^{+},z)x_{i,\hbar}^{-}-(q_{i}-q_{i}^{-1})^{-1}\left({{\bf
1}}z^{-1}-E(h_{i,\hbar})(z+2r\ell\hbar)^{-1}\right).$

It is easy to see that $A_{i}(z)=\operatorname{Sing}_{z}\bar{A}_{i}(z)$. From
(6.30), we have that
$\operatorname{Sing}_{z}z(z+2r\ell\hbar)\bar{A}_{i}(z)=0$. Then it follows
from Lemma 6.7 that

$\displaystyle
A_{i}(z)=\operatorname{Sing}_{z}\bar{A}_{i}(z)\in\bigoplus_{n=0}^{1}{\mathbb{C}}[z^{-1}][[\hbar]]\left(\operatorname{Res}_{z}z^{n}\bar{A}_{i}(z)\right).$

Recall from (6.18) and (6.19) that
$\operatorname{Res}_{z}z^{n}\bar{A}_{i}(z)=0$ for $n=0,1$. Therefore,
$A_{i}(z)=0$ in $V_{\hat{\mathfrak{g}},\hbar}(\ell,0)$.

From [Li6, (2.25)], we have that

$\displaystyle
z_{0}^{-1}\delta\left(\frac{z_{1}-z_{2}}{z_{0}}\right)Y_{\tau}(x_{i,\hbar}^{+},z_{1})Y_{\tau}(x_{j,\hbar}^{-},z_{2})$
$\displaystyle-
z_{0}^{-1}\delta\left(\frac{z_{2}-z_{1}}{-z_{0}}\right)\iota_{z_{2},z_{1}}g_{ij,\hbar}(z_{1}-z_{2})Y_{\tau}(x_{j,\hbar}^{-},z_{2})Y_{\tau}(x_{i,\hbar}^{+},z_{1})$
$\displaystyle=$ $\displaystyle
z_{1}^{-1}\delta\left(\frac{z_{2}+z_{0}}{z_{1}}\right)Y_{\tau}\left(Y_{\tau}(x_{i,\hbar}^{+},z_{0})x_{j,\hbar}^{-},z_{2}\right).$

Taking $\operatorname{Res}_{z_{0}}$, we get that

$\displaystyle
Y_{\tau}(x_{i,\hbar}^{+},z_{1})Y_{\tau}(x_{j,\hbar}^{-},z_{2})-\iota_{z_{2},z_{1}}g_{ij,\hbar}(z_{1}-z_{2})Y_{\tau}(x_{j,\hbar}^{-},z_{2})Y_{\tau}(x_{i,\hbar}^{+},z_{1})$
$\displaystyle=$
$\displaystyle\operatorname{Res}_{z_{0}}z_{1}^{-1}\delta\left(\frac{z_{2}+z_{0}}{z_{1}}\right)Y_{\tau}\left(Y_{\tau}(x_{i,\hbar}^{+},z_{0})x_{j,\hbar}^{-},z_{2}\right)$
$\displaystyle=$
$\displaystyle\operatorname{Res}_{z_{0}}z_{1}^{-1}\delta\left(\frac{z_{2}+z_{0}}{z_{1}}\right)Y_{\tau}\left(\operatorname{Sing}_{z_{0}}Y_{\tau}(x_{i,\hbar}^{+},z_{0})x_{j,\hbar}^{-},z_{2}\right)$
$\displaystyle=$
$\displaystyle\operatorname{Res}_{z_{0}}z_{1}^{-1}\delta\left(\frac{z_{2}+z_{0}}{z_{1}}\right)(q_{i}-q_{i}^{-1})^{-1}\left(z_{0}^{-1}-Y_{\tau}\left(E(h_{i,\hbar}),z_{2}\right)(z_{0}+2r\ell\hbar)^{-1}\right)$
$\displaystyle=$
$\displaystyle\delta_{ij}(q_{i}-q_{i}^{-1})^{-1}\left(z_{1}^{-1}\delta\left(\frac{z_{2}}{z_{1}}\right)-Y_{\tau}\left(E(h_{i,\hbar}),z_{2}\right)z_{1}^{-1}\delta\left(\frac{z_{2}-2r\ell\hbar}{z_{1}}\right)\right),$

where the third equation follows from the first statement. ∎

For any positive integer $k$ and $i_{1},\dots,i_{k}\in I$, we set

(6.32) $\displaystyle Y_{i_{1},\dots,i_{k}}^{\pm}(z_{1},\dots,z_{k})$
$\displaystyle=$ $\displaystyle\left(\prod_{1\leq a<b\leq
k}f_{i_{a},i_{b},\hbar}(z_{a}-z_{b})\right)Y_{\tau}(x_{i_{1},\hbar}^{\pm},z_{1})Y_{\tau}(x_{i_{2},\hbar}^{\pm},z_{2})\cdots
Y_{\tau}(x_{i_{k},\hbar}^{\pm},z_{k}).$

It is immediate from equations (6.11) and (6.9) that

######  Lemma 6.11.

For any positive integer $k$, $i_{1},\dots,i_{k}\in I$ and $\sigma\in S_{k}$,
we have that

(6.33) $\displaystyle
Y_{i_{\sigma(1)},\dots,i_{\sigma(k)}}^{\pm}(z_{\sigma(1)},\dots,z_{\sigma(k)})=\left(\prod_{\begin{subarray}{c}1\leq
a<b\leq k\\\
\sigma(a)>\sigma(b)\end{subarray}}C_{i_{a},i_{b}}\right)Y_{i_{1},\dots,i_{k}}^{\pm}(z_{1},\dots,z_{k}),$

where

(6.34) $\displaystyle C_{ij}=-(-1)^{\delta_{ij}}.$

Moreover,
$Y_{i_{1},\dots,i_{k}}^{\pm}(z_{1},\dots,z_{k})\in{\mathcal{E}}_{\hbar}^{(k)}(F_{\tau}(A,\ell))$.

######  Lemma 6.12.

Let $i,j\in I$ with $a_{ij}<0$ and $k\in{\mathbb{N}}$. Then in
$F_{\tau}(A,\ell)$, we have that

(6.35)
$\displaystyle\operatorname{Sing}_{z_{1},\dots,z_{k}}Y_{\tau}(x_{i,\hbar}^{\pm},z_{1})\cdots
Y_{\tau}(x_{i,\hbar}^{\pm},z_{k})x_{j,\hbar}^{\pm}$
$\displaystyle\quad\in{\mathbb{C}}[z_{1}^{-1},\dots,z_{k}^{-1}][[\hbar]]\left(\left(x_{i,\hbar}^{\pm}\right)_{0}^{k}x_{j,\hbar}^{\pm}\right).$

Moreover, we define $\bar{Y}_{ij,k}^{\pm}(z)$ to be

$\displaystyle
Y_{i,\dots,i,j}^{\pm}\left(z+r_{i}\big{(}(k-1)a_{ii}+a_{ij}\big{)}\hbar,z+r_{i}\big{(}(k-2)a_{ii}+a_{ij}\big{)}\hbar,\dots,z+r_{i}a_{ij}\hbar,z\right).$

Then we have that

(6.36) $\displaystyle
Y_{\tau}\left(\left(x_{i,\hbar}^{\pm}\right)_{0}^{k}x_{j,\hbar}^{\pm},z\right)=c_{k}\bar{Y}_{ij,k}^{\pm}(z)\quad\mbox{for
some }c_{k}\in{\mathbb{C}}[[\hbar]]^{\times}.$

###### Proof.

We prove the lemma by using induction on $k$. From equations (6.11) and (6.9),
we have that

$\displaystyle\left(\prod_{1\leq a\leq
k}f_{ii,\hbar}\left(z_{1}-z_{2}-r_{i}\big{(}(a-1)a_{ii}+a_{ij}\big{)}\hbar\right)\right)f_{ij,\hbar}(z_{1}-z_{2})Y_{\tau}\left(x_{i,\hbar}^{\pm},z_{1}\right)\bar{Y}_{ij,k}^{\pm}(z_{2})$
$\displaystyle=$ $\displaystyle
Y_{i,\dots,i,j}^{\pm}\left(z_{1},z_{2}+r_{i}((k-1)a_{ii}+a_{ij})\hbar,z_{2}+r_{i}((k-2)a_{ii}+a_{ij})\hbar,\dots,z_{2}+r_{i}a_{ij}\hbar,z_{2}\right)$

lies in ${\mathcal{E}}_{\hbar}^{(2)}\left(F_{\tau}(A,\ell)\right)$. Notice
that

$\displaystyle f_{ij,\hbar}(z_{1}-z_{2})\prod_{1\leq a\leq
k}f_{ii,\hbar}\left(z_{1}-z_{2}-r_{i}\big{(}(a-1)a_{ii}+a_{ij}\big{)}\hbar\right)$
$\displaystyle\quad=q_{i}^{-k-a_{ij}/2}e^{(z_{1}-z_{2})/2}-q_{i}^{k+a_{ij}/2}e^{-(z_{1}-z_{2})/2}.$

and that
$\left(q_{i}^{-k-a_{ij}/2}e^{(z_{1}-z_{2})/2}-q_{i}^{k+a_{ij}/2}e^{-(z_{1}-z_{2})/2}\right)/(z_{1}-z_{2}-r_{i}(ka_{ii}+a_{ij})\hbar)\in{\mathbb{C}}[[z_{1},z_{2}]]^{\times}$.
Then

$\displaystyle\left(z_{1}-z_{2}-r_{i}(ka_{ii}+a_{ij}\hbar)\right)Y_{\tau}\left(x_{i,\hbar}^{\pm},z_{1}\right)\bar{Y}_{ij,k}^{\pm}(z_{2})$
$\displaystyle=$ $\displaystyle
d_{k}(z_{1}-z_{2})Y_{i,\dots,i,j}^{\pm}\big{(}z_{1},z_{2}+r_{i}((k-1)a_{ii}+a_{ij})\hbar,$
$\displaystyle\qquad
z_{2}+r_{i}((k-2)a_{ii}+a_{ij})\hbar,\dots,z_{2}+r_{i}a_{ij}\hbar,z_{2}\big{)}$

lies in ${\mathcal{E}}_{\hbar}^{(2)}(F_{\tau}(A,\ell))$ for some
$d_{k}(z)\in{\mathbb{C}}[[z,\hbar]]^{\times}$. From induction assumptions
(6.35), (6.36) and Lemmas 6.7, 6.8, we complete the proof of (6.35) for $k+1$.

From induction assumption (6.36) and (3.4), we have that

$\displaystyle(z-r_{i}(ka_{ii}+a_{ij})\hbar)Y\left(Y\left(x_{i,\hbar}^{\pm},z\right)\left(\left(x_{i,\hbar}^{\pm}\right)_{0}^{k}x_{j,\hbar}^{\pm}\right),z_{2}\right)$
$\displaystyle=$ $\displaystyle
c_{k}(z-r_{i}(ka_{ii}+a_{ij})\hbar)Y_{\mathcal{E}}\left(Y(x_{i,\hbar}^{\pm},z_{2}),z\right)\bar{Y}_{ij,k}^{\pm}(z_{2})$
$\displaystyle=$ $\displaystyle
c_{k}\left.\left(\left(z_{1}-z-r_{i}(ka_{ii}+a_{ij}\hbar)\right)Y_{\tau}\left(x_{i,\hbar}^{\pm},z_{1}\right)\bar{Y}_{ij,k}^{\pm}(z_{2})\right)\right|_{z_{1}=z_{2}+z}$
$\displaystyle=$ $\displaystyle
c_{k}d_{k}(z)Y_{i,\dots,i,j}^{\pm}\big{(}z_{2}+z,z_{2}+r_{i}((k-1)a_{ii}+a_{ij})\hbar,$
$\displaystyle\qquad
z_{2}+r_{i}((k-2)a_{ii}+a_{ij})\hbar,\dots,z_{2}+r_{i}a_{ij}\hbar,z_{2}\big{)}.$

By multiplying $(z-r_{i}(ka_{ii}+a_{ij})\hbar)^{-1}$ and taking
$\operatorname{Sing}_{z}$ on both hand sides, we get that

(6.37) $\displaystyle
Y\left(\operatorname{Sing}_{z}Y\left(x_{i,\hbar}^{\pm},z\right)\left(x_{i,\hbar}^{\pm}\right)_{0}^{k}x_{j,\hbar}^{\pm},z_{2}\right)$
$\displaystyle=$
$\displaystyle(z-r_{i}(ka_{ii}+a_{ij})\hbar)^{-1}c_{k+1}\bar{Y}_{ij,k+1}^{\pm}(z_{2}),$

where $c_{k+1}=c_{k}d_{k}(r_{i}(ka_{ii}+a_{ij})\hbar)$. Then from the
induction assumption (6.35), we complete the proof of (6.35) for $k+1$. Notice
that both $c_{k}$ and $d_{k}(r_{i}(ka_{ii}+a_{ij})\hbar)$ are invertible in
${\mathbb{C}}[[\hbar]]$. We have that $c_{k+1}$ is invertible. By taking
$\operatorname{Res}_{z}$ on both hand sides of (6.37), we complete the proof
of (6.36) for $k+1$. Therefore, we complete the proof of the lemma. ∎

Similarly, we have that

######  Lemma 6.13.

Let $i\in I$ and $k\in{\mathbb{N}}$. Then in $F_{\tau}(A,\ell)$, we have that

(6.38) $\displaystyle\operatorname{Sing}_{z_{1},\dots,z_{k}}z_{1}^{-1}\cdots
z_{k}^{-1}Y_{\tau}(x_{i,\hbar}^{\pm},z_{1})\cdots
Y_{\tau}(x_{i,\hbar}^{\pm},z_{k})x_{i,\hbar}^{\pm}$
$\displaystyle\quad\in{\mathbb{C}}[z_{1}^{-1},\dots,z_{k}^{-1}][[\hbar]]\left(\left(x_{i,\hbar}^{\pm}\right)_{-1}^{k}x_{i,\hbar}^{\pm}\right).$

Moreover, there is $c_{k}^{\prime}\in{\mathbb{C}}[[\hbar]]^{\times}$, such
that

$\displaystyle
Y_{\tau}\left(\left(x_{i,\hbar}^{\pm}\right)_{-1}^{k}x_{i,\hbar}^{\pm},z\right)=c_{k}^{\prime}Y_{i,\dots,i}^{\pm}\left(z+kr_{i}a_{ii}\hbar,z+(k-1)r_{i}a_{ii}\hbar,\dots,z\right).$

It is immediate from Lemma 6.12 that

######  Proposition 6.14.

For $i,j\in I$ with $a_{ij}<0$, we let

$\displaystyle Q_{ij}^{\pm}(z_{1},\dots,z_{m_{ij}})$ $\displaystyle=$
$\displaystyle\operatorname{Sing}_{z_{1},z_{2},\dots,z_{m_{ij}}}Y_{\tau}(x_{i,\hbar}^{\pm},z_{1})Y_{\tau}(x_{i,\hbar}^{\pm},z_{2})\cdots
Y_{\tau}(x_{i,\hbar}^{\pm},z_{m_{ij}})x_{j,\hbar}^{\pm}.$

Then $Q_{ij}^{\pm}(z_{1},\dots,z_{m_{ij}})=0$ in
$V_{\hat{\mathfrak{g}},\hbar}(\ell,0)$.

And it is immediate from Lemma 6.13 that

######  Proposition 6.15.

For $\ell\in{\mathbb{Z}}_{+}$, we let

$\displaystyle M_{i}^{\pm}(z_{1},\dots,z_{r\ell/r_{i}})$ $\displaystyle=$
$\displaystyle\operatorname{Sing}_{z_{1},z_{2},\dots,z_{r\ell/r_{i}}}z_{1}^{-1}\cdots
z_{r\ell/r_{i}}^{-1}Y_{\tau}(x_{i,\hbar}^{\pm},z_{1})Y_{\tau}(x_{i,\hbar}^{\pm},z_{2})\cdots
Y_{\tau}(x_{i,\hbar}^{\pm},z_{r\ell/r_{i}})x_{i,\hbar}^{\pm}.$

Then $M_{i}^{\pm}(z_{1},\dots,z_{r\ell/r_{i}})=0$ in
$L_{\hat{\mathfrak{g}},\hbar}(\ell,0)$.

Combining Propositions 6.10, 6.14 and 6.15, we have that

######  Proposition 6.16.

$R_{1}^{\ell}$ is the minimal closed ideal of $F_{\tau}(A,\ell)$ such that
$[R_{1}^{\ell}]=R_{1}^{\ell}$, and contains all coefficients of $A_{i}(z)$ for
$i\in I$ and all coefficients of $Q_{ij}^{\pm}(z_{1},\dots,z_{m_{ij}})$ for
$i,j\in I$ with $a_{ij}<0$. Moreover, if $\ell\in{\mathbb{Z}}_{+}$, then
$R_{2}^{\ell}$ is the minimal closed ideal of
$V_{\hat{\mathfrak{g}},\hbar}(\ell,0)$ such that $[R_{2}^{\ell}]=R_{2}^{\ell}$
and contains all coefficients of $M_{i}^{\pm}(z_{1},\dots,z_{r\ell/r_{i}})$
for $i\in I$.

Theorem 6.6 is immediate from Lemma 3.3, Proposition 6.16 and the following
three technical results.

######  Lemma 6.17.

For $i,j\in I$, we have that

$\displaystyle S_{\tau}(z_{1})(A_{i}(z_{2})\otimes h_{j,\hbar})$
$\displaystyle=$ $\displaystyle A_{i}(z_{2})\otimes
h_{j,\hbar}+\operatorname{Sing}_{z_{2}}\left(A_{i}(z_{2})\otimes{{\bf
1}}\otimes\left(e^{z_{2}\frac{\partial}{\partial{z_{1}}}}-1\right)\left(\tau_{ij}^{1,+}(-z_{1})+\tau_{ji}^{2,+}(z_{1})\right)\right),$
$\displaystyle S_{\tau}(z_{1})(h_{j,\hbar}\otimes A_{i}(z_{2}))$
$\displaystyle=$ $\displaystyle h_{j,\hbar}\otimes
A_{i}(z_{2})+\operatorname{Sing}_{z_{2}}\left({{\bf 1}}\otimes
A_{i}(z_{2})\otimes\left(1-e^{-z_{2}\frac{\partial}{\partial{z_{1}}}}\right)\left(\tau_{ji}^{2,+}(-z_{1})+\tau_{ij}^{1,+}(z_{1})\right)\right),$
$\displaystyle S_{\tau}(z_{1})(A_{i}(z_{2})\otimes x_{j,\hbar}^{\pm})$
$\displaystyle=$
$\displaystyle\operatorname{Sing}_{z_{2}}\left(A_{i}(z_{2})\otimes
x_{j,\hbar}^{\pm}\otimes\exp\left(\left(e^{z_{2}\frac{\partial}{\partial{z_{1}}}}-1\right)\log\frac{\tau_{ij}^{+,\pm}(z_{1})}{\tau_{ji}^{\pm,+}(-z_{1})}\right)\right),$
$\displaystyle S_{\tau}(z_{1})(x_{j,\hbar}^{\pm}\otimes A_{i}(z_{2}))$
$\displaystyle=$
$\displaystyle\operatorname{Sing}_{z_{2}}\left(x_{j,\hbar}^{\pm}\otimes
A_{i}(z_{2})\otimes\exp\left(\left(1-e^{-z_{2}\frac{\partial}{\partial{z_{1}}}}\right)\log\frac{\tau_{ij}^{+,\pm}(-z_{1})}{\tau_{ji}^{\pm,+}(z_{1})}\right)\right).$

###### Proof.

From the relations (3.12), (5.20) and (6.13), we have that

$\displaystyle
S_{\tau}(z_{1})\left(\operatorname{Sing}_{z_{2}}Y_{\tau}(x_{i,\hbar}^{+},z_{2})x_{i,\hbar}^{-}\otimes
h_{j,\hbar}\right)$ $\displaystyle=$
$\displaystyle\operatorname{Sing}_{z_{2}}Y_{\tau}^{12}(z_{2})S_{\tau}^{23}(z_{1})S_{\tau}^{13}(z_{1}+z_{2})(x_{i,\hbar}^{+}\otimes
x_{i,\hbar}^{-}\otimes h_{j,\hbar})$ $\displaystyle=$
$\displaystyle\operatorname{Sing}_{z_{2}}Y_{\tau}^{12}(z_{2})S_{\tau}^{23}(z_{1})\Big{(}x_{i,\hbar}^{+}\otimes
x_{i,\hbar}^{-}\otimes h_{j,\hbar}$ $\displaystyle\quad+x_{i,\hbar}^{+}\otimes
x_{i,\hbar}^{-}\otimes{{\bf
1}}\otimes\otimes\left(\tau_{ji}^{1,+}(-z_{1}-z_{2})+\tau_{ij}^{2,+}(z_{1}+z_{2})\right)\Big{)}$
$\displaystyle=$
$\displaystyle\operatorname{Sing}_{z_{2}}Y_{\tau}^{12}(z_{2})\Big{(}x_{i,\hbar}^{+}\otimes
x_{i,\hbar}^{-}\otimes h_{j,\hbar}$ $\displaystyle\quad+x_{i,\hbar}^{+}\otimes
x_{i,\hbar}^{-}\otimes{{\bf
1}}\otimes\left(e^{z_{2}\frac{\partial}{\partial{z_{1}}}}-1\right)\left(\tau_{ji}^{1,+}(-z_{1})+\tau_{ij}^{2,+}(z_{1})\right)\Big{)}$
$\displaystyle=$
$\displaystyle\operatorname{Sing}_{z_{2}}Y_{\tau}(x_{i,\hbar}^{+},z_{2})x_{i,\hbar}^{-}\otimes
h_{j,\hbar}+\operatorname{Sing}_{z_{2},z}$
$\displaystyle\quad\left(Y_{\tau}(x_{i,\hbar}^{+},z_{2})x_{i,\hbar}^{-}\otimes{{\bf
1}}\otimes\left(e^{z_{2}\frac{\partial}{\partial{z_{1}}}}-1\right)\left(\tau_{ji}^{1,+}(-z_{1})+\tau_{ij}^{2,+}(z_{1})\right)\right).$

From the relations (3.16), (5.19) and (6.16), we have that

$\displaystyle
S_{\tau}(z)\left(q^{-r\ell\partial}F(\partial)h_{i,\hbar}\otimes
h_{j,\hbar}\right)$ $\displaystyle=$ $\displaystyle q^{-r\ell\partial\otimes
1-r\ell\frac{\partial}{\partial{z}}}F\left(\partial\otimes
1+\frac{\partial}{\partial{z}}\right)S(z)(h_{i,\hbar}\otimes h_{j,\hbar})$
$\displaystyle=$ $\displaystyle q^{-r\ell\partial\otimes
1-r\ell\frac{\partial}{\partial{z}}}F\left(\partial\otimes
1+\frac{\partial}{\partial{z}}\right)\left(h_{i,\hbar}\otimes
h_{j,\hbar}+{{\bf 1}}\otimes{{\bf
1}}\otimes\left(\tau_{ji}(-z)-\tau_{ij}(z)\right)\right)$ $\displaystyle=$
$\displaystyle q^{-r\ell\partial}F(\partial)h_{i,\hbar}\otimes
h_{j,\hbar}+{{\bf 1}}\otimes{{\bf 1}}\otimes
q^{-r\ell\frac{\partial}{\partial{z}}}F\left(\frac{\partial}{\partial{z}}\right)\left(\tau_{ji}(-z)-\tau_{ij}(z)\right)$
$\displaystyle=$ $\displaystyle
q^{-r\ell\partial}F(\partial)h_{i,\hbar}\otimes h_{j,\hbar}+{{\bf
1}}\otimes{{\bf
1}}\otimes\left(1-q^{-2r\ell\frac{\partial}{\partial{z}}}\right)\left(\tau_{ji}^{1,+}(-z)+\tau_{ij}^{2,+}(z)\right).$

Then by using Lemma 3.2, we get that

$\displaystyle
S_{\tau}(z)\left(\exp\left(\left(-q^{-r\ell\partial}F(\partial)h_{i,\hbar}\right)_{-1}\right){{\bf
1}}\otimes h_{j,\hbar}\right)$ $\displaystyle=$
$\displaystyle\exp\left(\left(-q^{-r\ell\partial}F(\partial)h_{i,\hbar}\right)_{-1}\right){{\bf
1}}\otimes h_{j,\hbar}$
$\displaystyle+\exp\left(\left(-q^{-r\ell\partial}F(\partial)h_{i,\hbar}\right)_{-1}\right){{\bf
1}}\otimes{{\bf
1}}\otimes\left(q^{-2r\ell\frac{\partial}{\partial{z}}}-1\right)\left(\tau_{ji}^{1,+}(-z)+\tau_{ij}^{2,+}(z)\right).$

Hence, we have that

$\displaystyle S_{\tau}(z_{1})\left(A_{i}(z_{2})\otimes
h_{j,\hbar}\right)=\operatorname{Sing}_{z_{2}}Y_{\tau}(x_{i,\hbar}^{+},z_{2})x_{i,\hbar}^{-}\otimes
h_{j,\hbar}+\operatorname{Sing}_{z_{2}}$
$\displaystyle\quad\left(\operatorname{Sing}_{z_{2}}Y_{\tau}(x_{i,\hbar}^{+},z_{2})x_{i,\hbar}^{-}\otimes{{\bf
1}}\otimes\left(e^{z_{2}\frac{\partial}{\partial{z_{1}}}}-1\right)\big{(}\tau_{ji}^{1,+}(-z_{1})+\tau_{ij}^{2,+}(z_{1})\big{)}\right)$
$\displaystyle\quad-\frac{1}{q_{i}-q_{i}^{-1}}\left({{\bf 1}}\otimes
h_{j,\hbar}z_{2}^{-1}-E(h_{i,\hbar})\otimes
h_{j,\hbar}(z_{2}+2r\ell\hbar)^{-1}\right)$
$\displaystyle\quad+\frac{(z_{2}+2r\ell\hbar)^{-1}}{q_{i}-q_{i}^{-1}}E(h_{i,\hbar})\otimes{{\bf
1}}\otimes\left(q^{-2r\ell\frac{\partial}{\partial{z_{1}}}}-1\right)\big{(}\tau_{ji}^{1,+}(-z_{1})+\tau_{ij}^{2,+}(z_{1})\big{)}$
(6.39) $\displaystyle=$ $\displaystyle A_{i}(z_{2})\otimes
h_{j,\hbar}+\operatorname{Sing}_{z_{2}}A_{i}(z_{2})\otimes{{\bf
1}}\otimes\left(e^{z_{2}\frac{\partial}{\partial{z_{1}}}}-1\right)\big{(}\tau_{ji}^{1,+}(-z_{1})+\tau_{ij}^{2,+}(z_{1})\big{)}$
$\displaystyle\quad+\frac{1}{q_{i}-q_{i}^{-1}}\operatorname{Sing}_{z_{2}}\left({{\bf
1}}\otimes{{\bf 1}}\otimes
z_{2}^{-1}\left(e^{z_{2}\frac{\partial}{\partial{z_{1}}}}-1\right)\big{(}\tau_{ji}^{1,+}(-z_{1})+\tau_{ij}^{2,+}(z_{1})\big{)}\right)$
$\displaystyle\quad-\frac{1}{q_{i}-q_{i}^{-1}}\operatorname{Sing}_{z_{2}}\left(E(h_{i,\hbar})\otimes{{\bf
1}}\otimes\frac{e^{z_{2}\frac{\partial}{\partial{z_{1}}}}-1}{z_{2}+2r\ell\hbar}\big{(}\tau_{ji}^{1,+}(-z_{1})+\tau_{ij}^{2,+}(z_{1})\big{)}\right)$
$\displaystyle\quad+\frac{(z_{2}+2r\ell\hbar)^{-1}}{q_{i}-q_{i}^{-1}}E(h_{i,\hbar})\otimes{{\bf
1}}\otimes\left(q^{-2r\ell\frac{\partial}{\partial{z_{1}}}}-1\right)\big{(}\tau_{ji}^{1,+}(-z_{1})+\tau_{ij}^{2,+}(z_{1})\big{)}.$

Notice that

$\displaystyle\operatorname{Sing}_{z}z^{-1}\left(e^{z\frac{\partial}{\partial{x}}}-1\right)=0,$
$\displaystyle\operatorname{Sing}_{z}(z+2r\ell\hbar)^{-1}\left(e^{z\frac{\partial}{\partial{x}}}-1\right)=\left(q^{-2r\ell\frac{\partial}{\partial{x}}}-1\right)(z+2r\ell\hbar)^{-1}.$

Combining these equations and (6), we get that

$\displaystyle S_{\tau}(z_{1})\left(A_{i}(z_{2})\otimes h_{j,\hbar}\right)$
$\displaystyle=$ $\displaystyle A_{i}(z_{2})\otimes
h_{j,\hbar}+\operatorname{Sing}_{z_{2}}A_{i}(z_{2})\otimes{{\bf
1}}\otimes\left(e^{z_{2}\frac{\partial}{\partial{z_{1}}}}-1\right)\left(\tau_{ji}^{1,+}(-z_{1})+\tau_{ij}^{2,+}(z_{1})\right).$

Therefore, we complete the proof of the first equation. The proof of the rest
equations are similar. ∎

Similar to the proof of Lemma 6.17, we have the following two results.

######  Lemma 6.18.

For $i,j,k\in I$ such that $a_{ij}<0$, we have that

$\displaystyle S_{\tau}(z)\left(Q_{ij}^{\pm}(z_{1},\dots,z_{m_{ij}})\otimes
h_{k,\hbar}\right)$ $\displaystyle=$ $\displaystyle
Q_{ij}^{\pm}(z_{1},\dots,z_{m_{ij}})\otimes
h_{k,\hbar}\pm\operatorname{Sing}_{z_{1},\dots,z_{m_{ij}}}\Bigg{(}Q_{ij}^{\pm}(z_{1},\dots,z_{m_{ij}})\otimes{{\bf
1}}$
$\displaystyle\quad\otimes\left(\tau_{kj}^{1,\pm}(-z)+\tau_{jk}^{2,\pm}(z)+\sum_{a=1}^{m_{ij}}\left(\tau_{ki}^{1,\pm}(-z-z_{a})+\tau_{ik}^{2,\pm}(z+z_{a})\right)\right)\Bigg{)},$
$\displaystyle S_{\tau}(z)\left(h_{k,\hbar}\otimes
Q_{ij}^{\pm}(z_{1},\dots,z_{m_{ij}})\right)$ $\displaystyle=$ $\displaystyle
h_{k,\hbar}\otimes
Q_{ij}^{\pm}(z_{1},\dots,z_{m_{ij}})\mp\operatorname{Sing}_{z_{1},\dots,z_{m_{ij}}}\Bigg{(}{{\bf
1}}\otimes Q_{ij}^{\pm}(z_{1},\dots,z_{m_{ij}})$
$\displaystyle\quad\otimes\left(\tau_{jk}^{2,\pm}(-z)+\tau_{kj}^{1,\pm}(z)+\sum_{a=1}^{m_{ij}}\left(\tau_{ik}^{2,\pm}(-z+z_{a})+\tau_{ki}^{1,\pm}(z-z_{a})\right)\right)\Bigg{)},$
$\displaystyle S_{\tau}(z)\left(Q_{ij}^{\pm}(z_{1},\dots,z_{m_{ij}})\otimes
x_{k,\hbar}^{\epsilon}\right)=\operatorname{Sing}_{z_{1},\dots,z_{m_{ij}}}\Bigg{(}Q_{ij}^{\pm}(z_{1},\dots,z_{m_{ij}})\otimes
x_{k,\hbar}^{\epsilon}$
$\displaystyle\quad\otimes\tau_{kj}^{\epsilon,\pm}(-z)^{-1}\tau_{jk}^{\pm,\epsilon}(z)\prod_{a=1}^{m_{ij}}\tau_{ki}^{\epsilon,\pm}(-z-z_{a})^{-1}\tau_{ik}^{\pm,\epsilon}(z+z_{a})\Bigg{)},$
$\displaystyle S_{\tau}(z)\left(x_{k,\hbar}^{\epsilon}\otimes
Q_{ij}^{\pm}(z_{1},\dots,z_{m_{ij}})\right)=\operatorname{Sing}_{z_{1},\dots,z_{m_{ij}}}\Bigg{(}x_{k,\hbar}^{\epsilon}\otimes
Q_{ij}^{\pm}(z_{1},\dots,z_{m_{ij}})$
$\displaystyle\quad\otimes\tau_{jk}^{\epsilon,\pm}(-z)^{-1}\tau_{kj}^{\pm,\epsilon}(z)\prod_{a=1}^{m_{ij}}\tau_{ik}^{\epsilon,\pm}(-z+z_{a})^{-1}\tau_{ki}^{\pm,\epsilon}(z-z_{a})\Bigg{)}.$

######  Lemma 6.19.

For any $\ell\in{\mathbb{Z}}_{+}$ and $i,j\in I$, we have that

$\displaystyle
S_{\tau}(z)\left(M_{i}^{\pm}(z_{1},\dots,z_{r\ell/r_{i}})\otimes
h_{j,\hbar}\right)=M_{i}^{\pm}(z_{1},\dots,z_{r\ell/r_{i}})\otimes
h_{j,\hbar}$
$\displaystyle\quad\pm\operatorname{Sing}_{z_{1},\dots,z_{r\ell/r_{i}}}z_{1}^{-1}\cdots
z_{r\ell/r_{i}}^{-1}$
$\displaystyle\quad\Bigg{(}M_{i}^{\pm}(z_{1},\dots,z_{r\ell/r_{i}})\otimes{{\bf
1}}\otimes\sum_{a=1}^{r\ell/r_{i}+1}\left(\tau_{ji}^{1,\pm}(-z-z_{a})+\tau_{ij}^{2,\pm}(z+z_{a})\right)\Bigg{)},$
$\displaystyle S_{\tau}(z)\left(h_{j,\hbar}\otimes
M_{i}^{\pm}(z_{1},\dots,z_{r\ell/r_{i}})\right)=h_{j,\hbar}\otimes
M_{i}^{\pm}(z_{1},\dots,z_{r\ell/r_{i}})$
$\displaystyle\quad\mp\operatorname{Sing}_{z_{1},\dots,z_{r\ell/r_{i}}}z_{1}^{-1}\cdots
z_{r\ell/r_{i}}^{-1}$ $\displaystyle\quad\left({{\bf 1}}\otimes
M_{i}^{\pm}(z_{1},\dots,z_{r\ell/r_{i}})\otimes\sum_{a=1}^{r\ell/r_{i}+1}\left(\tau_{ij}^{1,\pm}(-z+z_{a})+\tau_{ji}^{2,\pm}(z-z_{a})\right)\right),$
$\displaystyle
S_{\tau}(z)\left(M_{i}^{\pm}(z_{1},\dots,z_{r\ell/r_{i}})\otimes
x_{j,\hbar}^{\epsilon}\right)=\operatorname{Sing}_{z_{1},\dots,z_{r\ell/r_{i}}}z_{1}^{-1}\cdots
z_{r\ell/r_{i}}^{-1}$
$\displaystyle\quad\left(M_{i}^{\pm}(z_{1},\dots,z_{r\ell/r_{i}})\otimes
x_{j,\hbar}^{\epsilon}\otimes\prod_{a=1}^{r\ell/r_{i}+1}\tau_{ji}^{\epsilon,\pm}(-z-z_{a})^{-1}\tau_{ij}^{\pm,\epsilon}(z+z_{a})\right),$
$\displaystyle S_{\tau}(z)\left(x_{j,\hbar}^{\epsilon}\otimes
M_{i}^{\pm}(z_{1},\dots,z_{r\ell/r_{i}})\right)=\operatorname{Sing}_{z_{1},\dots,z_{r\ell/r_{i}}}z_{1}^{-1}\cdots
z_{r\ell/r_{i}}^{-1}$ $\displaystyle\quad\left(x_{j,\hbar}^{\epsilon}\otimes
M_{i}^{\pm}(z_{1},\dots,z_{r\ell/r_{i}})\otimes\prod_{a=1}^{r\ell/r_{i}+1}\tau_{ij}^{\epsilon,\pm}(-z+z_{a})^{-1}\tau_{ji}^{\pm,\epsilon}(z-z_{a})\right),$

where $z_{r\ell/r_{i}+1}=0$.

##  7\. $\phi$-coordinated modules

In this section, we recall the construction of $\hbar$-adic nonlocal vertex
algebras and their $\phi$-coordinated modules introduced in [Li7].

First, we fix an associate $\phi(z_{1},z)=z_{1}e^{z}$, which is a particular
associate of the additive formal group $F_{a}(z_{1},z_{2})=z_{1}+z_{2}$. Let
$V$ be a nonlocal vertex algebra. Recall from [Li7] that a _$\phi$
-coordinated $V$-module_ is a vector space $W$ equipped with a linear map
$Y_{W}^{\phi}(\cdot,z):V\to{\mathcal{E}}(W)$, satisfying the condition that
$Y_{W}^{\phi}({{\bf 1}},z)=1_{W}$ and that for $u,v\in V$, there exists
positive integer $k$ such that

(7.1)
$\displaystyle(1-z_{2}/z_{1})^{k}Y_{W}^{\phi}(u,z_{1})Y_{W}^{\phi}(v,z_{2})\in{\mathcal{E}}^{(2)}(W),$
(7.2)
$\displaystyle\left.\left((1-z_{2}/z_{1})^{k}Y_{W}^{\phi}(u,z_{1})Y_{W}^{\phi}(v,z_{2})\right)\right|_{z_{1}=z_{2}e^{z_{0}}}$
$\displaystyle\quad=(1-e^{-z_{0}})^{k}Y_{W}^{\phi}(Y(u,z_{0})v,z_{2}).$

Then we have that

######  Lemma 7.1.

Let $V$ be a nonlocal vertex algebra, and let $(W,Y_{W}^{\phi})$ be a
$\phi$-coordinated $V$-module. Suppose

$\displaystyle\sum_{i\geq 1}a_{i}\otimes b_{i}\otimes
f_{i}(z),\quad\sum_{j\geq 1}\alpha_{j}\otimes\beta_{j}\otimes g_{j}(z)\in
V\otimes V\otimes{\mathbb{C}}(z),$

such that

(7.3) $\displaystyle\sum_{i\geq
1}\iota_{z_{1},z_{2}}f_{i}(e^{z_{1}-z_{2}})Y(a_{i},z_{1})Y(b_{i},z_{2})$
$\displaystyle=$ $\displaystyle\sum_{j\geq
1}\iota_{z_{2},z_{1}}g_{j}(e^{z_{1}-z_{2}})Y(\alpha_{j},z_{2})Y(\beta_{j},z_{1}).$

Then we have that

$\displaystyle\sum_{i\geq
1}\iota_{z_{1},z_{2}}f_{i}(z_{1}/z_{2})Y_{W}^{\phi}(a_{i},z_{1})Y_{W}^{\phi}(b_{i},z_{2})$
$\displaystyle=$ $\displaystyle\sum_{j\geq
1}\iota_{z_{2},z_{1}}g_{j}(z_{1}/z_{2})Y_{W}^{\phi}(\alpha_{j},z_{2})Y_{W}^{\phi}(\beta_{j},z_{1}).$

###### Proof.

By letting the both hand sides of (7.3) act on ${{\bf 1}}$, we get that

(7.4) $\displaystyle\sum_{i\geq
1}\iota_{z_{1},z_{2}}f_{i}(e^{z_{1}-z_{2}})Y(a_{i},z_{1}-z_{2})b_{i}$
$\displaystyle=$ $\displaystyle\sum_{j\geq
1}\iota_{z_{2},z_{1}}g_{j}(e^{z_{1}-z_{2}})e^{(z_{1}-z_{2})\partial}Y(\alpha_{j},z_{2}-z_{1})\beta_{j}.$

Let $k$ be a positive integer such that ($i,j\geq 1$)

(7.5) $\displaystyle z^{k}f_{i}(e^{z}),\quad
z^{k}g_{j}(e^{z})\in{\mathbb{C}}[[z]],\quad z^{k}Y(a_{i},z)b_{i},\quad
z^{k}Y(\alpha_{j},z)\beta_{j}\in V[[z]].$

By multiplying $(z_{1}-z_{2})^{2k}$ on both hand sides of (7.4), we get that

$\displaystyle\sum_{i\geq
1}\left((z_{1}-z_{2})^{k}f_{i}(e^{z_{1}-z_{2}})\right)\left((z_{1}-z_{2})^{k}Y(a_{i},z_{1}-z_{2})b_{i}\right)$
(7.6) $\displaystyle=$ $\displaystyle\sum_{j\geq
1}\left((z_{1}-z_{2})^{k}g_{j}(e^{z_{1}-z_{2}})\right)e^{(z_{1}-z_{2})\partial}\left((z_{1}-z_{2})^{k}Y(\alpha_{j},z_{2}-z_{1})\beta_{j}\right).$

From (7.5), we can take $z_{2}=0$ in (7):

$\displaystyle z^{2k}\sum_{i\geq 1}f_{i}(e^{z})Y(a_{i},z)b_{i}=\sum_{j\geq
1}z^{k}g_{j}(e^{z})e^{z\partial}z^{k}Y(\alpha_{j},-z)\beta_{j}.$

Since $z^{k}$ is invertible, we get that

(7.7) $\displaystyle\sum_{i\geq 1}f_{i}(e^{z})Y(a_{i},z)b_{i}=\sum_{j\geq
1}g_{j}(e^{z})e^{z\partial}Y(\alpha_{j},-z)\beta_{j}.$

From (7.5), we get that

$\displaystyle\left.\left((z_{1}-z_{2})^{2k}\sum_{i\geq
1}f_{i}(e^{z_{1}-z_{2}})Y(a_{i},z_{1})Y(b_{i},z_{2})\right)\right|_{z_{1}=z_{2}+z_{0}}$
$\displaystyle=$ $\displaystyle\sum_{i\geq
1}z_{0}^{k}f_{i}(e^{z_{0}})z_{0}^{k}Y_{\mathcal{E}}\left(Y(a_{i},z_{2}),z_{0}\right)Y(b_{i},z_{2})$
$\displaystyle=$ $\displaystyle\sum_{i\geq
1}z_{0}^{k}f_{i}(e^{z_{0}})z_{0}^{k}Y\left(Y(a_{i},z_{0})b_{i},z_{2}\right),$

where the last equation follows from (3.4). We note that the condition (7.3)
shows that

$\displaystyle\sum_{i\geq
1}f_{i}(e^{z_{1}-z_{2}})Y(a_{i},z_{1})Y(b_{i},z_{2})\in{\mathcal{E}}^{(2)}(V).$

Combining these relations, we get that

$\displaystyle z_{0}^{2k}\left.\left(\sum_{i\geq
1}f_{i}(e^{z_{1}-z_{2}})Y(a_{i},z_{1})Y(b_{i},z_{2})\right)\right|_{z_{1}=z_{2}+z_{0}}$
$\displaystyle=$ $\displaystyle\sum_{i\geq
1}z_{0}^{k}f_{i}(e^{z_{0}})z_{0}^{k}Y\left(Y(a_{i},z_{0})b_{i},z_{2}\right).$

Then we have that

$\displaystyle\sum_{i\geq
1}f_{i}(e^{z_{0}})Y\left(Y(a_{i},z_{0})b_{i},z_{2}\right)$ $\displaystyle=$
$\displaystyle\left.\left(\sum_{i\geq
1}f_{i}(e^{z_{1}-z_{2}})Y(a_{i},z_{1})Y(b_{i},z_{2})\right)\right|_{z_{1}=z_{2}+z_{0}}\in{\mathcal{E}}(V)[[z_{0}]].$

Acting on ${{\bf 1}}$ and taking $z_{2}\to 0$, we get that

$\displaystyle\sum_{i\geq 1}f_{i}(e^{z_{0}})Y(a_{i},z_{0})b_{i}\in
V[[z_{0}]].$

Viewing $\log(1+z_{0})$ as an element in ${\mathbb{C}}[[z_{0}]]$, we get that

(7.8) $\displaystyle\sum_{i\geq 1}f_{i}(1+z_{0})Y(a_{i},\log(1+z_{0}))b_{i}\in
V[[z_{0}]].$

We replace $k$ with a larger one if necessary so that ($i,j\geq 1$):

$\displaystyle(1-z_{2}/z_{1})^{k}Y_{W}^{\phi}(a_{i},z_{1})Y_{W}^{\phi}(b_{i},z_{2}),\,\,(1-z_{2}/z_{1})^{k}Y_{W}^{\phi}(\alpha_{j},z_{2})Y_{W}^{\phi}(\beta_{j},z_{1})\in{\mathcal{E}}^{(2)}(W).$

From the weak associativity of $\phi$-coordinated modules (7.2), we get that

(7.9)
$\displaystyle\left.\left((1-z/z_{1})^{k}Y_{W}^{\phi}(a_{i},z_{1})Y_{W}^{\phi}(b_{i},z)\right)\right|_{z_{1}=\phi(z,z_{0})}$
$\displaystyle=$
$\displaystyle(1-e^{-z_{0}})^{k}Y_{W}^{\phi}(Y(a_{i},z_{0})b_{i},z),$ (7.10)
$\displaystyle\left.\left((1-z/z_{1})^{k}Y_{W}^{\phi}(\alpha_{j},z)Y_{W}^{\phi}(\beta_{j},z_{1})\right)\right|_{z=\phi(z_{1},-z_{0})}$
$\displaystyle=$
$\displaystyle(1-e^{-z_{0}})^{k}Y_{W}^{\phi}(Y(\alpha_{j},-z_{0})\beta_{j},z_{1}).$

From the equation (7.10) and [Li7, Remark 2.8], we have that

$\displaystyle\left.\left((1-z/z_{1})^{k}Y_{W}^{\phi}(\alpha_{j},z)Y_{W}^{\phi}(\beta_{j},z_{1})\right)\right|_{z_{1}=\phi(z,z_{0})}$
$\displaystyle=$
$\displaystyle\left.\left(\left.\left((1-z/z_{1})^{k}Y_{W}^{\phi}(\alpha_{j},z)Y_{W}^{\phi}(\beta_{j},z_{1})\right)\right|_{z=\phi(z_{1},-z_{0})}\right)\right|_{z_{1}=\phi(z,z_{0})}$
$\displaystyle=$
$\displaystyle\left.\left((1-e^{-z_{0}})^{k}Y_{W}^{\phi}(Y(\alpha_{j},-z_{0})\beta_{j},z_{1})\right)\right|_{z_{1}=\phi(z,z_{0})}$
$\displaystyle=$
$\displaystyle(1-e^{-z_{0}})^{k}Y_{W}^{\phi}(Y(\alpha_{j},-z_{0})\beta_{j},\phi(z,z_{0}))$
(7.11) $\displaystyle=$
$\displaystyle(1-e^{-z_{0}})^{k}Y_{W}^{\phi}(e^{z_{0}\partial}Y(\alpha_{j},-z_{0})\beta_{j},z),$

where the (7.11) follows from [Li7, Lemma 3.7]. Combining this with equations
(7.7) and (7.9), we get

$\displaystyle\left.\left((1-z/z_{1})^{2k}\sum_{i\geq
1}f_{i}(z_{1}/z)Y_{W}^{\phi}(a_{i},z_{1})Y_{W}^{\phi}(b_{i},z)\right)\right|_{z_{1}=\phi(z,z_{0})}$
$\displaystyle=$ $\displaystyle\sum_{i\geq
1}\left.\left(\left((1-z/z_{1})^{k}f_{i}(z_{1}/z)\right)\left((1-z/z_{1})^{k}Y_{W}^{\phi}(a_{i},z_{1})Y_{W}^{\phi}(b_{i},z)\right)\right)\right|_{z_{1}=\phi(z,z_{0})}$
(7.12) $\displaystyle=$ $\displaystyle(1-e^{-z_{0}})^{2k}\sum_{i\geq
1}f_{i}(e^{z_{0}})Y_{W}^{\phi}(Y(a_{i},z_{0})b_{i},z)$ $\displaystyle=$
$\displaystyle(1-e^{-z_{0}})^{2k}\sum_{j\geq
1}g_{j}(e^{z_{0}})Y_{W}^{\phi}(e^{z_{0}\partial}Y(\alpha_{j},-z_{0})\beta_{j},z)$
$\displaystyle=$ $\displaystyle\sum_{j\geq
1}\left.\left(\left((1-z/z_{1})^{k}g_{j}(z_{1}/z)\right)\left((1-z/z_{1})^{k}Y_{W}^{\phi}(\alpha_{j},z)Y_{W}^{\phi}(\beta_{j},z_{1})\right)\right)\right|_{z_{1}=\phi(z,z_{0})}$
$\displaystyle=$ $\displaystyle\left.\left((1-z/z_{1})^{2k}\sum_{j\geq
1}g_{j}(z_{1}/z)Y_{W}^{\phi}(\alpha_{j},z)Y_{W}^{\phi}(\beta_{j},z_{1})\right)\right|_{z_{1}=\phi(z,z_{0})}.$

Notice that

$\displaystyle(1-z_{2}/z_{1})^{2k}\sum_{i\geq
1}f_{i}(z_{1}/z_{2})Y_{W}^{\phi}(a_{i},z_{1})Y_{W}^{\phi}(b_{i},z_{2})\in{\mathcal{E}}^{(2)}(W),$
$\displaystyle(1-z_{2}/z_{1})^{2k}\sum_{j\geq
1}g_{j}(z_{1}/z_{2})Y_{W}^{\phi}(\alpha_{j},z_{2})Y_{W}^{\phi}(\beta_{j},z_{1})\in{\mathcal{E}}^{(2)}(W).$

Then we get from [Li7, Remark 2.8] that

$\displaystyle(1-z_{2}/z_{1})^{2k}\sum_{i\geq
1}f_{i}(z_{1}/z_{2})Y_{W}^{\phi}(a_{i},z_{1})Y_{W}^{\phi}(b_{i},z_{2})$
$\displaystyle=$ $\displaystyle(1-z_{2}/z_{1})^{2k}\sum_{j\geq
1}g_{j}(z_{1}/z_{2})Y_{W}^{\phi}(\alpha_{j},z_{2})Y_{W}^{\phi}(\beta_{j},z_{1}).$

Combining this with equation (7.12) and [Li7, Lemma 5.8], we get that

$\displaystyle(z_{2}z_{0})^{-1}\delta\left(\frac{z_{1}-z_{2}}{z_{2}z_{0}}\right)\sum_{i\geq
1}f_{i}(z_{1}/z_{2})Y_{W}^{\phi}(a_{i},z_{1})Y_{W}^{\phi}(b_{i},z_{2})$
$\displaystyle-$
$\displaystyle(z_{2}z_{0})^{-1}\delta\left(\frac{z_{2}-z_{1}}{-z_{2}z_{0}}\right)\sum_{j\geq
1}g_{j}(z_{1}/z_{2})Y_{W}^{\phi}(\alpha_{j},z_{2})Y_{W}^{\phi}(\beta_{j},z_{1})$
$\displaystyle=$ $\displaystyle
z_{1}^{-1}\delta\left(\frac{z_{2}(1+z_{0})}{z_{1}}\right)\sum_{i\geq
1}f_{i}(1+z_{0})Y_{W}^{\phi}(Y(a_{i},\log(1+z_{0}))b_{i},z_{2}).$

Taking $\operatorname{Res}_{z_{0}}$ on both hand sides, we get that

$\displaystyle\sum_{i\geq
1}f_{i}(z_{1}/z_{2})Y_{W}^{\phi}(a_{i},z_{1})Y_{W}^{\phi}(b_{i},z_{2})-\sum_{j\geq
1}g_{j}(z_{1}/z_{2})Y_{W}^{\phi}(\alpha_{j},z_{2})Y_{W}^{\phi}(\beta_{j},z_{1})$
$\displaystyle=$
$\displaystyle\operatorname{Res}_{z_{0}}z_{1}^{-1}\delta\left(\frac{z_{2}(1+z_{0})}{z_{1}}\right)\sum_{i\geq
1}f_{i}(1+z_{0})Y_{W}^{\phi}(Y(a_{i},\log(1+z_{0}))b_{i},z_{2})=0,$

where the last equation follows from (7.8). We complete the proof. ∎

######  Proposition 7.2.

Let $(W,Y_{W}^{\phi})$ be a $\phi$-coordinated module of a nonlocal vertex
algebra $V$. Let

$\displaystyle\sum_{i\geq 1}a_{i}\otimes b_{i}\otimes
f_{i}(z),\quad\sum_{j\geq 1}\alpha_{j}\otimes\beta_{j}\otimes g_{j}(z)\in
V\otimes V\otimes V\otimes{\mathbb{C}}(x),$
$\displaystyle\mbox{and}\quad\sum_{k\geq 0}\gamma_{k}\in V$

be finite sums, such that

$\displaystyle\sum_{i\geq
1}\iota_{z_{1},z_{2}}f_{i}(e^{z_{1}-z_{2}})Y(a_{i},z_{1})Y(b_{i},z_{2})$
$\displaystyle-$ $\displaystyle\sum_{j\geq
1}\iota_{z_{2},z_{1}}g_{j}(e^{z_{1}-z_{2}})Y(\alpha_{j},z_{2})Y(\beta_{j},z_{1})$
(7.13) $\displaystyle\qquad=\sum_{k\geq
0}Y(\gamma_{k},z_{2})\frac{1}{k!}\frac{\partial^{k}}{\partial
z_{2}^{k}}z_{1}^{-1}\delta\left(\frac{z_{2}}{z_{1}}\right).$

Then we have that

$\displaystyle\sum_{i\geq
1}\iota_{z_{1},z_{2}}f_{i}(z_{1}/z_{2})Y_{W}^{\phi}(a_{i},z_{1})Y_{W}^{\phi}(b_{i},z_{2})$
$\displaystyle-$ $\displaystyle\sum_{j\geq
1}\iota_{z_{2},z_{1}}g_{j}(z_{1}/z_{2})Y_{W}^{\phi}(\alpha_{j},z_{2})Y_{W}^{\phi}(\beta_{j},z_{1})$
(7.14) $\displaystyle\qquad=\sum_{k\geq
0}Y_{W}^{\phi}(\gamma_{k},z_{2})\frac{1}{k!}\left(z_{2}\frac{\partial}{\partial{z_{2}}}\right)^{k}\delta\left(\frac{z_{2}}{z_{1}}\right).$

###### Proof.

For any integer $i\geq 0$, we set $b_{-i}=\alpha_{-i}=\gamma_{i}$,
$a_{-i}=\beta_{-i}={{\bf 1}}$ and

$\displaystyle
f_{-i}(z)=g_{-i}(z)=\frac{1}{2i!}\left(-z\frac{\partial}{\partial{z}}\right)^{i}\frac{z+1}{z-1}\in{\mathbb{C}}(z).$

It is straightforward to verify that

$\displaystyle\iota_{z_{1},z_{2}}f_{-i}(z_{1}/z_{2})-\iota_{z_{2},z_{1}}g_{-i}(z_{1}/z_{2})=\frac{1}{i!}\left(z_{2}\frac{\partial}{\partial{z_{2}}}\right)^{i}\delta\left(\frac{z_{2}}{z_{1}}\right),$
$\displaystyle\iota_{z_{1},z_{2}}f_{-i}(e^{z_{1}-z_{2}})-\iota_{z_{2},z_{1}}g_{-i}(e^{z_{1}-z_{2}})=\frac{1}{i!}\frac{\partial^{i}}{\partial
z_{2}^{i}}z_{1}^{-1}\delta\left(\frac{z_{2}}{z_{1}}\right).$

Then the equation (7.13) is equivalent to

$\displaystyle\sum_{i\in{\mathbb{Z}}}\iota_{z_{1},z_{2}}f_{i}(e^{z_{1}-z_{2}})Y(a_{i},z_{1})Y(b_{i},z_{2})$
$\displaystyle=$
$\displaystyle\sum_{j\in{\mathbb{Z}}}\iota_{z_{2},z_{1}}g_{j}(e^{z_{1}-z_{2}})Y(\alpha_{j},z_{2})Y(\beta_{j},z_{1}),$

and the equation (7.14) is equivalent to

$\displaystyle\sum_{i\in{\mathbb{Z}}}\iota_{z_{1},z_{2}}f_{i}(z_{1}/z_{2})Y_{W}^{\phi}(a_{i},z_{1})Y_{W}^{\phi}(b_{i},z_{2})$
$\displaystyle=$
$\displaystyle\sum_{j\in{\mathbb{Z}}}\iota_{z_{2},z_{1}}g_{j}(z_{1}/z_{2})Y_{W}^{\phi}(\alpha_{j},z_{2})Y_{W}^{\phi}(\beta_{j},z_{1}).$

Therefore, this proposition follows immediate from Lemma 7.1. ∎

Let $V$ be an $\hbar$-adic nonlocal vertex algebra. A _$\phi$ -coordinated
$V$-module_ is a topologically free ${\mathbb{C}}[[\hbar]]$-module $W$
equipped with a ${\mathbb{C}}[[\hbar]]$-linear map
$Y_{W}^{\phi}(\cdot,x):V\to{\mathcal{E}}_{\hbar}(W)$, such that
$(W/\hbar^{n}W,Y_{W,n}^{\phi})$ is a $\phi$-coordinated $V/\hbar^{n}V$-module,
where $Y_{W,n}^{\phi}:V/\hbar^{n}V\to{\mathcal{E}}(W/\hbar^{n}W)$ is the
${\mathbb{C}}[[\hbar]]$-linear map induced from $Y_{W}^{\phi}$. As an
immediate $\hbar$-adic analogue of Proposition 7.2, we have that

######  Proposition 7.3.

Let $V$ be an $\hbar$-adic nonlocal vertex algebra, and let $(W,Y_{W}^{\phi})$
be a $\phi$-coordinated $V$-module. Suppose that

$\displaystyle\sum_{i\geq 1}a_{i}\otimes b_{i}\otimes
f_{i}(z),\quad\sum_{j\geq 1}\alpha_{j}\otimes\beta_{j}\otimes g_{j}(z)\in
V\widehat{\otimes}V\widehat{\otimes}V\widehat{\otimes}{\mathbb{C}}(z)[[\hbar]],$
$\displaystyle\mbox{and}\quad\sum_{k\geq 0}\gamma_{k}\in V,$

such that

$\displaystyle\sum_{i\geq
1}\iota_{z_{1},z_{2}}f_{i}(e^{z_{1}-z_{2}})Y(a_{i},z_{1})Y(b_{i},z_{2})$
$\displaystyle-$ $\displaystyle\sum_{j\geq
1}\iota_{z_{2},z_{1}}g_{j}(e^{z_{1}-z_{2}})Y(\alpha_{j},z_{2})Y(\beta_{j},z_{1})$
$\displaystyle\qquad=\sum_{k\geq
0}Y(\gamma_{k},z_{2})\frac{1}{k!}\frac{\partial^{k}}{\partial
z_{2}^{k}}z_{1}^{-1}\delta\left(\frac{z_{2}}{z_{1}}\right).$

Then we have that

$\displaystyle\sum_{i\geq
1}\iota_{z_{1},z_{2}}f_{i}(z_{1}/z_{2})Y_{W}^{\phi}(a_{i},z_{1})Y_{W}^{\phi}(b_{i},z_{2})$
$\displaystyle-$ $\displaystyle\sum_{j\geq
1}\iota_{z_{2},z_{1}}g_{j}(z_{1}/z_{2})Y_{W}^{\phi}(\alpha_{j},z_{2})Y_{W}^{\phi}(\beta_{j},z_{1})$
$\displaystyle\qquad=\sum_{k\geq
0}Y_{W}^{\phi}(\gamma_{k},z_{2})\frac{1}{k!}\left(z_{2}\frac{\partial}{\partial{z_{2}}}\right)^{k}\delta\left(\frac{z_{2}}{z_{1}}\right).$

Let
$Z^{\phi}(z_{1},z_{2}):{\mathcal{E}}_{\hbar}(W)\widehat{\otimes}{\mathcal{E}}_{\hbar}(W)\widehat{\otimes}{\mathbb{C}}(z)[[\hbar]]\to\operatorname{End}(W)[[z_{1}^{\pm
1},z_{2}^{\pm 1}]]$ be the ${\mathbb{C}}[[\hbar]]$-module map defined by

$\displaystyle Z^{\phi}(z_{1},z_{2})(a(z)\otimes b(z)\otimes
f(z))=f(z_{1}/z_{2})a(z_{1})b(z_{2}).$

Recall from [Li7] that a subset $U$ of ${\mathcal{E}}_{\hbar}(W)$ is said to
be _$\hbar$ -adically $S_{trig}$-local_ if for any $a(z),b(z)\in U$, there is
$A(z)\in\left({\mathbb{C}}U\otimes{\mathbb{C}}U\otimes{\mathbb{C}}(z)\right)[[\hbar]]$,
such that

$\displaystyle a(z_{1})b(z_{2})\sim Z^{\phi}(z_{2},z_{1})\left(A(z)\right).$

Let $U$ be an $\hbar$-adically $S_{trig}$-local subset of
${\mathcal{E}}_{\hbar}(W)$. For $a(z),b(z)\in U$, the locality implies that
for any positive integer $n$, there is positive integer $k_{n}$ such that

$\displaystyle(1-z_{1}/z_{2})^{k_{n}}\pi_{n}(a(z_{1}))\pi_{n}(b(z_{2}))\in{\mathcal{E}}^{(2)}(W/\hbar^{n}W).$

The following is a partial $\hbar$-adic analogue of [Li7, Definition 4.4]:

$\displaystyle
Y_{\mathcal{E}}^{\phi}(a(z),z_{0})b(z)=\sum_{n\in{\mathbb{Z}}}a(z)_{n}^{\phi}b(z)z_{0}^{-n-1}$
$\displaystyle=$
$\displaystyle\varprojlim_{n>0}\left(1-e^{z}_{0}\right)^{-k_{n}}\left.\left((1-z_{1}/z)^{k_{n}}a(z_{1})b(z)\right)\right|_{z_{1}=ze^{z_{0}}}.$

We have the partial $\hbar$-adic analogue of [Li7, Theorem 4.8]:

######  Theorem 7.4.

Let $U$ be an $\hbar$-adic $S_{trig}$-local subset of
${\mathcal{E}}_{\hbar}(W)$. Then there is a minimal $\hbar$-adically
$S_{trig}$-local subset
${\langle}U{\rangle}_{\phi}\subset{\mathcal{E}}_{\hbar}(W)$ containing $U$ and
$1_{W}$, such that

  * (1)

${\langle}U{\rangle}_{\phi}$ is topologically free and
$[{\langle}U{\rangle}_{\phi}]={\langle}U{\rangle}_{\phi}$.

  * (2)

${\langle}U{\rangle}_{\phi}$ is $Y_{\mathcal{E}}^{\phi}$ closed, that is, for
$a(x),b(x)\in{\langle}U{\rangle}_{\phi}$,
$a(x)_{n}b(x)\in{\langle}U{\rangle}_{\phi}$.

Then $({\langle}U{\rangle}_{\phi},Y_{\mathcal{E}}^{\phi},1_{W})$ carries the
structure of an $\hbar$-adic weak quantum vertex algebra and $W$ is a faithful
$\phi$-coordinated ${\langle}U{\rangle}_{\phi}$-module with
$Y_{W}(a(z),z_{0})=a(z_{0})$ for $a(z)\in{\langle}U{\rangle}_{\phi}$.

The following result is a partial $\hbar$-adic analogue of [JKLT1, Theorem
2.21]:

######  Proposition 7.5.

Let $W$ be a topologically free ${\mathbb{C}}[[\hbar]]$-module, and let
$V\subset{\mathcal{E}}_{\hbar}(W)$ be an $\hbar$-adic $S_{trig}$-local subset
such that $V={\langle}V{\rangle}_{\phi}$. Suppose that

$\displaystyle\sum_{i\geq 1}a_{i}(z)\otimes b_{i}(z)\otimes
f_{i}(z),\quad\sum_{j\geq 1}\alpha_{j}(z)\otimes\beta_{j}(z)\otimes
g_{j}(z)\in V\widehat{\otimes}V\widehat{\otimes}{\mathbb{C}}(z)[[\hbar]],$
$\displaystyle\mbox{and}\quad\sum_{k\geq 1}\gamma_{k}(z)\in V,$

such that the following relation holds on $W$:

$\displaystyle\sum_{i\geq
1}\iota_{z_{1},z_{2}}f_{i}(z_{1}/z_{2})a_{i}(z_{1})b_{i}(z_{2})-\sum_{j\geq
1}\iota_{z_{2},z_{1}}g_{j}(z_{1}/z_{2})\alpha_{j}(z_{2})\beta_{j}(z_{1})$
$\displaystyle\quad=\sum_{k\geq
1}\gamma_{k}(z_{2})\frac{1}{k!}\left(z_{2}\frac{\partial}{\partial{z_{2}}}\right)^{k}\delta\left(\frac{z_{2}}{z_{1}}\right).$

Then we have that

$\displaystyle\sum_{i\geq
1}\iota_{z_{1},z_{2}}f_{i}(e^{z_{1}-z_{2}})Y_{\mathcal{E}}^{\phi}(a_{i}(z),z_{1})Y_{\mathcal{E}}^{\phi}(b_{i}(z),z_{2})$
$\displaystyle-$ $\displaystyle\sum_{j\geq
1}\iota_{z_{2},z_{1}}g_{j}(e^{z_{1}-z_{2}})Y_{\mathcal{E}}^{\phi}(\alpha_{j}(z),z_{2})Y_{\mathcal{E}}^{\phi}(\beta_{j}(z),z_{1})$
$\displaystyle\qquad=\sum_{k\geq
1}Y_{\mathcal{E}}^{\phi}(\gamma_{k}(z),z_{2})\frac{1}{k!}\frac{\partial^{k}}{\partial
z_{2}^{k}}z_{1}^{-1}\delta\left(\frac{z_{2}}{z_{1}}\right).$

Now we start to determine the category of $\phi$-coordinated
$V_{\hat{\mathfrak{g}},\hbar}(\ell,0)$-modules.

######  Definition 7.6.

Let $\ell\in{\mathbb{C}}$. Define $\mathcal{M}_{\ell}^{\phi}$ to be the
category consisting of topologically free ${\mathbb{C}}[[\hbar]]$-modules $W$,
equipped with fields
$\psi_{i,q}(z),y_{i,q}^{\pm}(z)\in{\mathcal{E}}_{\hbar}(W)$ that satisfy the
following relations:

(7.15) $\displaystyle[\psi_{i,q}(z_{1}),\psi_{j,q}(z_{2})]$ $\displaystyle=$
$\displaystyle[r_{i}a_{ij}]_{q^{z_{2}\frac{\partial}{\partial{z_{2}}}}}[r\ell]_{q^{z_{2}\frac{\partial}{\partial{z_{2}}}}}\left(\iota_{z_{1},z_{2}}q^{-r\ell
z_{2}\frac{\partial}{\partial{z_{2}}}}-\iota_{z_{2},z_{1}}q^{r\ell
z_{2}\frac{\partial}{\partial{z_{2}}}}\right)\frac{z_{2}/z_{1}}{(1-z_{2}/z_{1})^{2}},$
(7.16) $\displaystyle[\psi_{i,q}(z_{1}),y_{j,q}^{\pm}(z_{2})]$
$\displaystyle=$ $\displaystyle\pm
y_{j,q}^{\pm}(z_{2})[r_{i}a_{ij}]_{q^{z_{2}\frac{\partial}{\partial{z_{2}}}}}\left(\iota_{z_{1},z_{2}}q^{-r\ell
z_{2}\frac{\partial}{\partial{z_{2}}}}-\iota_{z_{2},z_{1}}q^{r\ell
z_{2}\frac{\partial}{\partial{z_{2}}}}\right)\frac{1+z_{2}/z_{1}}{2-2z_{2}/z_{1}},$
(7.17)
$\displaystyle(1-z_{2}/z_{1})^{\delta_{ij}}(1-q^{-2r\ell}z_{2}/z_{1})^{\delta_{ij}}$
$\displaystyle\quad\times\left(y_{i,q}^{+}(z_{1})y_{j,q}^{-}(z_{2})-\iota_{z_{2},z_{1}}g_{ij,q}(z_{1}/z_{2})y_{j,q}^{-}(z_{2})y_{i,q}^{+}(z_{1})\right)=0,$
(7.18)
$\displaystyle\iota_{z_{1},z_{2}}f_{ij,q}(z_{1},z_{2})y_{i,q}^{\pm}(z_{1})y_{j,q}^{\pm}(z_{2})=C_{ij}\iota_{z_{2},z_{1}}f_{ji,q}(z_{2},z_{1})y_{j,q}^{\pm}(z_{2})y_{i,q}^{\pm}(z_{1}),$

where

(7.19) $\displaystyle
f_{ij,q}(z_{1},z_{2})=\iota_{z_{1},z_{2}}(z_{1}-q_{i}^{a_{ij}}z_{2})(z_{1}-z_{2})^{-\delta_{ij}},\quad
g_{ij,q}(z)=\frac{q_{i}^{a_{ij}}-z}{1-q_{i}^{a_{ij}}z}.$

For an object $(W,\psi_{i,q}(z),y_{i,q}^{\pm}(z))$ of
$\mathcal{M}_{\ell}^{\phi}$, we set

(7.20) $\displaystyle
U_{W}={\left.\left\\{{\psi_{i,q}(z),y_{i,q}^{\pm}(z)}\,\right|\,{i\in
I}\right\\}}.$

Then it is immediate from equations (7.15), (7.16), (7.17) and (7.18) that
$U_{W}$ is an $\hbar$-adically $S_{trig}$-local subset of
${\mathcal{E}}_{\hbar}(W)$.

######  Definition 7.7.

Let $\ell\in{\mathbb{C}}$. Define $\mathcal{R}_{\ell}^{\phi}$ to be the full
subcategory of $\mathcal{M}_{\ell}^{\phi}$ consisting of objects
$(W,\psi_{i,q}(z),y_{i,q}^{\pm}(z))$ such that

$\displaystyle
y_{i,q}^{+}(z_{1})y_{j,q}^{-}(z_{2})-\iota_{z_{2},z_{1}}g_{ij}(z_{1}/z_{2})y_{j,q}^{-}(z_{2})y_{i,q}^{+}(z_{1})$
(7.21)
$\displaystyle\qquad=\frac{\delta_{ij}}{q_{i}-q_{i}^{-1}}\left(\delta\left(\frac{z_{1}}{z_{2}}\right)-E(\psi_{i,q}(z))\delta\left(\frac{q^{-2r\ell}z_{2}}{z_{1}}\right)\right),$
(7.22)
$\displaystyle\left(\left(y_{i,q}^{\pm}(z)\right)_{0}^{\phi}\right)^{m_{ij}}y_{j,q}^{\pm}(z)=0,\quad\mbox{if
}a_{ij}<0.$

######  Proposition 7.8.

Let $(W,\psi_{i,q}(z),y_{i,q}^{\pm}(z))$ be an object of
$\mathcal{R}_{\ell}^{\phi}$. Then $W$ becomes a $\phi$-coordinated
$V_{\hat{\mathfrak{g}},\hbar}(\ell,0)$-module such that

$\displaystyle Y_{W}^{\phi}(h_{i,\hbar},z)=\psi_{i,q}(z),\quad
Y_{W}^{\phi}(x_{i,\hbar}^{\pm},z)=y_{i,q}^{\pm}(z),\quad i\in I.$

On the other hand, let $(W,Y_{W}^{\phi})$ be a $\phi$-coordinated
$V_{\hat{\mathfrak{g}},\hbar}(\ell,0)$-module. Then

$(W,Y_{W}^{\phi}(h_{i,\hbar},z),Y_{W}^{\phi}(x_{i,\hbar}^{\pm},z))$

is an object of $\mathcal{R}_{\ell}^{\phi}$.

###### Proof.

Let $(W,\psi_{i,q}(z),y_{i,q}^{\pm}(z))$ be an object of
$\mathcal{R}_{\ell}^{\phi}$. Then it is an object of
$\mathcal{M}_{\ell}^{\phi}$. Recall the $\hbar$-adically $S_{trig}$-local
subset $U_{W}$ (see (7.20) for details). From Theorem 7.4, we get an
$\hbar$-adic nonlocal vertex algebra ${\langle}U_{W}{\rangle}_{\phi}$, and $W$
becomes a $\phi$-coordinated ${\langle}U_{W}{\rangle}_{\phi}$-module with
module action $Y_{W}^{\phi}(a(z),z_{0})=a(z_{0})$ for
$a(z)\in{\langle}U_{W}{\rangle}_{\phi}$. From Proposition 7.5 and equations
(7.15), (7.16), (7.17), (7.18), we get that
$(W,Y_{\mathcal{E}}^{\phi}(\psi_{i,q}(z_{1}),z),Y_{\mathcal{E}}^{\phi}(y_{i,q}^{\pm}(z_{1}),z))$
is an object of $\mathcal{M}_{\tau}$ (see Definition 5.1). By using
Proposition 5.4, we get an $\hbar$-adic nonlocal vertex algebra homomorphism
$\varphi:F_{\tau}(A,\ell)\to{\langle}U_{W}{\rangle}_{\phi}$ such that

$\displaystyle\varphi(h_{i,\hbar})=\psi_{i,q}(z),\quad\varphi(x_{i,\hbar}^{\pm})=y_{i,q}^{\pm}(z),\quad
i\in I.$

Since $(W,\psi_{i,q}(z),y_{i,q}^{\pm}(z))$ be an object of
$\mathcal{R}_{\ell}^{\phi}$. We apply Proposition 7.5 to (7.21) and get that

$\displaystyle
Y_{\mathcal{E}}^{\phi}(y_{i,q}^{+}(z),z_{1})Y_{\mathcal{E}}^{\phi}(y_{j,q}^{-}(z),z_{2})-\iota_{z_{2},z_{1}}g_{ij,\hbar}(z_{1}-z_{2})Y_{\mathcal{E}}^{\phi}(y_{j,q}^{-}(z),z_{2})Y_{\mathcal{E}}^{\phi}(y_{i,q}^{+}(z),z_{1})$
$\displaystyle\quad=\delta_{ij}(q_{i}-q_{i}^{-1})^{-1}\left(z_{1}^{-1}\delta\left(\frac{z_{2}}{z_{1}}\right)-Y_{\mathcal{E}}^{\phi}\left(E(\psi_{i,q}(z)),z_{2}\right)z_{1}^{-1}\delta\left(\frac{z_{2}-2r\ell\hbar}{z_{1}}\right)\right).$

Let the both hand sides act on $1_{W}$, and take
$\operatorname{Sing}_{z_{1}}\operatorname{Res}_{z_{2}}z_{2}^{-1}$, we get that

$\displaystyle\operatorname{Sing}_{z_{1}}Y_{\mathcal{E}}^{\phi}(y_{i,q}^{+}(z),z_{1})y_{j,q}^{-}=\delta_{ij}(q_{i}-q_{i}^{-1})^{-1}\left(1_{W}z_{1}^{-1}-\theta_{i,q}(z)(z_{1}+2r\ell\hbar)^{-1}\right).$

Combining this with (7.22) and Definition 6.3, we get that $\varphi$ factor
through $V_{\hat{\mathfrak{g}},\hbar}(\ell,0)$. Therefore, $W$ becomes a
$\phi$-coordinated $V_{\hat{\mathfrak{g}},\hbar}(\ell,0)$-module such that
($i\in I$):

$\displaystyle
Y_{W}^{\phi}(h_{i,\hbar},z_{0})=Y_{W}^{\phi}(\varphi(h_{i,\hbar}),z_{0})=Y_{W}^{\phi}(\psi_{i,q}(z),z_{0})=\psi_{i,q}(z_{0}),$
$\displaystyle
Y_{W}^{\phi}(x_{i,\hbar}^{\pm},z_{0})=Y_{W}^{\phi}(\varphi(x_{i,\hbar}^{\pm}),z_{0})=Y_{W}^{\phi}(y_{i,q}(z)^{\pm},z_{0})=y_{i,q}^{\pm}(z_{0}).$

On the other hand, let $(W,Y_{W}^{\phi})$ be a $\phi$-coordinated