# Deforming vertex algebras by vertex bialgebras

Naihuan Jing1 Department of Mathematics, North Carolina State University,
Raleigh, NC 27695, USA<EMAIL_ADDRESS>,  Fei Kong2 Key Laboratory of
Computing and Stochastic Mathematics (Ministry of Education), School of
Mathematics and Statistics, Hunan Normal University, Changsha, China 410081
<EMAIL_ADDRESS>,  Haisheng Li  Department of Mathematical Sciences,
Rutgers University, Camden, NJ 08102, USA<EMAIL_ADDRESS>and  Shaobin
Tan4 School of Mathematical Sciences, Xiamen University, Xiamen, China 361005
<EMAIL_ADDRESS>
###### Abstract.

This is a continuation of a previous study initiated by one of us on nonlocal
vertex bialgebras and smash product nonlocal vertex algebras. In this paper,
we study a notion of right $H$-comodule nonlocal vertex algebra for a nonlocal
vertex bialgebra $H$ and give a construction of deformations of vertex
algebras with a right $H$-comodule nonlocal vertex algebra structure and a
compatible $H$-module nonlocal vertex algebra structure. We also give a
construction of $\phi$-coordinated quasi modules for smash product nonlocal
vertex algebras. As an example, we give a family of quantum vertex algebras by
deforming the vertex algebras associated to non-degenerate even lattices.

###### Key words and phrases:

vertex algebra, smash product, quantum vertex algebra, $phi$-coordinated quasi
module, lattice vertex operator algebra

###### 2010 Mathematics Subject Classification:

Primary 17B69, 17B68; Secondary 17B10, 17B37

1Partially supported by NSF of China (No.11531004) and Simons Foundation
(No.198129).

2Partially supported by NSF of China (No.11701183).

4Partially supported by NSF of China (No.11531004).

##  1\. Introduction

Vertex algebras (see [B1], [FLM]) are analogues and generalizations of
commutative associative algebras, while nonlocal vertex algebras (see [BK],
[Li2]; cf. [Li4]) are analogues of noncommutative associative algebras. The
(weak) quantum vertex algebras [Li4], which are certain variations of quantum
vertex operator algebras in the sense of Etingof-Kazhdan [EK], are a special
family of nonlocal vertex algebras. To a certain extent, the notion of
nonlocal vertex algebra plays the same role in the general vertex algebraic
theory as the notion of associative algebra does in the classical algebraic
theory. On the other hand, quantum vertex (operator) algebras, which from
various viewpoints are analogues of quantum groups (algebras), form a perfect
category.

In [Li8], certain vertex-algebra analogues of Hopf algebras were introduced
and studied. Specifically, notions of nonlocal vertex bialgebra and (left)
$H$-module nonlocal vertex algebra for a nonlocal vertex bialgebra $H$ were
introduced, and then the smash product nonlocal vertex algebra $V\sharp H$ of
an $H$-module nonlocal vertex algebra $V$ with $H$ was constructed. As an
application, a new construction of the lattice vertex algebras and their
modules was given.

In vertex algebra theory, among the most important notions are those of
modules and $\sigma$-twisted modules for a vertex algebra $V$ with $\sigma$ a
finite order automorphism. A notion of quasi module, generalizing that of
module, was introduced in [Li3] in order to associate vertex algebras to a
certain family of infinite-dimensional Lie algebras. Indeed, with this notion
vertex algebras can be associated to a wide variety of infinite-dimensional
Lie algebras. (From a certain point of view, the notion of quasi module also
generalizes that of twisted module (see [Li7]).) Furthermore, in order to
associate quantum vertex algebras to algebras such as quantum affine algebras,
a theory of $\phi$-coordinated quasi modules for nonlocal vertex algebras
(including quantum vertex algebras) was developed in [Li9, Li10], where weak
quantum vertex algebras were associated to quantum affine algebras
conceptually.

In the aforementioned notion of $\phi$-coordinated module, the symbol $\phi$
refers to an associate of the $1$-dimensional additive formal group (law),
which by definition is $F(x,y):=x+y$ (an element of ${\mathbb{C}}[[x,y]]$) and
which satisfies

$F(0,x)=x=F(x,0),\quad F(x,F(y,x))=F(F(x,y),z).$

This very formal group law (implicitly) plays an important role in the theory
of vertex algebras and their modules. In contrast, an associate of $F(x,y)$ is
a formal series $\phi(x,z)\in{\mathbb{C}}((x))[[z]]$, satisfying the condition

$\phi(x,0)=x,\ \ \ \phi(\phi(x,y),z)=\phi(x,y+z).$

It was proved therein that for any $p(x)\in{\mathbb{C}}((x))$, the formal
series $\phi(x,z)$ defined by $\phi(x,z)=e^{zp(x)\frac{d}{dx}}x$ is an
associate and every associate is of this form. The essence of [Li9] is that a
theory of $\phi$-coordinated modules for a general nonlocal vertex algebra is
attached to each associate $\phi(x,z)$, where the usual theory of modules
becomes a special case with $\phi$ taken to be the formal group law itself.
The importance lies in the fact that a wide variety of (quantum associative)
algebras can be associated to $\phi$-coordinated (quasi) modules for some
(weak) quantum vertex algebras by choosing a suitable associate $\phi$.

What is a $\phi$-coordinated quasi module? Let $\phi(x,z)$ be an associate of
$F(x,y)$. Note that in the definition of modules for an associative algebra
and for a (nonlocal) vertex algebra, the key ingredient is associativity. For
a general nonlocal vertex algebra $V$, the main defining property of a
$\phi$-coordinated quasi $V$-module $W$ with vertex operator map
$Y_{W}(\cdot,x)$ (the representation morphism) is that for any $u,v\in V$,
there exists a nonzero polynomial $q(x_{1},x_{2})$ such that

$q(x_{1},x_{2})Y_{W}(u,x_{1})Y_{W}(v,x_{2})\in\textnormal{{Hom}}(W,W((x_{1},x_{2}))),$
$\left(q(x_{1},x_{2})Y_{W}(u,x_{1})Y_{W}(v,x_{2})\right)|_{x_{1}=\phi(x_{2},z)}=q(\phi(x_{2},z),x_{2})Y_{W}(Y(u,z)v,x_{2}).$

The usual modules including the adjoint module are simply $\phi$-coordinated
(quasi) modules with $\phi(x,z)=F(x,z)=x+z$. In practice, for some algebras,
their modules of highest weight type cannot be associated with vertex algebras
in terms of usual modules, but can be viewed as $\phi$-coordinated (quasi)
modules for some vertex algebras or quantum vertex algebras with some $\phi$.

In this paper, we develop the theory of smash product nonlocal vertex algebras
further in several directions, with an ultimate goal to construct certain
desired quantum vertex (operator) algebras. Among the main results, we first
introduce a notion of right $H$-comodule nonlocal vertex algebra for a
nonlocal vertex bialgebra $H$ and then using a compatible right $H$-comodule
nonlocal vertex algebra structure on an $H$-module nonlocal vertex algebra
$V$, we construct a deformed nonlocal vertex algebra structure on $V$. We show
that under certain conditions, the deformed nonlocal vertex algebras are
quantum vertex algebras. In another direction (representation aspect), we
construct $\phi$-coordinated quasi modules for smash product nonlocal vertex
algebras and for the aforementioned deformed nonlocal vertex algebras. We also
apply the general results to the vertex algebras associated to non-degenerate
even lattices, to obtain a family of quantum vertex algebras.

Now, we give a more detailed description of the contents. Recall that a
nonlocal vertex bialgebra is simply a nonlocal vertex algebra $V$ equipped
with a classical coalgebra structure on $V$ such that the comultiplication
$\Delta$ and the counit $\epsilon$ are homomorphisms of nonlocal vertex
algebras. For a nonlocal vertex bialgebra $H$, an $H$-module nonlocal vertex
algebra is a nonlocal vertex algebra $V$ equipped with a module structure
$Y_{V}^{H}(\cdot,x)$ for $H$ viewed as a nonlocal vertex algebra such that

$\displaystyle Y_{V}^{H}(h,x)v\in V\otimes{\mathbb{C}}((x)),$ $\displaystyle
Y_{V}^{H}(h,x){\bf 1}_{V}=\epsilon(h){\bf 1}_{V},$ $\displaystyle
Y_{V}^{H}(h,x)Y(u,z)v=\sum
Y\left(Y_{V}^{H}(h_{(1)},x-z)u,z\right)Y_{V}^{H}(h_{(2)},x)v$

for $h\in H,\ u,v\in V$. (The first condition is technical, while the other
two are analogues of the classical counterparts.) Given an $H$-module nonlocal
vertex algebra $V$, we have a smash product nonlocal vertex algebra $V\sharp
H$, where $V\sharp H=V\otimes H$ as a vector space and

$Y^{\sharp}(u\otimes h,x)(v\otimes k)=\sum Y(u,x)Y_{V}^{H}(h_{(1)},x)v\otimes
Y(h_{(2)},x)k$

for $u,v\in V,\ h,k\in H$.

Let $H$ be a nonlocal vertex bialgebra. A right $H$-comodule nonlocal vertex
algebra is defined to be a nonlocal vertex algebra $V$ equipped with a comodue
structure $\rho:V\rightarrow V\otimes H$ for $H$ viewed as a coalgebra such
that $\rho$ is a homomorphism of nonlocal vertex algebras. Let $V$ be an
$H$-module nonlocal vertex algebra. We say a right $H$-comodule structure
$\rho:V\rightarrow V\otimes H$ is compatible with the (left) $H$-module
structure if $\rho$ is an $H$-module homomorphism with $H$ acting only on the
first factor of $V\otimes H$. Assuming that $\rho:V\rightarrow V\otimes H$ is
a compatible right $H$-comodule structure on an $H$-module nonlocal vertex
algebra $V$, we construct a new nonlocal vertex algebra
${\mathcal{D}}_{Y_{V}^{H}}^{\rho}(V)$ with $V$ as the underlying space, where
the vertex operator map, denoted here by $Y_{new}(\cdot,x)$, is given by

$Y_{new}(u,x)v=\sum Y(u_{(1)},x)Y_{V}^{H}(u_{(2)},x)v$

for $u,v\in V$, where $\rho(u)=\sum u_{(1)}\otimes u_{(2)}\in V\otimes H$ in
the Sweedler notation. (Recall that $Y_{V}^{H}(\cdot,x)$ denotes the vertex
operator map for $H$ on the module $V$.) Furthermore, we show that
${\mathcal{D}}_{Y_{V}^{H}}^{\rho}(V)$ is a quantum vertex algebra, assuming
that $H$ is cocommutative, $V$ is a vertex algebra, and $Y_{V}^{H}(\cdot,x)$
is invertible with respect to convolution (plus some technical condition).

In the theory of vertex algebras, an important family consists of the vertex
algebras $V_{L}$ associated to non-degenerate even lattices $L$, which are
rooted in the vertex operator realization of affine Kac-Moody algebras. These
structurally simple vertex algebras are often used to construct or study more
complicated vertex operator algebras (see [FLM] for example). Recall that
$V_{L}=S(\widehat{{\mathfrak{h}}}^{-})\otimes{\mathbb{C}}_{\varepsilon}[L]$ as
a vector space, where $\widehat{{\mathfrak{h}}}^{-}={\mathfrak{h}}\otimes
t^{-1}{\mathbb{C}}[t^{-1}]$ with
${\mathfrak{h}}={\mathbb{C}}\otimes_{{\mathbb{Z}}}L$ is an abelian Lie algebra
and ${\mathbb{C}}_{\varepsilon}[L]$ is the $\varepsilon$-twisted group algebra
of $L$ with $\varepsilon$ a certain particular $2$-cocycle of $L$. Set
$B_{L}=S(\widehat{{\mathfrak{h}}}^{-})\otimes{\mathbb{C}}[L]$ equipped with
the tensor product bialgebra structure, where ${\mathbb{C}}[L]$ is the
ordinary group algebra. Use a natural derivation on $B_{L}$ and the Borcherds
construction to make $B_{L}$ a commutative vertex algebra. In fact, $B_{L}$ is
a vertex bialgebra, which was exploited in [Li8].

In this paper, we explore the vertex bialgebra $B_{L}$ furthermore. We prove
that there exists a natural right $B_{L}$-comodule vertex algebra structure
$\rho:V_{L}\rightarrow V_{L}\otimes B_{L}$ on the vertex algebra $V_{L}$. In
fact, with $V_{L}$ identified with $B_{L}$ as a vector space, $\rho$ is simply
the comultiplication $\Delta:B_{L}\otimes B_{L}\rightarrow B_{L}$. (It was
proved in [Li8] that with canonical identifications, $\Delta$ is a vertex
algebra embedding of $V_{L}$ into $B_{L,\varepsilon}\sharp B_{L}$.) On the
other hand, for any linear map
$\eta(\cdot,x):{\mathfrak{h}}\rightarrow{\mathfrak{h}}\otimes
x{\mathbb{C}}[[x]]$, we give a compatible $B_{L}$-module vertex algebra
structure $Y^{\eta}_{M}(\cdot,x)$ on $V_{L}$. Consequently, we obtain a family
of quantum vertex algebras $V_{L}^{\eta}$ (with $V_{L}$ as the underlying
vector space).

In a sequel, we shall use $\hbar$-adic versions of the results obtained in
this paper to construct certain quantum vertex operator algebras (over
${\mathbb{C}}[[\hbar]]$) in the sense of Etingof-Kazhdan by deforming vertex
algebras $V_{L}$, where $\phi$-coordinated quasi modules for these quantum
vertex operator algebras are associated to highest weight modules for twisted
quantum affine algebras.

This paper is organized as follows: In Section 2, we first recall the basic
notions and results about smash product nonlocal vertex algebras, and then
study right comodule nonlocal vertex algebras, and compatible module nonlocal
vertex algebra structure. Furthermore, we give a deformation construction of
nonlocal vertex algebras. In Section 3, we continue to show that under certain
conditions, the deformed nonlocal vertex algebras are quantum vertex algebras.
In Section 4, we study $\phi$-coordinated quasi modules for smash product
nonlocal vertex algebras. In Section 5, for a general non-degenerate even
lattice $L$, we give a right $H$-comodule vertex algebra structure on $V_{L}$
and give a family of compatible $H$-module vertex algebra structures. Then we
give a family of deformations of the vertex algebra $V_{L}$.

In this paper, we work on the field ${\mathbb{C}}$ of complex numbers, and we
use ${\mathbb{Z}}_{+}$ for the set of positive integers, while we use
${\mathbb{N}}$ for the set of nonnegative integers. For a ring $R$, e.g.,
${\mathbb{Z}},\ {\mathbb{C}},\ {\mathbb{C}}[x,y]$, we use $R^{\times}$ for the
set of nonzero elements of $R$. We continue using the formal variable
notations and conventions (including formal delta functions) as established in
[FLM] and [FHL].

##  2\. Smash product nonlocal vertex algebras

In this section, we first recall the basic notions and results on smash
product nonlocal vertex algebras from [Li8], and we then study a notion of
right $H$-comodule nonlocal vertex algebra for a nonlocal vertex bialgebra
$H$. As the main result, for an $H$-module nonlocal vertex algebra $V$ with a
compatible right $H$-comodule nonlocal vertex algebra structure, we establish
a deformed nonlocal vertex algebra structure on $V$.

###  2.1. Nonlocal vertex algebras and quantum vertex algebras

Here, we recall some basic notions and results on nonlocal vertex algebras and
(weak) quantum vertex algebras from [Li4].

We start with the notion of nonlocal vertex algebra (see Remark 2.3).

######  Definition 2.1.

A nonlocal vertex algebra is a vector space $V$ equipped with a linear map

$\displaystyle Y(\cdot,x):$ $\displaystyle
V\rightarrow(\textnormal{{End}}V)[[x,x^{-1}]]$ $\displaystyle v\mapsto
Y(v,x)=\sum_{n\in{\mathbb{Z}}}v_{n}x^{-n-1}\ \ (\mbox{where
}v_{n}\in\textnormal{{End}}(V))$

and a distinguished vector ${\bf 1}\in V$, satisfying the conditions that

(2.1) $\displaystyle Y(u,x)v\in V((x))\ \ \ \mbox{ for }u,v\in V,$ (2.2)
$\displaystyle Y({\bf 1},x)v=v,\ \ \ \ Y(v,x){\bf 1}\in V[[x]]\ \mbox{ and }\
\lim_{x\rightarrow 0}Y(v,x){\bf 1}=v\ \ \ \mbox{ for }v\in V,$

and that for any $u,v\in V$, there exists a nonnegative integer $k$ such that

(2.3)
$\displaystyle(x_{1}-x_{2})^{k}Y(u,x_{1})Y(v,x_{2})\in\textnormal{{Hom}}(V,V((x_{1},x_{2})))$

and

(2.4)
$\displaystyle\left((x_{1}-x_{2})^{k}Y(u,x_{1})Y(v,x_{2})\right)|_{x_{1}=x_{2}+x_{0}}=x_{0}^{k}Y(Y(u,x_{0})v,x_{2}).$

The following was obtained in [JKLT] (cf. [DLMi], [LTW]):

######  Lemma 2.2.

Let $V$ be a nonlocal vertex algebra. Then the following weak associativity
holds: For any $u,v,w\in V$, there exists $l\in{\mathbb{N}}$ such that

(2.5)
$\displaystyle(x_{0}+x_{2})^{l}Y(u,x_{0}+x_{2})Y(v,x_{2})w=(x_{0}+x_{2})^{l}Y(Y(u,x_{0})v,x_{2})w.$

######  Remark 2.3.

Note that a notion of nonlocal vertex algebra was defined in [Li4] in terms of
weak associativity as in Lemma 2.2, which is the same as the notion of
axiomatic $G_{1}$-vertex algebra introduced in [Li2] and also the same as the
notion of field algebra introduced in [BK]. In view of Lemma 2.2, the notion
of nonlocal vertex algebra in the sense of Definition 2.1 is stronger than the
other notions.

Let $V$ be a nonlocal vertex algebra. Define a linear operator $\mathcal{D}$
by $\mathcal{D}(v)=v_{-2}{{\bf 1}}$ for $v\in V$. Then

(2.6)
$\displaystyle[\mathcal{D},Y(v,x)]=Y(\mathcal{D}(v),x)=\frac{d}{dx}Y(v,x)\ \
\textnormal{{for }}v\in V.$

The following notion singles out a family of nonlocal vertex algebras (see
[Li4]):

######  Definition 2.4.

A weak quantum vertex algebra is a nonlocal vertex algebra $V$ satisfying the
following ${\mathcal{S}}$-locality: For any $u,v\in V$, there exist

$u^{(i)},v^{(i)}\in V,\ f_{i}(x)\in{\mathbb{C}}((x))\ \ (i=1,\dots,r)$

and a nonnegative integer $k$ such that

(2.7)
$\displaystyle(x-z)^{k}Y(u,x)Y(v,z)=(x-z)^{k}\sum_{i=1}^{r}f_{i}(z-x)Y(v^{(i)},z)Y(u^{(i)},x).$

The following was proved in [Li4]:

######  Proposition 2.5.

A weak quantum vertex algebra can be defined equivalently by replacing the
conditions (2.3) and (2.4) in Definition 2.1 (for a nonlocal vertex algebra)
with the condition that for any $u,v\in V$, there exist

$u^{(i)},v^{(i)}\in V,\ f_{i}(x)\in{\mathbb{C}}((x))\ \ (i=1,\dots,r)$

such that

(2.8) $\displaystyle
x_{0}^{-1}\delta\left(\frac{x_{1}-x_{2}}{x_{0}}\right)Y(u,x_{1})Y(v,x_{2})$
$\displaystyle\ \ \ \
-x_{0}^{-1}\delta\left(\frac{x_{2}-x_{1}}{-x_{0}}\right)\sum_{i=1}^{r}f_{i}(x_{2}-x_{1})Y(v^{(i)},x_{2})Y(u^{(i)},x_{1})$
$\displaystyle=$ $\displaystyle
x_{2}^{-1}\delta\left(\frac{x_{1}-x_{0}}{x_{2}}\right)Y(Y(u,x_{0})v,x_{2})$

(the ${\mathcal{S}}$-Jacobi identity).

The following is a convenient technical result (cf. [LL, Proposition 5.6.7],
[Li1]):

######  Lemma 2.6.

Let $V$ be a nonlocal vertex algebra and let

$u,v,\ u^{(i)},v^{(i)},w^{(j)}\in V,\ f_{i}(x)\in{\mathbb{C}}((x))\ \ (1\leq
i\leq r,\ 0\leq j\leq s)$

such that

$\displaystyle(x_{1}-x_{2})^{k}Y(u,x_{1})Y(v,x_{2})=(x_{1}-x_{2})^{k}\sum_{i=1}^{r}f_{i}(x_{2}-x_{1})Y(v^{(i)},x_{2})Y(u^{(i)},x_{1})$

for some positive integer $k$. Then

$\displaystyle(x_{1}-x_{2})^{n}Y(u,x_{1})Y(v,x_{2})-(-x_{2}+x_{1})^{n}\sum_{i=1}^{r}f_{i}(x_{2}-x_{1})Y(v^{(i)},x_{2})Y(u^{(i)},x_{1})$
(2.9) $\displaystyle=\ $
$\displaystyle\sum_{j=0}^{s}Y(w^{(j)},x_{2})\frac{1}{j!}\left(\frac{\partial}{\partial
x_{2}}\right)^{j}x_{1}^{-1}\delta\left(\frac{x_{2}}{x_{1}}\right).$

if and only if $u_{n+j}v=w^{(j)}$ for $0\leq j\leq s$ and $u_{j+n}v=0$ for
$j>s$.

###### Proof.

From [Li4] the ${\mathcal{S}}$-Jacobi identity (2.8) holds, we have that

$\displaystyle(x_{1}-x_{2})^{n}Y(u,x_{1})Y(v,x_{2})-(-x_{2}+x_{1})^{n}\sum_{i=1}^{r}f_{i}(x_{2}-x_{1})Y(v^{(i)},x_{2})Y(u^{(i)},x_{1})$
$\displaystyle=\ $ $\displaystyle\sum_{j\geq
0}Y(u_{n+j}v,x_{2})\frac{1}{j!}\left(\frac{\partial}{\partial
x_{2}}\right)^{j}x_{1}^{-1}\delta\left(\frac{x_{2}}{x_{1}}\right).$

By using this, we complete the proof. ∎

Recall that a rational quantum Yang-Baxter operator on a vector space $U$ is a
linear map

${\mathcal{S}}(x):U\otimes U\rightarrow U\otimes U\otimes{\mathbb{C}}((x))$

such that

(2.10)
$\displaystyle{\mathcal{S}}^{12}(x){\mathcal{S}}^{13}(x+z){\mathcal{S}}^{23}(z)={\mathcal{S}}^{23}(z){\mathcal{S}}^{13}(x+z){\mathcal{S}}^{12}(x)$

(the quantum Yang-Baxter equation). Furthermore, a rational quantum Yang-
Baxter operator ${\mathcal{S}}(x)$ on $U$ is said to be unitary if

(2.11) $\displaystyle{\mathcal{S}}^{21}(x){\mathcal{S}}^{12}(-x)=1,$

where ${\mathcal{S}}(x)=\sigma{\mathcal{S}}(x)\sigma$ with $\sigma$ denoting
the flip operator on $U\otimes U$.

For any nonlocal vertex algebra $V$, follow [EK] to denote by $Y(x)$ the
linear map

$Y(x):V\otimes V\rightarrow V((x)),$

associated to the vertex operator map
$Y(\cdot,x):V\rightarrow\textnormal{{Hom}}(V,V((x)))$ of $V$. The following is
a variation of Etingof-Kazhdan’s notion of quantum vertex operator algebra
(see [EK], [Li4]):

######  Definition 2.7.

A quantum vertex algebra is a weak quantum vertex algebra $V$ equipped with a
unitary rational quantum Yang-Baxter operator ${\mathcal{S}}(x)$ on $V$,
satisfying the conditions that for any $u,v\in V$, there exists
$k\in{\mathbb{N}}$ such that

$\displaystyle(x-z)^{k}Y(x)\left(1\otimes
Y(z)\right)\left({\mathcal{S}}(x-z)(u\otimes v)\otimes
w\right)=(x-z)^{k}Y(z)\left(1\otimes Y(x)\right)(v\otimes u\otimes w)$

for all $w\in V$ and that

(2.12) $\displaystyle\left[\mathcal{D}\otimes 1,{\mathcal{S}}(x)\right]$
$\displaystyle=-\frac{d}{dx}{\mathcal{S}}(x),$ (2.13)
$\displaystyle{\mathcal{S}}(x)\left(Y(z)\otimes 1\right)$
$\displaystyle=\left(Y(z)\otimes
1\right){\mathcal{S}}^{23}(x){\mathcal{S}}^{13}(x+z)$

(the hexagon identity).

A nonlocal vertex algebra $V$ is said to be nondegenerate (see [EK]) if for
every positive integer $n$, the linear map

$Z_{n}:{\mathbb{C}}((x_{1}))\cdots((x_{n}))\otimes V^{\otimes n}\rightarrow
V((x_{1}))\cdots((x_{n}))),$

defined by

$Z_{n}(f\otimes v_{1}\otimes\cdots\otimes v_{n})=fY(v_{1},x_{1})\cdots
Y(v_{n},x_{n}){\bf 1},$

is injective.

We have (see [EK], [Li4]):

######  Proposition 2.8.

Let $V$ be a nondegenerate weak quantum vertex algebra. Then there exists a
linear map ${\mathcal{S}}(x):V\otimes V\rightarrow V\otimes
V\otimes{\mathbb{C}}((x))$ such that $V$ together with ${\mathcal{S}}(x)$ is a
quantum vertex algebra. Furthermore, such a linear map ${\mathcal{S}}(x)$ is
uniquely determined by the ${\mathcal{S}}$-locality.

The following notion (under a different name) was introduced in [Li3]:

######  Definition 2.9.

Let $G$ be a group, $\chi:G\rightarrow{\mathbb{C}}^{\times}$ a linear
character. A $(G,\chi)$-module nonlocal vertex algebra is a nonlocal vertex
algebra $V$ equipped with a representation
$R:G\rightarrow\textnormal{{GL}}(V)$ of $G$ on $V$ such that $R(g){{\bf
1}}={{\bf 1}}$ and

(2.14) $\displaystyle
R(g)Y(v,x)R(g)^{-1}=Y(R(g)v,\chi(g)x)\quad\textnormal{{for }}g\in G,\ v\in V.$

We sometimes denote a $(G,\chi)$-module nonlocal vertex algebra by a pair
$(V,R)$.

######  Remark 2.10.

Note that in Definition 2.9, for $g\in G$, if $\chi(g)=1$, then $g$ acts on
$V$ as an automorphism. Thus a $(G,\chi)$-module nonlocal vertex algebra with
$\chi=1$ (the trivial character) is simply a nonlocal vertex algebra on which
$G$ acts as an automorphism group. In this case, $V$ is a
${\mathbb{C}}[G]$-module nonlocal vertex algebra in the sense of [Li8] (see
Section 3).

Let $(U,R_{U})$ and $(V,R_{V})$ be $(G,\chi)$-module nonlocal vertex algebras.
A $(G,\chi)$-module nonlocal vertex algebra homomorphism is a nonlocal vertex
algebra homomorphism $f:U\rightarrow V$ which is also a $G$-module
homomorphism.

The following is a technical lemma formulated in [JKLT, Lemma 3.3]:

######  Lemma 2.11.

Let $G$ be a group equipped with a linear character
$\chi:G\rightarrow{\mathbb{C}}^{\times}$. Suppose that $V$ is a nonlocal
vertex algebra, $\rho:G\rightarrow\textnormal{{Aut}}(V)$ and
$L:G\rightarrow\textnormal{{GL}}(V)$ are group homomorphisms such that
$L(g){\bf 1}={\bf 1}$,

$\displaystyle\rho(g)L(h)=L(h)\rho(g)\quad\textnormal{{for }}g,h\in G,$
$\displaystyle L(g)Y(v,x)L(g)^{-1}=Y(L(g)v,\chi(g)x)\quad\textnormal{{for
}}g\in G,\ v\in S,$

where $S$ is a generating subset of $V$. Then $(V,R)$ is a $(G,\chi)$-module
nonlocal vertex algebra with $R$ defined by $R(g)=\rho(g)L(g)$ for $g\in G$.

###  2.2. Smash product nonlocal vertex algebras

We first recall from [Li8] the basic notions and results on smash product
nonlocal vertex algebras, and then we introduce a notion of right $H$-comodule
nonlocal vertex algebra with $H$ a nonlocal vertex bialgebra and we establish
a deformed nonlocal vertex algebra structure on a right $H$-comodule nonlocal
vertex algebra $V$ with a compatible (left) $H$-module nonlocal vertex algebra
structure.

We begin with the notion of nonlocal vertex bialgebra.

######  Definition 2.12.

A nonlocal vertex bialgebra is a nonlocal vertex algebra $V$ equipped with a
classical coalgebra structure ($\Delta,\varepsilon$) such that (the co-
multiplication) $\Delta:V\rightarrow V\otimes V$ and (the co-unit)
$\varepsilon:V\rightarrow{\mathbb{C}}$ are homomorphisms of nonlocal vertex
algebras.

The notion of homomorphism of nonlocal vertex bialgebras is defined in the
obvious way: For nonlocal vertex bialgebras ($V,\Delta,\varepsilon$) and
$(V^{\prime},\Delta^{\prime},\varepsilon^{\prime})$, a nonlocal vertex
bialgebra homomorphism from $V$ to $V^{\prime}$ is a homomorphism $f$ of
nonlocal vertex algebras such that

(2.15) $\displaystyle\Delta^{\prime}\circ f=(f\otimes f)\circ\Delta,\ \
\quad\varepsilon^{\prime}\circ f=\varepsilon.$

In other words, a nonlocal vertex bialgebra homomorphism is both a nonlocal
vertex algebra homomorphism and a coalgebra homomorphism.

######  Definition 2.13.

Let $(H,\Delta,\varepsilon)$ be a nonlocal vertex bialgebra. A (left)
$H$-module nonlocal vertex algebra is a nonlocal vertex algebra $V$ equipped
with a module structure $Y_{V}^{H}(\cdot,x)$ on $V$ for $H$ viewed as a
nonlocal vertex algebra such that

(2.16) $\displaystyle Y_{V}^{H}(h,x)v\in V\otimes{\mathbb{C}}((x)),$ (2.17)
$\displaystyle Y_{V}^{H}(h,x){{\bf 1}}_{V}=\varepsilon(h){{\bf 1}}_{V},$
(2.18) $\displaystyle Y_{V}^{H}(h,x_{1})Y(u,x_{2})v=\sum
Y(Y_{V}^{H}(h_{(1)},x_{1}-x_{2})u,x_{2})Y_{V}^{H}(h_{(2)},x_{1})v$

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.

The following two results were obtained in [Li8]:

######  Theorem 2.14.

Let $H$ be a nonlocal vertex bialgebra and let $V$ be an $H$-module nonlocal
vertex algebra. Set $V\sharp H=V\otimes H$ as a vector space. For $u,v\in V$,
$h,h^{\prime}\in H$, define

(2.19) $\displaystyle Y^{\sharp}(u\otimes h,x)(v\otimes h^{\prime})=\sum
Y(u,x)Y(h_{(1)},x)v\otimes Y(h_{(2)},x)h^{\prime}.$

Then $(V\sharp H,Y^{\sharp},{\bf 1}\otimes{\bf 1})$ carries the structure of a
nonlocal vertex algebra, which contains $V$ and $H$ canonically as subalgebras
such that for $h\in H$, $u\in V$,

(2.20) $\displaystyle Y^{\sharp}(h,x_{1})Y^{\sharp}(u,x_{2})=\sum
Y^{\sharp}(Y(h_{(1)},x_{1}-x_{2})u,x_{2})Y^{\sharp}(h_{(2)},x_{1}).$

######  Proposition 2.15.

Let $H$ be a nonlocal vertex bialgebra and let $V$ be an $H$-module nonlocal
vertex algebra. Let $W$ be a vector space and assume that $(W,Y_{W}^{V})$ is a
$V$-module and $(W,Y_{W}^{H})$ is an $H$-module such that for any $h\in H$,
$v\in V$, $w\in W$,

(2.21) $\displaystyle Y_{W}^{H}(h,x)w\in W\otimes{\mathbb{C}}((x)),$ (2.22)
$\displaystyle Y_{W}^{H}(h,x_{1})Y_{W}^{V}(v,x_{2})w=\sum
Y_{W}^{V}(Y_{V}^{H}(h_{(1)},x_{1}-x_{2})v,x_{2})Y_{W}^{H}(h_{(2)},x_{1})w.$

Then $W$ is a $V\sharp H$-module with the vertex operator map
$Y_{W}^{\sharp}(\cdot,x)$ given by

(2.23) $\displaystyle Y_{W}^{\sharp}(v\otimes
h,x)w=Y_{W}^{V}(v,x)Y_{W}^{H}(h,x)w\quad\textnormal{{ for }}h\in H,\ v\in V,\
w\in W.$

As an immediate consequence of Proposition 2.15, we have:

######  Proposition 2.16.

Let $H$ be a nonlocal vertex bialgebra and let $V$ be an $H$-module nonlocal
vertex algebra. Then $V$ is a $V\sharp H$-module with the vertex operator map
$Y_{V}^{\sharp}(\cdot,x)$ given by

$\displaystyle Y_{V}^{\sharp}(u\otimes h,x)v=Y(u,x)Y_{V}^{H}(h,x)v\ \ \ \
\textnormal{{for }}u,v\in V,\ h\in H.$

Furthermore, $1\otimes\varepsilon:V\sharp H\rightarrow V$ is a $V\sharp
H$-module epimorphism.

We shall need the following simple result:

######  Lemma 2.17.

Let $H$ be a nonlocal vertex bialgebra and let $(V,Y_{V}^{H})$ be an
$H$-module nonlocal vertex algebra. Then

(2.24) $\displaystyle
Y_{V}^{H}(Y(h,z)h^{\prime},x)=Y_{V}^{H}(h,x+z)Y_{V}^{H}(h^{\prime},x)\quad\textnormal{{for
}}h,h^{\prime}\in H.$

###### Proof.

Assume $h,h^{\prime}\in H$. Let $v\in V$. There exists a nonnegative integer
$l$ such that

$(z+x)^{l}Y_{V}^{H}(h,z+x)Y_{V}^{H}(h^{\prime},x)v=(z+x)^{l}Y_{V}^{H}(Y(h,z)h^{\prime},x)v.$

From the assumption (2.16), we have

$Y_{V}^{H}(h^{\prime},x)v,\ \ Y_{V}^{H}(h,x)u\in V\otimes{\mathbb{C}}((x))$

for any $u\in V$. Consequently, we have

$Y_{V}^{H}(h,x_{1})Y_{V}^{H}(h^{\prime},x)v\in
V\otimes{\mathbb{C}}((x,x_{1})).$

This implies that $Y_{V}^{H}(h,x+z)Y_{V}^{H}(h^{\prime},x)$ exists in
$(\textnormal{{Hom}}(V,V\otimes{\mathbb{C}}((x))))[[z]]$. It also implies that
we can choose $l$ so large that we also have

$x_{1}^{l}Y_{V}^{H}(h,x_{1})Y_{V}^{H}(h^{\prime},x)v\in
V\otimes{\mathbb{C}}((x))[[x_{1}]].$

Then

$(z+x)^{l}Y_{V}^{H}(h,z+x)Y_{V}^{H}(h^{\prime},x)v=(x+z)^{l}Y_{V}^{H}(h,x+z)Y_{V}^{H}(h^{\prime},x)v.$

Thus

$(x+z)^{l}Y_{V}^{H}(h,x+z)Y_{V}^{H}(h^{\prime},x)v=(z+x)^{l}Y_{V}^{H}(Y(h,z)h^{\prime},x)v.$

Consequently, by cancellation we get

$Y_{V}^{H}(h,x+z)Y_{V}^{H}(h^{\prime},x)v=Y_{V}^{H}(Y(h,z)h^{\prime},x)v,$

as desired. ∎

As in [Li8], by a differential bialgebra we mean a bialgebra
$(B,\Delta,\varepsilon)$ equipped with a derivation $\partial$ such that
$\varepsilon\circ\partial=0$ and $\Delta\partial=(\partial\otimes
1+1\otimes\partial)\Delta$. (That is, $\varepsilon$ and $\Delta$ are
homomorphisms of differential algebras.)

######  Remark 2.18.

Let $(B,\Delta,\varepsilon,\partial)$ be a differential bialgebra. In
particular, the associative algebra $B$ with derivation $\partial$ is a
differential algebra. Then we have a nonlocal vertex algebra structure on $B$
with

$Y(a,x)b=(e^{x\partial}a)b\ \ \ \mbox{ for }a,b\in B.$

Denote this nonlocal vertex algebra by $(B,\partial)$. Then $(B,\partial)$
equipped with $\Delta$ and $\varepsilon$ is naturally a nonlocal vertex
bialgebra (see [Li8, Example 4.2]).

Let $V$ be a nonlocal vertex algebra. A subset $U$ of
$\textnormal{{Hom}}(V,V\otimes{\mathbb{C}}((x)))$ is said to be
$\Delta$-closed (see [Li8]) if for any $a(x)\in U$, there exist
$a_{(1i)}(x),a_{(2i)}(x)\in U$ for $i=1,2,\dots,r$ such that

(2.25) $\displaystyle
a(x_{1})Y(v,x_{2})=\sum_{i=1}^{r}Y(a_{(1i)}(x_{1}-x_{2})v,x_{2})a_{(2i)}(x_{1})\
\ \ \ \textnormal{{for all }}v\in V.$

Let $B(V)$ be the sum of all $\Delta$-closed subspaces $U$ of
$\textnormal{{Hom}}(V,V\otimes{\mathbb{C}}((x)))$ such that

(2.26) $\displaystyle a(x){{\bf 1}}\in{\mathbb{C}}{{\bf
1}}\quad\textnormal{{for }}a(x)\in U.$

Note that
$\textnormal{{Hom}}(V,V\otimes{\mathbb{C}}((x)))\cong\textnormal{{End}}_{{\mathbb{C}}((x))}(V\otimes{\mathbb{C}}((x)))$
is an associative algebra.

The following is a summary of some results of [Li8]:

######  Proposition 2.19.

Let $V$ be a nonlocal vertex algebra. Then $B(V)$ is a $\Delta$-closed
subalgebra of $\textnormal{{Hom}}(V,V\otimes{\mathbb{C}}((x)))$ and is closed
under the derivation $\partial=\frac{d}{dx}$. Furthermore, if $V$ is
nondegenerate, then (a) $B(V)$ is a differential bialgebra with the coproduct
$\Delta$ and the counit $\varepsilon$, which are uniquely determined by

(2.27) $\displaystyle a(x){{\bf 1}}=\varepsilon(a(x)){{\bf
1}},\quad\Delta(a(x))=\sum a_{(1)}(x)\otimes a_{(2)}(x)$

for $a(x)\in B(V)$, where $a(x)Y(v,z)=\sum Y(a_{(1)}(x-z)v,z)a_{(2)}(x)$ for
all $v\in V$.

(b) $V$ is a $B(V)$-module nonlocal vertex algebra with
$Y_{V}(a(x),x_{0})=a(x_{0})$ for $a(x)\in B(V)$.

(c) For any nonlocal vertex bialgebra $H$, an $H$-module nonlocal vertex
algebra structure $Y_{V}^{H}(\cdot,x)$ on $V$ amounts to a nonlocal vertex
bialgebra homomorphism from $H$ to $B(V)$.

The following notions are due to [EK] and [Li6] (cf. [Li8]):

######  Definition 2.20.

Let $V$ be a nonlocal vertex algebra. A pseudo-derivation (resp. pseudo-
endomorphism) of $V$ is an element
$a(x)\in\textnormal{{Hom}}(V,V\otimes{\mathbb{C}}((x)))$ such that

(2.28) $\displaystyle[a(x),Y(u,z)]=Y(a(x-z)u,z)$

(resp. $a(x)Y(u,z)=Y(a(x-z)u,z)a(x)$) for all $u\in V$. Denote by
$\textnormal{{PDer}}(V)$ (resp. $\textnormal{{PEnd}}(V)$) the set of all
pseudo-derivations (resp. pseudo-endomorphisms).

From definition, we have

(2.29) $\displaystyle\textnormal{{PDer}}(V)\subset B(V),\ \ \ \
\textnormal{{PEnd}}(V)\subset B(V).$

The following construction of pseudo-derivations is due to [EK]:

######  Proposition 2.21.

Let $V$ be a vertex algebra. For $v\in V$, $f(x)\in{\mathbb{C}}((x))$, set

(2.30)
$\displaystyle\Phi(v,f)(x)=\textnormal{{Res}}_{z}f(x-z)Y(v,z)=\sum_{n\geq
0}\frac{(-1)^{n}}{n!}f^{(n)}(x)v_{n},$

where $f^{(n)}(x)=\left(\frac{d}{dx}\right)^{n}f(x)$. Then
$\Phi(v,f)(x)\in\textnormal{{PDer}}(V)$.

The following is an analogue of the notion of right comodule algebra:

######  Definition 2.22.

Let $H$ be a nonlocal vertex bialgebra. A right $H$-comodule nonlocal vertex
algebra is a nonlocal vertex algebra $V$ equipped with a homomorphism
$\rho:V\rightarrow V\otimes H$ of nonlocal vertex algebras such that

(2.31) $\displaystyle(\rho\otimes
1)\rho=(1\otimes\Delta)\rho,\quad(1\otimes\epsilon)\rho=\textnormal{{Id}}_{V},$

i.e., $\rho$ is also a right comodule structure on $V$ for $H$ viewed as a
coalgebra.

######  Remark 2.23.

Let $B$ be a coalgebra and let $U$ with the map $\rho:U\rightarrow U\otimes B$
be a right $B$-comodule. For $u\in U$, we have $(\rho\otimes
1)\rho(u)=(1\otimes\Delta)\rho(u)$, i.e.,

$\displaystyle\sum u_{(1),(1)}\otimes u_{(1),(2)}\otimes u_{(2)}=\sum
u_{(1)}\otimes u_{(2),(1)}\otimes u_{(2),(2)}.$

Note that by applying permutation operators on both sides we get five more
relations. If $B$ is cocommutative, we have
$P_{23}(1\otimes\Delta)\rho=(1\otimes\Delta)\rho$, so

(2.32) $\displaystyle P_{23}(\rho\otimes 1)\rho=(1\otimes\Delta)\rho,\quad
P_{23}(\rho\otimes 1)\rho=(\rho\otimes 1)\rho,$

where $P_{23}$ denotes the indicated flip operator on $B\otimes B\otimes B$.

######  Definition 2.24.

Let $H$ be a nonlocal vertex bialgebra. Assume that $V$ is an $H$-module
nonlocal vertex algebra with the module vertex operator map

$Y_{V}^{H}(\cdot,x):\
H\rightarrow\textnormal{{Hom}}(V,V\otimes{\mathbb{C}}((x)))$

and $V$ is also a right $H$-comodule nonlocal vertex algebra with the comodule
map $\rho:V\rightarrow V\otimes H$. We say $Y_{V}^{H}$ and $\rho$ are
compatible if $\rho$ is an $H$-module homomorphism with $V\otimes H$ viewed as
an $H$-module on which $H$ acts on the first factor only, i.e.,

(2.33) $\displaystyle\rho(Y_{V}^{H}(h,x)v)=(Y_{V}^{H}(h,x)\otimes
1)\rho(v)\quad\textnormal{{for }}h\in H,\ v\in V.$

The following is the main result of this section:

######  Theorem 2.25.

Let $H$ be a cocommutative nonlocal vertex bialgebra and let $V$ be a nonlocal
vertex algebra. Suppose that $(V,Y_{V}^{H})$ is an $H$-module nonlocal vertex
algebra and $(V,\rho)$ is a right $H$-comodule nonlocal vertex algebra such
that $Y_{V}^{H}$ and $\rho$ are compatible. For $a\in V$, set

(2.34) $\displaystyle\mathfrak{D}_{Y_{V}^{H}}^{\rho}(Y)(a,x)=\sum
Y(a_{(1)},x)Y_{V}^{H}(a_{(2)},x)$

on $V$, where $\rho(a)=\sum a_{(1)}\otimes a_{(2)}\in V\otimes H$. Then
$(V,\mathfrak{D}_{Y_{V}^{H}}^{\rho}(Y),{{\bf 1}})$ carries the structure of a
nonlocal vertex algebra. Denote this nonlocal vertex algebra by
$\mathfrak{D}_{Y_{V}^{H}}^{\rho}(V)$. Furthermore, $\rho$ is a nonlocal vertex
algebra homomorphism from $\mathfrak{D}_{Y_{V}^{H}}^{\rho}(V)$ to $V\sharp H$.

###### Proof.

Recall from Proposition 2.16 that $Y_{V}^{\sharp}(\cdot,x)$ denotes the
$V\sharp H$-module structure on $V$. Notice that for $a\in V$, we have

(2.35)
$\displaystyle\mathfrak{D}_{Y_{V}^{H}}^{\rho}(Y)(v,x)=Y_{V}^{\sharp}(\rho(v),x).$

It follows that
$\mathfrak{D}_{Y_{V}^{H}}^{\rho}(Y)(a,x)\in\textnormal{{Hom}}(V,V((x)))$. Let
$u,v\in V$. As $H$ is cocommutative, by (2.32) we have

$\displaystyle\sum u_{(1),(1)}\otimes u_{(2)}\otimes u_{(1),(2)}=\sum
u_{(1)}\otimes u_{(2),(1)}\otimes u_{(2),(2)}.$

Using this and (2.33), we get

$\displaystyle\rho\left(\mathfrak{D}_{Y_{V}^{H}}^{\rho}(Y)(u,x)v\right)=$
$\displaystyle\sum\rho\left(Y(u_{(1)},x)Y_{V}^{H}(u_{(2)},x)v\right)$
$\displaystyle=\ $ $\displaystyle\sum
Y^{\otimes}(\rho(u_{(1)}),x)\rho\left(Y_{V}^{H}(u_{(2)},x)v\right)$
$\displaystyle=\ $ $\displaystyle\sum
Y^{\otimes}(\rho(u_{(1)}),x)(Y_{V}^{H}(u_{(2)},x)\otimes 1)\rho(v)$
$\displaystyle=\ $ $\displaystyle\sum Y^{\otimes}(u_{(1),(1)}\otimes
u_{(1),(2)},x)(Y_{V}^{H}(u_{(2)},x)\otimes 1)(v_{(1)}\otimes v_{(2)})$
$\displaystyle=\ $ $\displaystyle\sum
Y(u_{(1),(1)},x)Y_{V}^{H}(u_{(2)},x)v_{(1)}\otimes Y(u_{(1),(2)},x)v_{(2)}$
$\displaystyle=\ $ $\displaystyle\sum
Y(u_{(1)},x)Y_{V}^{H}(u_{(2),(1)},x)v_{(1)}\otimes Y(u_{(2),(2)},x)v_{(2)}$
$\displaystyle=\ $ $\displaystyle\sum Y^{\sharp}(u_{(1)}\otimes
u_{(2)},x)(v_{(1)}\otimes v_{(2)})$ (2.36) $\displaystyle=\ $ $\displaystyle
Y^{\sharp}(\rho(u),x)\rho(v).$

Now, let $u,v\in V$. Then there exists a nonnegative integer $k$ such that

$\displaystyle(x_{1}-x_{2})^{k}Y_{V}^{\sharp}(\rho(u),x_{1})Y_{V}^{\sharp}(\rho(v),x_{2})\in\textnormal{{Hom}}(V,V((x_{1},x_{2}))),$
$\displaystyle\left((x_{1}-x_{2})^{k}Y_{V}^{\sharp}(\rho(u),x_{1})Y_{V}^{\sharp}(\rho(v),x_{2})\right)|_{x_{1}=x_{2}+x_{0}}=x_{0}^{k}Y_{V}^{\sharp}(Y^{\sharp}(\rho(u),x_{0})\rho(v),x_{2}).$

Using this we obtain

$(x_{1}-x_{2})^{k}\mathfrak{D}_{Y_{V}^{H}}^{\rho}(Y)(u,x_{1})\mathfrak{D}_{Y_{V}^{H}}^{\rho}(Y)(v,x_{2})\in\textnormal{{Hom}}(V,V((x_{1},x_{2})))$

(recalling (2.35)) and

$\displaystyle\left((x_{1}-x_{2})^{k}\mathfrak{D}_{Y_{V}^{H}}^{\rho}(Y)(u,x_{1})\mathfrak{D}_{Y_{V}^{H}}^{\rho}(Y)(v,x_{2})\right)|_{x_{1}=x_{2}+x_{0}}$
$\displaystyle=\ $
$\displaystyle\left((x_{1}-x_{2})^{k}Y_{V}^{\sharp}(\rho(u),x_{1})Y_{V}^{\sharp}(\rho(v),x_{2})\right)|_{x_{1}=x_{2}+x_{0}}$
$\displaystyle=\ $ $\displaystyle
x_{0}^{k}Y_{V}^{\sharp}(Y^{\sharp}(\rho(u),x_{0})\rho(v),x_{2})$
$\displaystyle=\ $ $\displaystyle
x_{0}^{k}Y_{V}^{\sharp}(\rho(\mathfrak{D}_{Y_{V}^{H}}^{\rho}(Y)(u,x_{0})v),x_{2})$
$\displaystyle=\ $ $\displaystyle
x_{0}^{k}\mathfrak{D}_{Y_{V}^{H}}^{\rho}(Y)(\mathfrak{D}_{Y_{V}^{H}}^{\rho}(Y)(u,x_{0})v,x_{2}).$

The vacuum and creation properties follow immediately from the counit property
and the vacuum property (2.17). Therefore,
$(V,\mathfrak{D}_{Y_{V}^{H}}^{\rho}(Y),{{\bf 1}})$ carries the structure of a
nonlocal vertex algebra and by (2.2) $\rho$ is a nonlocal vertex algebra
homomorphism from $\mathfrak{D}_{Y_{V}^{H}}^{\rho}(V)$ to $V\sharp H$. ∎

Furthermore, we have:

######  Proposition 2.26.

The nonlocal vertex algebra $\mathfrak{D}_{Y_{V}^{H}}^{\rho}(V)$ obtained in
Theorem 2.25 with the same map $\rho$ is also a right $H$-comodule nonlocal
vertex algebra.

###### Proof.

We only need to prove that $\rho:V\rightarrow V\otimes H$ is a homomorphism of
nonlocal vertex algebras from $\mathfrak{D}_{Y_{V}^{H}}^{\rho}(V)$ to
$\mathfrak{D}_{Y_{V}^{H}}^{\rho}(V)\otimes H$. For $u,v\in V$, we have

$\displaystyle\rho\left(\mathfrak{D}_{Y_{V}^{H}}^{\rho}(Y)(u,x)v\right)$
$\displaystyle=$ $\displaystyle\rho\left(\sum
Y(u_{(1)},x)Y_{V}^{H}(u_{(2)},x)v\right)$ $\displaystyle=$ $\displaystyle\sum
Y^{\otimes}(\rho(u_{(1)}),x)\rho\left(Y_{V}^{H}(u_{(2)},x)v\right)$
$\displaystyle=$ $\displaystyle\sum
Y^{\otimes}(\rho(u_{(1)}),x)(Y_{V}^{H}(u_{(2)},x)\otimes 1)(v_{(1)}\otimes
v_{(2)})$ $\displaystyle=$ $\displaystyle\sum
Y(u_{(1),(1)},x)Y_{V}^{H}(u_{(2)},x)v_{(1)}\otimes Y(u_{(1),(2)},x)v_{(2)}.$

On the other hand, denoting by $Y_{\otimes}^{\prime}(\cdot,x)$ the vertex
operator map of $\mathfrak{D}_{Y_{V}^{H}}^{\rho}(V)\otimes H$, we have

$\displaystyle Y_{\otimes}^{\prime}(\rho(u),x)\rho(v)$ $\displaystyle=$
$\displaystyle\sum Y_{\otimes}^{\prime}(u_{(1)}\otimes
u_{(2)},x)(v_{(1)}\otimes v_{(2)})$ $\displaystyle=$
$\displaystyle\sum\mathfrak{D}_{Y_{V}^{H}}^{\rho}(Y)(u_{(1)},x)v_{(1)}\otimes
Y(u_{(2)},x)v_{(2)}$ $\displaystyle=$ $\displaystyle\sum
Y(u_{(1),(1)},x)Y_{V}^{H}(u_{(1),(2)},x)v_{(1)}\otimes Y(u_{(2)},x)v_{(2)}.$

Meanwhile, since $H$ is cocommutative, (2.32) yields

$\sum u_{(1),(1)}\otimes u_{(2)}\otimes u_{(1),(2)}=\sum u_{(1),(1)}\otimes
u_{(1),(2)}\otimes u_{(2)}.$

Consequently, we get

(2.37)
$\displaystyle\rho\left(\mathfrak{D}_{Y_{V}^{H}}^{\rho}(Y)(u,x)v\right)=Y_{\otimes}^{\prime}(\rho(u),x)\rho(v).$

This proves that $\rho$ is also a homomorphism of nonlocal vertex algebras
from $\mathfrak{D}_{Y_{V}^{H}}^{\rho}(V)$ to
$\mathfrak{D}_{Y_{V}^{H}}^{\rho}(V)\otimes H$, concluding the proof. ∎

At the end of this section, we present some technical results. First, by a
straightforward argument we have:

######  Lemma 2.27.

Let $H$ be a nonlocal vertex bialgebra and let $(V,Y_{V}^{H})$ be an
$H$-module nonlocal vertex algebra. Suppose that $(W,Y_{W}^{V})$ is a
$V$-module and $(W,Y_{W}^{H})$ is an $H$-module such that

(2.38) $\displaystyle Y_{W}^{H}(h,x)w\in W\otimes{\mathbb{C}}((x)),$ (2.39)
$\displaystyle Y_{W}^{H}(h,x_{1})Y_{W}^{V}(v,x_{2})w=\sum
Y_{W}^{V}(Y_{V}^{H}(h_{(1)},x_{1}-x_{2})v,x_{2})Y_{W}^{H}(h_{(2)},x_{1})w$

for $h\in S$, $v\in V$, $w\in W$, where $S$ is a generating subset of $H$ as a
nonlocal vertex algebra. Then the two relations above hold for all $h\in H$,
$v\in V$, $w\in W$.

The following is the second technical result:

######  Lemma 2.28.

Let $H$ be a nonlocal vertex bialgebra and let $V$ be a nonlocal vertex
algebra. Suppose that
$Y_{V}^{H}(\cdot,x):H\rightarrow\textnormal{{Hom}}(V,V\otimes{\mathbb{C}}((x)))$
is an $H$-module nonlocal vertex algebra structure and $\rho:V\rightarrow
V\otimes H$ is an $H$-comodule nonlocal vertex algebra structure, satisfying

(2.40) $\displaystyle Y_{V}^{H}(h,x)v\in T\otimes{\mathbb{C}}((x)),$ (2.41)
$\displaystyle\rho(Y_{V}^{H}(h,x)v)=(Y_{V}^{H}(h,x)\otimes 1)\rho(v)$

for $h\in S,\ v\in T$, where $S$ and $T$ are generating subspaces of $H$ and
$V$ as nonlocal vertex algebras, respectively. Then $Y_{V}^{H}$ and $\rho$ are
compatible.

###### Proof.

First, we prove that (2.40) holds for all $h\in H,\ v\in T$. Let $K$ consist
of $h\in H$ such that $Y_{V}^{H}(h,x)v\in T\otimes{\mathbb{C}}((x))$ for $v\in
T$. Recall from Lemma 2.17 that for any $a,b\in H,\ w\in V$, we have

$Y_{V}^{H}(Y(a,x_{0})b,x_{2})w=Y_{V}^{H}(a,x_{2}+x_{0})Y_{V}^{H}(b,x_{2})w.$

Using this we get that $a_{m}b\in K$ for $a,b\in K,\ m\in{\mathbb{Z}}$. It
follows that $K$ is a nonlocal vertex subalgebra of $H$, containing $S$.
Consequently, we have $K=H$, confirming our assertion.

Second, we show that (2.41) holds for all $h\in H,\ v\in T$. Let $H^{\prime}$
consist of $a\in H^{\prime}$ such that (2.41) holds for all $v\in T$. Using
Lemma 2.17 and the first assertion, we can straightforwardly show that
$H^{\prime}$ is a nonlocal vertex subalgebra of $H$. Consequently, we have
$H^{\prime}=H$, confirming the second assertion.

Third, we prove that (2.41) holds for all $h\in H,\ v\in V$. Similarly, let
$V^{\prime}$ consist of $v\in V$ such that (2.41) holds for all $h\in H$ and
we then prove $V^{\prime}=V$ by showing that $V^{\prime}$ is a nonlocal vertex
subalgebra containing $T$. The closure of $V^{\prime}$ can be established
straightforwardly by using (2.18) and the fact that $\rho:V\rightarrow
V\otimes H$ is a homomorphism of nonlocal vertex algebras. Now that (2.41)
holds for all $h\in H,\ v\in V$, $Y_{V}^{H}$ and $\rho$ are compatible. ∎

##  3\. More on nonlocal vertex algebras $\mathfrak{D}_{Y_{V}^{H}}^{\rho}(V)$

In this section, we continue studying the deformed nonlocal vertex algebra
$\mathfrak{D}_{Y_{V}^{H}}^{\rho}(V)$. As the main result, we prove that
$\mathfrak{D}_{Y_{V}^{H}}^{\rho}(V)$ is a quantum vertex algebra under the
condition that $H$ is cocommutative, $V$ is a vertex algebra, and $Y_{V}^{H}$
is invertible in a certain sense.

Throughout this section, we assume that $H$ is a cocommutative nonlocal vertex
bialgebra. By coassociativity we have

$(\Delta\otimes\Delta)\Delta=(\Delta\otimes 1\otimes
1)(1\otimes\Delta)\Delta=(\Delta\otimes 1\otimes 1)(\Delta\otimes
1)\Delta=(1\otimes\Delta\otimes 1)(\Delta\otimes 1)\Delta,$

As $P_{23}(1\otimes\Delta\otimes 1)=(1\otimes\Delta\otimes 1)$ by
cocommutativity, consequently we have

(3.1) $\displaystyle
P_{23}(\Delta\otimes\Delta)\Delta=(\Delta\otimes\Delta)\Delta.$

######  Definition 3.1.

Let $(V,\rho)$ be a right $H$-comodule nonlocal vertex algebra. Denote by
$\mathfrak{L}^{\rho}_{H}(V)$ the set of all $H$-module nonlocal vertex algebra
structures on $V$ which are compatible with $\rho$.

######  Example 3.2.

Let $(V,\rho)$ be an $H$-comodule nonlocal vertex algebra. Define a linear map
$Y_{M}^{\varepsilon}(\cdot,x):H\rightarrow\textnormal{{End}}_{{\mathbb{C}}}(V)\subset(\textnormal{{End}}V)[[x,x^{-1}]]$
by

(3.2) $\displaystyle Y_{M}^{\varepsilon}(h,x)v=\varepsilon(h)v\ \ \ \mbox{ for
}h\in H,\ v\in V.$

Then it is straightforward to show that
$Y_{M}^{\varepsilon}\in\mathfrak{L}_{H}^{\rho}(V)$ and
$\mathfrak{D}_{Y_{M}^{\varepsilon}}^{\rho}(V)=V$.

From definition, $\mathfrak{L}^{\rho}_{H}(V)$ is a subset of the space
$\textnormal{{Hom}}\left(H,\textnormal{{Hom}}(V,V\otimes{\mathbb{C}}((x)))\right)$.
Note that $\textnormal{{Hom}}(V,V\otimes{\mathbb{C}}((x)))$ is naturally an
associative algebra. Furthermore, with $H$ a coalgebra
$\textnormal{{Hom}}\left(H,\textnormal{{Hom}}(V,V\otimes{\mathbb{C}}((x)))\right)$
is an associative algebra with respect to the convolution. More generally, it
is a classical fact that for any associative algebra $A$ with identity,
$\textnormal{{Hom}}(H,A)$ is an associative algebra with the operation $*$
defined by

(3.3) $\displaystyle(f*g)(h)=\sum f(h_{(1)})g(h_{(2)})$

for $h\in H,\ f,g\in\textnormal{{Hom}}(H,A)$, where $\Delta(h)=\sum
h_{(1)}\otimes h_{(2)}$, and with $\varepsilon$ as identity.

For

$Y_{M},Y_{M}^{\prime}\in\textnormal{{Hom}}(H,\textnormal{{Hom}}(V,V\otimes{\mathbb{C}}((x)))),$

we say that $Y_{M}$ and $Y_{M}^{\prime}$ commute if

(3.4) $\displaystyle
Y_{M}(h,x)Y_{M}^{\prime}(k,z)=Y_{M}^{\prime}(k,z)Y_{M}(h,x)\ \ \mbox{ for all
}h,k\in H.$

Furthermore, a subset $U$ of $\mathfrak{L}_{H}^{\rho}(V)$ is said to be
commutative if any two elements of $U$ commute.

######  Proposition 3.3.

Let $(V,\rho)$ be a right $H$-comodule nonlocal vertex algebra. For

$Y_{M}(\cdot,x),\
Y_{M}^{\prime}(\cdot,x)\in\textnormal{{Hom}}\left(H,\textnormal{{Hom}}(V,V\otimes{\mathbb{C}}((x)))\right),$

define a linear map $(Y_{M}\ast
Y_{M}^{\prime})(\cdot,x):H\to\textnormal{{Hom}}(V,V\otimes{\mathbb{C}}((x)))$
by

$\displaystyle(Y_{M}\ast Y_{M}^{\prime})(h,x)\mapsto\sum
Y_{M}(h_{(1)},x)Y^{\prime}_{M}(h_{(2)},x)\quad\textnormal{{for }}h\in H.$

Then
$\textnormal{{Hom}}\left(H,\textnormal{{Hom}}(V,V\otimes{\mathbb{C}}((x)))\right)$
equipped with the operation $*$ is an associative algebra with
$Y_{M}^{\varepsilon}$ as identity. Furthermore, if
$Y_{M},Y_{M}^{\prime}\in\mathfrak{L}_{H}^{\rho}(V)$ and if $Y_{M}$ and
$Y_{M}^{\prime}$ commute, then $Y_{M}\ast
Y_{M}^{\prime}\in\mathfrak{L}_{H}^{\rho}(V)$ and $Y_{M}\ast
Y_{M}^{\prime}=Y_{M}^{\prime}\ast Y_{M}$.

###### Proof.

The first assertion immediately follows from the aforementioned general fact,
so it remains to prove the second assertion. Let $(Y_{M},Y_{M}^{\prime})$ be a
commuting pair in $\mathfrak{L}_{H}^{\rho}(V)$. It follows immediately from
the cocommutativity of $H$ and the commutativity of $Y_{M}$ and
$Y_{M}^{\prime}$ that $Y_{M}\ast Y_{M}^{\prime}=Y_{M}^{\prime}\ast Y_{M}$. For
$h,k\in H$, using the homomorphism property of $\Delta$, Lemma 2.17, and the
commutativity of $Y_{M}$ with $Y_{M}^{\prime}$, we have

$\displaystyle(Y_{M}\ast Y_{M}^{\prime})(Y(h,z)k,x)$ $\displaystyle=\ $
$\displaystyle\sum
Y_{M}\Big{(}(Y(h,z)k)_{(1)},x\Big{)}Y_{M}^{\prime}\Big{(}(Y(h,z)k)_{(2)},x\Big{)}$
$\displaystyle=\ $ $\displaystyle\sum
Y_{M}\Big{(}Y(h_{(1)},z)k_{(1)},x\Big{)}Y_{M}^{\prime}\Big{(}Y(h_{(2)},z)k_{(2)},x\Big{)}$
$\displaystyle=\ $ $\displaystyle\sum
Y_{M}(h_{(1)},x+z)Y_{M}(k_{(1)},x)Y_{M}^{\prime}(h_{(2)},x+z)Y_{M}^{\prime}(k_{(2)},x)$
$\displaystyle=\ $ $\displaystyle\sum
Y_{M}(h_{(1)},x+z)Y_{M}^{\prime}(h_{(2)},x+z)Y_{M}(k_{(1)},x)Y_{M}^{\prime}(k_{(2)},x)$
$\displaystyle=\ $ $\displaystyle(Y_{M}\ast Y_{M}^{\prime})(h,x+z)(Y_{M}\ast
Y_{M}^{\prime})(k,x).$

Then it follows that $Y_{M}\ast Y_{M}^{\prime}$ is an $H$-module structure on
$V$. On the other hand, for $h\in H,\ u,v\in V$, using the relation
$P_{23}(\Delta\otimes\Delta)\Delta(h)=(\Delta\otimes\Delta)\Delta(h)$ from
(3.1), we get

$\displaystyle(Y_{M}\ast Y_{M}^{\prime})(h,x)Y(u,z)v$ $\displaystyle=\ $
$\displaystyle\sum Y_{M}(h_{(1)},x)Y_{M}^{\prime}(h_{(2)},x)Y(u,z)v$
$\displaystyle=\ $ $\displaystyle\sum
Y_{M}(h_{(1)},x)Y(Y_{M}^{\prime}(h_{(2),(1)},x-z)u,z)Y_{M}^{\prime}(h_{(2),(2)},x)v$
$\displaystyle=\ $ $\displaystyle\sum
Y\Big{(}Y_{M}(h_{(1),(1)},x-z)Y_{M}^{\prime}(h_{(2),(1)},x-z)u,z\Big{)}Y_{M}(h_{(1),(2)},x)Y_{M}^{\prime}(h_{(2),(2)},x)v$
$\displaystyle=\ $ $\displaystyle\sum
Y\Big{(}Y_{M}(h_{(1),(1)},x-z)Y_{M}^{\prime}(h_{(1),(2)},x-z)u,z\Big{)}Y_{M}(h_{(2),(1)},x)Y_{M}^{\prime}(h_{(2),(2)},x)v$
$\displaystyle=\ $ $\displaystyle\sum Y\Big{(}(Y_{M}\ast
Y_{M}^{\prime})(h_{(1)},x-z)u,z\Big{)}(Y_{M}\ast Y_{M}^{\prime})(h_{(2)},x)v.$

This proves that $(V,Y_{M}\ast Y_{M}^{\prime})$ is an $H$-module nonlocal
vertex algebra. It is straightforward to show that $Y_{M}\ast Y_{M}^{\prime}$
is compatible with $\rho$. Therefore, $Y_{M}\ast
Y_{M}^{\prime}\in\mathfrak{L}_{H}^{\rho}(V)$. ∎

Furthermore, we have:

######  Proposition 3.4.

Let $(V,\rho)$ be a right $H$-comodule nonlocal vertex algebra and let
$(Y_{M},Y_{M}^{\prime})$ be a commuting pair in $\mathfrak{L}_{H}^{\rho}(V)$.
Then
$Y_{M}\in\mathfrak{L}_{H}^{\rho}\left(\mathfrak{D}_{Y_{M}^{\prime}}^{\rho}(V)\right)$
and

(3.5)
$\displaystyle\mathfrak{D}_{Y_{M}}^{\rho}\left(\mathfrak{D}_{Y_{M}^{\prime}}^{\rho}(V)\right)=\mathfrak{D}_{Y_{M}\ast
Y_{M}^{\prime}}^{\rho}(V).$

###### Proof.

Let $h\in H,\ u\in V$. Recall that $\rho:V\rightarrow V\otimes H$ is an
$H$-module homomorphism, where $\rho(v)=\sum v_{(1)}\otimes v_{(2)}\in
V\otimes H$ for $v\in V$. Then we have

$\displaystyle Y_{M}(h,x)\mathfrak{D}_{Y_{M}^{\prime}}^{\rho}(Y)(u,z)=\sum
Y_{M}(h,x)Y(u_{(1)},z)Y_{M}^{\prime}(u_{(2)},z)$ $\displaystyle=$
$\displaystyle\sum
Y(Y_{M}(h_{(1)},x-z)u_{(1)},z)Y_{M}(h_{(2)},x)Y_{M}^{\prime}(u_{(2)},z)$
$\displaystyle=$ $\displaystyle\sum
Y(Y_{M}(h_{(1)},x-z)u_{(1)},z)Y_{M}^{\prime}(u_{(2)},z)Y_{M}(h_{(2)},x)$
$\displaystyle=$ $\displaystyle\sum
Y\left(\left(Y_{M}(h_{(1)},x-z)u\right)_{(1)},z\right)Y_{M}^{\prime}\left(\left(Y_{M}(h_{(1)},x-z)u\right)_{(2)},z\right)Y_{M}(h_{(2)},x)$
$\displaystyle=$
$\displaystyle\sum\mathfrak{D}_{Y_{M}^{\prime}}^{\rho}(Y)\left(Y_{M}(h_{(1)},x-z)u,z\right)Y_{M}(h_{(2)},x).$

Thus, $(\mathfrak{D}_{Y_{M}^{\prime}}^{\rho}(V),Y_{M})$ is an $H$-module
nonlocal vertex algebra.

From Proposition 2.26, $\mathfrak{D}_{Y_{M}^{\prime}}^{\rho}(V)$ with map
$\rho$ is a right $H$-comodule nonlocal vertex algebra. As $Y_{M}$ is
compatible with $\rho$, we have
$Y_{M}\in\mathfrak{L}_{H}^{\rho}\left(\mathfrak{D}_{Y_{M}^{\prime}}^{\rho}(V)\right)$.

Let $u\in V$. Note that from the cocommutativity we have

$\displaystyle\sum u_{(1),(1)}\otimes u_{(1),(2)}\otimes u_{(2)}=\sum
u_{(1)}\otimes u_{(2),(2)}\otimes u_{(2),(1)}.$

Then using the commutativity of $Y_{M}$ and $Y_{M}^{\prime}$, we get

$\displaystyle\mathfrak{D}^{\rho}_{Y_{M}}\left(\mathfrak{D}^{\rho}_{Y_{M}^{\prime}}(Y)\right)(u,x)$
$\displaystyle=\ $
$\displaystyle\sum\mathfrak{D}^{\rho}_{Y_{M}^{\prime}}(Y)(u_{(1)},x)Y_{M}(u_{(2)},x)$
$\displaystyle=\ $ $\displaystyle\sum
Y(u_{(1),(1)},x)Y_{M}^{\prime}(u_{(1),(2)},x)Y_{M}(u_{(2)},x)$
$\displaystyle=\ $ $\displaystyle\sum
Y(u_{(1)},x)Y_{M}^{\prime}(u_{(2),(2)},x)Y_{M}(u_{(2),(1)},x)$
$\displaystyle=\ $ $\displaystyle\sum
Y(u_{(1)},x)Y_{M}(u_{(2),(1)},x)Y_{M}^{\prime}(u_{(2),(2)},x)$
$\displaystyle=\ $ $\displaystyle\mathfrak{D}_{Y_{M}\ast
Y_{M}^{\prime}}^{\rho}(Y)(u,x).$

Therefore, we have
$\mathfrak{D}_{Y_{M}}^{\rho}\left(\mathfrak{D}_{Y_{M}^{\prime}}^{\rho}(V)\right)=\mathfrak{D}_{Y_{M}\ast
Y_{M}^{\prime}}^{\rho}(V)$. ∎

We also have the following result which is straightforward to prove:

######  Lemma 3.5.

Let $Y_{M}$ be an invertible element of $\mathfrak{L}_{H}^{\rho}(V)$ with
inverse $Y_{M}^{-1}$ with respect to the operation $*$. Then

(3.6) $\displaystyle
Y(u,x)v=\sum\mathfrak{D}_{Y_{M}}^{\rho}(Y)\left(u_{(1)},x\right)Y_{M}^{-1}\left(u_{(2)},x\right)v\
\ \quad\textnormal{{for }}u,v\in V,$

where $\rho(u)=\sum u_{(1)}\otimes u_{(2)}\in V\otimes H$.

As the main result of this section, we have:

######  Theorem 3.6.

Assume that $H$ is a nonlocal vertex cocommutative bialgebra. Let $V$ be a
vertex algebra with a right $H$-comodule vertex algebra structure $\rho$.
Suppose that $\\{Y_{M}^{\pm 1}\\}$ is a commutative subset of
$\mathfrak{L}_{H}^{\rho}(V)$ with $Y_{M}$ and $Y_{M}^{-}$ inverses each other
with respect to the operation $*$. Define a linear map
${\mathcal{S}}(x):V\otimes V\rightarrow V\otimes V\otimes{\mathbb{C}}((x))$ by

(3.7) $\displaystyle{\mathcal{S}}(x)(v\otimes u)=\sum
Y_{M}(u_{(2)},-x)v_{(1)}\otimes Y_{M}^{-1}(v_{(2)},x)u_{(1)}$

for $u,v\in V$. Then ${\mathcal{S}}(x)$ is a unitary rational quantum Yang-
Baxter operator and the nonlocal vertex algebra
$\mathfrak{D}_{Y_{M}}^{\rho}(V)$ which was obtained in Theorem 2.25 with
${\mathcal{S}}(x)$ is a quantum vertex algebra.

###### Proof.

Let $u,v\in V$. Note that as $V$ is a vertex algebra, the usual skew symmetry
holds. As $\rho:V\rightarrow V\otimes H$ is an $H$-module homomorphism, we
have

$\sum\left(Y_{M}(u_{(2)},x)v\right)_{(1)}\otimes\left(Y_{M}(u_{(2)},x)v\right)_{(2)}=\sum
Y_{M}(u_{(2)},x)v_{(1)}\otimes v_{(2)}.$

By using these properties and Lemma 3.5, it is straightforward to show that

$\displaystyle\mathfrak{D}_{Y_{M}}^{\rho}(Y)(u,x)v=\
e^{x{\mathcal{D}}}\mathfrak{D}_{Y_{M}}^{\rho}(Y)(-x){\mathcal{S}}(-x)(v\otimes
u).$

It then follows from [Li4] that ${\mathcal{S}}$-locality holds.

Recall that ${\mathcal{S}}^{21}(x)=R_{12}{\mathcal{S}}(x)R_{12}$. For any
$a,b\in V$, we have

$\displaystyle{\mathcal{S}}^{21}(-x)(b\otimes a)=\sum
Y_{M}^{-1}(a_{(2)},-x)b_{(1)}\otimes Y_{M}(b_{(2)},x)a_{(1)}.$

On the other hand, with $(\rho\otimes 1)\rho=(1\otimes\Delta)\rho$, it follows
that

$\displaystyle\sum u_{(1),(2)}\otimes u_{(2)}\otimes u_{(1),(1)}=\sum
u_{(2)(1)}\otimes u_{(2),(2)}\otimes u_{(1)},$ $\displaystyle\sum
v_{(1),(1)}\otimes v_{(1),(2)}\otimes v_{(2)}=\sum v_{(1)}\otimes
v_{(2),(1)}\otimes v_{(2),(2)}.$

The compatibilities of $\rho$ and $Y_{M}^{\pm 1}$ yield

$\sum\left(Y_{M}(u_{(2)},-x)v_{(1)}\right)_{(1)}\otimes\left(Y_{M}(u_{(2)},-x)v_{(1)}\right)_{(2)}=\sum
Y_{M}(u_{(2)},-x)v_{(1)(1)}\otimes v_{(1)(2)},$
$\sum\left(Y_{M}^{-1}(v_{(2)},x)u_{(1)}\right)_{(2)}\otimes\left(Y_{M}^{-1}(v_{(2)},x)u_{(1)}\right)_{(1)}=\sum
u_{(1)(2)}\otimes Y_{M}^{-1}(v_{(2)},x)u_{(1)(1)}.$

Then using all of these relations we get

$\displaystyle{\mathcal{S}}^{21}(-x){\mathcal{S}}(x)(v\otimes u)$
$\displaystyle=$
$\displaystyle\sum{\mathcal{S}}^{21}(-x)\left(Y_{M}(u_{(2)},-x)v_{(1)}\otimes
Y_{M}^{-1}(v_{(2)},x)u_{(1)}\right)$ $\displaystyle=$ $\displaystyle\sum
Y_{M}^{-1}\left(\left(Y_{M}^{-1}(v_{(2)},x)u_{(1)}\right)_{(2)},-x\right)\left(Y_{M}(u_{(2)},-x)v_{(1)}\right)_{(1)}$
$\displaystyle\quad\otimes
Y_{M}\left(\left(Y_{M}(u_{(2)},-x)v_{(1)}\right)_{(2)},x\right)\left(Y_{M}^{-1}(v_{(2)},x)u_{(1)}\right)_{(1)}$
$\displaystyle=$ $\displaystyle\sum
Y_{M}^{-1}\left(u_{(1),(2)},-x\right)Y_{M}(u_{(2)},-x)v_{(1),(1)}\otimes
Y_{M}\left(v_{(1),(2)},x\right)Y_{M}^{-1}(v_{(2)},x)u_{(1),(1)}$
$\displaystyle=$ $\displaystyle\sum
Y_{M}^{-1}\left(u_{(2),(1)},-x\right)Y_{M}(u_{(2),(2)},-x)v_{(1)}\otimes
Y_{M}\left(v_{(2),(1)},x\right)Y_{M}^{-1}(v_{(2),(2)},x)u_{(1)}$
$\displaystyle=$
$\displaystyle\sum\varepsilon(u_{(2)})v_{(1)}\otimes\varepsilon(v_{(2)})u_{(1)}=\sum\varepsilon(v_{(2)})v_{(1)}\otimes\varepsilon(u_{(2)})u_{(1)}=v\otimes
u.$

This proves that ${\mathcal{S}}(x)$ is unitary.

Furthermore, for $u,v,w\in V$, as $\rho$ is a vertex algebra homomorphism we
have

$\displaystyle{\mathcal{S}}(x)\left(\mathfrak{D}_{Y_{M}}^{\rho}(Y)(u,z)v\otimes
w\right)$ $\displaystyle=$
$\displaystyle\sum{\mathcal{S}}(x)\left(Y(u_{(1)},z)Y_{M}(u_{(2)},z)v\otimes
w\right)$ $\displaystyle=$ $\displaystyle\sum
Y_{M}\left(w_{(2)},-x\right)\left(Y(u_{(1)},z)Y_{M}(u_{(2)},z)v\right)_{(1)}\otimes
Y_{M}^{-1}\left(\left(Y(u_{(1)},z)Y_{M}(u_{(2)},z)v\right)_{(2)},x\right)w_{(1)}$
$\displaystyle=$ $\displaystyle\sum
Y_{M}\left(w_{(2)},-x\right)Y(u_{(1),(1)},z)(Y_{M}(u_{(2)},z)v)_{(1)}\otimes
Y_{M}^{-1}\left(Y(u_{(1),(2)},z)(Y_{M}(u_{(2)},z)v)_{(2)},x\right)w_{(1)}$
$\displaystyle=$ $\displaystyle\sum
Y_{M}\left(w_{(2)},-x\right)Y(u_{(1),(1)},z)Y_{M}(u_{(2)},z)v_{(1)}\otimes
Y_{M}^{-1}\left(Y(u_{(1),(2)},z)v_{(2)},x\right)w_{(1)}$ $\displaystyle=$
$\displaystyle\sum
Y(Y_{M}\left(w_{(2),(1)},-x-z\right)u_{(1),(1)},z)Y_{M}\left(w_{(2),(2)},-x\right)Y_{M}(u_{(2)},z)v_{(1)}$
$\displaystyle\quad\otimes
Y_{M}^{-1}\left(u_{(1),(2)},x+z\right)Y_{M}^{-1}\left(v_{(2)},x\right)w_{(1)}.$

On the other hand, we have

$\displaystyle\left(\mathfrak{D}_{Y_{M}}^{\rho}(Y)(z)\otimes
1\right){\mathcal{S}}^{23}(x){\mathcal{S}}^{13}(x+z)(u\otimes v\otimes w)$
$\displaystyle=$ $\displaystyle\left(\mathfrak{D}_{Y_{M}}^{\rho}(Y)(z)\otimes
1\right){\mathcal{S}}^{23}(x)\sum Y_{M}(w_{(2)},-x-z)u_{(1)}\otimes v\otimes
Y_{M}^{-1}(u_{(2)},x+z)w_{(1)}$ $\displaystyle=$
$\displaystyle\left(\mathfrak{D}_{Y_{M}}^{\rho}(Y)(z)\otimes 1\right)\sum
Y_{M}(w_{(2)},-x-z)u_{(1)}\otimes
Y_{M}\left(\left(Y_{M}^{-1}(u_{(2)},x+z)w_{(1)}\right)_{(2)},-x\right)v_{(1)}$
$\displaystyle\quad\otimes
Y_{M}^{-1}\left(v_{(2)},x\right)\left(Y_{M}^{-1}(u_{(2)},x+z)w_{(1)}\right)_{(1)}$
$\displaystyle=$ $\displaystyle\left(\mathfrak{D}_{Y_{M}}^{\rho}(Y)(z)\otimes
1\right)\sum Y_{M}(w_{(2)},-x-z)u_{(1)}\otimes
Y_{M}\left(w_{(1),(2)},-x\right)v_{(1)}$ $\displaystyle\quad\otimes
Y_{M}^{-1}\left(v_{(2)},x\right)Y_{M}^{-1}(u_{(2)},x+z)w_{(1),(1)}$
$\displaystyle=$ $\displaystyle\sum
Y\left(\left(Y_{M}(w_{(2)},-x-z)u_{(1)}\right)_{(1)},z\right)Y_{M}\left(\left(Y_{M}(w_{(2)},-x-z)u_{(1)}\right)_{(2)},z\right)Y_{M}\left(w_{(1),(2)},-x\right)v_{(1)}$
$\displaystyle\quad\otimes
Y_{M}^{-1}\left(v_{(2)},x\right)Y_{M}^{-1}(u_{(2)},x+z)w_{(1),(1)}$
$\displaystyle=$ $\displaystyle\sum
Y\left(Y_{M}(w_{(2)},-x-z)u_{(1),(1)},z\right)Y_{M}\left(u_{(1),(2)},z\right)Y_{M}\left(w_{(1),(2)},-x\right)v_{(1)}$
$\displaystyle\quad\otimes
Y_{M}^{-1}\left(v_{(2)},x\right)Y_{M}^{-1}(u_{(2)},x+z)w_{(1),(1)}$
$\displaystyle=$ $\displaystyle\sum
Y\left(Y_{M}(w_{(2)},-x-z)u_{(1),(1)},z\right)Y_{M}\left(u_{(1),(2)},z\right)Y_{M}\left(w_{(1),(2)},-x\right)v_{(1)}$
$\displaystyle\quad\otimes
Y_{M}^{-1}\left(v_{(2)},x\right)Y_{M}^{-1}(u_{(2)},x+z)w_{(1),(1)}$
$\displaystyle=$ $\displaystyle\sum
Y\left(Y_{M}(w_{(2)},-x-z)u_{(1),(1)},z\right)Y_{M}\left(w_{(1),(2)},-x\right)Y_{M}\left(u_{(1),(2)},z\right)v_{(1)}$
$\displaystyle\quad\otimes
Y_{M}^{-1}(u_{(2)},x+z)Y_{M}^{-1}\left(v_{(2)},x\right)w_{(1),(1)}$
$\displaystyle=$ $\displaystyle\sum
Y\left(Y_{M}(w_{(2),(1)},-x-z)u_{(1),(1)},z\right)Y_{M}\left(w_{(2),(2)},-x\right)Y_{M}\left(u_{(2)},z\right)v_{(1)}$
$\displaystyle\quad\otimes
Y_{M}^{-1}(u_{(1),(2)},x+z)Y_{M}^{-1}\left(v_{(2)},x\right)w_{(1)}.$

For the last equality we are also using the propertites

$\displaystyle\sum w_{(2)}\otimes w_{(1),(2)}\otimes w_{(1),(1)}=\sum
w_{(2),(1)}\otimes w_{(2),(2)}\otimes w_{(1)},$ $\displaystyle\sum
u_{(1),(1)}\otimes u_{(1),(2)}\otimes u_{(2)}=\sum u_{(1),(1)}\otimes
u_{(2)}\otimes u_{(1),(2)},$

i.e., $P_{13}(\rho\otimes 1)\rho(w)=P_{13}P_{23}(1\otimes\Delta)\rho(w)$, $\
P_{23}(\rho\otimes 1)\rho(u)=(\rho\otimes 1)\rho(u)$, which follow from the
cocommutativity as before. Thus we have

${\mathcal{S}}(x)\left(\mathfrak{D}_{Y_{M}}^{\rho}(Y)(u,z)v\otimes
w\right)=\left(\mathfrak{D}_{Y_{M}}^{\rho}(Y)(z)\otimes
1\right){\mathcal{S}}^{23}(x){\mathcal{S}}^{13}(x+z)(u\otimes v\otimes w).$

This proves that the hexagon identity holds.

Next, we show that the quantum Yang-Baxter equation holds. Let $u,v,w\in V$.
Recall that the compatibilities of $\rho$ with $Y_{M}$ and $Y_{M}^{-1}$ state
that

$\sum(Y_{M}^{\pm 1}(h,z)a)_{(1)}\otimes(Y_{M}^{\pm 1}(h,z)a)_{(2)}=\sum
Y_{M}^{\pm 1}(h,z)a_{(1)}\otimes a_{(2)}$

for $h\in H,\ a\in V$. Using these relations we get

$\displaystyle{\mathcal{S}}^{12}(x){\mathcal{S}}^{13}(x+z){\mathcal{S}}^{23}(z)(u\otimes
v\otimes w)$ $\displaystyle=$
$\displaystyle\sum{\mathcal{S}}^{12}(x){\mathcal{S}}^{13}(x+z)\left(u\otimes
Y_{M}(w_{(2)},-z)v_{(1)}\otimes Y_{M}^{-1}(v_{(2)},z)w_{(1)}\right)$
$\displaystyle=$
$\displaystyle\sum{\mathcal{S}}^{12}(x)Y_{M}\left(\left(Y_{M}^{-1}(v_{(2)},z)w_{(1)}\right)_{(2)},-x-z\right)u_{(1)}\otimes
Y_{M}(w_{(2)},-z)v_{(1)}$ $\displaystyle\quad\otimes
Y_{M}^{-1}(u_{(2)},x+z)\left(Y_{M}^{-1}(v_{(2)},z)w_{(1)}\right)_{(1)}$
$\displaystyle=$
$\displaystyle\sum{\mathcal{S}}^{12}(x)Y_{M}\left(w_{(1),(2)},-x-z\right)u_{(1)}\otimes
Y_{M}(w_{(2)},-z)v_{(1)}$ $\displaystyle\quad\otimes
Y_{M}^{-1}(u_{(2)},x+z)Y_{M}^{-1}(v_{(2)},z)w_{(1),(1)}$ $\displaystyle=$
$\displaystyle\sum
Y_{M}\left(\left(Y_{M}(w_{(2)},-z)v_{(1)}\right)_{(2)},-x\right)\left(Y_{M}\left(w_{(1),(2)},-x-z\right)u_{(1)}\right)_{(1)}$
$\displaystyle\quad\otimes
Y_{M}^{-1}\left(\left(Y_{M}\left(w_{(1),(2)},-x-z\right)u_{(1)}\right)_{(2)},x\right)\left(Y_{M}(w_{(2)},-z)v_{(1)}\right)_{(1)}$
$\displaystyle\quad\otimes
Y_{M}^{-1}(u_{(2)},x+z)Y_{M}^{-1}(v_{(2)},z)w_{(1),(1)}$ $\displaystyle=$
$\displaystyle\sum
Y_{M}\left(v_{(1),(2)},-x\right)Y_{M}\left(w_{(1),(2)},-x-z\right)u_{(1),(1)}$
$\displaystyle\quad\otimes
Y_{M}^{-1}\left(u_{(1),(2)},x\right)Y_{M}(w_{(2)},-z)v_{(1),(1)}$
$\displaystyle\quad\otimes
Y_{M}^{-1}(u_{(2)},x+z)Y_{M}^{-1}(v_{(2)},z)w_{(1),(1)}.$

Similarly, we have

$\displaystyle{\mathcal{S}}^{23}(z){\mathcal{S}}^{13}(x+z){\mathcal{S}}^{12}(x)(u\otimes
v\otimes w)$ $\displaystyle=$ $\displaystyle\sum
Y_{M}\left(w_{(2)},-x-z\right)Y_{M}(v_{(2)},-x)u_{(1),(1)}\otimes
Y_{M}\left(w_{(1),(2)},-z\right)Y_{M}^{-1}(u_{(2)},x)v_{(1),(1)}$
$\displaystyle\quad\otimes
Y_{M}^{-1}\left(v_{(1),(2)},z\right)Y_{M}^{-1}\left(u_{(1),(2)},x+z\right)w_{(1),(1)}$
$\displaystyle=$ $\displaystyle\sum
Y_{M}(v_{(2)},-x)Y_{M}\left(w_{(2)},-x-z\right)u_{(1),(1)}\otimes
Y_{M}^{-1}(u_{(2)},x)Y_{M}\left(w_{(1),(2)},-z\right)v_{(1),(1)}$
$\displaystyle\quad\otimes
Y_{M}^{-1}\left(u_{(1),(2)},x+z\right)Y_{M}^{-1}\left(v_{(1),(2)},z\right)w_{(1),(1)},$

where for the last equality we are also using the commutativity for
$(Y_{M}^{\pm 1},Y_{M}^{\pm 1})$ and $(Y_{M}^{\pm 1},Y_{M}^{\mp 1})$. Note that
with $H$ cocommutative we have $P_{12}(\rho\otimes
1)\rho(v)=P_{12}P_{23}(\rho\otimes 1)\rho(v),$ which states

$\sum v_{(1),(2)}\otimes v_{(1),(1)}\otimes v_{(2)}=\sum v_{(2)}\otimes
v_{(1),(1)}\otimes v_{(1),(2)}.$

Similarly, we have

$\displaystyle\sum u_{(1),(1)}\otimes u_{(1),(2)}\otimes u_{(2)}=\sum
u_{(1),(1)}\otimes u_{(2)}\otimes u_{(1),(2)},$ $\displaystyle\sum
w_{(1),(2)}\otimes w_{(2)}\otimes w_{(1),(1)}=\sum w_{(2)}\otimes
w_{(1),(2)}\otimes w_{(1),(1)}.$

Then we conclude

${\mathcal{S}}^{12}(x){\mathcal{S}}^{13}(x+z){\mathcal{S}}^{23}(z)(u\otimes
v\otimes
w)={\mathcal{S}}^{23}(z){\mathcal{S}}^{13}(x+z){\mathcal{S}}^{12}(x)(u\otimes
v\otimes w).$

Last, the shift condition can be proved straightforwardly. Therefore,
$\mathfrak{D}_{Y_{M}}^{\rho}(V)$ with ${\mathcal{S}}(x)$ is a quantum vertex
algebra. ∎

The following is a result about the generating subsets of
$\mathfrak{D}_{Y_{M}}^{\rho}(V)$:

######  Lemma 3.7.

Let $Y_{M}$ be an invertible element of $\mathfrak{L}_{H}^{\rho}(V)$. Suppose
that $S$ and $T$ are generating subspaces of $V$ and $H$ as nonlocal vertex
algebras, respectively, such that

(3.8) $\displaystyle\rho(S)\subset S\otimes T,\quad\Delta(T)\subset T\otimes
T,\quad Y_{M}^{-1}(T,x)S\subset S\otimes{\mathbb{C}}((x)).$

Then $S$ is also a generating subset of $\mathfrak{D}_{Y_{M}}^{\rho}(V)$.

###### Proof.

Let $X$ consist of all subspaces $U$ of $V$ such that

$\displaystyle Y_{M}^{-1}(T,x)U\subset U\otimes{\mathbb{C}}((x)).$

Denote by $K$ the sum of all such subspaces. It is clear that $K$ is the
(unique) largest in $X$. We see that $S$ and ${\mathbb{C}}{\bf 1}$ are in $X$,
so that ${\mathbb{C}}{\bf 1}+S\subset K$. Let $K^{2}$ denote the linear span
of $u_{n}v$ in $\mathfrak{D}_{Y_{M}}^{\rho}(V)$ for $u,v\in K,\
n\in{\mathbb{Z}}$. For $u,v\in K$ and $t\in T$, from the first part of the
proof of Proposition 3.4, we see that

$\displaystyle
Y_{M}^{-1}(t,x)\mathfrak{D}_{Y_{M}}^{\rho}(Y)(u,z)v=\sum\mathfrak{D}_{Y_{M}}^{\rho}(Y)(Y_{M}^{-1}(t_{(1)},x-z)u,z)Y_{M}^{-1}(t_{(2)},x)v.$

It follows that $K^{2}$ is in $X$. As $K$ is the largest, we have
$K^{2}\subset K$. Thus $K$ is a nonlocal vertex subalgebra of
$\mathfrak{D}_{Y_{M}}^{\rho}(V)$ with $S\subset K$.

Now, let $H^{\prime}$ be the maximal subspace of $H$ such that

$\displaystyle Y_{M}^{-1}(H^{\prime},x)K\subset K\otimes{\mathbb{C}}((x)).$

We have $T+{\mathbb{C}}{\bf 1}\subset H^{\prime}$. Note that for $h,k\in
H^{\prime}$, we have

$\displaystyle
Y_{M}^{-1}(Y(h,z)k,x)V^{\prime}=Y_{M}^{-1}(h,x+z)Y_{M}^{-1}(k,x)K\subset
K\otimes{\mathbb{C}}((x))[[z]].$

It follows that $H^{\prime}$ is a nonlocal vertex subalgebra of $H$ and hence
$H^{\prime}=H$. Thus $Y_{M}^{-1}(H,x)K\subset K\otimes{\mathbb{C}}((x))$.

Let $V^{\prime\prime}$ be any subspace of $V$ such that

(3.9) $\displaystyle Y_{M}(H,x)V^{\prime\prime}\subset
V^{\prime\prime}\otimes{\mathbb{C}}((x))\quad\textnormal{{and}}\quad\rho(V^{\prime\prime})\subset
V^{\prime\prime}\otimes H.$

That is, $V^{\prime\prime}$ is an $H$-submodule and a sub-comodule. For
$u,v\in V^{\prime\prime}$, we have

$\displaystyle\rho\left(\mathfrak{D}_{Y_{M}}^{\rho}(V)(u,z)v\right)=Y^{\sharp}(\rho(u),z)\rho(v)$
$\displaystyle=$ $\displaystyle\sum
Y(u_{(1)},z)Y_{M}(u_{(2),(1)},z)v_{(1)}\otimes Y(u_{(2),(2)},z)v_{(2)}$
$\displaystyle=$ $\displaystyle\sum
Y(u_{(1),(1)},z)Y_{M}(u_{(1),(2)},z)v_{(1)}\otimes Y(u_{(2)},z)v_{(2)}$
$\displaystyle=$
$\displaystyle\sum\mathfrak{D}_{Y_{M}}^{\rho}(Y)(u_{(1)},z)v_{(1)}\otimes
Y(u_{(2)},z)v_{(2)}.$

It follows that (3.9) holds for the nonlocal vertex subalgebra of
$\mathfrak{D}_{Y_{M}}^{\rho}(V)$ generated by $V^{\prime\prime}$. Since (3.9)
holds for $V^{\prime\prime}=S$, it holds for $V^{\prime\prime}=K$.

From Lemma 3.5, for $u\in S,\ v\in K$ we have

$\displaystyle
Y(u,x)v=\sum\mathfrak{D}_{Y_{M}}^{\rho}(Y)(u_{(1)},x)Y_{M}^{-1}(u_{(2)},x)v\in
K[[x,x^{-1}]].$

As ${\mathbb{C}}{\bf 1}+S\subset K$, it follows that $K$ contains the nonlocal
vertex subalgebra of $V$ generated by $S$. Thus $K=V$. Therefore, $S$ is also
a generating subspace of the quantum vertex algebra
$\mathfrak{D}_{Y_{M}}^{\rho}(V)$. ∎

##  4\. $\phi$-coordinated quasi modules for smash product nonlocal vertex
algebras

In this section, we first recall from [Li9] the basic notions and results on
$\phi$-coordinated quasi modules for nonlocal vertex algebras and then study
(equivariant) $\phi$-coordinated modules for smash product nonlocal vertex
algebras.

We begin with formal group. A one-dimensional formal group (law) over
${\mathbb{C}}$ (see [H]) is a formal power series
$F(x,y)\in{\mathbb{C}}[[x,y]]$ such that

$F(0,y)=y,\ \ F(x,0)=x,\ \ F(F(x,y),z)=F(x,F(y,z)).$

A particular example is the additive formal group $F_{a}(x,y):=x+y$.

Let $F(x,y)$ be a one-dimensional formal group over ${\mathbb{C}}$. An
associate of $F(x,y)$ (see [Li9]) is a formal series
$\phi(x,z)\in{\mathbb{C}}((x))[[z]]$, satisfying the condition that

$\displaystyle\phi(x,0)=x,\ \ \ \ \phi(\phi(x,y),z)=\phi(x,F(y,z)).$

The following result was obtained therein:

######  Proposition 4.1.

Let $p(x)\in{\mathbb{C}}((x))$. Set

$\displaystyle\phi(x,z)=e^{zp(x)\frac{d}{dx}}x=\sum_{n\geq
0}\frac{z^{n}}{n!}\left(p(x)\frac{d}{dx}\right)^{n}x\in{\mathbb{C}}((x))[[z]].$

Then $\phi(x,z)$ is an associate of $F_{a}(x,y)$. On the other hand, every
associate of $F_{a}(x,y)$ is of this form with $p(x)$ uniquely determined.

######  Example 4.2.

Taking $p(x)=1$ in Proposition 4.1, we get $\phi(x,z)=x+z=F_{a}(x,z)$ (the
formal group itself), and taking $p(x)=x$ we get $\phi(x,z)=xe^{z}$. More
generally, for $r\in{\mathbb{Z}}$, from [FHL] we have

(4.1)
$\displaystyle\phi_{r}(x,z):=e^{zx^{r+1}\frac{d}{dx}}x=\begin{cases}x\left(1-rzx^{r}\right)^{\frac{-1}{r}}&\
\ \mbox{ if }r\neq 0\\\ xe^{z}&\ \ \mbox{ if }r=0.\end{cases}$

From now on, we shall always assume that $\phi(x,z)$ is an associate of
$F_{a}(x,y)$ with $\phi(x,z)\neq x$, or equivalently,
$\phi(x,z)=e^{zp(x)\frac{d}{dx}}x$ with $p(x)\neq 0$.

From [Li9], for $f(x_{1},x_{2})\in{\mathbb{C}}((x_{1},x_{2}))$,
$f(\phi(x,z),x)$ exists in ${\mathbb{C}}((x))[[z]]$ and the correspondence
$f(x_{1},x)\mapsto f(\phi(x,z),x)$ gives a ring embedding

$\iota_{x_{1}=\phi(x,z)}:\
{\mathbb{C}}((x_{1},x))\rightarrow{\mathbb{C}}((x))[[z]]\subset{\mathbb{C}}((x))((z)).$

Denote by ${\mathbb{C}}_{\ast}((x_{1},x_{2}))$ the fraction field of
${\mathbb{C}}((x_{1},x_{2}))$. Then $\iota_{x_{1}=\phi(x,z)}$ naturally
extends to a field embedding

(4.2) $\displaystyle\iota_{x_{1}=\phi(x,z)}^{*}:\
{\mathbb{C}}_{\ast}((x_{1},x))\rightarrow{\mathbb{C}}((x))((z)).$

As a convention, for $F(x_{1},x_{2})\in{\mathbb{C}}_{\ast}((x_{1},x_{2}))$, we
write

(4.3) $\displaystyle
F(\phi(x,z),x)=\iota_{x_{1}=\phi(x,z)}^{*}(F(x_{1},x))\in{\mathbb{C}}((x))((z)).$

For $f(x_{1},x_{2})\in{\mathbb{C}}((x_{1},x_{2}))$, by definition

$f(\phi(x,z_{1}),\phi(x,z_{2}))=\left(e^{z_{1}p(x_{1})\partial_{x_{1}}}e^{z_{2}p(x_{2})\partial_{x_{2}}}f(x_{1},x_{2})\right)|_{x_{1}=x_{2}=x},$

which exists in ${\mathbb{C}}((x))[[z_{1},z_{2}]]$, and the correspondence

$f(x_{1},x_{2})\mapsto f(\phi(x,z_{1}),\phi(x,z_{2}))$

is a ring embedding of ${\mathbb{C}}((x_{1},x_{2}))$ into
${\mathbb{C}}((x))[[z_{1},z_{2}]]$ (see [JKLT]). Then this ring embedding
gives rise to a field embedding

(4.4)
$\displaystyle{\mathbb{C}}_{*}((x_{1},x_{2}))\rightarrow{\mathbb{C}}((x))_{*}((z_{1},z_{2})),$

where ${\mathbb{C}}((x))_{*}((z_{1},z_{2}))$ denotes the fraction field of
${\mathbb{C}}((x))((z_{1},z_{2}))$. As a convention, for
$F(x_{1},x_{2})\in{\mathbb{C}}_{*}((x_{1},x_{2}))$, we view
$F(\phi(x,z_{1}),\phi(x,z_{2}))$ as an element of
${\mathbb{C}}((x))_{*}((z_{1},z_{2}))$.

######  Definition 4.3.

Let $V$ be a nonlocal vertex algebra. A $\phi$-coordinated quasi $V$-module is
a vector space $W$ equipped with a linear map

$\displaystyle Y_{W}(\cdot,x):$ $\displaystyle
V\rightarrow(\textnormal{{End}}W)[[x,x^{-1}]]$ $\displaystyle v\mapsto
Y_{W}(v,x),$

satisfying the conditions that

$Y_{W}(u,x)w\in W((x))\ \ \ \textnormal{{for }}u\in V,\ w\in W,$ $Y_{W}({\bf
1},x)=1_{W}\ \ (\textnormal{{the identity operator on }}W),$

and that for any $u,v\in V$, there exists
$f(x_{1},x_{2})\in{\mathbb{C}}((x_{1},x_{2}))^{\times}$ such that

(4.5) $\displaystyle
f(x_{1},x_{2})Y_{W}(u,x_{1})Y_{W}(v,x_{2})\in\textnormal{{Hom}}(W,W((x_{1},x_{2}))),$
(4.6)
$\displaystyle\left(f(x_{1},x_{2})Y_{W}(u,x_{1})Y_{W}(v,x_{2})\right)|_{x_{1}=\phi(x_{2},z)}=f(\phi(x_{2},z),x_{2})Y_{W}(Y(u,z)v,x_{2}).$

The following notion was introduced in [JKLT] (cf. [Li10]):

######  Definition 4.4.

Let $V$ be a $(G,\chi)$-module nonlocal vertex algebra and let $\chi_{\phi}$
be a linear character of $G$ such that

(4.7)
$\displaystyle\phi(x,\chi(g)z)=\chi_{\phi}(g)\phi(\chi_{\phi}(g)^{-1}x,z)\ \
\mbox{ for }g\in G.$

A $(G,\chi_{\phi})$-equivariant $\phi$-coordinated quasi $V$-module is a
$\phi$-coordinated quasi $V$-module $(W,Y_{W})$ satisfying the conditions that

(4.8) $\displaystyle Y_{W}(R(g)v,x)=Y_{W}(v,\chi_{\phi}(g)^{-1}x)\ \ \ \
\mbox{ for }g\in G,\ v\in V$

and that for $u,v\in V$, there exists
$q(x)\in{\mathbb{C}}_{\chi_{\phi}(G)}[x]$ such that

(4.9) $\displaystyle
q(x_{1}/x_{2})Y_{W}(u,x_{1})Y_{W}(v,x_{2})\in\textnormal{{Hom}}(W,W((x_{1},x_{2}))),$

where ${\mathbb{C}}_{\chi_{\phi}(G)}[x]$ denotes the multiplicative monoid
generated in ${\mathbb{C}}[x]$ by $x-\chi_{\phi}(g)$ for $g\in G$.

######  Remark 4.5.

It was proved (see loc. cit.) that the condition (4.7) is equivalent to

(4.10) $\displaystyle p(\chi_{\phi}(g)x)=\chi(g)^{-1}\chi_{\phi}(g)p(x)\ \
\mbox{ for }g\in G,$

where $\phi(x,z)=e^{zp(x)\frac{d}{dx}}x$. In case $p(x)=x^{r+1}$ with
$r\in{\mathbb{Z}}$, the compatibility condition (4.7) is equivalent to
$\chi=\chi_{\phi}^{-r}$.

The following is a straightforward analogue of Lemma 2.11 for
$(G,\chi_{\phi})$-equivariant $\phi$-coordinated quasi $V$-modules:

######  Lemma 4.6.

Under the setting of Definition 4.4, assume that all the conditions hold
except (4.8), and instead assume

(4.11) $\displaystyle
Y_{W}^{\phi}(R(g)v,x)=Y_{W}^{\phi}(v,\chi_{\phi}(g)^{-1}x)\quad{for}\ g\in G,\
v\in S,$

where $S$ is a generating subset of $V$. Then $W$ is a
$(G,\chi_{\phi})$-equivariant $\phi$-coordinated quasi $V$-module.

Next, we study $(G,\chi)$-module nonlocal vertex algebra structures on smash
product nonlocal vertex algebra $V\sharp H$ and equivariant $\phi$-coordinated
quasi modules for $V\sharp H$. For the rest of this section, we fix a group
$G$ with a linear character $\chi$. We first formulate the following notion:

######  Definition 4.7.

A _$(G,\chi)$ -module nonlocal vertex bialgebra_ $H$ is a nonlocal vertex
bialgebra and a $(G,\chi)$-module nonlocal vertex algebra on which $G$ acts as
an automorphism group with $H$ viewed as a coalgebra.

The following two propositions can be proved straightforwardly:

######  Proposition 4.8.

Let $H$ be a $(G,\chi)$-module nonlocal vertex bialgebra. Suppose that $V$ is
a $(G,\chi)$-module nonlocal vertex algebra and an $H$-module nonlocal vertex
algebra with the $H$-module structure denoted by $Y_{V}^{H}(\cdot,x)$ such
that

(4.12) $\displaystyle
R_{V}(g)Y_{V}^{H}(h,x)v=Y_{V}^{H}(R_{H}(g)h,\chi(g)x)R_{V}(g)v\quad{for}\ g\in
G,\ h\in H,\ v\in V,$

where $R_{H}$ and $R_{V}$ denote the representations of $G$ on $H$ and $V$,
respectively. Then $V\sharp H$ is a $(G,\chi)$-module nonlocal vertex algebra
with $R=R_{V}\otimes R_{H}$.

######  Proposition 4.9.

Let $H,V$ be given as in Proposition 4.8. In addition, assume that $V$ is a
right $H$-comodule nonlocal vertex algebra whose right $H$-comodule structure
$\rho:V\rightarrow V\otimes H$ is compatible with $Y_{V}^{H}$ and is a
$G$-module homomorphism. Then $\mathfrak{D}_{Y_{V}^{H}}^{\rho}(V)$ is a
$(G,\chi)$-module nonlocal vertex algebra with the same representation $R$ of
$G$ on $V$. Furthermore, the map $\rho:V\rightarrow V\otimes H$ is a
homomorphism of $(G,\chi)$-module nonlocal vertex algebras from
$\mathfrak{D}_{Y_{V}^{H}}^{\rho}(V)$ to $V\sharp H$.

######  Remark 4.10.

Let $H$ be a $(G,\chi)$-module nonlocal vertex bialgebra and let $V$ be a
$(G,\chi)$-module nonlocal vertex algebra. As a convention, we shall always
assume that the condition (4.12) holds for an $H$-module nonlocal vertex
algebra structure $Y_{V}^{H}$ on $V$ and that $\rho$ is also a $G$-module
homomorphism for an $H$-comodule nonlocal vertex algebra structure $\rho$.

Recall that ${\mathbb{C}}_{*}((x_{1},x_{2}))$ denotes the fraction field of
${\mathbb{C}}((x_{1},x_{2}))$ and that for
$F(x_{1},x_{2})\in{\mathbb{C}}_{*}((x_{1},x_{2}))$, $F(\phi(x,z),x)$ was
defined as an element of ${\mathbb{C}}((x))((z))$. Here, we interpret this
definition in a different way. For
$f(x_{1},x_{2})\in{\mathbb{C}}((x_{1},x_{2}))$, we alternatively define
$f(\phi(x,z),x)$ in two steps via substitutions

${\mathbb{C}}((x_{1},x_{2}))\overset{s_{x_{1}=\phi(x,z)}}{\longrightarrow}{\mathbb{C}}((x,x_{2}))[[z]]\overset{s_{x_{2}=x}}{\longrightarrow}{\mathbb{C}}((x))[[z]]\subset{\mathbb{C}}((x))((z)),$

where $s_{x_{1}=\phi(x,z)}$ and $s_{x_{2}=x}$ denote the indicated
substitution maps. The composition of the two substitutions was proved to be
one-to-one. But, note that the substitution
${\mathbb{C}}((x,x_{2}))[[z]]\overset{s_{x_{2}=x}}{\longrightarrow}{\mathbb{C}}((x))[[z]]$
is not one-to-one. Set

(4.13)
$\displaystyle{\mathbb{C}}((x,x_{2}))[[z]]^{o}=\\{f(x,x_{2},z)\in{\mathbb{C}}((x,x_{2}))[[z]]\
|\ f(x,x,z)\neq 0\\}.$

Note that $f(\phi(x,z),x_{2})\in{\mathbb{C}}((x,x_{2}))[[z]]^{o}$ for
$f(x_{1},x_{2})\in{\mathbb{C}}((x_{1},x_{2}))^{\times}$. Then set

(4.14) $\displaystyle\Lambda(x,x_{2},z)=\left\\{\frac{f}{g}\ |\
f\in{\mathbb{C}}((x,x_{2}))[[z]],\
g\in{\mathbb{C}}((x,x_{2}))[[z]]^{o}\right\\}.$

The substitution map $s_{x_{2}=x}$ extends uniquely to a ring embedding

(4.15) $\displaystyle\tilde{s}_{x_{2}=x}:\
\Lambda(x,x_{2},z)\longrightarrow{\mathbb{C}}((x))((z)).$

Now, for any $F(x_{1},x_{2})\in{\mathbb{C}}_{*}((x_{1},x_{2}))$, we have
$F(\phi(x,z),x_{2})\in\Lambda(x,x_{2},z)$ and

(4.16) $\displaystyle
F(\phi(x,z),x_{2})|_{x_{2}=x}:=\tilde{s}_{x_{2}=x}(F(\phi(x,z),x_{2}))=F(\phi(x,z),x).$

We have the following result:

######  Lemma 4.11.

Let $\phi(x,z)=e^{zp(x)\frac{d}{dx}}x$ with
$p(x)\in{\mathbb{C}}((x))^{\times}$ and let
$F(x_{1},x_{2})\in{\mathbb{C}}_{*}((x_{1},x_{2}))$. Then

(4.17) $\displaystyle F(\phi(x_{1},z),x_{2})=F(x_{1},\phi(x_{2},-z))$

holds in ${\mathbb{C}}_{*}((x_{1},x_{2}))[[z]]$ if and only if

(4.18) $\displaystyle p(x_{1})\frac{\partial}{\partial
x_{1}}F(x_{1},x_{2})=-p(x_{2})\frac{\partial}{\partial x_{2}}F(x_{1},x_{2}).$

Furthermore, assuming either one of the two equivalent conditions, we have

(4.19) $\displaystyle F(\phi(x,z),x)=f(z),$ (4.20)
$\displaystyle\iota_{x,z_{1},z_{2}}F(\phi(x,z_{1}),\phi(x,z_{2}))=f(z_{1}-z_{2})$

for some $f(z)\in{\mathbb{C}}((z))$.

###### Proof.

From definition we have

$F(\phi(x_{1},z),x_{2})=e^{zp(x_{1})\partial_{x_{1}}}F(x_{1},x_{2})\ \mbox{
and }\ F(x_{1},\phi(x_{2},-z))=e^{-zp(x_{2})\partial_{x_{2}}}F(x_{1},x_{2}).$

This immediately confirms the first assertion. For the second assertion,
assuming $F(\phi(x_{1},z),x_{2})=F(x_{1},\phi(x_{2},-z))$, we have

$\displaystyle p(x)\frac{\partial}{\partial x}F(\phi(x,z),x)$
$\displaystyle=\left(p(x)\frac{\partial}{\partial
x}F(\phi(x,z),x_{2})+p(x_{2})\frac{\partial}{\partial
x_{2}}F(\phi(x,z),x_{2})\right)|_{x_{2}=x}$
$\displaystyle=\left(\frac{\partial}{\partial
z}F(\phi(x,z),x_{2})+p(x_{2})\frac{\partial}{\partial
x_{2}}F(x,\phi(x_{2},-z))\right)|_{x_{2}=x}$
$\displaystyle=\left(\frac{\partial}{\partial
z}F(\phi(x,z),x_{2})-\frac{\partial}{\partial
z}F(x,\phi(x_{2},-z))\right)|_{x_{2}=x}$
$\displaystyle=\frac{\partial}{\partial
z}\left(F(\phi(x,z),x_{2})-F(x,\phi(x_{2},-z))\right)|_{x_{2}=x}$
$\displaystyle=0.$

As $p(x)\neq 0$, we get $\frac{\partial}{\partial x}F(\phi(x,z),x)=0$. Thus
$F(\phi(x,z),x)\in{\mathbb{C}}((z))$. Setting $f(z)=F(\phi(x,z),x)$, using
(4.17) we have

$\displaystyle
F(\phi(x,z_{1}),\phi(x,z_{2}))=F(\phi(\phi(x,z_{1}),-z_{2}),x)=F(\phi(x,z_{1}-z_{2}),x)=f(z_{1}-z_{2}),$

where we are using obvious expansion conventions. This completes the proof. ∎

######  Definition 4.12.

Let $\phi(x,z)=e^{zp(x)\frac{d}{dx}}x$ with
$p(x)\in{\mathbb{C}}((x))^{\times}$. Denote by
${\mathbb{C}}_{\phi}((x_{1},x_{2}))$ the set of all
$F(x_{1},x_{2})\in{\mathbb{C}}_{*}((x_{1},x_{2}))$ such that (4.18) holds.

Note that with $\phi(x,z)=e^{zp(x)\frac{d}{dx}}x$, we have

(4.21)
$\displaystyle\partial_{z}\phi(x,z)=\partial_{z}e^{zp(x)\frac{d}{dx}}x=e^{zp(x)\frac{d}{dx}}p(x)=p(\phi(x,z)).$

For $F(x_{1},x_{2})\in{\mathbb{C}}_{\phi}((x_{1},x_{2}))$, we get

$\displaystyle\partial_{z}F(\phi(x_{2},z),x_{2})$ $\displaystyle=$
$\displaystyle\left(\partial_{x_{1}}F(x_{1},x_{2})\right)|_{x_{1}=\phi(x_{2},z)}\partial_{z}\phi(x_{2},z)$
$\displaystyle=$
$\displaystyle\left(p(x_{1})\partial_{x_{1}}F(x_{1},x_{2})\right)|_{x_{1}=\phi(x_{2},z)}.$

Then by a straightforward argument we have:

######  Lemma 4.13.

The set ${\mathbb{C}}_{\phi}((x_{1},x_{2}))$ is a subalgebra of
${\mathbb{C}}_{*}((x_{1},x_{2}))$ with $p(x_{1})\partial_{x_{1}}$ as a
derivation and the map
$\pi_{\phi}:{\mathbb{C}}_{\phi}((x_{1},x_{2}))\rightarrow{\mathbb{C}}((z))$
defined by

(4.22) $\displaystyle\pi_{\phi}(F(x_{1},x_{2}))=F(\phi(x,z),x)$

for $F(x_{1},x_{2})\in{\mathbb{C}}_{\phi}((x_{1},x_{2}))$ is an embedding of
differential algebras, where ${\mathbb{C}}((z))$ is viewed as a differential
algebra with derivation $\frac{d}{dz}$.

With Lemma 4.13 we define a differential subalgebra of ${\mathbb{C}}((z))$

(4.23)
$\displaystyle{\mathbb{C}}_{\phi}((z))=\pi_{\phi}({\mathbb{C}}_{\phi}((x_{1},x_{2})))\subset{\mathbb{C}}((z)).$

We also have the following result, whose proof is straightforward:

######  Lemma 4.14.

Let $F(x_{1},x_{2})\in{\mathbb{C}}_{\phi}((x_{1},x_{2})),\
\lambda\in{\mathbb{C}}^{\times}$ such that $p(\lambda x)=\mu p(x)$ for some
$\mu\in{\mathbb{C}}^{\times}$. Then $F(\lambda x_{1},\lambda
x_{2})\in{\mathbb{C}}_{\phi}((x_{1},x_{2}))$ with

(4.24) $\displaystyle\pi_{\phi}(F(\lambda x_{1},\lambda
x_{2}))(z)=\pi_{\phi}(F(x_{1},x_{2}))(\lambda\mu^{-1}z).$

######  Remark 4.15.

Let $G$ be a group with two linear characters $\chi$ and $\chi_{\phi}$ such
that

$p(\chi_{\phi}(g)x)=\chi(g)^{-1}\chi_{\phi}(g)p(x)\quad\textnormal{{ for
}}g\in G$

(see (4.10)). Assume $F(x_{1},x_{2})\in{\mathbb{C}}_{\phi}((x_{1},x_{2}))$. By
Lemma 4.14 we have

(4.25)
$\displaystyle\pi_{\phi}(F(\chi_{\phi}(g)x_{1},\chi_{\phi}(g)x_{2}))(z)=\pi_{\phi}(F(x_{1},x_{2}))(\chi(g)z).$

Let $G$ act on ${\mathbb{C}}[[x,x^{-1}]]$ by

(4.26) $\displaystyle(\sigma f)(x)=f(\chi(\sigma)^{-1}x)\quad\textnormal{{for
}}\sigma\in G,\ f(x)\in{\mathbb{C}}[[x,x^{-1}]].$

On the other hand, let $G$ act on ${\mathbb{C}}[[x_{1}^{\pm 1},x_{2}^{\pm
1}]]$ by

(4.27) $\displaystyle(\sigma
F)(x_{1},x_{2})=F(\chi_{\phi}(\sigma)^{-1}x_{1},\chi_{\phi}(\sigma)^{-1}x_{2})$

for $\sigma\in G,\ F\in{\mathbb{C}}[[x_{1}^{\pm 1},x_{2}^{\pm 1}]]$. It is
straightforward to see that ${\mathbb{C}}_{\phi}((x))$ and
$C_{\phi}((x_{1},x_{2}))$ are $G$-submodules of ${\mathbb{C}}[[x^{\pm 1}]]$
and ${\mathbb{C}}[[x_{1}^{\pm 1},x_{2}^{\pm 1}]]$, respectively, and
$\pi_{\phi}$ is a $G$-module isomorphism from
${\mathbb{C}}_{\phi}((x_{1},x_{2}))$ onto ${\mathbb{C}}_{\phi}((x))$. For
$f(x)\in{\mathbb{C}}_{\phi}((x))$, set

(4.28)
$\displaystyle\widehat{f}=\pi_{\phi}^{-1}(f)\in{\mathbb{C}}_{\phi}((x_{1},x_{2})).$

Then $\widehat{\sigma f}=\sigma\widehat{f}$ for
$f\in{\mathbb{C}}_{\phi}((x)),\ \sigma\in G$.

######  Remark 4.16.

Let $r\in{\mathbb{Z}}$. Recall (4.1):

$\displaystyle\phi_{r}(x,z)=e^{zx^{r+1}\frac{d}{dx}}x=\begin{cases}x\left(1-rzx^{r}\right)^{\frac{-1}{r}}&\
\ \mbox{ if }r\neq 0\\\ xe^{z}&\ \ \mbox{ if }r=0.\end{cases}$

Set

(4.29) $\displaystyle
F_{r}(z_{1},z_{2})=\begin{cases}-\frac{1}{r}(z_{1}^{-r}-z_{2}^{-r})&\ \ \mbox{
if }r\neq 0\\\ z_{1}/z_{2}&\ \ \mbox{ if }r=0,\end{cases}$

which are elements of ${\mathbb{C}}[z_{1}^{\pm 1},z_{2}^{\pm 1}]$, and set

(4.30) $\displaystyle f_{r}(z)=\begin{cases}z&\ \ \mbox{ if }r\neq 0\\\
e^{z}&\ \ \mbox{ if }r=0.\end{cases}$

It can be readily seen that
$F_{r}(x_{1},x_{2})\in{\mathbb{C}}_{\phi_{r}}((x_{1},x_{2}))$ with
$\pi_{\phi_{r}}(F_{r})=f_{r}$.

######  Remark 4.17.

Consider the special case $\phi(x,z)=e^{zx\frac{d}{dx}}x=xe^{z}$. It is
straightforward to show that for $F(x_{1},x_{2})\in{\mathbb{C}}[[x_{1}^{\pm
1},x_{2}^{\pm 1}]]$,

$x_{1}\frac{\partial}{\partial
x_{1}}F(x_{1},x_{2})=-x_{2}\frac{\partial}{\partial x_{2}}F(x_{1},x_{2})$

if and only if $F(x_{1},x_{2})=f(x_{1}/x_{2})$ for some
$f(z)\in{\mathbb{C}}[[z,z^{-1}]]$. From this we get

(4.31)
$\displaystyle{\mathbb{C}}((x_{1},x_{2}))\cap{\mathbb{C}}_{\phi}((x_{1},x_{2}))={\mathbb{C}}[x_{12}^{\pm
1}]$

(the Laurent polynomial ring in $x_{12}$), where $x_{12}={x_{1}}/{x_{2}}$. It
is clear that

(4.32)
$\displaystyle{\mathbb{C}}(x_{1}/x_{2})\subset{\mathbb{C}}_{\phi}((x_{1},x_{2})),$

where ${\mathbb{C}}(x_{1}/x_{2})$ denotes the field of rational functions in
$x_{1}/x_{2}$. Notice that
${\mathbb{C}}(x_{1}/x_{2})={\mathbb{C}}(x_{2}/x_{1})$ inside
${\mathbb{C}}(x_{1},x_{2})$ $(\subset{\mathbb{C}}_{*}((x_{1},x_{2})))$.

Note that in case $\phi(x,z)=e^{z\frac{d}{dx}}x=x+z$, we have

(4.33)
$\displaystyle{\mathbb{C}}_{\phi}((x_{1},x_{2}))\supset{\mathbb{C}}((x_{1}-x_{2})),\
\ {\mathbb{C}}((x_{2}-x_{1})),\ \ {\mathbb{C}}(x_{1}-x_{2}).$

For any subgroup $\Gamma$ of ${\mathbb{C}}^{\times}$, set

(4.34)
$\displaystyle{\mathbb{C}}_{*}^{\Gamma}((x_{1},x_{2}))=\left\\{\frac{F(x_{1},x_{2})}{q(x_{2}/x_{1})}\mid
F(x_{1},x_{2})\in{\mathbb{C}}((x_{1},x_{2})),\
q(x)\in{\mathbb{C}}_{\Gamma}[x]\right\\},$

recalling that ${\mathbb{C}}_{\Gamma}[x]$ denotes the monoid generated
multiplicatively by polynomials $x-\alpha$ for $\alpha\in\Gamma$. Furthermore,
set

(4.35)
$\displaystyle{\mathbb{C}}^{\Gamma}_{\phi}((x_{1},x_{2}))={\mathbb{C}}_{\phi}((x_{1},x_{2}))\cap{\mathbb{C}}^{\Gamma}_{*}((x_{1},x_{2})).$

It is straightforward to see that
${\mathbb{C}}^{\Gamma}_{\phi}((x_{1},x_{2}))$ is a differential subalgebra of
${\mathbb{C}}_{\phi}((x_{1},x_{2}))$. Consequently,
$\pi_{\phi}\left({\mathbb{C}}^{\Gamma}_{\phi}((x_{1},x_{2}))\right)$ is a
differential subalgebra of ${\mathbb{C}}_{\phi}((x))$.

######  Definition 4.18.

Let $H$ be a nonlocal vertex bialgebra. A $\phi$-compatible $H$-module
nonlocal vertex algebra is an $H$-module nonlocal vertex algebra
$(V,Y_{V}^{H})$ such that

(4.36) $\displaystyle Y_{V}^{H}(h,z)v\in
V\otimes{\mathbb{C}}_{\phi}((z))\quad\textnormal{{for }}h\in H,\,v\in V.$

If $H$ is a $(G,\chi)$-module nonlocal vertex bialgebra, we in addition assume
that for $h\in H,\ v\in V$, there exists
$q(x)\in{\mathbb{C}}_{\chi_{\phi}(G)}[x]$ such that

(4.37) $\displaystyle q(x_{2}/x_{1})\widehat{Y}_{V}^{H}(h,x_{1},x_{2})v\in$
$\displaystyle V\otimes{\mathbb{C}}((x_{1},x_{2})),$

where
$\widehat{Y}_{V}^{H}(h,x_{1},x_{2})v=(1\otimes\pi_{\phi}^{-1})Y_{V}^{H}(h,x)v$.

Note that we can alternatively combine the conditions (4.36) and (4.37) to
write

(4.38) $\displaystyle Y_{V}^{H}(h,z)v\in
V\otimes\pi_{\phi}({\mathbb{C}}_{\phi}^{\chi_{\phi}(G)}((x_{1},x_{2})))\quad\textnormal{{
for }}h\in H,\ v\in V.$

For convenience, we formulate the following technical result:

######  Lemma 4.19.

Let $(V,Y_{V}^{H})$ be an $H$-module nonlocal vertex algebra. Assume that
(4.36) holds for $h\in S,\ v\in U$, where $S$ is a generating subspace of $H$
as a nonlocal vertex algebra such that $\Delta(S)\subset S\otimes S$ and $U$
is a generating subset of $V$. Then $(V,Y_{V}^{H})$ is $\phi$-compatible.

###### Proof.

Set $V^{\prime}=\\{v\in V\mid Y_{V}^{H}(h,x)v\in
V\otimes{\mathbb{C}}_{\phi}((x))\ \textnormal{{for }}h\in S\\}$. Then
$U\cup\\{{\bf 1}\\}\subset V^{\prime}$. Let $h\in S,\ u,v\in V^{\prime}$. With
$V$ an $H$-module nonlocal vertex algebra, we have

$Y_{V}^{H}(h,x)Y(u,z)v=\sum
Y(Y_{V}^{H}(h_{(1)},x-z)u,z)Y_{V}^{H}(h_{(2)},x)v.$

Writing

$Y_{V}^{H}(h_{(1)},x)u=\sum_{i=1}^{r}a^{(i)}\otimes f_{i}(x),\quad
Y_{V}^{H}(h_{(2)},x)v=\sum_{j=1}^{s}b^{(j)}\otimes g_{i}(x),$

where $a^{(i)},b^{(j)}\in V,\ f_{i}(x),g_{j}(x)\in{\mathbb{C}}_{\phi}((x))$,
we have

$\displaystyle
Y_{V}^{H}(h,x)Y(u,z)v=\sum_{i,j}g_{j}(x)e^{-z\frac{\partial}{\partial
x}}f_{i}(x)Y(a^{(i)},z)b^{(j)}.$

From this we get $Y_{V}^{H}(h,x)u_{m}v\in V\otimes{\mathbb{C}}_{\phi}((x))$
for all $m\in{\mathbb{Z}}$, recalling that ${\mathbb{C}}_{\phi}((x))$ is a
subalgebra of ${\mathbb{C}}((x))$, which is closed under the derivation
$\frac{d}{dx}$. As $U$ generates $V$, it follows that $V^{\prime}=V$. Thus
$Y_{V}^{H}(h,x)v\in V\otimes{\mathbb{C}}_{\phi}((x))$ for all $h\in S,\ v\in
V$.

Furthermore, for $h,h^{\prime}\in H,\ v\in V$, by Lemma 2.17 we have

$Y_{V}^{H}(Y(h,z)h^{\prime},x)v=Y_{V}^{H}(h,x+z)Y_{V}^{H}(h^{\prime},x)v.$

Using a similar argument we get $Y_{V}^{H}(h_{m}h^{\prime},x)v\in
V\otimes{\mathbb{C}}_{\phi}((x))$ for all $m\in{\mathbb{Z}}$. It follows that
(4.36) holds for all $h\in H,\ v\in V$. Therefore, $(V,Y_{V}^{H})$ is
$\phi$-compatible. ∎

Let $(V,Y_{V}^{H})$ be a $\phi$-compatible $H$-module nonlocal vertex algebra.
From definition, we have

(4.39) $\displaystyle\widehat{Y}_{V}^{H}(h,x_{1},x_{2})v\in
V\otimes{\mathbb{C}}_{\phi}((x_{1},x_{2}))\quad\textnormal{{for }}h\in H,\
v\in V.$

The following are immediate consequences:

######  Lemma 4.20.

Let $(V,Y_{V}^{H})$ be a $\phi$-compatible $H$-module nonlocal vertex algebra.
For $h,h^{\prime}\in H,\ v\in V$, we have

(4.40)
$\displaystyle\widehat{Y}_{V}^{H}(h,\phi(x_{1},z),x_{2})=\widehat{Y}_{V}^{H}(h,x_{1},\phi(x_{2},-z)),$
(4.41)
$\displaystyle\widehat{Y}_{V}^{H}\left(Y(h,z)h^{\prime},x_{1},x_{2}\right)=\widehat{Y}_{V}^{H}(h,\phi(x_{1},z),x_{2})\widehat{Y}_{V}^{H}(h^{\prime},x_{1},x_{2}),$
(4.42) $\displaystyle\widehat{Y}_{V}^{H}(h,x_{1},x_{2})Y(v,x_{3})=\sum
Y(\widehat{Y}_{V}^{H}(h_{(1)},x_{1},\phi(x_{2},x_{3}))v,x_{3})\widehat{Y}_{V}^{H}(h_{(2)},x_{1},x_{2}).$

Furthermore, we have

(4.43) $\displaystyle
g(\phi(x,z),x)\widehat{Y}_{V}^{H}(h,\phi(x,z),x)v=g(\phi(x,z),x)Y_{V}^{H}(h,z)v$

for any $g(x_{1},x_{2})\in{\mathbb{C}}((x_{1},x_{2}))$ such that

$g(x_{1},x_{2})\widehat{Y}_{V}^{H}(h,x_{1},x_{2})v\in
V\otimes{\mathbb{C}}((x_{1},x_{2})).$

######  Definition 4.21.

Let $H$ be a $(G,\chi)$-module nonlocal vertex bialgebra, let $(V,Y_{V}^{H})$
be a $\phi$-compatible $H$-module nonlocal vertex algebra, and let
$\chi_{\phi}$ be a linear character of $G$ satisfying (4.7). A
$(G,\chi_{\phi})$-equivariant $\phi$-coordinated quasi $(H,V)$-module is a
$(G,\chi_{\phi})$-equivariant $\phi$-coordinated quasi $H$-module
$(W,Y_{W}^{H})$, equipped with a $(G,\chi_{\phi})$-equivariant
$\phi$-coordinated quasi $V$-module structure $Y_{W}^{V}(\cdot,x)$, such that

(4.44) $\displaystyle Y_{W}^{H}(h,x)w\in W\otimes{\mathbb{C}}((x)),$ (4.45)
$\displaystyle Y_{W}^{H}(h,x_{1})Y_{W}^{V}(v,x_{2})=\sum
Y_{W}^{V}\left(\iota_{x_{2},x_{1}}\widehat{Y}_{V}^{H}(h_{(1)},x_{1},x_{2})v,x_{2}\right)Y_{W}^{H}(h_{(2)},x_{1})$

for $h\in H$, $v\in V,$ $w\in W$.

The following lemma follows immediately from the proof of Lemma 2.17:

######  Lemma 4.22.

Let $(W,Y_{W}^{H})$ be a $\phi$-coordinated quasi $H$-module such that (4.44)
holds for any $h\in H,\ w\in W$. Then

(4.46) $\displaystyle
Y_{W}^{H}(h,x_{1})Y_{W}^{H}(k,x_{2})|_{x_{1}=\phi(x_{2},z)}=Y_{W}^{H}(Y(h,z)k,x_{2})$

for $h,k\in H$.

As the main result of this section, we have (cf. Proposition 2.15):

######  Theorem 4.23.

Let $H$ be a $(G,\chi)$-module nonlocal vertex bialgebra, let $(V,Y_{V}^{H})$
be a $\phi$-compatible $H$-module nonlocal vertex algebra, and let
$(W,Y_{W}^{H},Y_{W}^{V})$ be a $(G,\chi_{\phi})$-equivariant
$\phi$-coordinated quasi $(H,V)$-module. In addition, assume

$\displaystyle R_{V}(g)Y_{V}^{H}(h,x)v=Y_{V}^{H}(R_{H}(g)h,\chi(g)x)R_{V}(g)v\
\ \ \mbox{ for }g\in G,\ h\in H,\ v\in V$

(see Proposition 4.8). For $h\in H,\ v\in V$, set

(4.47) $\displaystyle Y_{W}^{\sharp}(v\otimes
h,x)=Y_{W}^{V}(v,x)Y_{W}^{H}(h,x)\in({\rm End}W)[[x,x^{-1}]].$

Then $(W,Y_{W}^{\sharp})$ is a $(G,\chi_{\phi})$-equivariant
$\phi$-coordinated quasi $V\sharp H$-module.

###### Proof.

First of all, for $v\in V,\ h\in H$, with the property (4.44) we have

$Y_{W}^{\sharp}(v\otimes
h,x)=Y_{W}^{V}(v,x)Y_{W}^{H}(h,x)\in\textnormal{{Hom}}(W,W((x))).$

We also have

$Y_{W}^{\sharp}({\bf 1}\otimes{\bf 1},x)=Y_{W}^{V}({\bf 1},x)Y_{W}^{H}({\bf
1},x)=1.$

Let $u,v\in V$, $h,k\in H$. From definition, there exists
$q(x)\in{\mathbb{C}}_{\chi(G)}[x]$ such that

$\displaystyle
q(x_{1}/x_{2})\widehat{Y}_{V}^{H}\left(h_{(1)},x_{1},x_{2}\right)v\in
V\otimes{\mathbb{C}}((x_{1},x_{2})),$ $\displaystyle
q(x_{1}/x_{2})Y_{W}^{V}(u,x_{1})Y_{W}^{V}\left(\widehat{Y}_{V}^{H}(h_{(1)},x_{1},x_{2})v,x_{2}\right)\in\textnormal{{Hom}}(W,W((x_{1},x_{2}))).$

On the other hand, from (4.44) we have

$Y_{W}^{H}(h_{(2)},x_{1})Y_{W}^{H}(k,x_{2})\in\textnormal{{Hom}}(W,W\otimes{\mathbb{C}}((x_{1},x_{2}))).$

Then

$\displaystyle q(x_{1}/x_{2})Y_{W}^{\sharp}(u\otimes
h,x_{1})Y_{W}^{\sharp}(v\otimes k,x_{2})$ $\displaystyle=\ $ $\displaystyle
q(x_{1}/x_{2})\sum
Y_{W}^{V}(u,x_{1})Y_{W}^{V}\left(\iota_{x_{2},x_{1}}\widehat{Y}_{V}^{H}(h_{(1)},x_{1},x_{2})v,x_{2}\right)Y_{W}^{H}(h_{(2)},x_{1})Y_{W}^{H}(k,x_{2}),$

which implies that the common quantity on both sides lies in
$\textnormal{{Hom}}(W,W((x_{1},x_{2})))$. Using all of these, Lemma 4.22, and
(4.43), we get

$\displaystyle\left(q(x_{1}/x_{2})Y_{W}^{\sharp}(u\otimes
h,x_{1})Y_{W}^{\sharp}(v\otimes k,x_{2})\right)|_{x_{1}=\phi(x_{2},z)}$
$\displaystyle=\ $ $\displaystyle\Big{(}q(x_{1}/x_{2})\sum
Y_{W}^{V}(u,x_{1})Y_{W}^{V}\left(\widehat{Y}_{V}^{H}(h_{(1)},x_{1},x_{2})v,x_{2}\right)Y_{W}^{H}(h_{(2)},x_{1})Y_{W}^{H}(k,x_{2})\Big{)}|_{x_{1}=\phi(x_{2},z)}$
$\displaystyle=\ $ $\displaystyle
q(\phi(x_{2},z)/x_{2})Y_{W}^{V}\left(Y(u,z)\widehat{Y}_{V}^{H}(h_{(1)},\phi(x_{2},z),x_{2})v,x_{2}\right)Y_{W}^{H}(Y(h_{(2)},z)k,x_{2})$
$\displaystyle=\ $ $\displaystyle
q(\phi(x_{2},z)/x_{2})Y_{W}^{V}(Y(u,z)Y_{V}^{H}(h_{(1)},z)v,x_{2})Y_{W}^{H}(Y(h_{(2)},z)k,x_{2})$
$\displaystyle=\ $ $\displaystyle
q(\phi(x_{2},z)/x_{2})Y_{W}^{\sharp}(Y^{\sharp}(u\otimes h,z)(v\otimes
k),x_{2}).$

This proves that $(W,Y_{W}^{\sharp})$ carries the structure of a
$\phi$-coordinated quasi $V\sharp H$-module.

Furthermore, for $g\in G,\ v\in V,\ h\in H$, we have

$\displaystyle Y_{W}^{\sharp}$ $\displaystyle\left((R_{V}(g)\otimes
R_{H}(g))(v\otimes h),x\right)=\ Y_{W}^{V}(R_{V}(g)v,x)Y_{W}^{H}(R_{H}(g)h,x)$
$\displaystyle=\
Y_{W}^{V}(v,\chi_{\phi}(g)^{-1}x)Y_{W}^{H}(h,\chi_{\phi}(g)^{-1}x)=Y_{W}^{\sharp}(u\otimes
h,\chi_{\phi}(g)^{-1}x).$

This proves that $(W,Y_{W}^{\sharp})$ is $(G,\chi_{\phi})$-equivariant. ∎

Recall from Theorem 2.25 that the right $H$-comodule map $\rho:V\rightarrow
V\otimes H$ is also a nonlocal vertex algebra homomorphism from
$\mathfrak{D}_{Y_{V}^{H}}^{\rho}(V)$ to $V\sharp H$. Note that
$\mathfrak{D}_{Y_{V}^{H}}^{\rho}(V)$ is a $(G,\chi)$-module nonlocal vertex
algebra. As an immediate consequence of Theorem 4.23 we have:

######  Corollary 4.24.

Under the setting of Theorem 4.23, for $v\in V$, set

(4.48) $\displaystyle Y_{W}^{\sharp\rho}(v,x)=\sum
Y_{W}^{V}(v_{(1)},x)Y_{W}^{H}(v_{(2)},x)\
\left(=Y_{W}^{\sharp}(\rho(v),x)\right).$

In addition we assume that $H$ is cocommutative. Then $(W,Y_{W}^{\sharp\rho})$
is a $(G,\chi_{\phi})$-equivariant $\phi$-coordinated quasi
$\mathfrak{D}_{Y_{V}^{H}}^{\rho}(V)$-module.

For any subgroup $\Gamma$ of ${\mathbb{C}}^{\times}$, set

(4.49) $\displaystyle{\mathbb{C}}_{\Gamma}[x_{1},x_{2}]=\langle x_{1}-\alpha
x_{2}\ |\ \alpha\in\Gamma\rangle,$

the multiplicative moniod generated by $x_{1}-\alpha x_{2}$ for
$\alpha\in\Gamma$ in ${\mathbb{C}}[x_{1},x_{2}]$.

######  Lemma 4.25.

Let $V$ be a $(G,\chi)$-module nonlocal vertex algebra and let
$(W,Y_{W}^{\phi})$ be a $(G,\chi_{\phi})$-equivariant $\phi$-coordinated quasi
$V$-module. Suppose that $u,v\in V$, and

$\displaystyle u^{(i)},v^{(i)}\in V,\ f_{i}(x)\in{\mathbb{C}}_{\phi}((x))\ \
(i=1,\dots,r)$

such that for some nonnegative integer $k$,

$\displaystyle(x_{1}-x_{2})^{k}Y(u,x_{1})Y(v,x_{2})$ (4.50) $\displaystyle=$
$\displaystyle(x_{1}-x_{2})^{k}\sum_{i=1}^{r}f_{i}(-x_{2}+x_{1})Y(v^{(i)},x_{2})Y(u^{(i)},x_{1}).$

Then there exists
$q(x_{1},x_{2})\in{\mathbb{C}}_{\chi_{\phi}(G)}[x_{1},x_{2}]$ such that

$\displaystyle q(x_{1},x_{2})Y_{W}^{\phi}(u,x_{1})Y_{W}^{\phi}(v,x_{2})$
(4.51) $\displaystyle=\ $ $\displaystyle
q(x_{1},x_{2})\sum_{i=1}^{r}\iota_{x_{2},x_{1}}\left(\widehat{f_{i}}(x_{1},x_{2})\right)Y_{W}^{\phi}(v^{(i)},x_{2})Y_{W}^{\phi}(u^{(i)},x_{1}).$

Furthermore, (4.25) holds for any
$q(x_{1},x_{2})\in{\mathbb{C}}((x_{1},x_{2}))$ such that

$\displaystyle
q(x_{1},x_{2})Y_{W}^{\phi}(u,x_{1})Y_{W}^{\phi}(v,x_{2})\in\textnormal{{Hom}}(W,W((x_{1},x_{2}))).$

###### Proof.

With (4.25), from [Li4, Corollary 5.3] we have

$\displaystyle
Y(u,x)v=\sum_{i=1}^{r}f_{i}(x)e^{x\mathcal{D}}Y(v^{(i)},-x)u^{(i)}.$

From definition, there exists
$h(x_{1},x_{2})\in{\mathbb{C}}_{\chi_{\phi}(G)}[x_{1},x_{2}]$ such that

$\displaystyle
h(x_{1},x_{2})Y_{W}^{\phi}(u,x_{1})Y_{W}^{\phi}(v,x_{2})\in\textnormal{{Hom}}(W,W((x_{1},x_{2}))),$

$h(x_{1},x_{2})\widehat{f}_{i}(x_{1},x_{2})\in{\mathbb{C}}((x_{1},x_{2}))$,
and

$\displaystyle(h(x_{1},x_{2})\widehat{f}_{i}(x_{1},x_{2}))Y_{W}^{\phi}(v^{(i)},x_{2})Y_{W}(u^{(i)},x_{1})\in\textnormal{{Hom}}(W,W((x_{1},x_{2})))$

for $i=1,\dots,r$. Then using [Li9, Lemma 3.7] we get

$\displaystyle(h(x_{1},x_{2})Y_{W}^{\phi}(u,x_{1})Y_{W}^{\phi}(v,x_{2}))|_{x_{1}=\phi(x_{2},z)}$
$\displaystyle=\ $ $\displaystyle
h(\phi(x_{2},z),x_{2})Y_{W}^{\phi}(Y(u,z)v,x_{2})$ $\displaystyle=\ $
$\displaystyle\sum_{i=1}^{r}(h(\phi(x_{2},z),x_{2})f_{i}(z)Y_{W}^{\phi}(e^{z\mathcal{D}}(v^{(i)},-z)u^{(i)},x_{2})$
$\displaystyle=\ $
$\displaystyle\sum_{i=1}^{r}(h(\phi(x_{2},z),x_{2})f_{i}(z)Y_{W}^{\phi}(Y(v^{(i)},-z)u^{(i)},\phi(x_{2},z)).$

On the other hand, we have

$\displaystyle\left(h(x_{1},x_{2})\sum_{i=1}^{r}\iota_{x_{2},x_{1}}(\widehat{f}_{i}(x_{1},x_{2}))Y_{W}^{\phi}(v^{(i)},x_{2})Y_{W}^{\phi}(u^{(i)},x_{1})\right)|_{x_{2}=\phi(x_{1},-z)}$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{r}h(x_{1},\phi(x_{1},-z))\widehat{f}_{i}(x_{1},\phi(x_{1},-z))Y_{W}^{\phi}(Y(v^{(i)},-z)u^{(i)},x_{1})$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{r}h(x_{1},\phi(x_{1},-z))f_{i}(z)Y_{W}^{\phi}(Y(v^{(i)},-z)u^{(i)},x_{1}).$

Then using [Li9, Remark 2.8] we have

$\displaystyle(h(x_{1},x_{2})Y_{W}^{\phi}(u,x_{1})Y_{W}^{\phi}(v,x_{2}))|_{x_{1}=\phi(x_{2},z)}$
$\displaystyle=$
$\displaystyle\left(\left(h(x_{1},x_{2})\sum_{i=1}^{r}\widehat{f}_{i}(x_{1},x_{2})Y_{W}^{\phi}(v^{(i)},x_{2})Y_{W}^{\phi}(u^{(i)},x_{1})\right)|_{x_{2}=\phi(x_{1},-z)}\right)|_{x_{1}=\phi(x_{2},z)}$
$\displaystyle=$
$\displaystyle\left(h(x_{1},x_{2})\sum_{i=1}^{r}\iota_{x_{2},x_{1}}(\widehat{f}_{i}(x_{1},x_{2}))Y_{W}^{\phi}(v^{(i)},x_{2})Y_{W}^{\phi}(u^{(i)},x_{1})\right)|_{x_{1}=\phi(x_{2},z)},$

noticing that $\phi(\phi(x_{1},z),-z)=\phi(x_{1},0)=x_{1}$. By [Li9, Remark
2.8] again we get

$\displaystyle
h(x_{1},x_{2})Y_{W}^{\phi}(u,x_{1})Y_{W}^{\phi}(v,x_{2})=h(x_{1},x_{2})\sum_{i=1}^{r}\iota_{x_{2},x_{1}}(\widehat{f}_{i}(x_{1},x_{2}))Y_{W}^{\phi}(v^{(i)},x_{2})Y_{W}^{\phi}(u^{(i)},x_{1}).$

This proves the first assertion. As for the second assertion we have

$\displaystyle
h(x_{1},x_{2})\left[q(x_{1},x_{2})Y_{W}^{\phi}(u,x_{1})Y_{W}^{\phi}(v,x_{2})\right]$
$\displaystyle=\ $ $\displaystyle
h(x_{1},x_{2})q(x_{1},x_{2})\sum_{i=1}^{r}\widehat{f}_{i}(x_{1},x_{2})Y_{W}^{\phi}(v^{(i)},x_{2})Y_{W}^{\phi}(u^{(i)},x_{1}).$

Multiplying both sides by the inverse of $h(x_{1},x_{2})$ in
${\mathbb{C}}((x_{2}))((x_{1}))$ we obtain the desired relation. ∎

######  Remark 4.26.

Let $p(x)\in{\mathbb{C}}((x))^{\times}$. It follows from induction that for
every positive integer $n$, there exist
$A_{n,0}(x),\dots,A_{n,n}(x)\in{\mathbb{C}}((x))$ with $A_{n,n}(x)=p(x)^{-n}$
such that

(4.52)
$\displaystyle\left(\frac{d}{dx}\right)^{n}=A_{n,0}(x)+A_{n,1}(x)p(x)\frac{d}{dx}+\cdots+A_{n,n}(x)\left(p(x)\frac{d}{dx}\right)^{n}$

on ${\mathbb{C}}((x))$. For $B_{n,0}(x),\dots,B_{n,n}(x)\in{\mathbb{C}}((x))$,
it can be readily seen that

$B_{n,0}(x)+B_{n,1}(x)\frac{d}{dx}+\cdots+B_{n,n}(x)\left(\frac{d}{dx}\right)^{n}=0$

if and only if $B_{n,i}(x)=0$ for $i=0,\dots,n$. Then it follows that those
coefficients $A_{n,i}(x)$ for $0\leq i\leq n$ in (4.52) are uniquely
determined.

The following result generalizes the corresponding results in [Li4] and
[Li10]:

######  Proposition 4.27.

Let $V$ be a $(G,\chi)$-module nonlocal vertex algebra, $(W,Y_{W}^{\phi})$ a
$(G,\chi_{\phi})$-equivariant $\phi$-coordinated quasi $V$-module, and let
$G^{0}$ be a complete set of coset representatives of $\ker\chi_{\phi}$ in
$G$. Assume $u,v\in V$, and

$\displaystyle u^{(i)},v^{(i)}\in V,\ f_{i}(x)\in{\mathbb{C}}_{\phi}((x))\ \
(i=1,\dots,r)$

such that

$\displaystyle(x_{1}-x_{2})^{k}Y(u,x_{1})Y(v,x_{2})=(x_{1}-x_{2})^{k}\sum_{i=1}^{r}f_{i}(-x_{2}+x_{1})Y(v^{(i)},x_{2})Y(u^{(i)},x_{1})$

for some nonnegative integer $k$. Then

$\displaystyle
Y_{W}^{\phi}(u,x_{1})Y_{W}^{\phi}(v,x_{2})-\sum_{i=1}^{r}\iota_{x_{2},x_{1}}(\widehat{f_{i}}(x_{1},x_{2}))Y_{W}^{\phi}(v^{(i)},x_{2})Y_{W}^{\phi}(u^{(i)},x_{1})$
(4.53) $\displaystyle=$ $\displaystyle\sum_{g\in G^{0}}\sum_{j\geq
0}Y_{W}^{\phi}\left((R(g)u)_{j}v,x_{2}\right)\frac{1}{j!}(p(x_{2})\partial_{x_{2}})^{j}p(x_{1})x_{1}^{-1}\delta\left(\chi_{\phi}(g)^{-1}\frac{x_{2}}{x_{1}}\right).$

###### Proof.

From Lemma 4.25, there exist distinct (nonzero complex numbers)
$c_{1},\dots,c_{s}\in\chi_{\phi}(G)$ and positive integers $k_{1},\dots,k_{s}$
such that (4.25) holds with

$q(x_{1},x_{2})=(x_{1}-c_{1}x_{2})^{k_{1}}\cdots(x_{1}-c_{s}x_{2})^{k_{s}}.$

By [Li3, Lemma 2.5], we have

$\displaystyle
p(x_{1})^{-1}Y_{W}^{\phi}(u,x_{1})Y_{W}^{\phi}(v,x_{2})-p(x_{1})^{-1}\sum_{i=1}^{r}\iota_{x_{2},x_{1}}(\widehat{f_{i}}(x_{1},x_{2}))Y_{W}^{\phi}(v^{(i)},x_{2})Y_{W}^{\phi}(u^{(i)},x_{1})$
$\displaystyle=\ $
$\displaystyle\sum_{j=1}^{s}\sum_{t=0}^{k_{j}-1}C_{j,t}(x_{2})\left(\frac{\partial}{\partial
x_{2}}\right)^{t}x_{1}^{-1}\delta\left(\frac{c_{j}x_{2}}{x_{1}}\right),$

where $C_{j,t}(x)\in{\mathcal{E}}(W)$. Furthermore, using Remark 4.26, we get

$\displaystyle
Y_{W}^{\phi}(u,x_{1})Y_{W}^{\phi}(v,x_{2})-\sum_{i=1}^{r}\iota_{x_{2},x_{1}}(\widehat{f_{i}}(x_{1},x_{2}))Y_{W}^{\phi}(v^{(i)},x_{2})Y_{W}^{\phi}(u^{(i)},x_{1})$
(4.54) $\displaystyle=\ $
$\displaystyle\sum_{j=1}^{s}\sum_{t=0}^{k_{j}-1}A_{j,t}(x_{2})\frac{1}{t!}\left(p(x_{2})\frac{\partial}{\partial
x_{2}}\right)^{t}p(x_{1})x_{1}^{-1}\delta\left(\frac{c_{j}x_{2}}{x_{1}}\right),$

where $A_{j,t}(x)\in{\mathcal{E}}(W)$. By [JKLT, Theorems 2.19, 2.21] and
Lemma 2.6 we have

(4.55) $\displaystyle
Y_{W}^{\phi}(u,c_{j}x)_{t}^{\phi}Y_{W}^{\phi}(v,x)=\begin{cases}A_{j,t}(x)&\quad\textnormal{{
for }}1\leq j\leq s,\ 0\leq t\leq k_{j}-1,\\\ 0&\quad\textnormal{{ for }}1\leq
j\leq s,\ t\geq k_{j}.\end{cases}$

Let $g_{1},\dots,g_{r}\in G^{0}$ such that $c_{j}=\chi_{\phi}(g_{j})^{-1}$ for
$1\leq j\leq s$. Then using [Li9, Proposition 4.11] we get

$\displaystyle
Y_{W}^{\phi}(u,c_{j}x)_{t}^{\phi}Y_{W}^{\phi}(v,x)=Y_{W}^{\phi}(R(g_{j})u,x)_{t}^{\phi}Y_{W}^{\phi}(v,x)=Y_{W}^{\phi}((R(g_{j})u)_{t}v,x).$

Consequently, we have

$\displaystyle
Y_{W}^{\phi}(u,x_{1})Y_{W}^{\phi}(v,x_{2})-\sum_{i=1}^{r}\iota_{x_{2},x_{1}}(\widehat{f}_{i}(x_{1},x_{2}))Y_{W}^{\phi}(v^{(i)},x_{2})Y_{W}^{\phi}(u^{(i)},x_{1})$
$\displaystyle=\ $
$\displaystyle\sum_{j=1}^{s}\sum_{t=0}^{k_{j}-1}Y_{W}^{\phi}((R(g_{j})u)_{t}v,x_{2})\frac{1}{t!}(p(x_{2})\partial_{x_{2}})^{t}p(x_{1})x_{1}^{-1}\delta\left(\chi_{\phi}(g_{j})^{-1}\frac{x_{2}}{x_{1}}\right).$

Suppose $g\in G^{0}$ such that
$\chi_{\phi}(g)^{-1}\notin\\{c_{1},\dots,c_{s}\\}$. By [JKLT, Theorems 2.19,
2.21] also (by adding a term with zero coefficients corresponding to $c$ on
the right-hand side of (4)) we have $Y_{W}^{\phi}((R(g)u)_{t}v,x)=0$ for
$t\geq 0$. Therefore (4.27) holds. This completes the proof. ∎

##  5\. Deformations of lattice vertex algebras $V_{L}$

In this section, we use the vertex bialgebra $B_{L}$ introduced in [Li8] to
study the lattice vertex algebra $V_{L}$. More specifically, we give a right
$B_{L}$-comodule nonlocal vertex algebra structure and a family of compatible
$B_{L}$-module nonlocal vertex algebra structures on $V_{L}$. As an
application, we obtain a family of deformations of $V_{L}$.

###  5.1. Vertex algebra $V_{L}$ and associative algebra $A(L)$

We first recall the lattice vertex algebra $V_{L}$ (see [B1], [FLM]). Let $L$
be a finite rank even lattice, i.e., a free abelian group of finite rank
equipped with a symmetric ${\mathbb{Z}}$-valued bilinear form
${\langle}\cdot,\cdot{\rangle}$ such that ${\langle}\alpha,\alpha{\rangle}\in
2{\mathbb{Z}}$ for $\alpha\in L$. We assume that $L$ is nondegenerate in the
obvious sense. Set

(5.1) $\displaystyle{\mathfrak{h}}={\mathbb{C}}\otimes_{{\mathbb{Z}}}L$

and extend ${\langle}\cdot,\cdot{\rangle}$ to a symmetric
${\mathbb{C}}$-valued bilinear form on ${\mathfrak{h}}$. View ${\mathfrak{h}}$
as an abelian Lie algebra with ${\langle}\cdot,\cdot{\rangle}$ as a
nondegenerate symmetric invariant bilinear form. Then we have an affine Lie
algebra $\widehat{{\mathfrak{h}}}$, where

$\widehat{{\mathfrak{h}}}={\mathfrak{h}}\otimes{\mathbb{C}}[t,t^{-1}]+{\mathbb{C}}{\bf
k}$

as a vector space, and where ${\bf k}$ is central and

(5.2)
$\displaystyle[\alpha(m),\beta(n)]=m\delta_{m+n,0}{\langle}\alpha,\beta{\rangle}{\bf
k}$

for $\alpha,\beta\in{\mathfrak{h}},\ m,n\in{\mathbb{Z}}$ with $\alpha(m)$
denoting $\alpha\otimes t^{m}$. Set

(5.3) $\displaystyle\widehat{{\mathfrak{h}}}^{\pm}={\mathfrak{h}}\otimes
t^{\pm 1}{\mathbb{C}}[t^{\pm 1}],$

which are abelian Lie subalgebras. Identify ${\mathfrak{h}}$ with
${\mathfrak{h}}\otimes t^{0}$. Furthermore, set

(5.4)
$\displaystyle\widehat{{\mathfrak{h}}}^{\prime}=\widehat{{\mathfrak{h}}}^{+}+\widehat{{\mathfrak{h}}}^{-}+{\mathbb{C}}{\bf
k},$

which is a Heisenberg algebra. Then
$\widehat{{\mathfrak{h}}}=\widehat{{\mathfrak{h}}}^{\prime}\oplus{\mathfrak{h}}$,
which is a direct sum of Lie algebras.

Let $\varepsilon:L\times L\rightarrow{\mathbb{C}}^{\times}$ be a 2-cocycle
such that

$\displaystyle\varepsilon(\alpha,\beta)\varepsilon(\beta,\alpha)^{-1}=(-1)^{{\langle}\alpha,\beta{\rangle}},\
\ \ \varepsilon(\alpha,0)=1=\varepsilon(0,\alpha)\quad\textnormal{{ for
}}\alpha,\beta\in L.$

Denote by ${\mathbb{C}}_{\varepsilon}[L]$ the $\varepsilon$-twisted group
algebra of $L$, which by definition has a designated basis $\\{e_{\alpha}\ |\
\alpha\in L\\}$ with relations

(5.5) $\displaystyle e_{\alpha}\cdot
e_{\beta}=\varepsilon(\alpha,\beta)e_{\alpha+\beta}\quad\textnormal{{ for
}}\alpha,\beta\in L.$

Make ${\mathbb{C}}_{\varepsilon}[L]$ an $\widehat{{\mathfrak{h}}}$-module by
letting $\widehat{{\mathfrak{h}}}^{\prime}$ act trivially and letting
${\mathfrak{h}}$ $(={\mathfrak{h}}(0))$ act by

(5.6) $\displaystyle h(0)e_{\beta}={\langle}h,\beta{\rangle}e_{\beta}\ \
\mbox{ for }h\in{\mathfrak{h}},\ \beta\in L.$

Note that $S(\widehat{{\mathfrak{h}}}^{-})$ is naturally an
$\widehat{\mathfrak{h}}$-module of level $1$. Set

(5.7) $\displaystyle
V_{L}=S(\widehat{{\mathfrak{h}}}^{-})\otimes{\mathbb{C}}_{\varepsilon}[L],$

the tensor product of $\widehat{\mathfrak{h}}$-modules, which is an
$\widehat{\mathfrak{h}}$-module of level $1$. Set

${{\bf 1}}=1\otimes e_{0}\in V_{L}.$

Identify ${\mathfrak{h}}$ and ${\mathbb{C}}_{\varepsilon}[L]$ as subspaces of
$V_{L}$ via the correspondence

$a\mapsto a(-1)\otimes 1\ (a\in{\mathfrak{h}})\quad\textnormal{{and}}\quad
e_{\alpha}\mapsto 1\otimes e_{\alpha}\ (\alpha\in L).$

For $h\in{\mathfrak{h}}$, set

(5.8) $\displaystyle h(z)=\sum_{n\in{\mathbb{Z}}}h(n)z^{-n-1}.$

On the other hand, for $\alpha\in L$ set

(5.9) $\displaystyle
E^{\pm}(\alpha,z)=\exp\left(\sum_{n\in{\mathbb{Z}}_{+}}\frac{\alpha(\pm
n)}{\pm n}z^{\mp n}\right)$

on $V_{L}$. For $\alpha\in L$, define a linear operator $z^{\alpha}:\
{\mathbb{C}}_{\varepsilon}[L]\rightarrow{\mathbb{C}}_{\varepsilon}[L][z,z^{-1}]$
by

(5.10) $\displaystyle z^{\alpha}\cdot
e_{\beta}=z^{{\langle}\alpha,\beta{\rangle}}e_{\beta}\ \ \ \mbox{ for
}\beta\in L.$

Then there exists a vertex algebra structure on $V_{L}$, which is uniquely
determined by the conditions that ${{\bf 1}}$ is the vacuum vector and that
$(h\in{\mathfrak{h}},\,\alpha\in L)$:

(5.11) $\displaystyle Y(h,z)=h(z),\quad
Y(e_{\alpha},z)=E^{-}(-\alpha,z)E^{+}(-\alpha,z)e_{\alpha}z^{\alpha}.$

Next, we recall from [LL, Section 6.5] the associative algebra $A(L)$. By
definition, $A(L)$ is the associative algebra over ${\mathbb{C}}$ with unit
$1$, generated by

${\left.\left\\{{h[n],\ e_{\alpha}[n]}\,\right|\,{h\in{\mathfrak{h}},\
\alpha\in L,\ n\in{\mathbb{Z}}}\right\\}},$

where $h[n]$ is linear in $h$, subject to a set of relations written in terms
of generating functions

$\displaystyle h[z]=\sum_{n\in{\mathbb{Z}}}h[n]z^{-n-1},\ \ \ \
e_{\alpha}[z]=\sum_{n\in{\mathbb{Z}}}e_{\alpha}[n]z^{-n-1}.$

The relations are

$\displaystyle\textnormal{{(AL1) }}\ \ \ \ e_{0}[z]=1,$
$\displaystyle\textnormal{{(AL2) }}\ \ \ \
\left[h[z_{1}],h^{\prime}[z_{2}]\right]={\langle}h,h^{\prime}{\rangle}\frac{\partial}{\partial
z_{2}}z_{1}^{-1}\delta\left(\frac{z_{2}}{z_{1}}\right),$
$\displaystyle\textnormal{{(AL3) }}\ \ \ \
\left[h[z_{1}],e_{\alpha}[z_{2}]\right]={\langle}\alpha,h{\rangle}e_{\alpha}[z_{2}]z_{1}^{-1}\delta\left(\frac{z_{2}}{z_{1}}\right),$
$\displaystyle\textnormal{{(AL4) }}\ \ \ \
\left[e_{\alpha}[z_{1}],e_{\beta}[z_{2}]\right]=0\ \textnormal{{ if
}}{\langle}\alpha,\beta{\rangle}\geq 0,$ $\displaystyle\textnormal{{(AL5) }}\
\ \ \
(z_{1}-z_{2})^{-{\langle}\alpha,\beta{\rangle}}\left[e_{\alpha}[z_{1}],e_{\beta}[z_{2}]\right]=0\
\textnormal{{ if }}{\langle}\alpha,\beta{\rangle}<0,$

for $h,h^{\prime}\in{\mathfrak{h}},\ \alpha,\beta\in L$. An $A(L)$-module $W$
is said to be restricted if for every $w\in W$, we have $h[z]w,\
e_{\alpha}[z]w\in W((z))$ for all $h\in{\mathfrak{h}},\ \alpha\in L$.

The following result was obtained in [LL]:

######  Proposition 5.1.

For any $V_{L}$-module $(W,Y_{W})$, $W$ is a restricted $A(L)$-module with

$\displaystyle h[z]=Y_{W}(h,z),\ \ \ \,\,e_{\alpha}[z]=Y_{W}(e_{\alpha},z)$

for $h\in{\mathfrak{h}},\ \alpha\in L$, such that the following relations hold
on $W$ for $\alpha,\beta\in L$:

$\displaystyle\textnormal{{(AL6) }}\ \ \ \
\partial_{z}e_{\alpha}[z]=\alpha[z]^{+}e_{\alpha}[z]+e_{\alpha}[z]\alpha[z]^{-},$
$\displaystyle\textnormal{{(AL7) }}\ \ \ \
\textnormal{{Res}}_{x}\big{(}(x-z)^{-{\langle}\alpha,\beta{\rangle}-1}e_{\alpha}[x]e_{\beta}[z]-(-z+x)^{-{\langle}\alpha,\beta{\rangle}-1}e_{\beta}[z]e_{\alpha}[x]\big{)}$
$\displaystyle\hskip 49.79231pt=\varepsilon(\alpha,\beta)e_{\alpha+\beta}[z],$

where $\alpha[z]^{+}=\sum_{n<0}\alpha[n]z^{-n-1}$ and
$\alpha[z]^{-}=\sum_{n\geq 0}\alpha[n]z^{-n-1}$. On the other hand, if $W$ is
a restricted $A(L)$-module satisfying (AL6) and (AL7), then $W$ admits a
$V_{L}$-module structure $Y_{W}(\cdot,z)$ which is uniquely determined by

$Y_{W}(h,z)=h[z],\ \ \ Y_{W}(e_{\alpha},z)=e_{\alpha}[z]\ \ \ \mbox{ for
}h\in{\mathfrak{h}},\ \alpha\in L.$

Let $(W,Y_{W})$ be any $V_{L}$-module. From [LL, Proposition 6.5.2], the
following relations hold on $W$ for $\alpha,\beta\in L$:

$\displaystyle(x-z)^{-{\langle}\alpha,\beta{\rangle}-1}Y_{W}(e_{\alpha},x)Y_{W}(e_{\beta},z)-(-z+x)^{-{\langle}\alpha,\beta{\rangle}-1}Y_{W}(e_{\beta},z)Y_{W}(e_{\alpha},x)$
(5.12)
$\displaystyle\quad\quad\quad=\varepsilon(\alpha,\beta)Y_{W}(e_{\alpha+\beta},z)x^{-1}\delta\left(\frac{z}{x}\right).$

It can be readily seen that this implies the relations (AL4-5) and (AL7) with
$e_{\alpha}[z]=Y_{W}(e_{\alpha},z)$ for $\alpha\in L$. In fact, (5.1) is
equivalent to (AL7) together with

(5.13)
$\displaystyle(x-z)^{-{\langle}\alpha,\beta{\rangle}}Y_{W}(e_{\alpha},x)Y_{W}(e_{\beta},z)=(-z+x)^{-{\langle}\alpha,\beta{\rangle}}Y_{W}(e_{\beta},z)Y_{W}(e_{\alpha},x).$

The following is a universal property of the vertex algebra $V_{L}$ (see
[JKLT]):

######  Proposition 5.2.

Let $V$ be a nonlocal vertex algebra and let
$\psi:{\mathfrak{h}}\oplus{\mathbb{C}}_{\varepsilon}[L]\rightarrow V$ be a
linear map such that $\psi(e_{0})={\bf 1}$, the relations (AL1-3) and (AL6)
hold with

$\displaystyle h[z]=Y(\psi(h),z),\quad
e_{\alpha}[z]=Y(\psi(e_{\alpha}),z)\quad\textnormal{{for
}}h\in{\mathfrak{h}},\ \alpha\in L,$

and such that the following relation holds for $\alpha,\beta\in L$:

$\displaystyle(x-z)^{-{\langle}\alpha,\beta{\rangle}-1}e_{\alpha}[x]e_{\beta}[z]-(-z+x)^{-{\langle}\alpha,\beta{\rangle}-1}e_{\beta}[z]e_{\alpha}[x]$
(5.14)
$\displaystyle\quad\quad\quad=\varepsilon(\alpha,\beta)e_{\alpha+\beta}[z]x^{-1}\delta\left(\frac{z}{x}\right).$

Then $\psi$ can be extended uniquely to a nonlocal vertex algebra homomorphism
from $V_{L}$ to $V$.

###  5.2. Vertex bialgebra $B_{L}$

View $L$ as an abelian group and let ${\mathbb{C}}[L]$ be its group algebra
with basis ${\left.\left\\{{e^{\alpha}}\,\right|\,{\alpha\in L}\right\\}}$.
Recall that ${\mathfrak{h}}={\mathbb{C}}\otimes_{{\mathbb{Z}}}L$, a vector
space, and $\widehat{{\mathfrak{h}}}^{-}={\mathfrak{h}}\otimes
t^{-1}{\mathbb{C}}[t^{-1}]$, an abelian Lie algebra. Note that both
${\mathbb{C}}[L]$ and $S(\widehat{{\mathfrak{h}}}^{-})$
$(=U(\widehat{{\mathfrak{h}}}^{-}))$ are Hopf algebras. Set

(5.15) $\displaystyle
B_{L}=S(\widehat{{\mathfrak{h}}}^{-})\otimes{\mathbb{C}}[L],$

which is a Hopf algebra. In particular, $B_{L}$ is a bialgebra, where the
comultiplication $\Delta$ and counit $\varepsilon$ are uniquely determined by

(5.16) $\displaystyle\Delta(h(-n))=h(-n)\otimes 1+1\otimes
h(-n),\quad\Delta(e^{\alpha})=e^{\alpha}\otimes e^{\alpha},$ (5.17)
$\displaystyle\varepsilon(h(-n))=0,\quad\varepsilon(e^{\alpha})=1$

for $h\in{\mathfrak{h}},\ n\in{\mathbb{Z}}_{+},\ \alpha\in L$. On the other
hand, $B_{L}$ admits a derivation $\partial$ which is uniquely determined by

(5.18) $\displaystyle\partial(h(-n))=nh(-n-1),\quad\quad\partial(u\otimes
e^{\alpha})=\alpha(-1)u\otimes e^{\alpha}+\partial u\otimes e^{\alpha}$

for $h\in{\mathfrak{h}}$, $n\in{\mathbb{Z}}_{+}$, $u\in
S(\widehat{{\mathfrak{h}}}^{-}),\ \alpha\in L$. Then $B_{L}$ becomes a
commutative vertex algebra where the vertex operator map, denoted by
$Y_{B_{L}}(\cdot,x)$, is given by

$Y_{B_{L}}(a,x)b=(e^{x\partial}a)b\ \ \ \mbox{ for }a,b\in B_{L}.$

We sometimes denote this vertex algebra by $(B_{L},\partial)$.

The following result was obtained in [Li8]:

######  Proposition 5.3.

For $h\in{\mathfrak{h}},\ \alpha\in L$, we have

(5.19) $\displaystyle Y_{B_{L}}(h(-1),x)=\sum_{n\geq 1}h(-n)x^{n-1},$ (5.20)
$\displaystyle Y_{B_{L}}(e^{\alpha},x)=e^{\alpha}\exp\left(\sum_{n\geq
1}\frac{\alpha(-n)}{n}x^{n}\right)=e^{\alpha}E^{-}(-\alpha,x).$

On the other hand, $(B_{L},\partial)$ is a vertex bialgebra with the
comultiplication $\Delta$ and counit $\varepsilon$ of $B_{L}$, which is both
commutative and cocommutative.

Recall that the vertex algebra
$V_{L}=S(\widehat{{\mathfrak{h}}}^{-})\otimes{\mathbb{C}}_{\epsilon}[L]$
contains ${\mathfrak{h}}$ and ${\mathbb{C}}_{\epsilon}[L]$ as subspaces. As
the main result of this section, we have:

######  Theorem 5.4.

There exists a right $B_{L}$-comodule vertex algebra structure $\rho$ on
vertex algebra $V_{L}$, which is uniquely determined by

(5.21) $\displaystyle\rho(h)=h\otimes 1+1\otimes
h(-1),\quad\rho(e_{\alpha})=e_{\alpha}\otimes e^{\alpha}\quad\textnormal{{for
}}h\in{\mathfrak{h}},\ \alpha\in L.$

###### Proof.

The uniqueness is clear as $V_{L}$ as a vertex algebra is generated by the
subspace ${\mathfrak{h}}+{\mathbb{C}}_{\epsilon}[L]$. Denote by $Y^{\otimes}$
the vertex operator map of the tensor product vertex algebra $V_{L}\otimes
B_{L}$. Let $\rho^{0}$ be the linear map from
${\mathfrak{h}}\oplus{\mathbb{C}}_{\epsilon}[L]$ to $V_{L}\otimes B_{L}$,
defined by

$\displaystyle\rho^{0}(h)=h\otimes 1+1\otimes
h(-1),\quad\rho^{0}(e_{\alpha})=e_{\alpha}\otimes
e^{\alpha}\quad\textnormal{{for }}h\in{\mathfrak{h}},\ \alpha\in L.$

We are going to apply Proposition 5.2 with $V=V_{L}\otimes B_{L}$ and
$\psi=\rho^{0}$. As $B_{L}$ is a commutative vertex algebra, for
$h,h^{\prime}\in{\mathfrak{h}},\ \alpha\in L$ we have

$\displaystyle[Y^{\otimes}(\rho^{0}(h),x),Y^{\otimes}(\rho^{0}(h^{\prime}),z)]={\langle}h,h^{\prime}{\rangle}\frac{\partial}{\partial
z}x^{-1}\delta\left(\frac{z}{x}\right),$
$\displaystyle[Y^{\otimes}(\rho^{0}(h),x),Y^{\otimes}(\rho^{0}(e_{\alpha}),z)]={\langle}h,\alpha{\rangle}x^{-1}\delta\left(\frac{z}{x}\right)Y^{\otimes}(\rho^{0}(e_{\alpha}),z).$

Let $\alpha,\beta\in L$. Using Proposition 5.3 we get

$\displaystyle
Y^{\otimes}(\rho^{0}(e_{\alpha}),x_{1})Y^{\otimes}(\rho^{0}(e_{\beta}),x_{2})$
$\displaystyle=\ $ $\displaystyle Y(e_{\alpha},x_{1})Y(e_{\beta},x_{2})\otimes
e^{\alpha}E^{-}(-\alpha,x_{1})e^{\beta}E^{-}(-\beta,x_{2})$ $\displaystyle=\ $
$\displaystyle Y(e_{\alpha},x_{1})Y(e_{\beta},x_{2})\otimes
e^{\alpha+\beta}E^{-}(-\alpha,x_{1})E^{-}(-\beta,x_{2})$

and

$\displaystyle
Y^{\otimes}(\rho^{0}(e_{\beta}),x_{2})Y^{\otimes}(\rho^{0}(e_{\alpha}),x_{1})=Y(e_{\beta},x_{2})Y(e_{\alpha},x_{1})\otimes
e^{\alpha+\beta}E^{-}(-\beta,x_{2})E^{-}(-\alpha,x_{1}).$

Note that
$E^{-}(-\alpha,x_{1})E^{-}(-\beta,x_{2})=E^{-}(-\beta,x_{2})E^{-}(-\alpha,x_{1})$.
Furthermore, we have

$\displaystyle(x_{1}-x_{2})^{-{\langle}\alpha,\beta{\rangle}-1}Y^{\otimes}(\rho^{0}(e_{\alpha}),x_{1})Y^{\otimes}(\rho^{0}(e_{\beta}),x_{2})$
$\displaystyle\ \ \ \
-(-x_{2}+x_{1})^{-{\langle}\alpha,\beta{\rangle}-1}Y^{\otimes}(\rho^{0}(e_{\beta}),x_{2})Y^{\otimes}(\rho^{0}(e_{\alpha}),x_{1})$
$\displaystyle=\ $
$\displaystyle\epsilon(\alpha,\beta)Y(e_{\alpha+\beta},x_{2})x_{1}^{-1}\delta\left(\frac{x_{2}}{x_{1}}\right)\otimes
e^{\alpha+\beta}E^{-}(-\alpha,x_{1})E^{-}(-\beta,x_{2})$ $\displaystyle=\ $
$\displaystyle\epsilon(\alpha,\beta)Y(e_{\alpha+\beta},x_{2})\otimes
e^{\alpha+\beta}E^{-}(-\alpha,x_{2})E^{-}(-\beta,x_{2})x_{1}^{-1}\delta\left(\frac{x_{2}}{x_{1}}\right)$
$\displaystyle=\ $
$\displaystyle\epsilon(\alpha,\beta)Y(e_{\alpha+\beta},x_{2})\otimes
e^{\alpha+\beta}E^{-}(-\alpha-\beta,x_{2})x_{1}^{-1}\delta\left(\frac{x_{2}}{x_{1}}\right)$
$\displaystyle=\ $
$\displaystyle\epsilon(\alpha,\beta)Y^{\otimes}(e_{\alpha+\beta},x_{2})x_{1}^{-1}\delta\left(\frac{x_{2}}{x_{1}}\right)$

and

$\displaystyle\frac{d}{dx}Y^{\otimes}(e_{\alpha},x)=\
\frac{d}{dx}Y(e_{\alpha},x)\otimes
e^{\alpha}E^{-}(-\alpha,x)+Y(e_{\alpha},x)\otimes
e^{\alpha}\alpha(x)^{+}E^{-}(-\alpha,x)$ $\displaystyle=\ $
$\displaystyle\left(\alpha(x)^{+}Y(e_{\alpha},x)+Y(e_{\alpha},x)\alpha(x)^{-}\right)\otimes
Y(e^{\alpha},x)+Y(e_{\alpha},x)\otimes\alpha(x)^{+}Y(e^{\alpha},x)$
$\displaystyle=\ $ $\displaystyle
Y^{\otimes}(\rho^{0}(\alpha),x)^{+}Y^{\otimes}(\rho^{0}(e_{\alpha}),x)+Y^{\otimes}(\rho^{0}(e_{\alpha}),x)Y^{\otimes}(\rho^{0}(\alpha),x)^{-},$

where
$Y^{\otimes}(\rho^{0}(\alpha),x)^{+}=\sum_{n<0}\rho^{0}(\alpha)_{n}x^{-n-1}$
and $Y^{\otimes}(\rho^{0}(\alpha),x)^{-}=\sum_{n\geq
0}\rho^{0}(\alpha)_{n}x^{-n-1}$. (Notice that $Y_{B_{L}}(h,x)^{-}=0$ for
$h\in{\mathfrak{h}}$.) Then by Proposition 5.2 there exists a vertex algebra
homomorphism $\rho$ from $V_{L}$ to $V_{L}\otimes B_{L}$, which extends
$\rho^{0}$ uniquely.

Recall that $\Delta:B_{L}\rightarrow B_{L}\otimes B_{L}$ is a homomorphism of
vertex algebras. Notice that both $(\rho\otimes 1)\rho$ and
$(1\otimes\Delta)\rho$ are vertex algebra homomorphisms from $V_{L}$ to
$V_{L}\otimes B_{L}\otimes B_{L}$. It can be readily seen that $(\rho\otimes
1)\rho$ and $(1\otimes\Delta)\rho$ agree on the subspace
${\mathfrak{h}}+{\mathbb{C}}_{\epsilon}[L]$ of $V_{L}$. Since $V_{L}$ is
generated by ${\mathfrak{h}}+{\mathbb{C}}_{\epsilon}[L]$, we conclude

$\displaystyle(\rho\otimes 1)\rho=(1\otimes\Delta)\rho$

(on $V_{L}$). Similarly, we can prove

$\displaystyle(1\otimes\varepsilon)\rho(u)=u\quad\textnormal{{for }}u\in
V_{L}.$

Therefore, $\rho$ is a right $B_{L}$-comodule vertex algebra structure on
$V_{L}$. ∎

Recall the $\epsilon$-twisted group algebra ${\mathbb{C}}_{\epsilon}[L]$. Set

(5.22) $\displaystyle
B_{L,\epsilon}=S(\widehat{{\mathfrak{h}}}^{-})\otimes{\mathbb{C}}_{\epsilon}[L],$

viewed as an associative algebra. From [Li8], $B_{L,\epsilon}$ admits a
derivation $\partial$ such that

$\partial e_{\alpha}=\alpha(-1)\otimes e_{\alpha},\ \ \ \partial
h(-n)=nh(-n-1)$

for $\alpha\in L,\ h\in{\mathfrak{h}},\ n\geq 1$. Then we have a nonlocal
vertex algebra $(B_{L,\epsilon},\partial)$, where

$Y(e_{\alpha},x)=E^{-}(-\alpha,x)e_{\alpha}\ \ \ \ \mbox{ for }\alpha\in L.$

Furthermore, it was proved in [Li8] that $B_{L,\epsilon}$ is a $B_{L}$-module
nonlocal vertex algebra where the $B_{L}$-module structure
$Y_{B_{L,\epsilon}}^{B_{L}}(\cdot,x)$ on $B_{L,\epsilon}$ is uniquely
determined by

(5.23) $\displaystyle
Y_{B_{L,\epsilon}}^{B_{L}}(e^{\alpha},x)=E^{+}(-\alpha,x)x^{\alpha(0)}\ \ \
\mbox{ for }\alpha\in L.$

It was proved therein that the comultiplication $\Delta:B_{L}\rightarrow
B_{L}\otimes B_{L}$ gives rise to a vertex algebra homomorphism from $V_{L}$
into $B_{L,\epsilon}\sharp B_{L}$, with $V_{L}$ and $B_{L,\epsilon}$ being
canonically identified with $B_{L}$.

The following is an analogue of Theorem 5.4:

######  Proposition 5.5.

There exists a nonlocal vertex algebra homomorphism
$\rho:B_{L,\epsilon}\rightarrow B_{L,\epsilon}\otimes B_{L}$, which is
uniquely determined by

(5.24) $\displaystyle\rho(h(-n))=h(-n)\otimes 1+1\otimes
h(-n),\quad\rho(e_{\alpha})=e_{\alpha}\otimes e^{\alpha}$

for $h\in{\mathfrak{h}},\ n\in{\mathbb{Z}}_{+},\ \alpha\in L$. Furthermore,
$B_{L,\epsilon}$ with $\rho$ becomes a right $B_{L}$-comodule nonlocal vertex
algebra and $\rho$ is compatible with
$Y_{B_{L,\varepsilon}}^{B_{L}}(\cdot,x)$.

###### Proof.

Let $\Delta$ be the comultiplication map of $S(\widehat{{\mathfrak{h}}}^{-})$,
which is an algebra homomorphism. Define a linear map
$\rho:B_{L,\epsilon}\rightarrow B_{L,\epsilon}\otimes B_{L}$ by

$\rho(u\otimes e_{\alpha})=\Delta(u)(e_{\alpha}\otimes e^{\alpha})\ \ \ \mbox{
for }u\in S(\widehat{{\mathfrak{h}}}^{-}),\ \alpha\in L.$

As $B_{L}$ is commutative, it can be readily seen that the linear map from
${\mathbb{C}}_{\varepsilon}[L]$ to
${\mathbb{C}}_{\varepsilon}[L]\otimes{\mathbb{C}}[L]$, sending $e_{\alpha}$ to
$e_{\alpha}\otimes e^{\alpha}$ for $\alpha\in L$, is an algebra homomorphism.
Consequently, $\rho$ is an algebra homomorphism. It is straightforward to see
that

$\rho(\partial h(-n))=(\partial\otimes 1+1\otimes\partial)\rho(h(-n)),\ \ \ \
\rho(\partial e_{\alpha})=(\partial\otimes
1+1\otimes\partial)\rho(e_{\alpha})$

for $h\in{\mathfrak{h}},\ n\in{\mathbb{Z}}_{+},\ \alpha\in L$. Since
$B_{L,\varepsilon}$ as an algebra is generated by
$\widehat{{\mathfrak{h}}}^{-}+{\mathbb{C}}_{\varepsilon}[L]$, it follows that
$\rho$ is a homomorphism of differential algebras. Thus $\rho$ is a nonlocal
vertex algebra homomorphism. The same argument at the end of the proof of
Theorem 5.4 shows that $B_{L,\epsilon}$ with $\rho$ is a right
$B_{L}$-comodule. Therefore, $B_{L,\epsilon}$ with $\rho$ becomes a right
$B_{L}$-comodule nonlocal vertex algebra.

Let $\alpha,\beta\in L$. As
$Y_{B_{L,\varepsilon}}^{B_{L}}(e^{\alpha},x)e_{\beta}=E^{+}(-\alpha,x)x^{\alpha(0)}e_{\beta}=x^{{\langle}\alpha,\beta{\rangle}}e_{\beta}$,
we have

$\rho(Y_{B_{L,\varepsilon}}^{B_{L}}(e^{\alpha},x)e_{\beta})=\rho\left(x^{{\langle}\alpha,\beta{\rangle}}e_{\beta}\right)=x^{{\langle}\alpha,\beta{\rangle}}(e_{\beta}\otimes
e^{\beta})=(Y_{B_{L,\varepsilon}}^{B_{L}}(e^{\alpha},x)\otimes
1)\rho(e_{\beta}).$

On the other hand, we have

$\displaystyle
Y_{B_{L,\varepsilon}}^{B_{L}}(e^{\alpha},x)e_{\beta}\in{\mathbb{C}}_{\epsilon}[L]\otimes{\mathbb{C}}((x)).$

Since $B_{L}$ as a vertex algebra is generated by ${\mathbb{C}}[L]$ and
$B_{L,\varepsilon}$ as a nonlocal vertex algebra is generated by
${\mathbb{C}}_{\epsilon}[L]$, from Lemma 2.28
$Y_{B_{L,\varepsilon}}^{B_{L}}(\cdot,x)$ is compatible with $\rho$. ∎

######  Remark 5.6.

With Proposition 5.5, by Theorem 2.25 we get a new nonlocal vertex algebra
structure on $B_{L,\epsilon}$. It was essentially proved in [Li8] that vertex
algebra $V_{L}$ coincides with (is canonically isomorphic to) this deformation
of $B_{L,\epsilon}$.

Recall that for $a\in{\mathfrak{h}},\ f(x)\in{\mathbb{C}}((x))$,

$\Phi(a,f)(z)=\sum_{n\geq 0}\frac{(-1)^{n}}{n!}f^{(n)}(z)a_{n}$

on $V_{L}$. We have

(5.25)
$\displaystyle\Phi(a,f)(z)e_{\beta}={\langle}\beta,a{\rangle}f(z)e_{\beta}\ \
\ \mbox{ for }\beta\in L.$

Now we define $\Phi(G(x))(z)$ for
$G(x)\in{\mathfrak{h}}\otimes{\mathbb{C}}((x))$ by linearity. On the other
hand, we extend the bilinear form ${\langle}\cdot,\cdot{\rangle}$ on
${\mathfrak{h}}$ to a ${\mathbb{C}}((x))$-valued bilinear form on
${\mathfrak{h}}\otimes{\mathbb{C}}((x))$. Then we have

$\Phi(G(x))(z)e_{\beta}={\langle}\beta,G(z){\rangle}e_{\beta}$

for $G(x)\in{\mathfrak{h}}\otimes{\mathbb{C}}((x)),\ \beta\in L$.

Note that for any $g(x)\in x{\mathbb{C}}[[x]]$, $e^{g(x)}$ exists in
${\mathbb{C}}[[x]]$. Recall from Theorem 5.4 the right $B_{L}$-comodule vertex
algebra structure $\rho:V_{L}\rightarrow V_{L}\otimes B_{L}$ on $V_{L}$.

######  Proposition 5.7.

Let $f:{\mathfrak{h}}\rightarrow{\mathfrak{h}}\otimes x{\mathbb{C}}[[x]];\
\alpha\mapsto f(\alpha,x)$ be any linear map. Then there exists a
$B_{L}$-module structure $Y_{M}^{f}(\cdot,x)$ on $V_{L}$, which is uniquely
determined by

(5.26) $\displaystyle Y_{M}^{f}(e^{\alpha},z)=\exp(\Phi(f(\alpha,x))(z))\ \ \
\mbox{ for }\alpha\in L,$

and $V_{L}$ with this $B_{L}$-module structure becomes a $B_{L}$-module vertex
algebra. Furthermore, $Y_{M}^{f}(\cdot,x)$ is compatible with $\rho$ and
$Y_{M}^{f}$ is an invertible element of $\mathfrak{L}_{B_{L}}^{\rho}(V_{L})$.

###### Proof.

Since $V_{L}$ as a $V_{L}$-module is irreducible, by [Li5, Corollary 3.10]
$V_{L}$ is non-degenerate. By Proposition 2.19, $B(V_{L})$ is a vertex
bialgebra and $V_{L}$ is naturally a $B(V_{L})$-module vertex algebra.

For $\alpha\in{\mathfrak{h}}$, as $f(\alpha,x)\in{\mathfrak{h}}\otimes
x{\mathbb{C}}[[x]]$, $\exp(\Phi(f(\alpha,x))(z))$ exists and from [Li8,
Proposition 2.10], we have

$\displaystyle\exp(\Phi(f(\alpha,x))(z))\in\textnormal{{PEnd}}(V_{L})\subset
B(V_{L}),$

a group-like element, i.e.,
$\Delta(\exp(\Phi(f(\alpha,x))))=\exp(\Phi(f(\alpha,x)))\otimes\exp(\Phi(f(\alpha,x)))$.
Since $\Phi$ is linear and $f$ is linear in $\alpha\in{\mathfrak{h}}$, we have

$\displaystyle\exp(\Phi(f(\alpha,x)))\exp\left(\Phi\left(f(\beta,x)\right)\right)=\exp(\Phi(f(\alpha+\beta,x)))\quad\textnormal{{for
}}\alpha,\beta\in L.$

Then the linear map $\psi:{\mathbb{C}}[L]\rightarrow B(V_{L})$ defined by
$\psi(e^{\alpha})=\exp(\Phi(f(\alpha,x)))$ for $\alpha\in L$ is an algebra
homomorphism. Note that though $B(V_{L})$ is not necessarily commutative, the
subalgebra generated by

$\Phi(f^{(r)}(\alpha,x)),\ \exp(\Phi(f(\alpha,x)))\ \ \mbox{ for }r\geq 0,\
\alpha\in{\mathfrak{h}}$

is a commutative differential subalgebra of $B(V_{L})$. Then by [Li8, Lemma
5.4], $\psi$ can be extended uniquely to a homomorphism $\bar{\psi}$ of
differential algebras from $B_{L}$ to $B(V_{L})$. Furthermore, $\bar{\psi}$ is
a homomorphism of vertex algebras. We have

$\displaystyle(\bar{\psi}\otimes\bar{\psi})\Delta_{B_{L}}(e^{\alpha})=\exp(\Phi(f(\alpha,x)))\otimes\exp(\Phi(f(\alpha,x)))=\Delta_{B}\bar{\psi}(e^{\alpha}),$
$\displaystyle\varepsilon_{B}(\bar{\psi}(e^{\alpha}))=\varepsilon_{B}(\exp(\Phi(f(\alpha,x))))=1=\varepsilon_{B_{L}}(e^{\alpha})$

for $\alpha\in L$, where $(\Delta_{B_{L}},\varepsilon_{B_{L}})$ and
$(\Delta_{B},\varepsilon_{B})$ denote the coalgebra structures of $B_{L}$ and
$B(V_{L})$, respectively. Since ${\mathbb{C}}[L]$ generates $B_{L}$ as a
differential algebra, it follows that $\bar{\psi}$ is a homomorphism of
differential bialgebras. Consequently, $\bar{\psi}$ is a homomorphism of
vertex bialgebras. Then, by [Li8, Proposition 4.8], $\bar{\psi}$ gives rise to
a $B_{L}$-module structure $Y_{M}^{f}(\cdot,x)$ on $V_{L}$, which makes
$V_{L}$ a $B_{L}$-module vertex algebra.

Next, we prove that $Y_{M}^{f}(\cdot,x)$ is compatible with $\rho$, i.e.,

(5.27) $\displaystyle\rho(Y_{M}^{f}(b,x)v)=\left(Y_{M}^{f}(b,x)\otimes
1\right)\rho(v)\ \ \mbox{ for }b\in B_{L},\ v\in V_{L}.$

Let $\alpha,\beta\in L$. By (5.25) we have
$\Phi(f(\alpha,x))e_{\beta}={\langle}\beta,f(\alpha,x){\rangle}e_{\beta}$, so
that

$\displaystyle
Y_{M}^{f}(e^{\alpha},x)e_{\beta}=\exp(\Phi(f(\alpha,x)))e_{\beta}=e^{{\langle}\beta,f(\alpha,x){\rangle}}e_{\beta}.$

Then

$\rho(Y_{M}^{f}(e^{\alpha},x)e_{\beta})=\rho\left(e^{{\langle}\beta,f(\alpha,x){\rangle}}e_{\beta}\right)=e^{{\langle}\beta,f(\alpha,x){\rangle}}(e_{\beta}\otimes
e^{\beta})=(Y_{M}^{f}(e^{\alpha},x)\otimes 1)\rho(e_{\beta}).$

This shows that (5.27) holds for $b=e^{\alpha},\ v=e_{\beta}\in L$. On the
other hand, we have

$\displaystyle
Y_{M}^{f}(e^{\alpha},x)e_{\beta}\in{\mathbb{C}}_{\epsilon}[L]\otimes{\mathbb{C}}((x)).$

Since $B_{L}$ as a vertex algebra is generated by ${\mathbb{C}}[L]$, and
$V_{L}$ as a vertex algebra is generated by ${\mathbb{C}}_{\epsilon}[L]$, it
follows from Lemma 2.28 that (5.27) holds for all $u\in B_{L},\ v\in V_{L}$.
Thus $Y_{M}^{f}(\cdot,x)$ is compatible with $\rho$. Namely,
$Y_{M}^{f}\in\mathfrak{L}_{B_{L}}^{\rho}(V_{L})$.

Note that with $-f$ in place of $f$ we get another $B_{L}$-module vertex
algebra structure $Y_{M}^{-f}$ on $V_{L}$ such that
$Y_{M}^{-f}\in\mathfrak{L}_{B_{L}}^{\rho}(V_{L})$, where

(5.28) $\displaystyle
Y_{M}^{-f}(e^{\alpha},x)=\exp(\Phi(-f(\alpha,x))=\exp(-\Phi(f(\alpha,x))$

for $\alpha\in L$. It follows that $(Y_{M}^{f}\ast
Y_{M}^{-f})(e^{\alpha})=1=\varepsilon(e^{\alpha})$. Again, since $B_{L}$ as a
vertex algebra is generated by ${\mathbb{C}}[L]$, we have $Y_{M}^{f}\ast
Y_{M}^{-f}=Y_{M}^{\varepsilon}$. That is, $Y_{M}^{-f}$ is an inverse of
$Y_{M}^{f}$ in $\mathfrak{L}_{B_{L}}^{\rho}(V_{L})$. ∎

######  Remark 5.8.

Let $\phi(x,z)$ be an associate given as before. Note that if the linear map