#  $(G,\chi_{\phi})$-equivariant $\phi$-coordinated quasi modules for vertex
algebras

Fulin Chen1 School of Mathematical Sciences, Xiamen University, Xiamen, China
361005<EMAIL_ADDRESS>,  Xiaoling Liao  School of Mathematical Sciences,
Xiamen University, Xiamen, China 361005<EMAIL_ADDRESS>,  Shaobin
Tan2 School of Mathematical Sciences, Xiamen University, Xiamen, China 361005
<EMAIL_ADDRESS>and  Qing Wang3 School of Mathematical Sciences, Xiamen
University, Xiamen, China 361005<EMAIL_ADDRESS>
###### Abstract.

To give a unified treatment on the association of Lie algebras and vertex
algebras, we study $(G,\chi_{\phi})$-equivariant $\phi$-coordinated quasi
modules for vertex algebras, where $G$ is a group with $\chi_{\phi}$ a linear
character of $G$ and $\phi$ is an associate of the one-dimensional additive
formal group. The theory of $(G,\chi_{\phi})$-equivariant $\phi$-coordinated
quasi modules for nonlocal vertex algebra is established in [JKLT]. In this
paper, we concentrate on the context of vertex algebras. We establish several
conceptual results, including a generalized commutator formula and a general
construction of vertex algebras and their $(G,\chi_{\phi})$-equivariant
$\phi$-coordinated quasi modules. Furthermore, for any conformal algebra
$\mathcal{C}$, we construct a class of Lie algebras
$\widehat{\mathcal{C}}_{\phi}[G]$ and prove that restricted
$\widehat{\mathcal{C}}_{\phi}[G]$-modules are exactly
$(G,\chi_{\phi})$-equivariant $\phi$-coordinated quasi modules for the
universal enveloping vertex algebra of $\mathcal{C}$. As an application, we
determine the $(G,\chi_{\phi})$-equivariant $\phi$-coordinated quasi modules
for affine and Virasoro vertex algebras.

###### 2010 Mathematics Subject Classification:

17B65 & 17B69

1Partially supported by China NSF grant (No.11971397) and the Fundamental
Research Funds for the Central Universities (No.20720190069).

2Partially supported by China NSF grants (No.11971397)

3Partially supported by China NSF grants (No.12071385) and the Fundamental
Research Funds for the Central Universities (No.20720200067).

##  1\. Introduction

Modules and twisted modules have been extensively studied in vertex algebra
theory. We usually associate Lie algebras with vertex algebras through modules
or twisted modules (see [FLM, FZ, Li1, Li6], etc). But these theories are not
valid for some algebras which generating functions are not “local”, we need
generalize the notion of modules. A notion of quasi module of vertex algebra
was introduced in [Li5] to associate more infinite dimensional Lie algebras
like quantum torus Lie algebras with vertex algebras. In order to associate
quantum affine algebras with quantum vertex algebras, a theory of
$\phi$-coordinated module for the quantum vertex algebra was developed in
[Li2]. The parameter $\phi$ is an associate of the one-dimensional additive
formal group $F(x,y)=x+y$. When $\phi=x+z$, a $\phi$-coordinated module of
vertex algebra is just a module. To establish the explicit connections of
quantum vertex algebras to various algebraic structures like quantum affine
algebras, a notion of $(G,\chi_{\phi})$-equivariant $\phi$-coordinated quasi
module for nonlocal vertex algebras was introduced in [JKLT], where $G$ is a
group and $\chi_{\phi}$ is a linear character of $G$. In [Li2], the author
proved that for every $p(x)\in\mathbb{C}((x))$,
$\phi=e^{zp(x)\frac{d}{dx}}(x)$ is an associate of $F(x,y)$ and every
associate is of this form. Moreover, an explicit association between quantum
vertex algebra and certain quantum $\beta\gamma$-system was established in
[Li2] through $\phi$-coordinated modules, where $\phi=xe^{z}$, i.e., $p(x)=x$.
Recently, the $\phi_{\epsilon}$-coordinated modules of vertex algebras have
been studied precisely in [BLP] for $p(x)=x^{\epsilon}$ with $\epsilon$ an
integer, and they use the theory of $\phi_{\epsilon}$-coordinated modules to
associate vertex algebras with Novikov algebras. Specifically, for
$\phi=xe^{z}$, a notion of $G$-equivariant $\phi$-coordinated quasi module for
quantum vertex algebras was introduced in [Li4]. A commutator formula for
$G$-equivariant $\phi$-coordinated quasi module was obtained and a natural
connection between the deformed Virasoro algebra $Vir_{p,-1}$ of a certain
Clifford vertex superalgebra through $G$-equivariant $\phi$-coordinated quasi
module was established therein. Later in [JKLT], a generalized commutator
formula and a general construction of $(G,\chi_{\phi})$-equivariant
$\phi$-coordinated quasi modules for nonlocal vertex algebras were
established, and by using these results, they give a construction of
(equivariant) $\phi$-coordinated quasi modules for lattice vertex algebras. We
note that if $\phi=xe^{z}$, the notion of $(G,\chi_{\phi})$-equivariant
$\phi$-coordinated quasi module coincides with the notion of $G$-equivariant
$\phi$-coordinated quasi module. Motivated by these works, in this paper, we
focus on the frame of vertex algebra. We expect to unify the association of
Lie algebras and vertex algebras though $(G,\chi_{\phi})$-equivariant
$\phi$-coordinated quasi modules.

Next, we briefly outline our ideas. Let $\phi=e^{zp(x)\frac{d}{dx}}(x)$ for
$0\neq p(x)\in\mathbb{C}((x))$. A $\phi$-coordinated module for a vertex
algebra $V$ is a vector space $W$ with a linear map $Y_{W}(\cdot,z)$ from $V$
to $\mathrm{Hom}(W,W((z)))$ satisfying the conditions that
$Y_{W}(\mathbf{1},z)=1_{W}$ and that for $u,v\in V$, there exists a
nonnegative integer $k$ such that

$\displaystyle(z_{1}-z_{2})^{k}Y_{W}(u,z_{1})Y_{W}(v,z_{2})\in\mathrm{Hom}(W,W((z_{1},z_{2}))),$
$\displaystyle(\phi(z_{2},z_{0})-z_{2})^{k}Y_{W}(Y(u,z_{0})v,z_{2})=((z_{1}-z_{2})^{k}Y_{W}(u,z_{1})Y_{W}(v,z_{2}))\mid_{z_{1}=\phi(z_{2},z_{0})}.$

By generalizing the corresponding result in [BLP], we prove that if
$(W,Y_{W})$ is a $\phi$-coordinated $V$-module, then for $u,v\in V$, we have
the commutator formula

$\begin{split}[Y_{W}(u,z_{1}),Y_{W}(v,z_{2})]=\sum_{i\geq
0}\frac{1}{i!}Y_{W}(u_{i}v,z_{2})\left(p(z_{2})\frac{\partial}{\partial
z_{2}}\right)^{i}p(z_{1})z_{1}^{-1}\delta\left(\frac{z_{2}}{z_{1}}\right).\end{split}$

For any vector space $W$, we prove that any (quasi) local subset of
${\mathcal{E}}(W)(=\mathrm{Hom}(W,W((z))))$ can generate a vertex algebra with
$W$ is naturally a $\phi$-coordinated (quasi) module of it. To give a unified
treatment on the association of Lie algebras and vertex algebras, we need the
notion of conformal algebra which was introduced in [K] (also known as a
vertex Lie algebra in [DLM, P]) as a bridge. A conformal algebra $\mathcal{C}$
is a $\mathbb{C}[\partial]$-module with a $\mathbb{C}$-bilinear product
$a_{n}b$ for each $n\in\mathbb{N}$ satisfying some axioms. Define a
multiplication on the formal loop space $\mathcal{C}\otimes\mathbb{C}((x))$ by

$\displaystyle[a\otimes f(x),b\otimes g(x)]=\sum_{i\geq
0}\frac{1}{i!}(a_{i}b)\otimes\left(\left(p(x)\frac{d}{dx}\right)^{i}f(x)\right)g(x)$

for $a,b\in\mathcal{C}$ and $f(x),g(x)\in\mathbb{C}((x))$. Then the quotient
space

$\displaystyle\widehat{\mathcal{C}}_{\phi}:=\mathcal{C}\otimes\mathbb{C}((x))/\mathrm{Im}\,(\partial\otimes
1+1\otimes p(x)\frac{d}{dx})$

with this multiplication form a Lie algebra [K]. For $a\in\mathcal{C}$, we set

$a_{\phi}(z)=\sum_{n\in\mathbb{Z}}\overline{a\otimes
x^{n}p(x)}z^{-n-1}\in\widehat{\mathcal{C}}_{\phi}[[z,z^{-1}]],$

where $\overline{a\otimes x^{n}p(x)}$ is the image of $a\otimes x^{n}p(x)$ in
$\widehat{\mathcal{C}}_{\phi}$. Then for $a,b\in\mathcal{C}$,

$\displaystyle[a_{\phi}(z),b_{\phi}(w)]=\sum_{i\geq
0}\frac{1}{i!}(a_{i}b)_{\phi}(w)\left(p(w)\frac{\partial}{\partial
w}\right)^{i}p(z)z^{-1}\delta\left(\frac{w}{z}\right).$

We prove that restricted $\widehat{\mathcal{C}}_{\phi}$-modules are exactly
$\phi$-coordinated modules for the universal enveloping vertex algebra
$V_{\mathcal{C}}$ of $\mathcal{C}$.

Let $G$ be a group with a linear character $\chi$. Following [JKLT], a
$(G,\chi)$-vertex algebra is a vertex algebra $V$ with a $G$-action $R$ on $V$
such that $R_{g}(\mathbf{1})=\mathbf{1}$ and

$R_{g}Y(v,z)R_{g}^{-1}=Y(R_{g}(v),\chi(g)z)\quad\textnormal{{for }}g\in G,v\in
V.$

The notion of $(G,\chi)$-vertex algebra was introduced in [Li3] (cf. [Li5]). A
$(G,\chi_{\phi})$-equivariant $\phi$-coordinated quasi module is defined for a
$(G,\chi)$-vertex algebra and from [JKLT], we may always assume that

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

In this paper, we obtain a general construction of vertex algebras and their
$(G,\chi_{\phi})$-equivariant $\phi$-coordinated quasi modules based on
[JKLT]. Let $(W,Y_{W})$ be a $(G,\chi_{\phi})$-equivariant $\phi$-coordinated
quasi $V$-module, we prove the following generalized commutator formula:

$\begin{split}&[Y_{W}(u,z_{1}),Y_{W}(v,z_{2})]\\\
=\mathrm{Res}_{z_{0}}&\sum_{g\in\psi(\chi_{\phi}(G))}Y_{W}(Y(R_{g^{-1}}u,z_{0})v,z_{2})e^{z_{0}p(z_{2})\frac{\partial}{\partial
z_{2}}}\left(\chi(g)p(z_{1})z_{1}^{-1}\delta\left(\frac{\chi_{\phi}(g)z_{2}}{z_{1}}\right)\right),\end{split}$

where $\psi:\chi_{\phi}(G)\rightarrow G$ is a section of $\chi_{\phi}$. The
notion of $(G,\chi)$-conformal algebra was introduced in [Li3] which
generalizes the notion of $\Gamma$-conformal algebra in [G-K-K]. A
$(G,\chi)$-conformal algebra is a conformal algebra $\mathcal{C}$ with a
$G$-action on $\mathcal{C}$ satisfying

$R_{g}(a_{m}b)=\chi(g)^{-(m+1)}(R_{g}a)_{m}(R_{g}b)\quad\textnormal{{ for any
}}a,b\in\mathcal{C},m\in\mathbb{N},$

and some other compatible conditions. Let $(\mathcal{C},R)$ be a
$(G,\chi)$-conformal algebra, for any $g\in G$, we define a linear
automorphism $\hat{R}_{g}$ on $\mathcal{C}\otimes\mathbb{C}((x))$ by

$\displaystyle\hat{R}_{g}(a\otimes f(x))=\chi(g)^{-1}R_{g}(a)\otimes
f(\chi_{\phi}(g)^{-1}x),\quad a\in\mathcal{C},f(x)\in\mathbb{C}((x)).$

Then we define a multiplication on $\widehat{\mathcal{C}}_{\phi}$ by

$\displaystyle(u,v)\mapsto\sum_{g\in G}[\hat{R}_{g}(u),v],\quad
u,v\in\widehat{\mathcal{C}}_{\phi}.$

It follows that the quotient space

$\displaystyle\widehat{\mathcal{C}}_{\phi}[G]:=\widehat{\mathcal{C}}_{\phi}/\textnormal{{span}}\\{\hat{R}_{g}(u)-u\mid
g\in G,u\in\widehat{\mathcal{C}}_{\phi}\\}$

is a Lie algebra with this new multiplication. For $a\in\mathcal{C}$,
$f(x)\in\mathbb{C}((x))$, we still denote the image of $\overline{a\otimes
f(x)}$ in $\widehat{\mathcal{C}}_{\phi}[G]$ by $\overline{a\otimes f(x)}$. For
$a\in\mathcal{C}$, we set

$a^{G}_{\phi}(z)=\sum_{n\in\mathbb{Z}}\overline{a\otimes
x^{n}p(x)}z^{-n-1}\in\widehat{\mathcal{C}}_{\phi}[G][[z,z^{-1}]].$

Then for $a,b\in\mathcal{C}$, we have

$\begin{split}&[a^{G}_{\phi}(z),b^{G}_{\phi}(w)]\\\ =&\sum_{g\in G}\sum_{i\geq
0}\frac{1}{i!}\left((R_{g^{-1}}a)_{i}b\right)^{G}_{\phi}(w)\left(p(w)\frac{\partial}{\partial
w}\right)^{i}\left(\chi(g)p(z)z^{-1}\delta\left(\frac{\chi_{\phi}(g)w}{z}\right)\right).\end{split}$

Let $H=\ker\chi_{\phi}$ and let $\bar{\chi}_{\phi}$ be the linear character of
$G/H$ induced by $\chi_{\phi}$. From [Li3], the quotient space

$\mathcal{C}/H=\mathcal{C}/\textnormal{{span}}\\{R_{h}a-a\mid h\in H,\
a\in\mathcal{C}\\}$

has a natural $(G/H,\bar{\chi})$-conformal algebra structure, where
$\bar{\chi}$ is a linear character of $G/H$ induced by $\chi$. Then we prove
that restricted $\widehat{\mathcal{C}}_{\phi}[G]$-modules are exactly
$(G/H,\bar{\chi}_{\phi})$-equivariant $\phi$-coordinated quasi
$V_{\mathcal{C}/H}$-modules. Finally, we apply these theories on affine and
Virasoro vertex algebras, and the vertex algebras associated to Novikov
algebras.

This paper is organized as follows. In Section 2, we recall $\phi$-operations
on ${\mathcal{E}}(W)$ and establish some technical lemmas. In Section 3, we
recall the notion of $\phi$-coordinated (quasi) module for a vertex algebra
and establish some conceptual results. In Section 4, we prove that restricted
$\widehat{\mathcal{C}}_{\phi}$-modules are exactly $\phi$-coordinated
$V_{\mathcal{C}}$-modules. In Section 5, we establish a generalized commutator
formula and a general construction of $(G,\chi_{\phi})$-equivariant
$\phi$-coordinated quasi modules of vertex algebras. Furthermore, we construct
a class of Lie algebras $\widehat{\mathcal{C}}_{\phi}[G]$ and prove that
restricted $\widehat{\mathcal{C}}_{\phi}[G]$-modules are exactly
$(G/H,\bar{\chi}_{\phi})$-equivariant $\phi$-coordinated quasi
$V_{\mathcal{C}/H}$-modules. In Section 6, we determine a class of
$(G,\chi_{\phi})$-equivariant $\phi$-coordinated quasi modules for affine
vertex algebras, and determine the $\phi$-coordinated modules for Virasoro
vertex algebra and the vertex algebras associated to Novikov algebras.

In this paper, we let $\mathbb{Z}$, $\mathbb{Z}^{\times}$, $\mathbb{Z}_{+}$,
$\mathbb{N}$ and $\mathbb{C}$ be the sets of integers, nonzero integers,
positive integers, nonnegative integers and complex numbers, respectively. All
the Lie algebras in this paper are over the field of complex numbers. For any
linear endomorphism $\varphi$ of a vector space, we set
$\varphi^{(k)}=\frac{\varphi^{k}}{k!}$.

##  2\. $\phi$-operations on ${\mathcal{E}}(W)$

In this section, we recall the $\phi$-operations following [Li2] and prove
some technical Lemmas. We begin by recalling the notion of an associate for
the one-dimensional additive formal group (law) $F(x,y)=x+y$. Following [Li2],
an associate $\phi(x,z)$ (or simply $\phi$) of $F(x,y)$ by definition is a
formal series in $\mathbb{C}((x))[[z]]$ satisfying

$\phi(x,0)=x,\quad\phi(\phi(x,z_{1}),z_{2})=\phi(x,z_{1}+z_{2}).$

It was proved therein that for every $p(x)\in\mathbb{C}((x))$,
$\phi(x,z)=e^{zp(x)\frac{d}{dx}}(x)$ is an associate of $F(x,y)$ and every
associate is of this form with $p(x)$ uniquely determined. Throughout this
paper, we fix an associate

(2.1)
$\displaystyle\phi=\phi(x,z)=e^{zp(x)\frac{d}{dx}}(x)\quad\textnormal{{with}}\quad
0\neq p(x)\in\mathbb{C}((x)).$

Throughout this section let $W$ be a fixed vector space and set

${\mathcal{E}}(W)=\mathrm{Hom}(W,W((z)))\subset(\mathrm{End}W)[[z,z^{-1}]].$

A pair $(a(z),b(z))$ in ${\mathcal{E}}(W)$ is said to be quasi local if there
exists a nonzero polynomial $q(z_{1},z_{2})\in\mathbb{C}[z_{1},z_{2}]$ such
that

(2.2) $\displaystyle q(z_{1},z_{2})\ [a(z_{1}),b(z_{2})]=0,$

and is said to be local if the polynomial $q(z_{1},z_{2})$ is assumed to a
polynomial of the form $(z_{1}-z_{2})^{n}$ for some $n\in\mathbb{N}$. The main
goal of this section is to establish a commutator formula for quasi local
pairs in ${\mathcal{E}}(W)$ in terms of the following $\phi$-operations
introduced in [Li2]:

######  Definition 2.1.

Let $(a(z),b(z))$ be a quasi local pair in ${\mathcal{E}}(W)$ such that (2.2)
holds. We define

$Y_{\mathcal{E}}^{\phi}(a(z),z_{0})b(z)=\sum\limits_{n\in\mathbb{Z}}a(z)_{n}^{\phi}b(z)z_{0}^{-n-1}\in{\mathcal{E}}(W)((z_{0}))$

by the following rule:

$Y_{\mathcal{E}}^{\phi}(a(z),z_{0})b(z)=q(\phi(z,z_{0}),z)^{-1}(q(z_{1},z)a(z_{1})b(z))\mid_{z_{1}=\phi(z,z_{0})}.$

It was proved therein that the definition of $Y_{\mathcal{E}}^{\phi}$ dose not
depend on the choice of $q(z_{1},z_{2})$. In what follows, we establish some
technical lemmas.

######  Lemma 2.2.

Let $j,k$ be two nonnegative integers with $j>k$. Then for any nonzero complex
number $\lambda$, we have

(2.3) $\displaystyle(z_{1}-\lambda
z_{2})^{j}\left(p(z_{2})\frac{\partial}{\partial
z_{2}}\right)^{(k)}\delta\left(\frac{\lambda z_{2}}{z_{1}}\right)=0.$

###### Proof.

Note that we may write

$\left(p(z_{2})\frac{\partial}{\partial
z_{2}}\right)^{(k)}=f_{k}(z_{2})\left(\frac{\partial}{\partial
z_{2}}\right)^{(k)}+\cdots+f_{1}(z_{2})\frac{\partial}{\partial
z_{2}}+f_{0}(z_{2})$

for some uniquely determined
$f_{0}(z_{2}),\cdots,f_{k}(z_{2})\in\mathbb{C}((z_{2}))$. Then the assertion
follows from the fact that $(z_{1}-\lambda
z_{2})^{j}\left(\frac{\partial}{\partial
z_{2}}\right)^{(i)}\delta\left(\frac{\lambda z_{2}}{z_{1}}\right)=0$ for
$i=0,\cdots,k$. ∎

For convenience, we set

$\bar{p}(x)=x^{-1}p(x)\in\mathbb{C}((x)).$

######  Lemma 2.3.

Let $\lambda_{1},\cdots,\lambda_{r}$ be distinct nonzero complex numbers,
$k_{1},\cdots,k_{r}\in\mathbb{Z}_{+}$ and $A_{ij}(z)\in{\mathcal{E}}(W)$,
where $i=1,\cdots,r,$ and for a fixed $i$, $0\leq j\leq k_{i}-1$. Then

(2.4)
$\displaystyle\sum_{i=1}^{r}\sum_{j=0}^{k_{i}-1}A_{ij}(z_{2})\left(p(z_{2})\frac{\partial}{\partial
z_{2}}\right)^{(j)}\bar{p}(\lambda_{i}^{-1}z_{1})\delta\left(\frac{\lambda_{i}z_{2}}{z_{1}}\right)=0$

if and only if $A_{ij}(z)=0$ for all $i,j$.

###### Proof.

For $1\leq i\leq r$, set

$f_{i}(z)=\frac{\prod_{j=1}^{r}(z-\lambda_{j})^{k_{j}}}{(z-\lambda_{i})^{k_{i}}}\in\mathbb{C}[z].$

In view of Lemma 2.2, for $1\leq i\neq s\leq r$ and $0\leq j\leq k_{s}-1$ we
have

(2.5) $\displaystyle f_{i}(z_{1}/z_{2})\left(p(z_{2})\frac{\partial}{\partial
z_{2}}\right)^{(j)}\bar{p}(\lambda_{s}^{-1}z_{1})\delta\left(\frac{\lambda_{s}z_{2}}{z_{1}}\right)=0.$

As $f_{1}(z),\cdots,f_{r}(z)$ are pairwise relatively prime, we may take
suitable polynomials $q_{1}(z),\cdots,q_{r}(z)\in\mathbb{C}[z]$ such that

$q_{1}(z)f_{1}(z)+\cdots+q_{r}(z)f_{r}(z)=1.$

This together with (2.5) gives that for $1\leq s\leq r$, $j=0,\cdots,k_{s}-1$,

(2.6) $\begin{split}&A_{sj}(z_{2})\left(p(z_{2})\frac{\partial}{\partial
z_{2}}\right)^{(j)}\bar{p}(\lambda_{s}^{-1}z_{1})\delta\left(\frac{\lambda_{s}z_{2}}{z_{1}}\right)\\\
=\,&\left(\sum_{i=1}^{r}f_{i}(z_{1}/z_{2})q_{i}(z_{1}/z_{2})\right)A_{sj}(z_{2})\left(p(z_{2})\frac{\partial}{\partial
z_{2}}\right)^{(j)}\bar{p}(\lambda_{s}^{-1}z_{1})\delta\left(\frac{\lambda_{s}z_{2}}{z_{1}}\right)\\\
=\,&A_{sj}(z_{2})f_{s}(z_{1}/z_{2})q_{s}(z_{1}/z_{2})\left(p(z_{2})\frac{\partial}{\partial
z_{2}}\right)^{(j)}\bar{p}(\lambda_{s}^{-1}z_{1})\delta\left(\frac{\lambda_{s}z_{2}}{z_{1}}\right).\end{split}$

Now we assume that (2.4) holds and fix $s$ with $1\leq s\leq r$. Then from
(2.5) and (2.6) we have

$\displaystyle 0=$
$\displaystyle\sum_{i=1}^{r}\sum_{j=0}^{k_{i}-1}A_{ij}(z_{2})f_{s}(z_{1}/z_{2})q_{s}(z_{1}/z_{2})\left(p(z_{2})\frac{\partial}{\partial
z_{2}}\right)^{(j)}\bar{p}(\lambda_{i}^{-1}z_{1})\delta\left(\frac{\lambda_{i}z_{2}}{z_{1}}\right)$
$\displaystyle=$
$\displaystyle\sum_{j=0}^{k_{s}-1}A_{sj}(z_{2})f_{s}(z_{1}/z_{2})q_{s}(z_{1}/z_{2})\left(p(z_{2})\frac{\partial}{\partial
z_{2}}\right)^{(j)}\bar{p}(\lambda_{s}^{-1}z_{1})\delta\left(\frac{\lambda_{s}z_{2}}{z_{1}}\right)$
$\displaystyle=$
$\displaystyle\sum_{j=0}^{k_{s}-1}A_{sj}(z_{2})\left(p(z_{2})\frac{\partial}{\partial
z_{2}}\right)^{(j)}\bar{p}(\lambda_{s}^{-1}z_{1})\delta\left(\frac{\lambda_{s}z_{2}}{z_{1}}\right),$

which amounts to

(2.7)
$\displaystyle\sum_{j=0}^{k_{s}-1}A_{sj}(z_{2})\left(p(z_{2})\frac{\partial}{\partial
z_{2}}\right)^{(j)}p(z_{1})z_{1}^{-1}\delta\left(\frac{z_{2}}{z_{1}}\right)=0.$

For $j=0,\cdots,k_{s}-1$, write

$\displaystyle A_{sj}(z_{2})\left(p(z_{2})\frac{\partial}{\partial
z_{2}}\right)^{(j)}=\sum_{n=0}^{j}g_{jn}(z_{2})\left(\frac{\partial}{\partial
z_{2}}\right)^{(n)}$

for some uniquely determined $g_{jn}(z_{2})\in\mathbb{C}((z_{2}))$. In view of
(2.7) we have

$\sum_{n=0}^{k_{s}-1}\left(\sum_{j=n}^{k_{s}-1}g_{jn}(z_{2})\right)\left(\frac{\partial}{\partial
z_{2}}\right)^{(n)}z_{1}^{-1}\delta\left(\frac{z_{2}}{z_{1}}\right)=0.$

By [Li1, Lemma 2.1.4], this implies $\sum_{j=n}^{k_{s}-1}g_{jn}(z_{2})=0$ for
$0\leq n\leq k_{s}-1$, and hence implies $A_{sj}(z)=0$ for $0\leq j\leq
k_{s}-1$, as desired. ∎

The following result is a slight generalization of [Li4, Lemma 2.3].

######  Lemma 2.4.

Let $a(z),b(z)\in{\mathcal{E}}(W)$ and let

(2.8) $\displaystyle
q(z)=(z-\lambda_{1})^{k_{1}}\cdots(z-\lambda_{r})^{k_{r}}\in\mathbb{C}[z],$

where $\lambda_{1},\cdots,\lambda_{r}\in\mathbb{C}^{\times}$ are distinct and
$k_{1},\cdots,k_{r}\in\mathbb{Z}_{+}$. Then

(2.9) $\displaystyle q(z_{1}/z_{2})\,[a(z_{1}),b(z_{2})]=0$

if and only if

(2.10)
$\begin{split}[a(z_{1}),b(z_{2})]=\sum_{i=1}^{r}\sum_{j=0}^{k_{i}-1}A_{ij}(z_{2})\left(p(z_{2})\frac{\partial}{\partial
z_{2}}\right)^{(j)}\bar{p}(\lambda_{i}^{-1}z_{1})\delta\left(\frac{\lambda_{i}z_{2}}{z_{1}}\right)\end{split}$

for some uniquely determined $A_{ij}(z)\in{\mathcal{E}}(W)$.

###### Proof.

The proof is similar to that of [Li4, Lemma 2.3] and we give a sketch of it.
The uniqueness of $A_{ij}(z)$ follows from Lemma 2.3. For the existence, write

$\frac{1}{q(x)}=\sum_{i=1}^{r}\sum_{j=1}^{k_{i}}\frac{a_{ij}}{(x-\lambda_{i})^{j}}$

for some $a_{ij}\in\mathbb{C}$. Then from (2.9) we obtain (cf. [Li4, Lemma
2.3])

(2.11)
$\begin{split}[a(z_{1}),b(z_{2})]=\sum_{i=1}^{r}\sum_{j=0}^{k_{i}-1}a_{i,j+1}\lambda_{i}^{-j}z_{2}^{j+1}A(z_{1},z_{2})\left(\frac{\partial}{\partial
z_{2}}\right)^{(j)}z_{1}^{-1}\delta\left(\frac{\lambda_{i}z_{2}}{z_{1}}\right),\end{split}$

where
$A(z_{1},z_{2})=q(z_{1}/z_{2})a(z_{1})b(z_{2})\in\mathrm{Hom}(W,W((z_{1},z_{2})))$.

Note that for any $n\in\mathbb{N}$, there exist
$f_{0}(z_{2}),\cdots,f_{n}(z_{2})\in\mathbb{C}((z_{2}))$ such that

$\left(\frac{\partial}{\partial
z_{2}}\right)^{(n)}=f_{n}(z_{2})\left(p(z_{2})\frac{\partial}{\partial
z_{2}}\right)^{(n)}+f_{n-1}(z_{2})\left(p(z_{2})\frac{\partial}{\partial
z_{2}}\right)^{(n-1)}+\cdots+f_{0}(z_{2}).$

An induction argument shows that for any
$B(z_{1},z_{2})\in\mathrm{Hom}(W,W((z_{1},z_{2})))$ and $n\in\mathbb{N}$,
there exist $B_{0}(z),\cdots,B_{n}(z)\in{\mathcal{E}}(W)$ such that

$\begin{split}&B(z_{1},z_{2})\left(\frac{\partial}{\partial
z_{2}}\right)^{(n)}z_{1}^{-1}\delta\left(\frac{\lambda_{i}z_{2}}{z_{1}}\right)=\sum_{i=0}^{n}B_{i}(z_{2})\left(\frac{\partial}{\partial
z_{2}}\right)^{(i)}\bar{p}(\lambda_{i}^{-1}z_{1})\delta\left(\frac{\lambda_{i}z_{2}}{z_{1}}\right).\end{split}$

Thus (2.10) follows. ∎

We fix a $\hat{p}(x)\in\mathbb{C}((x))$ such that
$\frac{d}{dx}\hat{p}(x)=p^{-1}(x)$, and set

$f_{p(x)}(x,z):=\hat{p}(x(1+z))-\hat{p}(x).$

Then we can deduce that

(2.12) $\displaystyle\phi(x,f_{p(x)}(x,z))=x(1+z).$

We have (cf. [BLP, Li2]):

######  Lemma 2.5.

Let

$\displaystyle
A(z_{1},z_{2})\in\mathrm{Hom}\big{(}W,W((z_{1}))((z_{2}))\big{)},\quad
B(z_{1},z_{2})\in\mathrm{Hom}\big{(}W,W((z_{2}))((z_{1}))\big{)},$
$\displaystyle
C(z_{0},z_{2})\in\big{(}\mathrm{Hom}\big{(}W,W((z_{2}))\big{)}\big{)}((z_{0})).$

If there exists a nonnegative integer $k$ such that

$\displaystyle(z_{1}-z_{2})^{k}A(z_{1},z_{2})=(z_{1}-z_{2})^{k}B(z_{1},z_{2}),$
$\displaystyle\left((z_{1}-z_{2})^{k}A(z_{1},z_{2})\right)|_{z_{1}=\phi(z_{2},z_{0})}=(\phi(z_{2},z_{0})-z_{2})^{k}C(z_{0},z_{2}),$

then

$\begin{split}&(z_{2}z)^{-1}\delta\left(\frac{z_{1}-z_{2}}{z_{2}z}\right)A(z_{1},z_{2})-(z_{2}z)^{-1}\delta\left(\frac{z_{2}-z_{1}}{-z_{2}z}\right)B(z_{1},z_{2})\\\
&\quad\quad=z_{1}^{-1}\delta\left(\frac{z_{2}(1+z)}{z_{1}}\right)C\left(f_{p(z_{2})}(z_{2},z),z_{2}\right).\end{split}$

###### Proof.

When $p(x)=x^{\epsilon}$ for some $\epsilon\in\mathbb{Z}$, the assertion is
proved in [BLP, Lemma 3.16], where $f_{p(x)}(x,z)$ is denoted as
$f_{\epsilon}(x,z)$. In view of (2.12), the proof therein also valid for a
general $p(x)$. ∎

Now we are in a position to state the main result of this section, which says
that those $A_{ij}(z)$ in (2.10) are nothing but the $\phi$-products
$a(\lambda_{i}z)_{j}^{\phi}b(z)$.

######  Proposition 2.6.

Let $(a(z),b(z))$ be a pair in ${\mathcal{E}}(W)$ satisfying

$\displaystyle q(z_{1}/z_{2})[a(z_{1}),b(z_{2})]=0$

for some $q(z)\in\mathbb{C}[z]$. Then

$\displaystyle[a(z_{1}),b(z_{2})]=\sum_{\lambda\in\mathbb{C}^{\times}}\sum_{j\geq
0}a(\lambda z_{2})_{j}^{\phi}b(z_{2})\left(p(z_{2})\frac{\partial}{\partial
z_{2}}\right)^{(j)}\bar{p}(\lambda^{-1}z_{1})\delta\left(\frac{\lambda
z_{2}}{z_{1}}\right),$

which is a finite sum. Furthermore, the coefficients $a(\lambda
z_{2})_{j}^{\phi}b(z_{2})\neq 0$ only if $\lambda$ is a root of $q(z)$ and $j$
is less than the multiplicity of $\lambda$.

###### Proof.

We assume that $q(z)$ has the form as in (2.8) and fix an $s$ with $1\leq
s\leq r$. From (2.10), we only need to prove that

(2.13) $\displaystyle
A_{sj}(z_{2})=a(\lambda_{s}z_{2})_{j}^{\phi}b(z_{2}),\quad
j=0,\cdots,k_{s}-1.$

Let us introduce the polynomials

$\displaystyle q_{s}(z)=\prod_{1\leq i\neq s\leq
r}((z-\lambda_{s})^{k_{s}}-(\lambda_{i}-\lambda_{s})^{k_{s}})^{k_{i}}.$

Note that $(z-\lambda_{i})^{k_{i}}$ divides $q_{s}(z)$ and so there is a
$g_{s}(z)\in\mathbb{C}[z]$ such that

(2.14) $\displaystyle q_{s}(z)(z-\lambda_{s})^{k_{s}}=g_{s}(z)q(z).$

Then from (2.14) and (2.3), we have

(2.15) $\begin{split}&q_{s}(z_{1}/z_{2})[a(z_{1}),b(z_{2})]\\\
=\,&\sum_{j=0}^{k_{s}-1}q_{s}(z_{1}/z_{2})A_{sj}(z_{2})\left(p(z_{2})\frac{\partial}{\partial
z_{2}}\right)^{(j)}\bar{p}(\lambda_{s}^{-1}z_{1})\delta\left(\frac{\lambda_{s}z_{2}}{z_{1}}\right).\end{split}$

By replacing $z_{1}$ with $\lambda_{s}z_{1}$ in (LABEL:eq:firstdec) and
multiplying $(z_{1}/z_{2}-1)^{k_{s}}$ on both sides of (LABEL:eq:firstdec) we
have

$\displaystyle(z_{1}/z_{2}-1)^{k_{s}}q_{s}(\lambda_{s}z_{1}/z_{2})[a(\lambda_{s}z_{1}),b(z_{2})]=0.$

Thus we have (see Definition 2.1)

$\displaystyle\quad(\phi(z,z_{0})/z-1)^{k_{s}}q_{s}(\lambda_{s}\phi(z,z_{0})/z)Y_{\mathcal{E}}^{\phi}(a(\lambda_{s}z),z_{0})b(z)$
$\displaystyle=(z_{1}/z-1)^{k_{s}}q_{s}(\lambda_{s}z_{1}/z)a(\lambda_{s}z_{1})b(z)\mid_{z_{1}=\phi(z,z_{0})}.$

In view of Lemma 2.5, this implies that

$\begin{split}&(z_{2}z)^{-1}\delta\left(\frac{z_{1}-z_{2}}{z_{2}z}\right)q_{s}(\lambda_{s}z_{1}/z_{2})a(\lambda_{s}z_{1})b(z_{2})\\\
&\quad\quad\quad\quad\quad\quad-(z_{2}z)^{-1}\delta\left(\frac{z_{2}-z_{1}}{-z_{2}z}\right)q_{s}(\lambda_{s}z_{1}/z_{2})b(z_{2})a(\lambda_{s}z_{1})\\\
&=z_{1}^{-1}\delta\left(\frac{z_{2}(1+z)}{z_{1}}\right)q_{s}(\lambda_{s}z_{1}/z_{2})Y_{\mathcal{E}}^{\phi}(a(\lambda_{s}z_{2}),f_{p(z_{2})}(z_{2},z))b(z_{2}).\end{split}$

By taking $\mathrm{Res}_{z}z_{2}$ in the above identity we find

$\displaystyle\quad
q_{s}(\lambda_{s}z_{1}/z_{2})[a(\lambda_{s}z_{1}),b(z_{2})]$
$\displaystyle=\mathrm{Res}_{z}z_{1}^{-1}\delta\left(\frac{z_{2}(1+z)}{z_{1}}\right)z_{2}q_{s}(\lambda_{s}z_{1}/z_{2})Y_{\mathcal{E}}^{\phi}(a(\lambda_{s}z_{2}),f_{p(z_{2})}(z_{2},z))b(z_{2})$
$\displaystyle=\mathrm{Res}_{z_{0}}\frac{\partial}{\partial
z_{0}}\left(\frac{\phi(z_{2},z_{0})}{z_{2}}-1\right)z_{1}^{-1}\delta\left(\frac{\phi(z_{2},z_{0})}{z_{1}}\right)z_{2}q_{s}(\lambda_{s}z_{1}/z_{2})Y_{\mathcal{E}}^{\phi}(a(\lambda_{s}z_{2}),z_{0})b(z_{2})$
$\displaystyle=\mathrm{Res}_{z_{0}}p(\phi(z_{2},z_{0}))z_{1}^{-1}\delta\left(\frac{\phi(z_{2},z_{0})}{z_{1}}\right)q_{s}(\lambda_{s}z_{1}/z_{2})Y_{\mathcal{E}}^{\phi}(a(\lambda_{s}z_{2}),z_{0})b(z_{2})$
$\displaystyle=\mathrm{Res}_{z_{0}}p(z_{1})z_{1}^{-1}\delta\left(\frac{\phi(z_{2},z_{0})}{z_{1}}\right)q_{s}(\lambda_{s}z_{1}/z_{2})Y_{\mathcal{E}}^{\phi}(a(\lambda_{s}z_{2}),z_{0})b(z_{2})$
$\displaystyle=\sum_{j\geq 0}\left(p(z_{2})\frac{\partial}{\partial
z_{2}}\right)^{(j)}\left(\bar{p}(z_{1})\delta\left(\frac{z_{2}}{z_{1}}\right)\right)q_{s}(\lambda_{s}z_{1}/z_{2})a(\lambda_{s}z_{2})_{j}^{\phi}b(z_{2}),$

where in the second equality we have used the fact (2.12). Combining this with
(LABEL:eq:firstdec) (replacing $z_{1}$ with $\lambda_{s}z_{1}$ therein) we
obtain

(2.16)
$\displaystyle\sum_{j=0}^{k_{s}-1}q_{s}\left(\lambda_{s}\frac{z_{1}}{z_{2}}\right)(A_{sj}(z_{2})-a(\lambda_{s}z_{2})_{j}^{\phi}b(z_{2}))\left(p(z_{2})\frac{\partial}{\partial
z_{2}}\right)^{(j)}\bar{p}(z_{1})\delta\left(\frac{z_{2}}{z_{1}}\right)=0.$

Set

$\alpha=\prod_{1\leq i\neq s\leq
r}(-1)^{k_{i}}(\lambda_{i}-\lambda_{s})^{k_{s}k_{i}}\in\mathbb{C}^{\times}.$

Note that there is an $h(z)\in\mathbb{C}[z]$ such that

$q_{s}(\lambda_{s}z)=\prod_{1\leq i\neq s\leq
r}((\lambda_{s}z-\lambda_{s})^{k_{s}}-(\lambda_{i}-\lambda_{s})^{k_{s}})^{k_{i}}=(z-1)^{k_{s}}h(z)+\alpha.$

In view of this and (2.3), (2.16) can be rewritten as

$\begin{split}\sum_{j=0}^{k_{s}-1}\alpha(A_{sj}(z_{2})-a(\lambda_{s}z_{2})_{j}^{\phi}b(z_{2}))\left(p(z_{2})\frac{\partial}{\partial
z_{2}}\right)^{(j)}\bar{p}(z_{1})\delta\left(\frac{z_{2}}{z_{1}}\right)=0.\end{split}$

Note that $\alpha\neq 0$, this together with Lemma 2.3 proves (2.13). ∎

######  Remark 2.7.

For $p(x)=1$ and $p(x)=x$, Proposition 2.6 was proved in [Li5] and [Li4]
respectively.

##  3\. $\phi$-coordinated modules of vertex algebras

In this section we establish some conceptual results on $\phi$-coordinated
(quasi) modules for vertex algebras. Let $(V,Y,\mathbf{1})$ be a vertex
algebra, the following notion was introduced in [Li2].

######  Definition 3.1.

A $\phi$-coordinated quasi $V$-module is a vector space $W$ equipped with a
linear map

$\displaystyle Y_{W}(\cdot,z):V$ $\displaystyle\rightarrow$
$\displaystyle\mathrm{Hom}(W,W((z)))\subset(\mathrm{End}W)[[z,z^{-1}]],$
$\displaystyle v$ $\displaystyle\mapsto$ $\displaystyle
Y_{W}(v,z)=\sum_{n\in\mathbb{Z}}v_{n}z^{-n-1},$

satisfying the conditions that $Y_{W}(\mathbf{1},z)=1_{W}$ and that for
$u,v\in V$, there exists a polynomial
$f(z_{1},z_{2})\in\mathbb{C}[z_{1},z_{2}]$ such that

$\displaystyle
f(z_{1},z_{2})Y_{W}(u,z_{1})Y_{W}(v,z_{2})\in\mathrm{Hom}(W,W((z_{1},z_{2}))),$
$\displaystyle
f(\phi(z_{2},z_{0}),z_{2})Y_{W}(Y(u,z_{0})v,z_{2})=(f(z_{1},z_{2})Y_{W}(u,z_{1})Y_{W}(v,z_{2}))\mid_{z_{1}=\phi(z_{2},z_{0})}.$

A $\phi$-coordinated $V$-module is defined as above except that
$f(z_{1},z_{2})$ is assumed to be a polynomial of the form $(z_{1}-z_{2})^{k}$
with $k\in\mathbb{N}$.

######  Remark 3.2.

When $p(x)=1$, a $\phi$-coordinated $V$-module is just a $V$-module.

In what follows we collect some basic properties of $\phi$-coordinated quasi
$V$-modules for later use.

######  Lemma 3.3.

[Li2] Let $(W,Y_{W})$ be a $\phi$-coordinated quasi $V$-module and let $u,v\in
V$. Assume that there exists a nonzero polynomial
$q(z_{1},z_{2})\in\mathbb{C}[z_{1},z_{2}]$ such that

$\displaystyle
q(z_{1},z_{2})Y_{W}(u,z_{1})Y_{W}(v,z_{2})\in\mathrm{Hom}(W,W((z_{1},z_{2}))).$

Then

(3.1) $\displaystyle q(z_{1},z_{2})[Y_{W}(u,z_{1}),Y_{W}(v,z_{2})]$
$\displaystyle=0,$ (3.2) $\displaystyle
q(\phi(z_{2},z_{0}),z_{2})Y_{W}(Y(u,z_{0})v,z_{2})$
$\displaystyle=(q(z_{1},z_{2})Y_{W}(u,z_{1})Y_{W}(v,z_{2}))\mid_{z_{1}=\phi(z_{2},z_{0})}.$

We define the linear operator $\mathcal{D}$ on $V$ by
$\mathcal{D}v=v_{-2}\mathbf{1}$ for $v\in V$.

######  Lemma 3.4.

[Li2] Let $(W,Y_{W})$ be a $\phi$-coordinated quasi $V$-module. Then

(3.3) $\displaystyle Y_{W}(\mathcal{D}v,z)$
$\displaystyle=\left(p(z)\frac{d}{dz}\right)Y_{W}(v,z),\quad v\in V,$ (3.4)
$\displaystyle Y_{W}(Y(u,z_{0})v,z)$
$\displaystyle=Y_{\mathcal{E}}^{\phi}(Y_{W}(u,z),z_{0})Y_{W}(v,z),\quad u,v\in
V.$

The following is an analogue of Borcherd’s commutator formula (cf. [BLP]).

######  Proposition 3.5.

Let $(W,Y_{W})$ be a $\phi$-coordinated $V$-module. Then for $u,v\in V$,

$\begin{split}[Y_{W}(u,z_{1}),Y_{W}(v,z_{2})]=\sum_{j\geq
0}Y_{W}(u_{j}v,z_{2})\left(p(z_{2})\frac{\partial}{\partial
z_{2}}\right)^{(j)}\bar{p}(z_{1})\delta\left(\frac{z_{2}}{z_{1}}\right).\end{split}$

Conversely, if $W$ is faithful and

$\begin{split}[Y_{W}(u,z_{1}),Y_{W}(v,z_{2})]=\sum_{j\geq
0}Y_{W}(c_{j},z_{2})\left(p(z_{2})\frac{\partial}{\partial
z_{2}}\right)^{(j)}\bar{p}(z_{1})\delta\left(\frac{z_{2}}{z_{1}}\right),\end{split}$

for some $c_{j}\in V$, then $c_{j}=u_{j}v$ for all $j\geq 0$.

###### Proof.

The second assertion follows from Lemma 2.3. For the first one, let $u,v\in
V$. In view of (3.1), there is a nonnegative integer $k$ such that

$(z_{1}-z_{2})^{k}\,[Y_{W}(u,z_{1}),Y_{W}(v,z_{2})]=0.$

It then follows from Proposition 2.6 that

$\begin{split}[Y_{W}(u,z_{1}),Y_{W}(v,z_{2})]&=\sum\limits_{j\geq
0}Y_{W}(u,z_{2})^{\phi}_{j}Y_{W}(v,z_{2})\left(p(z_{2})\frac{\partial}{\partial
z_{2}}\right)^{(j)}\bar{p}(z_{1})\delta\left(\frac{z_{2}}{z_{1}}\right).\end{split}$

This together with (3.4) proves the first assertion. ∎

Next we are going to give a conceptual construction of $\phi$-coordinated
quasi modules for vertex algebras. The following is a $\phi$-generalization of
[Li5, Proposition 4.3].

######  Proposition 3.6.

Let $a(z),b(z),c(z)\in{\mathcal{E}}(W)$. Assume that

$\begin{split}f(z_{1},z_{2})[a(z_{1}),b(z_{2})]=g(z_{1},z_{2})[a(z_{1}),c(z_{2})]=h(z_{1},z_{2})[b(z_{1}),c(z_{2})]=0\end{split}$

for some nonzero polynomials $f(z_{1},z_{2}),g(z_{1},z_{2}),h(z_{1},z_{2})$.
Then for any $n\in\mathbb{Z}$, there exists a nonnegative integer $k$ such
that

$g(z_{1},z_{2})^{k}h(z_{1},z_{2})[a(z_{1})_{n}^{\phi}b(z_{1}),c(z_{2})]=0.$

###### Proof.

Fix an $n\in\mathbb{Z}$ and take a nonnegative integer $k$ such that

$z_{0}^{k+n}f(\phi(z,z_{0}),z)^{-1}\in\mathbb{C}((z))[[z_{0}]].$

Using this and the fact

$\phi(\phi(z,z_{0}),-z_{0})=\phi(z,0)=z=\phi(\phi(z,-z_{0}),z_{0}),$

we obtain

$\displaystyle\quad
g(z_{2},z_{3})^{k}h(z_{2},z_{3})(a(z_{2})_{n}^{\phi}b(z_{2}))c(z_{3})$
$\displaystyle=\mathrm{Res}_{z_{0}}z_{0}^{n}g(z_{2},z_{3})^{k}h(z_{2},z_{3})\left(Y_{\mathcal{E}}^{\phi}(a(z_{2}),z_{0})b(z_{2})\right)c(z_{3})$
$\displaystyle=\mathrm{Res}_{z_{1}}\mathrm{Res}_{z_{0}}z_{0}^{n}g(z_{2},z_{3})^{k}h(z_{2},z_{3})f(\phi(z_{2},z_{0}),z_{2})^{-1}$
$\displaystyle\quad
z_{1}^{-1}\delta\left(\frac{\phi(z_{2},z_{0})}{z_{1}}\right)(f(z_{1},z_{2})a(z_{1})b(z_{2}))c(z_{3})$
$\displaystyle=\mathrm{Res}_{z_{1}}\mathrm{Res}_{z_{0}}z_{0}^{n}g(\phi(z_{1},-z_{0}),z_{3})^{k}h(z_{2},z_{3})f(\phi(z_{2},z_{0}),z_{2})^{-1}$
$\displaystyle\quad
z_{1}^{-1}\delta\left(\frac{\phi(z_{2},z_{0})}{z_{1}}\right)(f(z_{1},z_{2})a(z_{1})b(z_{2}))c(z_{3})$
$\displaystyle=\mathrm{Res}_{z_{1}}\mathrm{Res}_{z_{0}}z_{0}^{n}\left(e^{-z_{0}p(z_{1})\frac{\partial}{\partial
z_{1}}}g(z_{1},z_{3})^{k}\right)h(z_{2},z_{3})f(\phi(z_{2},z_{0}),z_{2})^{-1}$
$\displaystyle\quad
z_{1}^{-1}\delta\left(\frac{\phi(z_{2},z_{0})}{z_{1}}\right)(f(z_{1},z_{2})a(z_{1})b(z_{2}))c(z_{3})$
$\displaystyle=\mathrm{Res}_{z_{1}}\mathrm{Res}_{z_{0}}\sum_{i=0}^{k-1}z_{0}^{n+i}\left(\left(-p(z_{1})\frac{\partial}{\partial
z_{1}}\right)^{(i)}g(z_{1},z_{3})^{k}\right)h(z_{2},z_{3})f(\phi(z_{2},z_{0}),z_{2})^{-1}$
$\displaystyle\quad
z_{1}^{-1}\delta\left(\frac{\phi(z_{2},z_{0})}{z_{1}}\right)(f(z_{1},z_{2})a(z_{1})b(z_{2}))c(z_{3})$
$\displaystyle=\mathrm{Res}_{z_{1}}\mathrm{Res}_{z_{0}}\sum_{i=0}^{k-1}z_{0}^{n+i}\left(\left(-p(z_{1})\frac{\partial}{\partial
z_{1}}\right)^{(i)}g(z_{1},z_{3})^{k}\right)h(z_{2},z_{3})c(z_{3})$
$\displaystyle\quad
f(\phi(z_{2},z_{0}),z_{2})^{-1}z_{1}^{-1}\delta\left(\frac{\phi(z_{2},z_{0})}{z_{1}}\right)(f(z_{1},z_{2})a(z_{1})b(z_{2}))$
$\displaystyle=\mathrm{Res}_{z_{1}}\mathrm{Res}_{z_{0}}z_{0}^{n}\left(e^{-z_{0}p(z_{1})\frac{\partial}{\partial
z_{1}}}g(z_{1},z_{3})^{k}\right)h(z_{2},z_{3})c(z_{3})f(\phi(z_{2},z_{0}),z_{2})^{-1}$
$\displaystyle\quad
z_{1}^{-1}\delta\left(\frac{\phi(z_{2},z_{0})}{z_{1}}\right)(f(z_{1},z_{2})a(z_{1})b(z_{2}))$
$\displaystyle=g(z_{2},z_{3})^{k}h(z_{2},z_{3})c(z_{3})(a(z_{2})_{n}^{\phi}b(z_{2})).$

∎

A subset $S$ of ${\mathcal{E}}(W)$ is said to be (resp. quasi) local if any
pair in $S$ is (resp. quasi) local, and a (resp. quasi) local subspace $U$ of
${\mathcal{E}}(W)$ is said to be $Y_{\mathcal{E}}^{\phi}$-closed if

$a(z)^{\phi}_{n}b(z)\in U\quad\textnormal{{for all }}a(z),b(z)\in U,\
n\in\mathbb{Z}.$

In view of Proposition 3.6, we can formulate the following notion.

######  Definition 3.7.

Let $S$ be a quasi local subset of ${\mathcal{E}}(W)$. We define
${\langle}S{\rangle}_{\phi}$ to be the smallest
$Y_{\mathcal{E}}^{\phi}$-closed quasi local subspace of ${\mathcal{E}}(W)$
which contains $S$ and $1_{W}$.

We need the following $\phi$-generalization of [Li5, Proposition 4.8] for
later use.

######  Proposition 3.8.

For $a(z),b(z)\in{\mathcal{E}}(W)$, if

$\displaystyle f(z_{1},z_{2})\,[a(z_{1}),b(z_{2})]=0$

for some $f(z_{1},z_{2})\in\mathbb{C}[z_{1},z_{2}]$, then

$(z_{1}-z_{2})^{k}\,[Y_{\mathcal{E}}^{\phi}(a(z),z_{1}),Y_{\mathcal{E}}^{\phi}(b(z),z_{2})]=0,$

where $k$ is the order of the zero of $f(z_{1},z_{2})$ at $z_{1}=z_{2}$.

###### Proof.

Let $a(z),b(z)$, $f(z_{1},z_{2})$ and $k$ be as in the proposition. Then it
follows from [JKLT, Proposition 2.18] that

(3.5) $\displaystyle
f(\phi(z,z_{1}),\phi(z,z_{2}))[Y_{\mathcal{E}}^{\phi}(a(z),z_{1}),Y_{\mathcal{E}}^{\phi}(b(z),z_{2})]=0.$

Write

$f(z_{1},z_{2})=(z_{1}-z_{2})^{k}f_{0}(z_{1},z_{2})$

for some polynomial $f_{0}(z_{1},z_{2})$ with $f_{0}(z_{1},z_{1})\neq 0$. Note
that $\phi(z,0)=z$. This implies that $f_{0}(\phi(z,0),\phi(z,0))$ is nonzero
and hence $f_{0}(\phi(z,z_{1}),\phi(z,z_{2}))$ is invertible in
$\mathbb{C}((z))[[z_{1},z_{2}]]$. On the other hand, from the definition of
$\phi(z,z_{1})=e^{z_{1}p(z)\frac{d}{dz}}(z)$, we have

$\displaystyle\phi(z,z_{1})-\phi(z,z_{2})=(z_{1}-z_{2})g(z,z_{1},z_{2})$

for some $g(z,z_{1},z_{2})\in\mathbb{C}((z))[[z_{1},z_{2}]]$. Furthermore, we
have $g(z,0,0)=p(z)\neq 0$ and hence $g(z,z_{1},z_{2})$ is also invertible in
$\mathbb{C}((z))[[z_{1},z_{2}]]$. Thus, we have shown that
$f(\phi(z,z_{1}),\phi(z,z_{2}))$ is a product of $(z_{1}-z_{2})^{k}$ and an
invertible element in $\mathbb{C}((z))[[z_{1},z_{2}]]$. This together with
(3.5) proves the proposition. ∎

As the main result of this section we have:

######  Theorem 3.9.

Let $S$ be a quasi local subset of ${\mathcal{E}}(W)$. Then $(\langle
S\rangle_{\phi},Y_{\mathcal{E}}^{\phi},1_{W})$ is a vertex algebra with $S$ as
a generating set and with $W$ as a faithful $\phi$-coordinated quasi module on
which $Y_{W}(a(z),z_{0})=a(z_{0})$ for $a(z)\in\langle S\rangle_{\phi}$.
Furthermore, if $S$ is local, then $W$ is a $\phi$-coordinated $\langle
S\rangle_{\phi}$-module.

###### Proof.

It suffices to prove the case that $S$ is quasi local. Recall from [Li2] that
$\langle S\rangle_{\phi}$ is a nonlocal vertex algebra with $W$ as a faithful
$\phi$-coordinated quasi $\langle S\rangle_{\phi}$-module. On the other hand,
it follows from Proposition 3.8 that

$\\{Y_{{\mathcal{E}}}^{\phi}(a(z),z_{0})\mid a(z)\in\langle
S\rangle_{\phi}\\}$

is a local subset of ${\mathcal{E}}(\langle S\rangle_{\phi})$. This implies
that ${\langle}S{\rangle}_{\phi}$ is indeed a vertex algebra. ∎

##  4\. $\phi$-coordinated $V_{\mathcal{C}}$-modules

In this section, we first recall the notion of a conformal algebra
$\mathcal{C}$ [K] (also known as a vertex Lie algebra in [DLM, P]) and then
determine the $\phi$-coordinated modules for the universal enveloping vertex
algebra of $\mathcal{C}$ in terms of restricted modules of a Lie algebra
$\widehat{\mathcal{C}}_{\phi}$ arising from $\mathcal{C}$ and $\phi$.

######  Definition 4.1.

A conformal algebra $\mathcal{C}$ is a $\mathbb{C}[\partial]$-module with a
$\mathbb{C}$-bilinear product $a_{n}b$ for each $n\in\mathbb{N}$ such that the
following axioms hold $(a,b,c\in\mathcal{C},\ m,n\in\mathbb{N})$:

(C0) $\displaystyle a_{n}b=0\quad\textnormal{{for }}n\gg 0,$ (C1)
$\displaystyle(\partial a)_{n}b=-na_{n-1}b,$ (C2) $\displaystyle
a_{n}b=-\sum_{j=0}^{\infty}(-1)^{j+n}\partial^{(j)}(b_{n+j}a),$ (C3)
$\displaystyle
a_{m}(b_{n}c)=\sum_{j=0}^{m}\binom{m}{j}(a_{j}b)_{m+n-j}c+b_{n}(a_{m}c).$

A linear map $\varphi:\mathcal{C}\rightarrow\mathcal{C}^{\prime}$ of conformal
algebras is called a _homomorphism_ if

$\varphi\partial=\partial\varphi\quad\textnormal{{and}}\quad\varphi(a_{{n}}b)=\varphi(a)_{n}\varphi(b)\quad\textnormal{{
for }}a,b\in\mathcal{C},\ n\in\mathbb{N}.$

Let $\mathcal{C}$ be a conformal algebra. Recall that $\phi$ is an associate
of $F(x,y)$, which is determined by a nonzero $p(x)\in\mathbb{C}((x))$. We
define a multiplication on the formal loop space
$\mathcal{C}\otimes\mathbb{C}((x))$ by

(4.1) $\displaystyle[a\otimes f(x),b\otimes g(x)]=\sum_{i\geq
0}(a_{i}b)\otimes\left(\left(p(x)\frac{d}{dx}\right)^{(i)}f(x)\right)g(x)$

for $a,b\in\mathcal{C}$ and $f(x),g(x)\in\mathbb{C}((x))$. Form the quotient
space

$\displaystyle\widehat{\mathcal{C}}_{\phi}:=\mathcal{C}\otimes\mathbb{C}((x))/\mathrm{Im}\,(\partial\otimes
1+1\otimes p(x)\frac{d}{dx}).$

It was known from [K] that the multiplication (4.1) defines a Lie algebra
structure on the quotient space $\widehat{\mathcal{C}}_{\phi}$. When
$\phi=x+z$, i.e., $p(x)=1$, we also write
$\widehat{\mathcal{C}}=\widehat{\mathcal{C}}_{\phi}.$ For $a\in\mathcal{C}$
and $f(x)\in\mathbb{C}((x))$, we set

$\overline{a\otimes f(x)}=\rho(a\otimes f(x)),$

where
$\rho:\mathcal{C}\otimes\mathbb{C}((x))\rightarrow\widehat{\mathcal{C}}_{\phi}$
is the quotient map.

For $a\in\mathcal{C}$, we set

$a_{\phi}(z)=\sum_{n\in\mathbb{Z}}\overline{a\otimes
x^{n}p(x)}z^{-n-1}\in\widehat{\mathcal{C}}_{\phi}[[z,z^{-1}]].$

######  Definition 4.2.

A $\widehat{\mathcal{C}}_{\phi}$-module $W$ is called restricted if
$a_{\phi}(z)\in{\mathcal{E}}(W)$ for any $a\in\mathcal{C}$.

One notices from definition that

(4.2) $\displaystyle\left(p(z)\frac{d}{dz}\right)a_{\phi}(z)=(\partial
a)_{\phi}(z),\quad a\in\mathcal{C}.$

Furthermore, in view of (4.1), we have for $a,b\in\mathcal{C}$,

(4.3) $\displaystyle[a_{\phi}(z),b_{\phi}(w)]=\sum_{i\geq
0}(a_{i}b)_{\phi}(w)\left(p(w)\frac{\partial}{\partial
w}\right)^{(i)}\bar{p}(z)\delta\left(\frac{w}{z}\right).$

We have the following useful lemma.

######  Lemma 4.3.

Let $W$ be a vector space and $\mathcal{C}$ be a conformal algebra. Assume
that there is a linear map

$\pi:\mathcal{C}\rightarrow{\mathcal{E}}(W),\quad a\mapsto
a(z)=\sum_{n\in\mathbb{Z}}a_{[n]}z^{-n-1},$

such that for $a\in\mathcal{C}$,

(4.4) $\displaystyle\left(p(z)\frac{d}{dz}\right)a(z)=(\partial a)(z),$

and for $a,b\in\mathcal{C}$,

(4.5) $\displaystyle[a(z),b(w)]=\sum_{i\geq
0}(a_{i}b)(w)\left(p(w)\frac{\partial}{\partial_{w}}\right)^{(i)}\bar{p}(z)\delta\left(\frac{w}{z}\right).$

Then there is a restricted $\widehat{\mathcal{C}}_{\phi}$-module structure on
$W$ with $a_{\phi}(z)=a(z)$ for $a\in\mathcal{C}$.

###### Proof.

Let $a\in\mathcal{C}$ and $f(x)\in\mathbb{C}((x))$. Set

$\displaystyle f_{0}(x)=p(x)^{-1}f(x)=\sum_{n\geq n_{0}}b_{n}x^{n},$

where $n_{0}\in\mathbb{Z}$. We define an action of $\overline{a\otimes f(x)}$
on $W$ by

(4.6) $\displaystyle\overline{a\otimes
f(x)}w=\sum_{n=n_{0}}^{n_{a,w}}b_{n}a_{[n]}w,\quad w\in W,$

where $n_{a,w}$ is a sufficiently large integer such that $a_{[m]}w=0$ for all
$m>n_{a,w}$. It follows from (4.4) that this action is well-defined.
Furthermore, by comparing (4.3) with (4.5), one can check that (4.6) defines a
restricted $\widehat{\mathcal{C}}_{\phi}$-module structure on $W$ with
$a_{\phi}(z)=a(z)$ for $a\in\mathcal{C}$. ∎

Now we recall the notion of universal enveloping vertex algebra of a conformal
algebra $\mathcal{C}$. Set
$\widehat{\mathcal{C}}_{+}=\rho(\mathcal{C}\otimes\mathbb{C}[[x]]),$ which is
a Lie subalgebra of $\widehat{\mathcal{C}}$. Let $\mathbb{C}$ be the one-
dimensional trivial $\widehat{\mathcal{C}}_{+}$-module. Then we have the
induced $\widehat{\mathcal{C}}$-module

$V_{\mathcal{C}}=\mathcal{U}(\widehat{\mathcal{C}})\otimes_{\mathcal{U}(\widehat{\mathcal{C}}_{+})}\mathbb{C}.$

Set $\mathbf{1}=1\otimes 1$. It is known that $\mathcal{C}$ can be identified
as a subspace of $V_{\mathcal{C}}$ through the linear map

$a\mapsto\overline{a\otimes x^{-1}}\mathbf{1},\quad a\in\mathcal{C}.$

Note that each vertex algebra is naturally a conformal algebra by setting
$\partial=\mathcal{D}$ and forgetting the $n$-products with $n<0$. The
following result is well-known (cf. [K, DLM, P, FB]).

######  Proposition 4.4.

There is a vertex algebra structure on $V_{\mathcal{C}}$ such that
$\mathbf{1}$ is the vacuum vector and that

$Y(a,z)=a(z)=\sum_{n\in\mathbb{Z}}\overline{a\otimes x^{n}}z^{-n-1}$

for $a\in\mathcal{C}$. Moreover, for any vertex algebra $V$, if
$\varphi:\mathcal{C}\rightarrow V$ is a homomorphism of conformal algebras,
then $\varphi$ extends uniquely to a vertex algebra homomorphism from
$V_{\mathcal{C}}$ to $V$.

The following is the main result of this section.

######  Theorem 4.5.

Let $\mathcal{C}$ be a conformal algebra. Then any restricted
$\widehat{\mathcal{C}}_{\phi}$-module $W$ is naturally a $\phi$-coordinated
$V_{\mathcal{C}}$-module with $Y_{W}(a,z)=a_{\phi}(z)$ for $a\in\mathcal{C}$.
On the other hand, any $\phi$-coordinated $V_{\mathcal{C}}$-module $W$ is
naturally a restricted $\widehat{\mathcal{C}}_{\phi}$-module with
$a_{\phi}(z)=Y_{W}(a,z)$ for $a\in\mathcal{C}.$

###### Proof.

Let $W$ be a restricted $\widehat{\mathcal{C}}_{\phi}$-module. Then it follows
from (4.3) that

$S=\\{a_{\phi}(z)\mid a\in\mathcal{C}\\}$

is a local subset of $\mathcal{E}(W)$. In view of Theorem 3.9, there is a
vertex algebra ${\langle}S{\rangle}_{\phi}$ with $W$ as a faithful
$\phi$-coordinated ${\langle}S{\rangle}_{\phi}$-module on which
$Y_{W}(b(z),z_{0})=b(z_{0})$ for $b(z)\in{\langle}S{\rangle}_{\phi}$. This
together with (4.3) implies that for $a,b\in\mathcal{C}$,

$\displaystyle[Y_{W}(a_{\phi}(z_{0}),z),Y_{W}(b_{\phi}(z_{0}),w)]=[a_{\phi}(z),b_{\phi}(w)]$
$\displaystyle=$ $\displaystyle\sum_{i\geq
0}(a_{i}b)_{\phi}(w)\left(p(w)\frac{\partial}{\partial
w}\right)^{(i)}\bar{p}(z)\delta\left(\frac{w}{z}\right).$

As $W$ is faithful, it follows from Proposition 3.5 that

(4.7) $\displaystyle
a_{\phi}(z)_{i}^{\phi}b_{\phi}(z)=(a_{i}b)_{\phi}(z),\quad
a,b\in\mathcal{C},i\in\mathbb{N}.$

We define a linear map

$\varphi:\ \mathcal{C}\rightarrow\langle S\rangle_{\phi},\quad a\mapsto
a_{\phi}(z).$

It follows from (3.3) and (4.2) that for $a\in\mathcal{C}$,

$\mathcal{D}\varphi(a)=\left(p(z)\frac{d}{dz}\right)a_{\phi}(z)=(\partial
a)_{\phi}(z)=\varphi\partial(a).$

This together with (4.7) implies that $\varphi$ is a homomorphism of conformal
algebras. Then Proposition 4.4 says that $\varphi$ can be extended to a vertex
algebra homomorphism from $V_{\mathcal{C}}$ to $\langle S\rangle_{\phi}$.
Thus, $W$ becomes a $\phi$-coordinated $V_{\mathcal{C}}$-module with

$Y_{W}(a,w)=Y_{W}(\varphi(a),w)=Y_{W}(a_{\phi}(z),w)=a_{\phi}(w)\quad\textnormal{{for}}\
a\in\mathcal{C}.$

On the other hand, let $W$ be a $\phi$-coordinated $V_{\mathcal{C}}$-module.
Recall from (3.3) that for $a\in\mathcal{C}$,

$\displaystyle Y_{W}(\partial
a,z)=Y_{W}(\mathcal{D}a,z)=\left(p(z)\frac{d}{dz}\right)Y_{W}(a,z),$

and recall from Proposition 3.5 that for $a,b\in\mathcal{C}$,

$\begin{split}[Y_{W}(a,z),Y_{W}(b,w)]=\sum_{i\geq
0}Y_{W}(a_{i}b,w)\left(p(w)\frac{\partial}{\partial_{w}}\right)^{(i)}\bar{p}(z)\delta\left(\frac{w}{z}\right).\end{split}$

Combining this with Lemma 4.3, we see that $W$ is a restricted
$\widehat{\mathcal{C}}_{\phi}$-module with $a_{\phi}(z)=Y_{W}(a,z)$ for
$a\in\mathcal{C}$, as desired. ∎

######  Remark 4.6.

When $p(x)\in\mathbb{C}[x,x^{-1}]$ is a Laurent polynomial, the Lie algebra
$\widehat{\mathcal{C}}_{\phi}$ can be defined in the following usual way:

$\widehat{\mathcal{C}}_{\phi}:=\mathcal{C}\otimes\mathbb{C}[x,x^{-1}]/\textnormal{{Im}}(\partial\otimes
1+1\otimes p(x)\frac{d}{dx}).$

However, for the case $p(x)\in\mathbb{C}((x))$, the derivation
$p(x)\frac{d}{dx}$ may be not well-defined on $\mathbb{C}[x,x^{-1}]$. So we
need to enlarge the usual loop algebra
$\mathcal{C}\otimes\mathbb{C}[x,x^{-1}]$ to the formal loop algebra
$\mathcal{C}\otimes\mathbb{C}((x))$.

##  5\. $(G,\chi_{\phi})$-equivariant $\phi$-coordinated quasi modules

Let $G$ be a group and let $\chi,\chi_{\phi}:G\rightarrow\mathbb{C}^{\times}$
be two linear characters on $G$. In this section we generalize the previous
results on $\phi$-coordinated modules for vertex algebras to the
$(G,\chi_{\phi})$-equivariant $\phi$-coordinated quasi modules for
$(G,\chi)$-vertex algebras. We start with the notion of $(G,\chi)$-vertex
algebra in [JKLT] (cf. [Li3, Li5]).

######  Definition 5.1.

A $(G,\chi)$-vertex algebra is a pair $(V,R)$ consisting of a vertex algebra
$V$ and a group homomorphism $R:G\rightarrow\mathrm{GL}(V)$ such that
$R_{g}(\mathbf{1})=\mathbf{1}$ and

$R_{g}Y(v,z)R_{g}^{-1}=Y(R_{g}(v),\chi(g)z)\quad\textnormal{{for }}g\in G,v\in
V.$

For $(G,\chi)$-vertex algebras $(V,R)$, $(V^{\prime},R^{\prime})$, a linear
map $\psi:V\rightarrow V^{\prime}$ is called a $(G,\chi)$-vertex algebra
homomorphism if $\psi$ is a vertex algebra homomorphism and $\psi\circ
R_{g}=R^{\prime}_{g}\circ\psi$ for $g\in G$.

For a subgroup $\Gamma$ of $\mathbb{C}^{\times}$, we denote by
$\mathbb{C}_{\Gamma}[x]$ the set of all monic polynomials in $\mathbb{C}[x]$
whose roots contained in $\Gamma$. The following notion was introduced in
[JKLT].

######  Definition 5.2.

Let $(V,R)$ be a $(G,\chi)$-vertex algebra. A $(G,\chi_{\phi})$-equivariant
$\phi$-coordinated quasi $V$-module is a $\phi$-coordinated quasi $V$-module
$(W,Y_{W})$ such that

(5.1) $\displaystyle
Y_{W}(R_{g}v,z)=Y_{W}(v,\chi_{\phi}(g)^{-1}z)\quad\textnormal{{for }}g\in
G,v\in V,$

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

(5.2) $\displaystyle
q(z_{1}/z_{2})Y_{W}(u,z_{1})Y_{W}(v,z_{2})\in\mathrm{Hom}(W,W((z_{1},z_{2}))).$

Let $(V,R)$ be a $(G,\chi)$-vertex algebra and let $(W,Y_{W})$ be a
$(G,\chi_{\phi})$-equivariant $\phi$-coordinated quasi $V$-module. If
$\frac{d}{dz}Y_{W}(v,z)\neq 0$ for some $v\in V$, then it was shown in [JKLT]
that

(5.3)
$\begin{split}\phi(x,\chi(g)z)=\chi_{\phi}(g)\phi(\chi_{\phi}(g)^{-1}x,z)\quad\textnormal{{for
}}g\in G.\end{split}$

Note that this relation is equivalent to

(5.4)
$\begin{split}p(\chi_{\phi}(g)x)=\chi(g)^{-1}\chi_{\phi}(g)p(x)\quad\textnormal{{for
}}g\in G.\end{split}$

On the other hand, if $\frac{d}{dz}Y_{W}(v,z)=0$ for any $v\in V$, then the
quotient vertex algebra $V/\ker Y_{W}$ is simply an associative algebra and
$G$ acts trivially on $V/\ker Y_{W}$. In view of this, we always assume that
the characters $\chi$ and $\chi_{\phi}$ satisfy the relation (5.3). Note that
we have $\chi=\chi_{\phi}$ if $p(x)=1$ and $\chi=1$ if $p(x)=x$.

As a generalization of Proposition 3.5, we have:

######  Proposition 5.3.

Let $(V,R)$ be a $(G,\chi)$-vertex algebra and let $(W,Y_{W})$ be a
$(G,\chi_{\phi})$-equivariant $\phi$-coordinated quasi $V$-module. Let
$\psi:\chi_{\phi}(G)\rightarrow G$ be a section of $\chi_{\phi}$. Then for
$u,v\in V$,

(5.5) $\begin{split}&[Y_{W}(u,z_{1}),Y_{W}(v,z_{2})]\\\
=\mathrm{Res}_{z_{0}}&\sum_{g\in\psi(\chi_{\phi}(G))}Y_{W}(Y(R_{g^{-1}}u,z_{0})v,z_{2})e^{z_{0}p(z_{2})\frac{\partial}{\partial
z_{2}}}\left(\chi(g)\bar{p}(z_{1})\delta\left(\frac{\chi_{\phi}(g)z_{2}}{z_{1}}\right)\right).\end{split}$

###### Proof.

Let $u,v\in V$. From (5.2) and (3.1), it follows that there exists a
$q(z)\in\mathbb{C}_{\chi_{\phi}(G)}[z]$ such that

$q(z_{1}/z_{2})[Y_{W}(u,z_{1}),Y_{W}(v,z_{2})]=0.$

Then by applying Proposition 2.6 we have

$\displaystyle[Y_{W}(u,z_{1}),Y_{W}(v,z_{2})]$ $\displaystyle=$
$\displaystyle\sum_{\lambda\in\chi_{\phi}(G)}\sum_{j\geq 0}Y_{W}(u,\lambda
z_{2})_{j}^{\phi}Y_{W}(v,z_{2})\left(p(z_{2})\frac{\partial}{\partial
z_{2}}\right)^{(j)}\bar{p}(\lambda^{-1}z_{1})\delta\left(\frac{\lambda
z_{2}}{z_{1}}\right).$

Note that (5.4) implies that for $g\in G$,

$\displaystyle\bar{p}(\chi_{\phi}(g)^{-1}z_{1})=\chi_{\phi}(g)z_{1}^{-1}p(\chi_{\phi}(g^{-1})z_{1})=\chi(g)z_{1}^{-1}p(z_{1})=\chi(g)\bar{p}(z_{1}),$

together with $\psi$ a section of $\chi_{\phi}$, we obtain

$\begin{split}&[Y_{W}(u,z_{1}),Y_{W}(v,z_{2})]\\\
=\mathrm{Res}_{z_{0}}&\sum_{g\in\psi(\chi_{\phi}(G))}Y_{\mathcal{E}}^{\phi}\left(Y_{W}(u,\chi_{\phi}(g)z_{2}),z_{0}\right)Y_{W}(v,z_{2})e^{z_{0}p(z_{2})\frac{\partial}{\partial
z_{2}}}\left(\chi(g)\bar{p}(z_{1})\delta\left(\frac{\chi_{\phi}(g)z_{2}}{z_{1}}\right)\right).\end{split}$

From (5.1) and (3.4), we have

$\begin{split}Y_{\mathcal{E}}^{\phi}\left(Y_{W}(u,\chi_{\phi}(g)z),z_{0}\right)Y_{W}(v,z)=Y_{\mathcal{E}}^{\phi}\left(Y_{W}(R_{g^{-1}}u,z),z_{0}\right)Y_{W}(v,z)=Y_{W}(Y(R_{g^{-1}}u,z_{0})v,z)\end{split}$

for $g\in G$. Thus (5.5) follows. ∎

######  Definition 5.4.

Let $W$ be a vector space and let $\Gamma$ be a subgroup of
$\mathbb{C}^{\times}$. A subset $S$ of ${\mathcal{E}}(W)$ is said to
$\Gamma$-quasi local if for any pair $a(z),b(z)$ in $S$, there exists
$q(z)\in\mathbb{C}_{\Gamma}[z]$ such that

$q(z_{1}/z_{2})[a(z_{1}),b(z_{2})]=0,$

and is said to be $\Gamma$-stable if for any $a(z)\in S$ and
$\lambda\in\Gamma$, $a(\lambda z)\in S$.

For a $\Gamma$-quasi local and $\Gamma$-stable subset $S$ of
${\mathcal{E}}(W)$, ${\langle}S{\rangle}_{\phi}$ is also $\Gamma$-quasi local
(Theorem 3.6) and $\Gamma$-stable ([JKLT]). Furthermore, from [JKLT, Theorem
3.11] and Theorem 3.9, we immediately have the following result.

######  Theorem 5.5.

Let $S$ be a $\chi_{\phi}(G)$-stable and $\chi_{\phi}(G)$-quasi local subset
of ${\mathcal{E}}(W)$. Assume that the character $\chi_{\phi}$ is injective.
Then the vertex algebra $\langle S\rangle_{\phi}$ together with the group
homomorphism

$R:G\rightarrow\mathrm{GL}({\langle}S{\rangle}_{\phi}),\quad g\mapsto R_{g}$

is a $(G,\chi)$-vertex algebra and $W$ is a faithful
$(G,\chi_{\phi})$-equivariant $\phi$-coordinated quasi $\langle
S\rangle_{\phi}$-module with $Y_{W}(a(z),z_{0})=a(z_{0})$ for $a(z)\in\langle
S\rangle_{\phi}$, where $R_{g}(a(z))=a(\chi_{\phi}(g)^{-1}z)$.

The following notion was introduced in [Li3]:

######  Definition 5.6.

A $(G,\chi)$-conformal algebra $(\mathcal{C},R)$ is a conformal algebra
$\mathcal{C}$ together with a group homomorphism

$\displaystyle R:G\rightarrow\mathrm{GL}(\mathcal{C}),\quad g\mapsto R_{g},$

such that for any $a,b\in\mathcal{C}$,

(CR1) $\displaystyle\partial R_{g}=\chi(g)^{-1}R_{g}\partial,$ (CR2)
$\displaystyle
R_{g}(a_{m}b)=\chi(g)^{-(m+1)}(R_{g}a)_{m}(R_{g}b)\quad\textnormal{{for
}}m\in\mathbb{N},$ (CR3) $\displaystyle(R_{g}a)_{m}b=0\quad\textnormal{{for
all but finitely many }}m\in\mathbb{N},g\in G.$

For $(G,\chi)$-conformal algebras $(\mathcal{C},R)$,
$(\mathcal{C}^{\prime},R^{\prime})$, a linear map
$\psi:\mathcal{C}\rightarrow\mathcal{C}^{\prime}$ is called a
$(G,\chi)$-conformal algebra homomorphism if $\psi$ is a conformal algebra
homomorphism and $\psi\circ R_{g}=R^{\prime}_{g}\circ\psi$ for $g\in G$.

Let $(\mathcal{C},R)$ be a $(G,\chi)$-conformal algebra. For any $g\in G$, we
define a linear automorphism $\hat{R}_{g}$ on
$\mathcal{C}\otimes\mathbb{C}((x))$ by

(5.6) $\displaystyle\hat{R}_{g}(a\otimes f(x))=\chi(g)^{-1}R_{g}(a)\otimes
f(\chi_{\phi}(g)^{-1}x),\quad a\in\mathcal{C},f(x)\in\mathbb{C}((x)).$

######  Lemma 5.7.

For $g\in G$, one has

$\displaystyle\hat{R}_{g}\circ(\partial\otimes 1+1\otimes
p(x)\frac{d}{dx})=\chi(g)(\partial\otimes 1+1\otimes
p(x)\frac{d}{dx})\circ\hat{R}_{g}.$

###### Proof.

For $a\in\mathcal{C}$, $f(x)\in\mathbb{C}((x))$, we have

$\begin{split}&\hat{R}_{g}\big{(}\partial a\otimes f(x)+a\otimes
p(x)\frac{df(x)}{dx}\big{)}\\\ =&\chi(g)^{-1}\Big{(}R_{g}(\partial a)\otimes
f(\chi_{\phi}(g)^{-1}x)+R_{g}a\otimes
p(\chi_{\phi}(g)^{-1}x)\big{(}\frac{df(x)}{dx}|_{x=\chi_{\phi}(g)^{-1}x}\big{)}\Big{)}\\\
=&(\partial R_{g}a)\otimes
f(\chi_{\phi}(g)^{-1}x)+\chi_{\phi}(g)^{-1}R_{g}a\otimes
p(x)\big{(}\frac{df(x)}{dx}|_{x=\chi_{\phi}(g)^{-1}x}\big{)}\\\
=&\chi(g)(\partial\otimes 1+1\otimes p(x)\frac{d}{dx})\hat{R}_{g}(a\otimes
f(x)),\end{split}$

where in the second equality we have used the relations (CR1) and (5.4). ∎

In view of Lemma 5.7, $\hat{R}_{g}$ induces a linear automorphism on
$\widehat{\mathcal{C}}_{\phi}$, which we still denote as $\hat{R}_{g}$. Then
we have:

######  Lemma 5.8.

For $g\in G$, $\hat{R}_{g}$ is a Lie automorphism of
$\widehat{\mathcal{C}}_{\phi}$. Furthermore, for any
$u,v\in\widehat{\mathcal{C}}_{\phi}$, $[\hat{R}_{g}(u),v]=0$ for all but
finitely many $g\in G$.

###### Proof.

For $a,b\in\mathcal{C}$ and $f(x),g(x)\in\mathbb{C}((x))$ we have

$\begin{split}&[\hat{R}_{g}(a\otimes f(x)),\hat{R}_{g}(b\otimes g(x))]\\\
=&\chi(g)^{-2}[R_{g}(a)\otimes f(\chi_{\phi}(g)^{-1}x),R_{g}(b)\otimes
g(\chi_{\phi}(g)^{-1}x)]\\\ =&\chi(g)^{-2}\sum_{i\geq
0}((R_{g}a)_{i}(R_{g}b))\otimes\left(\left(p(x)\frac{d}{dx}\right)^{(i)}f(\chi_{\phi}(g)^{-1}x)\right)g(\chi_{\phi}(g)^{-1}x)\\\
=&\chi(g)^{-1+i}\sum_{i\geq
0}(R_{g}(a_{i}b))\otimes\left(\left(p(x)\frac{d}{dx}\right)^{(i)}f(\chi_{\phi}(g)^{-1}x)\right)g(\chi_{\phi}(g)^{-1}x)\\\
=&\hat{R}_{g}([a\otimes f(x),b\otimes g(x)]),\end{split}$

where in the third and last equality we have used the relations (CR2) and
(5.4), respectively. This proves that $\hat{R}_{g}$ is a Lie automorphism and
the second assertion follows from (CR3). ∎

Following [Li3], we define a multiplication on $\widehat{\mathcal{C}}_{\phi}$
by

(5.7) $\displaystyle(u,v)\mapsto\sum_{g\in G}[\hat{R}_{g}(u),v],\quad
u,v\in\widehat{\mathcal{C}}_{\phi}.$

In view of Lemma 5.8, it was known that under this new multiplication, the
quotient space

$\displaystyle\widehat{\mathcal{C}}_{\phi}[G]:=\widehat{\mathcal{C}}_{\phi}/\textnormal{{span}}\\{\hat{R}_{g}(u)-u\mid
g\in G,u\in\widehat{\mathcal{C}}_{\phi}\\}$

is a Lie algebra [Li3]. For convenience, for $a\in\mathcal{C}$,
$f(x)\in\mathbb{C}((x))$, we still denote the image of $\overline{a\otimes
f(x)}$ in $\widehat{\mathcal{C}}_{\phi}[G]$ by $\overline{a\otimes f(x)}$.

######  Remark 5.9.

When $G$ is a finite group, $\widehat{\mathcal{C}}_{\phi}[G]$ is obviously
isomorphic to the subalgebra of $\widehat{\mathcal{C}}_{\phi}$ fixed by
$\hat{R}_{g}$ for any $g\in G$.

For $a\in\mathcal{C}$, we set

$a^{G}_{\phi}(z)=\sum_{n\in\mathbb{Z}}\overline{a\otimes
x^{n}p(x)}z^{-n-1}\in\widehat{\mathcal{C}}_{\phi}[G][[z,z^{-1}]].$

We have

######  Definition 5.10.

A $\widehat{\mathcal{C}}_{\phi}[G]$-module $W$ is called restricted if
$a^{G}_{\phi}(z)\in{\mathcal{E}}(W)$ for any $a\in\mathcal{C}$.

From (5.4), we have that for $a\in\mathcal{C}$,

(5.8)
$\displaystyle(R_{g}a)^{G}_{\phi}(z)=a^{G}_{\phi}(\chi_{\phi}(g)^{-1}z).$

Furthermore, for $a,b\in\mathcal{C}$, we have

(5.9) $\begin{split}&[a^{G}_{\phi}(z),b^{G}_{\phi}(w)]=\sum_{g\in
G}[(R_{g}a)^{G}_{\phi}(\chi_{\phi}(g)z),b^{G}_{\phi}(w)]\\\ =&\sum_{g\in
G}\sum_{i\geq
0}\left((R_{g^{-1}}a)_{i}b\right)^{G}_{\phi}(w)\left(p(w)\frac{\partial}{\partial
w}\right)^{(i)}\left(\chi(g)\bar{p}(z)\delta\left(\frac{\chi_{\phi}(g)w}{z}\right)\right).\end{split}$

The same proof as that of Lemma 4.3, we have

######  Lemma 5.11.

Let $W$ be a vector space. Assume that there is a linear map

$\pi:\mathcal{C}\rightarrow{\mathcal{E}}(W),\quad a\mapsto
a(z)=\sum_{n\in\mathbb{Z}}a_{[n]}z^{-n-1},$

such that for $a\in\mathcal{C}$,

(5.10) $\displaystyle\left(p(z)\frac{\partial}{\partial
z}\right)a(z)=(\partial a)(z),\quad(R_{g}a)(z)=a(\chi_{\phi}(g)^{-1}z),$

and for $a,b\in\mathcal{C}$,

$\displaystyle[a(z),b(w)]=\sum_{g\in G}\sum_{i\geq
0}\left((R_{g^{-1}}a)_{i}b\right)(w)\left(p(w)\frac{\partial}{\partial
w}\right)^{(i)}\left(\chi(g)\bar{p}(z)\delta\left(\frac{\chi_{\phi}(g)w}{z}\right)\right).$

Then there is a restricted $\widehat{\mathcal{C}}_{\phi}[G]$-module structure
on $W$ with $a^{G}_{\phi}(z)=a(z)$ for $a\in\mathcal{C}$.

Let $(\mathcal{C},R)$ be a $(G,\chi)$-conformal algebra and set
$H=\ker\chi_{\phi}$. One notices from (5.4) that
$\ker\chi_{\phi}\subset\ker\chi$, so $\chi$ induces a linear character on
$G/H$, denoted as $\bar{\chi}$. Following [Li3], we have a
$(G/H,\bar{\chi})$-conformal algebra $(\mathcal{C}/H,R)$ induced from
$(\mathcal{C},R)$. Explicitly, as a vector space,

$\mathcal{C}/H=\mathcal{C}/\textnormal{{span}}\\{R_{h}a-a\mid h\in H,\
a\in\mathcal{C}\\}.$

Define $\partial(aH)=(\partial a)H$ for $aH\in\mathcal{C}/H$, and the
$n$-products on $\mathcal{C}/H$ by

$(uH)_{n}(vH)=\sum_{h\in H}((R_{h}u)_{n}v)H\quad\textnormal{{ for
}}u,v\in\mathcal{C},\ n\in\mathbb{N},$

we also define the $G/H$-action on $\mathcal{C}/H$ by $R_{gH}(uH)=(R_{g}u)H$
for $g\in G,u\in\mathcal{C}$. Denote by $\bar{\chi}_{\phi}$ the character on
$G/H$ induced by $\chi_{\phi}$. Note that the characters
$\bar{\chi},\bar{\chi}_{\phi}$ still satisfy the relation (5.4). It is known
that the $G/H$-action $R$ on $\mathcal{C}/H$ can be extended uniquely to a
$G/H$-action on $V_{\mathcal{C}/H}$, we still denote it by $R$, then
$(V_{\mathcal{C}/H},R)$ becomes a $(G/H,\bar{\chi})$-vertex algebra [Li3]. In
particular we have a notion of $(G/H,\bar{\chi}_{\phi})$-equivariant
$\phi$-coordinated quasi module for the $(G/H,\bar{\chi})$-vertex algebra
$V_{\mathcal{C}/H}$. The following theorem is the main result of this section.

######  Theorem 5.12.

Let $(\mathcal{C},R)$ be a $(G,\chi)$-conformal algebra and set
$H=\ker\chi_{\phi}$. Then any restricted
$\widehat{\mathcal{C}}_{\phi}[G]$-module $W$ is a
$(G/H,\bar{\chi}_{\phi})$-equivariant $\phi$-coordinated quasi
$V_{\mathcal{C}/H}$-module with $Y_{W}(aH,z)=a^{G}_{\phi}(z)$ for
$a\in\mathcal{C}$. On the other hand, any
$(G/H,\bar{\chi}_{\phi})$-equivariant $\phi$-coordinated quasi
$V_{\mathcal{C}/H}$-module $W$ is a restricted
$\widehat{\mathcal{C}}_{\phi}[G]$-module with $a^{G}_{\phi}(z)=Y_{W}(aH,z)$
for $a\in\mathcal{C}$.

###### Proof.

Let $W$ be a restricted $\widehat{\mathcal{C}}_{\phi}[G]$-module. Then
$S=\\{a^{G}_{\phi}(z)\mid a\in\mathcal{C}\\}$ is a $\chi_{\phi}(G)$-quasi
local and $\chi_{\phi}(G)$-stable subset of ${\mathcal{E}}(W)$. Note that the
character $\bar{\chi}_{\phi}:G/H\rightarrow\mathbb{C}^{\times}$ is injective
and its image is $\chi_{\phi}(G)$. In view of the Theorem 5.5, $\langle
S\rangle_{\phi}$ is a $(G/H,\bar{\chi})$-vertex algebra and $W$ is a faithful
$(G/H,\bar{\chi}_{\phi})$-equivariant $\phi$-coordinated quasi $\langle
S\rangle_{\phi}$-module with $Y_{W}(a(z),z_{0})=a(z_{0})$ for $a(z)\in\langle
S\rangle_{\phi}$. Note that for $h\in H$ and $a\in\mathcal{C}$, from (5.8) we
have that

$(R_{h}a)^{G}_{\phi}(z)=a^{G}_{\phi}(\chi_{\phi}(h)^{-1}z)=a^{G}_{\phi}(z).$

Thus the linear map

$\varphi:\ \mathcal{C}/H\rightarrow\langle S\rangle_{\phi},\quad aH\mapsto
a^{G}_{\phi}(z)$

is well-defined. Take a section $\psi:\chi_{\phi}(G)\rightarrow G$ of
$\chi_{\phi}$. For $a,b\in\mathcal{C}$, we have

$\displaystyle[a^{G}_{\phi}(z),b^{G}_{\phi}(w)]$ $\displaystyle=$
$\displaystyle\sum_{g\in G}\sum_{i\geq
0}\left((R_{g^{-1}}a)_{i}b\right)^{G}_{\phi}(w)\left(p(w)\frac{\partial}{\partial
w}\right)^{(i)}\left(\chi(g)\bar{p}(z)\delta\left(\frac{\chi_{\phi}(g)w}{z}\right)\right)$
$\displaystyle=$ $\displaystyle\sum_{g\in\psi(\chi_{\phi}(G))}\sum_{i\geq
0}\sum_{h\in
H}\left((R_{(gh)^{-1}}a)_{i}b\right)^{G}_{\phi}(w)\left(p(w)\frac{\partial}{\partial
w}\right)^{(i)}\left(\chi(g)\bar{p}(z)\delta\left(\frac{\chi_{\phi}(g)w}{z}\right)\right).$

By comparing this with (5.5) we have

$\displaystyle\sum_{h\in
H}\left((R_{(gh)^{-1}}a)_{i}b\right)^{G}_{\phi}(w)=(R_{g^{-1}}a)^{G}_{\phi}(w)_{i}^{\phi}b^{G}_{\phi}(w).$

This particular implies that

$\displaystyle\varphi(aH)^{\phi}_{i}\varphi(bH)=\varphi((aH)_{i}(bH)),\quad
a,b\in\mathcal{C},i\in\mathbb{N}.$

Furthermore, for any $a\in\mathcal{C}$, we have

$\begin{split}&(\mathcal{D}\circ\varphi)(aH)=\mathcal{D}(a^{G}_{\phi}(z))=\left(p(z)\frac{\partial}{\partial
z}\right)a^{G}_{\phi}(z)=(\partial
a)^{G}_{\phi}(z)=(\varphi\circ\partial)(aH).\end{split}$

Thus, $\varphi$ is a conformal algebra homomorphism. From the Proposition 4.4,
there is a vertex algebra homomorphism from $V_{\mathcal{C}/H}$ to
${\langle}S{\rangle}_{\phi}$ which extends $\varphi$, we also denote it as
$\varphi$. Note that for $a\in\mathcal{C}$ and $g\in G$,

$\begin{split}&(R_{gH}\circ\varphi)(aH)=R_{gH}a^{G}_{\phi}(z)=a^{G}_{\phi}(\bar{\chi}_{\phi}(gH)^{-1}z)\\\
=&a^{G}_{\phi}(\chi_{\phi}(g)^{-1}z)=(R_{g}a)^{G}_{\phi}(z)=\varphi((R_{g}a)H)=(\varphi\circ
R_{gH})(aH).\end{split}$

Then for $a,b\in\mathcal{C}$ and $g\in G$, we have

$\displaystyle(R_{gH}\circ\varphi)(Y(aH,z)bH)=R_{gH}\big{(}Y_{\mathcal{E}}(\varphi(aH),z)\varphi(bH)\big{)}$
$\displaystyle=$ $\displaystyle
Y_{\mathcal{E}}(R_{gH}(\varphi(aH)),\bar{\chi}(gH)^{-1}z)R_{gH}(\varphi(bH))=Y_{\mathcal{E}}(\varphi(R_{gH}aH),\bar{\chi}(gH)^{-1}z)\varphi(R_{gH}bH)$
$\displaystyle=$
$\displaystyle\varphi\big{(}Y(R_{gH}(aH),\bar{\chi}(gH)^{-1}z)R_{gH}(bH)\big{)}=(\varphi\circ
R_{gH})(Y(aH,z)bH).$

This together with the fact that $\mathcal{C}/H$ is a generating set of
$V_{\mathcal{C}/H}$ proves that $R_{gH}\circ\varphi=\varphi\circ R_{gH}$ on
$V_{\mathcal{C}/H}$. Thus $\varphi$ is a $(G/H,\bar{\chi})$-vertex algebra
homomorphism from $V_{\mathcal{C}/H}$ to ${\langle}S{\rangle}_{\phi}$. Via
this homomorphism, $W$ is naturally a $(G/H,\bar{\chi}_{\phi})$-equivariant
$\phi$-coordinated quasi $V_{\mathcal{C}/H}$-module.

On the other hand, let $W$ be a $(G/H,\bar{\chi}_{\phi})$-equivariant
$\phi$-coordinated quasi $V_{\mathcal{C}/H}$-module. From (3.3) and (5.8), for
$a\in\mathcal{C}$, we have

$\displaystyle
Y_{W}(\partial(aH),z)=Y_{W}(\mathcal{D}(aH),z)=\left(p(z)\frac{d}{dz}\right)Y_{W}(aH,z),$
$\displaystyle(R_{g}a)^{G}_{\phi}(z)=a^{G}_{\phi}(\chi_{\phi}(g)^{-1}z).$

From Proposition 5.3, for $a,b\in\mathcal{C}$, we have

$\begin{split}&[Y_{W}(aH,z),Y_{W}(bH,w)]\\\
=&\sum_{g\in\psi(\chi_{\phi}(G))}\sum_{i\geq 0}\sum_{h\in
H}Y_{W}(((R_{hg^{-1}}a)_{i}b)H,w)\left(p(w)\frac{\partial}{\partial
w}\right)^{(i)}\left(\chi(g)\bar{p}(z)\delta\left(\frac{\chi_{\phi}(g)w}{z}\right)\right)\\\
=&\sum_{g\in G}\sum_{i\geq
0}Y_{W}(((R_{g^{-1}}a)_{i}b)H,w)\left(p(w)\frac{\partial}{\partial
w}\right)^{(i)}\left(\chi(g)\bar{p}(z)\delta\left(\frac{\chi_{\phi}(g)w}{z}\right)\right).\end{split}$

Thus, from Lemma 5.11, we obtain that $W$ is a restricted
$\widehat{\mathcal{C}}_{\phi}[G]$-module with $a^{G}_{\phi}(z)=Y_{W}(aH,z)$
for $a\in\mathcal{C}$. ∎

##  6\. Applications

In this section, as an application, a class of $(G,\chi_{\phi})$-equivariant
$\phi$-coordinated quasi modules for affine vertex algebras is determined, and
the $\phi$-coordinated modules for Virasoro vertex algebra and the vertex
algebras associated to Novikov algebras are determined.

Firstly, let $\mathfrak{g}$ be a (possibly infinite dimensional) Lie algebra,
which is equipped with an invariant symmetric bilinear form $(\cdot,\cdot)$.
Then we have the (formal) untwisted affine Lie algebra

$\displaystyle\widehat{\mathfrak{g}}:=\mathfrak{g}\otimes\mathbb{C}((x))\oplus\mathbb{C}\mathbf{c},$

where $\mathbf{c}$ is central and for $a,b\in\mathfrak{g}$,
$f(x),g(x)\in\mathbb{C}((x))$,

$\displaystyle[a\otimes f(x),b\otimes g(x)]=[a,b]\otimes
f(x)g(x)+(a,b)\mathrm{Res}_{x}((\frac{d}{dx}f(x))g(x))\mathbf{c}.$

On the other hand, we have the conformal algebra ([K])

$\mathcal{C}_{\mathfrak{g}}=(\mathbb{C}[\partial]\otimes\mathfrak{g})\oplus\mathbb{C}\mathrm{c},$

where for $a,b\in\mathfrak{g}$ and $n\in\mathbb{N}$,
$\partial(\partial^{n}\otimes a)=\partial^{n+1}\otimes a,$
$\partial(\mathrm{c})=0,$ and

$\displaystyle a_{n}b=\delta_{n,0}[a,b]+\delta_{n,1}(a,b)\mathrm{c},\quad
a_{n}\mathrm{c}=0=\mathrm{c}_{n}\mathrm{c}.$

We have

######  Lemma 6.1.

The assignment $(a\in\mathfrak{g},f(x)\in\mathbb{C}((x)))$

$\displaystyle\overline{a\otimes f(x)}\mapsto a\otimes
f(x),\quad\overline{\mathrm{c}\otimes x^{-1}p(x)}\mapsto\mathbf{c}$

determines a Lie algebra isomorphism from
$\widehat{\mathcal{C}_{\mathfrak{g}}}_{\phi}$ to $\widehat{\mathfrak{g}}$.

###### Proof.

It is obvious that this assignment gives rise to a linear isomorphism, say
$\varphi$, from $\widehat{\mathcal{C}_{\mathfrak{g}}}_{\phi}$ to
$\widehat{\mathfrak{g}}$. For any $a,b\in\mathfrak{g}$,
$f(x),g(x)\in\mathbb{C}((x))$, we have

$\begin{split}&\varphi([\overline{a\otimes f(x)},\overline{b\otimes g(x)}])\\\
=&\varphi(\overline{[a,b]\otimes f(x)g(x)}+(a,b)\overline{\mathrm{c}\otimes
p(x)(\frac{d}{dx}f(x))g(x)})\\\ =&[a,b]\otimes
f(x)g(x)+(a,b)\mathrm{Res}_{x}((\frac{d}{dx}f(x))g(x))\mathbf{c}\\\
=&[\varphi(\overline{a\otimes f(x)}),\varphi(\overline{b\otimes
g(x)})],\end{split}$

noticing that

$\overline{\mathrm{c}\otimes
p(x)(\frac{d}{dx}f(x))g(x)}=\overline{\mathrm{c}\otimes
x^{-1}p(x)}\mathrm{Res}_{x}((\frac{d}{dx}f(x))g(x)).$

Thus $\varphi$ is a Lie algebra isomorphism. ∎

For any $\ell\in\mathbb{C}$, we denote by

$V_{\widehat{\mathfrak{g}}}(\ell,0)=\mathcal{U}(\widehat{\mathfrak{g}})\otimes_{\mathcal{U}(\mathfrak{g}\otimes\mathbb{C}[[x]]\oplus\mathbb{C}\mathbf{c})}\mathbb{C}$

the affine vertex algebra of $\mathfrak{g}$ with the central charge $\ell$,
where $\mathfrak{g}\otimes\mathbb{C}[[x]]$ acts trivially on $\mathbb{C}$ and
$\mathbf{c}$ acts as the scalar $\ell$ on $\mathbb{C}$. Note that
$V_{\widehat{\mathfrak{g}}}(\ell,0)$ is naturally isomorphic to the quotient
vertex algebra of $V_{\mathcal{C}_{\mathfrak{g}}}$ modulo the ideal generated
by $\mathrm{c}-\ell$. We say that a $\widehat{\mathfrak{g}}$-module $W$ is
restricted if for any $a\in\mathfrak{g},w\in W$, $(a\otimes x^{n}).w=0$ for
$n\gg 0$, and is of level $\ell$ if $\mathbf{c}$ acts as the scaler $\ell$ on
$W$. From Lemma 6.1 and Theorem 4.5, we have the following characterization of
$\phi$-coordinated $V_{\widehat{\mathfrak{g}}}(\ell,0)$-modules.

######  Proposition 6.2.

$\phi$-coordinated modules for the affine vertex algebra
$V_{\widehat{\mathfrak{g}}}(\ell,0)$ are exactly restricted
$\widehat{\mathfrak{g}}$-modules of level $\ell$.

More generally, assume that $G$ is a group acting on $\mathfrak{g}$ as an
automorphism group preserving the bilinear form $(\cdot,\cdot)$ such that for
$a,b\in\mathfrak{g}$,

(6.1)
$\displaystyle[ga,b]=0\quad\textnormal{{and}}\quad(ga,b)=0\quad\textnormal{{for
all but finitely many}}\ g\in G.$

Let $\chi,\chi_{\phi}:G\rightarrow\mathbb{C}^{\times}$ be any linear
characters satisfy the relation (5.3). We lift the $G$-action from
$\mathfrak{g}$ to $\widehat{\mathfrak{g}}$ by

(6.2) $\displaystyle g(a\otimes f(x)+\beta\mathbf{c})=(ga\otimes
f(\chi_{\phi}(g)^{-1}x))+\beta\mathbf{c}$

for $g\in G$, $a\in\mathfrak{g}$, $f(x)\in\mathbb{C}((x))$,
$\beta\in\mathbb{C}$. From [Li3], the $(G,\chi_{\phi})$-covariant algebra of
$\widehat{\mathfrak{g}}$ is the Lie algebra $\widehat{\mathfrak{g}}[G]$, where

(6.3) $\displaystyle\widehat{\mathfrak{g}}[G]=\widehat{\mathfrak{g}}/{\rm
span}\\{gu-u\mid g\in G,u\in\widehat{\mathfrak{g}}\\}$

as a vector space and the Lie bracket is given by

$\displaystyle[\overline{u},\overline{v}]=\sum_{g\in
G}\overline{[gu,v]}\quad\textnormal{{for}}\ u,v\in\widehat{\mathfrak{g}},$

where for $u\in\widehat{\mathfrak{g}}$, $\overline{u}$ stands for the image of
$u$ in $\widehat{\mathfrak{g}}[G]$ under the natural quotient map
$\widehat{\mathfrak{g}}\rightarrow\widehat{\mathfrak{g}}[G]$. We define a
$G$-action on $\mathcal{C}_{\mathfrak{g}}$ by

$R_{g}(\partial^{n}\otimes a)=\chi(g)^{n+1}\partial^{n}\otimes
ga\quad\textnormal{{and}}\quad
R_{g}(\mathrm{c})=\mathrm{c}\quad\textnormal{{for }}g\in
G,a\in\mathfrak{g},n\in\mathbb{N}.$

Then $(\mathcal{C}_{\mathfrak{g}},R)$ is a $(G,\chi)$-conformal algebra and so
$(V_{\mathcal{C}_{\mathfrak{g}}},R)$ is a $(G,\chi)$-vertex algebra. A similar
proof as that in Lemma 6.1, we have

######  Lemma 6.3.

The assignment $(a\in\mathfrak{g},f(x)\in\mathbb{C}((x)))$

$\displaystyle\overline{a\otimes f(x)}\mapsto\overline{a\otimes
f(x)},\quad\overline{\mathrm{c}\otimes
x^{-1}p(x)}\mapsto\overline{\mathbf{c}}$

determines a Lie algebra isomorphism from
$\widehat{\mathcal{C}_{\mathfrak{g}}}_{\phi}[G]$ to
$\widehat{\mathfrak{g}}[G]$.

As $R_{g}(\mathrm{c})=\mathrm{c}$ for $g\in G$,
$V_{\widehat{\mathfrak{g}}}(\ell,0)$ becomes a $(G,\chi)$-vertex algebra with
the $G$-action $R$ transferred from that of $V_{\mathcal{C}_{\mathfrak{g}}}$.
As before, we can define the notion “restricted
$\widehat{\mathfrak{g}}[G]$-module of level $\ell$”. Then from Lemma 6.3 and
Theorem 5.12, we immediately have the following generalization of Proposition
6.2.

######  Proposition 6.4.

Let $\chi,\chi_{\phi}$ be linear characters of $G$ with $\chi_{\phi}$ is
injective and the relation (5.3) hold. Then $(G,\chi_{\phi})$-equivariant
$\phi$-coordinated quasi modules for the affine $(G,\chi)$-vertex algebra
$V_{\widehat{\mathfrak{g}}}(\ell,0)$ are exactly restricted
$\widehat{\mathfrak{g}}[G]$-modules of level $\ell$.

######  Remark 6.5.

We illustrate two examples for the Proposition 6.4(cf.[Li3]). First, let
$\sigma$ be an automorphism of $\mathfrak{g}$ which has order $M$ and
preserves the form $(\cdot,\cdot)$, and let $\omega$ be a primitive $M$-th
root of unity. Set $G={\langle}\sigma{\rangle}$ and take $\chi_{\phi}$ to be
the character of $G$ determined by $\chi_{\phi}(\sigma)=\omega$. Then
$\widehat{\mathfrak{g}}[G]$ is isomorphic to the $\sigma$-twisted affine Lie
algebra

$\widehat{\mathfrak{g}}[\sigma]=\\{u\in\widehat{\mathfrak{g}}\mid\hat{\sigma}(u)=u\\},$

where $\hat{\sigma}$ is the automorphism of $\widehat{\mathfrak{g}}$ defined
by $\hat{\sigma}(\mathbf{c})=\mathbf{c}$ and $\hat{\sigma}(a\otimes
f(x))=\sigma(a)\otimes f(\omega^{-1}x),$
$a\in\mathfrak{g},f(x)\in\mathbb{C}((x))$.

Second, let $\mathfrak{g}=\mathfrak{gl}_{\infty}$ be the Lie algebra of
$\mathbb{Z}\times\mathbb{Z}$-matrices which have only finitely many nonzero
entries and set $(x,y)=Tr(xy)$. For any positive integer $N$, define an
automorphism $\sigma_{N}$ on $\mathfrak{gl}_{\infty}$ by
$\sigma_{N}(E_{i,j})=E_{i+N,j+N}$ for $i,j\in\mathbb{Z}$, where $E_{i,j}$
stands for the usual unit matrix. Set $G={\langle}\sigma_{N}{\rangle}$ and
take $\chi_{\phi}$ to be the character of $G$ determined by
$\chi_{\phi}(\sigma_{N})=q^{-1}$ with $q$ is not a root of unit. As
$G\cong\mathbb{Z}$ is an infinite group, the relation (5.3) forces that $p(x)$
must be a homogenous polynomial in $\mathbb{C}[x,x^{-1}]$ ([JKLT]). Thus we
may define $\widehat{\mathfrak{g}}$ to be the usual affine Lie algebra
$\mathfrak{g}\otimes\mathbb{C}[x,x^{-1}]\oplus\mathbb{C}\mathbf{c}$. Then
$\widehat{\mathfrak{g}}[G]$ is isomorphic to a one dimensional central
extension of the $N\times N$-matrix algebra
$\mathfrak{gl}_{N}(\mathbb{C}_{q})$ over a $2$-dimensional quantum torus
$\mathbb{C}_{q}$, where $\mathbb{C}_{q}=\mathbb{C}[x^{\pm{1}},y^{\pm 1}]$ as
vector space with the multiplication given by
$(x^{m}y^{n})(x^{s}y^{t})=q^{ns}x^{m+s}y^{n+t}$ (cf.[Li3, CLTW]).

Next, we consider the $\phi$-coordinated modules for the Virasoro vertex
algebra. Let

$\displaystyle\mathcal{V}=\mathbb{C}((x))\frac{d}{dx}\oplus\mathbb{C}\mathbf{c}$

be (formal) Virasoro Lie algebra, where $\mathbf{c}$ is central and for
$f(x),g(x)\in\mathbb{C}((x))$,

$\displaystyle[f(x)\frac{d}{dx},g(x)\frac{d}{dx}]=$ $\displaystyle
f(x)(\frac{d}{dx}g(x))\frac{d}{dx}-g(x)(\frac{d}{dx}f(x))\frac{d}{dx}$
$\displaystyle+\frac{\mathbf{c}}{2}\mathrm{Res}_{x}((\frac{d}{dx})^{(3)}f(x))g(x).$

On the other hand, we have the Virasoro conformal algebra ([K])

$\mathcal{C}_{\mathcal{V}}=(\mathbb{C}[\partial]\otimes\mathbb{C}L)\oplus\mathbb{C}\mathrm{c},$

where for $n\in\mathbb{N}$, $\partial(\partial^{n}\otimes
L)=\partial^{n+1}\otimes L,$ $\partial\mathrm{c}=0$ and

$\displaystyle L_{n}L=\delta_{n,0}\partial
L+\delta_{n,1}2L+\delta_{n,3}\frac{\mathrm{c}}{2},\quad
L_{n}\mathrm{c}=0=\mathrm{c}_{n}\mathrm{c}.$

We identify $L$ as $1\otimes L\in\mathcal{C}_{\mathcal{V}}$. It is
straightforward to see that the Lie algebra
$\widehat{\mathcal{C}_{\mathcal{V}}}_{\phi}$ is spanned by the following
elements

$\displaystyle\overline{\mathrm{c}\otimes x^{-1}p(x)},\quad\overline{L\otimes
f(x)},\ f(x)\in\mathbb{C}((x)).$

Furthermore, we have

######  Lemma 6.6.

The assignment

$\displaystyle\overline{L\otimes f(x)}\mapsto
p(x)f(x)\frac{d}{dx}+\alpha_{\phi}(f(x))\mathbf{c},\quad\overline{\mathrm{c}\otimes
x^{-1}p(x)}\mapsto\mathbf{c}$

determines a Lie algebra isomorphism from
$\widehat{\mathcal{C}_{\mathcal{V}}}_{\phi}$ to $\mathcal{V}$, where
$\alpha_{\phi}:\mathbb{C}((x))\rightarrow\mathbb{C}$ is a linear map defined
by

$f(x)\mapsto\frac{1}{24}\mathrm{Res}_{x}f(x)\big{(}2(\frac{d}{dx})^{2}p(x)-p(x)^{-1}(\frac{d}{dx}p(x))^{2}\big{)}.$

###### Proof.

Obviously, this assignment gives rise to a linear isomorphism
$\varphi:\widehat{\mathcal{C}_{\mathcal{V}}}_{\phi}\rightarrow\mathcal{V}$.
Now we prove that $\varphi$ is a Lie algebra homomorphism. For any
$f(x),g(x)\in\mathbb{C}((x))$, we have

$\displaystyle\quad\varphi\Big{(}[\overline{L\otimes f(x)},\overline{L\otimes
g(x)}]\Big{)}$ $\displaystyle=\varphi\Big{(}-\overline{L\otimes
p(x)\frac{d}{dx}(f(x)g(x))}+2\overline{L\otimes p(x)(\frac{d}{dx}f(x))g(x)}$
$\displaystyle\quad+\frac{1}{2}\overline{\mathrm{c}\otimes((p(x)\frac{d}{dx})^{(3)}f(x))g(x)}\Big{)}$
$\displaystyle=\varphi\Big{(}\overline{L\otimes
p(x)g(x)\frac{d}{dx}f(x)}-\overline{L\otimes p(x)f(x)\frac{d}{dx}g(x)}$
$\displaystyle\quad+\frac{1}{12}\overline{\mathrm{c}\otimes
x^{-1}p(x)}\mathrm{Res}_{x}\Big{(}(\frac{d}{dx}p(x))^{2}\frac{d}{dx}f(x)+3p(x)(\frac{d}{dx}p(x))(\frac{d}{dx})^{2}f(x)$
$\displaystyle\quad+p(x)((\frac{d}{dx})^{2}p(x))\frac{d}{dx}f(x)+p(x)^{2}(\frac{d}{dx})^{3}f(x)\Big{)}g(x)\Big{)}$
$\displaystyle=p(x)^{2}f(x)(\frac{d}{dx}g(x))\frac{d}{dx}-p(x)^{2}g(x)(\frac{d}{dx}f(x))\frac{d}{dx}$
$\displaystyle\quad+\frac{1}{12}\mathbf{c}\mathrm{Res}_{x}\Big{(}f(x)(\frac{d}{dx})^{3}p(x)+3(\frac{d}{dx}p(x))(\frac{d}{dx})^{2}f(x)$
$\displaystyle\quad+3((\frac{d}{dx})^{2}p(x))\frac{d}{dx}f(x)+p(x)(\frac{d}{dx})^{3}f(x)\Big{)}p(x)g(x)$
$\displaystyle=p(x)^{2}f(x)(\frac{d}{dx}g(x))\frac{d}{dx}-p(x)^{2}g(x)(\frac{d}{dx}f(x))\frac{d}{dx}+\frac{\mathbf{c}}{2}\mathrm{Res}_{x}(\frac{d}{dx})^{(3)}(p(x)f(x))p(x)g(x)$
$\displaystyle=[p(x)f(x)\frac{d}{dx},p(x)g(x)\frac{d}{dx}]=[\varphi(\overline{L\otimes
f(x)}),\varphi(\overline{L\otimes g(x)})],$

noticing that

$\overline{\mathrm{c}\otimes p(x)h(x)}=\overline{\mathrm{c}\otimes
x^{-1}p(x)}\mathrm{Res}_{x}h(x)\quad\textnormal{{for any
}}h(x)\in\mathbb{C}((x)).$

Thus $\varphi$ is a Lie algebra isomorphism. ∎

For any $\ell\in\mathbb{C}$, we denote by

$V_{\mathcal{V}}(\ell,0)=\mathcal{U}(\mathcal{V})\otimes_{\mathcal{U}(\mathbb{C}[[x]]\frac{d}{dx}\oplus\mathbb{C}\mathbf{c})}\mathbb{C}$

the Virasoro vertex algebra with central charge $\ell$, where
$\mathbb{C}[[x]]\frac{d}{dx}$ acts trivially on $\mathbb{C}$ and $\mathbf{c}$
acts as the scalar $\ell$ on $\mathbb{C}$. Then $V_{\mathcal{V}}(\ell,0)$ is
naturally isomorphic to the quotient vertex algebra of
$V_{\mathcal{C}_{\mathcal{V}}}$ modulo the ideal generated by
$\mathrm{c}-\ell$. We say that a $\mathcal{V}$-module $W$ is restricted if for
any $w\in W$, $(x^{n}\frac{d}{dx}).w=0$ for $n\gg 0$, and is of level $\ell$
if $\mathbf{c}$ acts as the scaler $\ell$ on $W$. From Lemma 6.6 and Theorem
4.5, we immediately have the following characterization of $\phi$-coordinated
$V_{\mathcal{V}}(\ell,0)$-modules.

######  Proposition 6.7.

$\phi$-coordinated modules for the Viraroso vertex algebra
$V_{\mathcal{V}}(\ell,0)$ are exactly restricted $\mathcal{V}$-modules of
level $\ell$.

######  Remark 6.8.

When $p(x)=x$, Propositions 6.2 and 6.7 were proved in [CLT] by a different
method.

Finally we consider the $\phi$-coordinated modules of the vertex algebras
associated to Novikov algebras. Let $\mathcal{A}$ be a Novikov algebra, in the
sense that $\mathcal{A}$ is a non-associative algebra satisfying

$(ab)c-a(bc)=(ba)c-b(ac),\quad(ab)c=(ac)b\quad\textnormal{{for
}}a,b,c\in\mathcal{A},$

and let $\langle\cdot,\cdot\rangle$ be a symmetric bilinear form of
$\mathcal{A}$ satisfying

(6.4) $\begin{split}\langle ab,c\rangle=\langle a,bc\rangle,\quad\langle
ab,c\rangle=\langle ba,c\rangle\quad\textnormal{{for
}}a,b,c\in\mathcal{A}.\end{split}$

Similar to the algebra $\widehat{\mathfrak{g}}$, we also have an “untwisted
affine Lie algebra”

$\displaystyle\widehat{\mathcal{A}}=\mathcal{A}\otimes\mathbb{C}((x))\oplus\mathbb{C}\mathbf{c}$

of $\mathcal{A}$ ([BLP]), where $\mathbf{c}$ is central and for
$a,b\in\mathcal{A},f(x),g(x)\in\mathbb{C}((x))$,

$\begin{split}[a\otimes f(x),b\otimes
g(x)]=&ab\otimes\left(\frac{d}{dx}f(x)\right)g(x)-ba\otimes\left(\frac{d}{dx}g(x)\right)f(x)\\\
&+\frac{\mathbf{c}}{2}{\langle}a,b{\rangle}\mathrm{Res}_{x}\
\big{(}(\frac{d}{dx})^{(3)}f(x)\big{)}g(x).\end{split}$

On the other hand, following [P], we have the conformal algebra

$\mathcal{C}_{\mathcal{A}}=\mathbb{C}[\partial]\otimes\mathcal{A}\oplus\mathbb{C}\mathrm{c},$

where for $a,b\in\mathcal{A}$, $n\in\mathbb{N}$, $\partial(\partial^{n}\otimes
a)=\partial^{n+1}\otimes a,$ $\partial\mathrm{c}=0$ and

$\displaystyle
a_{n}b=\delta_{n,0}\partial(ba)+\delta_{n,1}(ab+ba)+\delta_{n,3}{\langle}a,b{\rangle}\frac{\mathrm{c}}{2},\quad
a_{n}\mathrm{c}=0=\mathrm{c}_{n}\mathrm{c}.$

Note that there is a linear isomorphism from
$\widehat{\mathcal{C}_{\mathcal{A}}}_{\phi}\rightarrow\widehat{\mathcal{A}}$
such that $\overline{a\otimes f(x)}\rightarrow a\otimes f(x)$ for
$a\in\mathcal{A},f(x)\in\mathbb{C}((x))$ and that $\overline{\mathrm{c}\otimes
x^{-1}p(x)}\mapsto\mathbf{c}$. Via this isomorphism, we obtain a new Lie
algebra structure on $\widehat{\mathcal{A}}$ such that

$\begin{split}[a\otimes f(x),b\otimes
g(x)]=&ab\otimes\left(p(x)\frac{d}{dx}f(x)\right)g(x)-ba\otimes\left(p(x)\frac{d}{dx}g(x)\right)f(x)\\\
&+\frac{\mathbf{c}}{2}{\langle}a,b{\rangle}\mathrm{Res}_{x}\
p(x)^{-1}\big{(}(p(x)\frac{d}{dx})^{(3)}f(x)\big{)}g(x)\end{split}$

for $a,b\in\mathcal{A},f(x),g(x)\in\mathbb{C}((x))$ and that $\mathbf{c}$ is
central. We denote this Lie algebra by $\widehat{\mathcal{A}}_{\phi}$.

For any $\ell\in\mathbb{C}$, just as $V_{\widehat{\mathfrak{g}}}(\ell,0)$, we
have an affine type vertex algebra $V_{\widehat{\mathcal{A}}}(\ell,0)$ ([BLP])
which is isomorphic to the quotient vertex algebra
$V_{\mathcal{C}_{\mathcal{A}}}/{\langle}\mathrm{c}-\ell{\rangle}$. We say that
a $\widehat{\mathcal{A}}_{\phi}$-module $W$ is restricted if for any
$a\in\mathcal{A},w\in W$, $(a\otimes x^{n}).w=0$ for $n\gg 0$, and is of level
$\ell$ if $\mathbf{c}$ acts as the scaler $\ell$ on $W$. In view of Theorem
4.5, we have the following slight generalization of a result in [BLP]:

######  Proposition 6.9.

$\phi$-coordinated $V_{\widehat{\mathcal{A}}}(\ell,0)$-modules are exactly
restricted $\widehat{\mathcal{A}}_{\phi}$-modules of level $\ell$.

######  Remark 6.10.

Equip $\mathcal{A}=\mathbb{C}[y,y^{-1}]$ with a bilinear operator

$\circ:\mathcal{A}\times\mathcal{A}\rightarrow\mathcal{A},\quad(y^{i},y^{j})\mapsto
jy^{i+j}.$

Then $(\mathcal{A},\circ)$ is a Novikov algebra. Unlike the usual affine Lie
algebra case (cf. Lemma 6.1), it follows from [DZ] that the Lie algebra
$\widehat{\mathcal{A}}$ is not isomorphic to $\widehat{\mathcal{A}}_{\phi}$
with $\phi=xe^{z}$.

## References

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