# Extended affine Lie algebras, vertex algebras and equivariant
$\phi$-coordinated quasi modules

Fulin Chen  Shaobin Tan and Nina Yu

###### Abstract

For any nullity $2$ extended affine Lie algebra $\mathcal{E}$ of maximal type
and $\ell\in\mathbb{C}$, we prove that there exist a vertex algebra
$V_{\mathcal{E}}(\ell)$ and an automorphism group $G$ of
$V_{\mathcal{E}}(\ell)$ equipped with a linear character $\chi$, such that the
category of restricted $\mathcal{E}$-modules of level $\ell$ is canonically
isomorphic to the category of $(G,\chi)$-equivariant $\phi$-coordinated quasi
$V_{\mathcal{E}}(\ell)$-modules. Moreover, when $\ell$ is a nonnegative
integer, there is a quotient vertex algebra $L_{\mathcal{E}}(\ell)$ of
$V_{\mathcal{E}}(\ell)$ modulo by a $G$-stable ideal, and we prove that the
integrable restricted $\mathcal{E}$-modules of level $\ell$ are exactly the
$(G,\chi)$-equivariant $\phi$-coordinated quasi
$L_{\mathcal{E}}(\ell)$-modules.

##  1 Introduction

_Extended affine Lie algebra_ (EALA for short) was introduced by Hoegh-Krohn
and Torresani in [H-KT] with applications to quantum gauge theory, and since
then it has been studied extensively in literature (see [N] and the references
therein). An EALA by definition is a complex Lie algebra $\mathcal{E}$,
together with a finite-dimensional ad-diagonalizable subalgebra and a
nondegenerate invariant symmetric bilinear form, satisfying a list of natural
axioms. The isotropic roots (i.e., roots of length $0$) in $\mathcal{E}$
generate a free abelian group of finite rank called the _nullity_ of
$\mathcal{E}$. EALAs of nullity $0$ and $1$ precisely coincide with the
finite-dimensional simple Lie algebras and affine Kac-Moody algebras
respectively [ABGP]. Meanwhile, the structure of EALAs with positive nullity
are like affine Kac-Moody algebras in many ways [ABFP, BGK, CNPY]. It is well-
known that affine Kac-Moody algebras through their restricted modules can be
naturally associated with vertex algebras and (twisted) modules [FZ, FLM, Li1,
Li2]. This association plays an important role in both affine Kac-Moody
algebra theory and vertex (operator) algebra theory. It is a natural question
to ask whether EALAs of nullity $\geq 2$ can be associated with vertex
algebras in a similar way as the affine Kac-Moody algebras.

The main goal of this paper is to associate the nullity $2$ EALAs and their
modules to vertex algebras. We would like to point out that the representation
theory of nullity $2$ EALAs is totally different from that for EALAs with
nullity $\geq 3$ (see [ESB] and [CLT2]), and the nullity 2 EALAs arose
naturally in the work of Saito [Sa] and Slodowy [Sl] on simple elliptic
singularities and can be connected with Ringel-Hall algebras [LP]. The
subalgebra of an EALA generated by its nonisotropic root vectors is called the
_core_ of the EALA, and the core modulo its center is often called the
_centerless core_ of the algebra. It is known that the classification of EALAs
can be reduced to the classification of their centerless cores [N]. And the
centerless cores of nullity $2$ EALAs were classified by Allison-Berman-
Pianzola in [ABP] (see also [GP]).

In this paper we deal with the nullity $2$ EALAs of maximal type in the sense
that their cores are centrally closed [BGK]. The representation theory of
nullity $2$ EALAs of maximal type has been extensively studied (see [G1, G2,
G3, GZ, B2, CLT1, CLT2] for example). By applying the theory of _equivariant
$\phi$-coordinated quasi modules_ for vertex algebras developed by Li (see
[Li6, Li8]), we associate all nullity $2$ EALAs of maximal type and their
restricted modules to vertex algebras.

Li introduced the notion of _$\left(G,\chi\right)$ -equivariant quasi modules_
for vertex algebras in [Li3, Li4] to associate certain infinite-dimensional
Lie algebras to vertex algebras, where $G$ is a group and $\chi$ is a linear
character of $G$. Li [Li6, Li8] also developed a theory of
_$\left(G,\chi\right)$ -equivariant $\phi$-coordinated quasi modules_ for
nonlocal vertex algebras to associate quantum affine algebras with quantum
vertex algebras (see also [JKLiT, CLTW]), where $\phi$ is an _associate_ of
the $1$-dimensional additive formal group $F(z,w)=z+w$ [Li6]. In this paper we
assume the _associate_ $\phi=ze^{w}$, which is indeed the associate appearing
in the quantum vertex algebra theory [Li6, Li8].

If $\dot{\mathfrak{g}}$ is a finite-dimensional simple Lie algebra with a
diagram automorphism $\dot{\nu}$, we denote by
$\widetilde{\mathcal{L}}(\dot{\mathfrak{g}},\dot{\nu})$ the corresponding
affine Kac-Moody algebra (i.e. nullity 1 EALA), and
$\widehat{\mathcal{L}}(\dot{\mathfrak{g}},\dot{\nu})=[\widetilde{\mathcal{L}}(\dot{\mathfrak{g}},\dot{\nu}),\widetilde{\mathcal{L}}(\dot{\mathfrak{g}},\dot{\nu})]$
the derived Lie subalgebra of
$\widetilde{\mathcal{L}}(\dot{\mathfrak{g}},\dot{\nu})$. We know that the
restricted modules for the affine Lie algebra
$\widehat{\mathcal{L}}(\dot{\mathfrak{g}},\dot{\nu})$ of level
$\ell\in\mathbb{C}$ can be associated with the $\dot{\nu}$-twisted modules for
the universal affine vertex algebra
$V_{\widehat{\mathcal{L}}(\dot{\mathfrak{g}})}(\ell,0)$ of the untwisted
affine Lie algebra
$\widehat{\mathcal{L}}(\dot{\mathfrak{g}})=\widehat{\mathcal{L}}(\dot{\mathfrak{g}},\mathrm{id})$
(see [FZ, Li1, FLM, Li2]). And the integrable restricted modules for
$\widehat{\mathcal{L}}(\dot{\mathfrak{g}},\dot{\nu})$ of level
$\ell\in\mathbb{N}$ can be associated with the $\dot{\nu}$-twisted modules for
the corresponding simple affine vertex algebras
$L_{\widehat{\mathcal{L}}(\dot{\mathfrak{g}})}(\ell,0)$ (see [FZ, Li1, Li2]).

Similar to the association of the affine Lie algebra
$\widehat{\mathcal{L}}(\dot{\mathfrak{g}},\dot{\nu})$ and its restricted
modules with the affine vertex algebras and their twisted modules, we can
apply the equivariant $\phi$-coordinated quasi modules for the affine vertex
algebras to associate with the restricted modules for the affine Kac-Moody
algebra $\widetilde{\mathcal{L}}(\dot{\mathfrak{g}},\dot{\nu})$. For this
purpose, we first, by using results from [Z] and [Li5], investigate the
natural connections among equivariant $\phi$-coordinated quasi modules,
equivariant quasi modules and twisted modules for the general vertex operator
algebras (see Proposition 3.4). And then we prove that the category of
restricted (resp. integrable restricted) modules for the affine Kac-Moody
algebra $\widetilde{\mathcal{L}}(\dot{\mathfrak{g}},\dot{\nu})$ of level
$\ell$ is isomorphic to the category of equivariant $\phi$-coordinated quasi
modules for the universal (resp. simple) affine vertex algebra
$V_{\widehat{\mathcal{L}}(\dot{\mathfrak{g}})}(\ell,0)$ (resp.
$L_{\widehat{\mathcal{L}}(\dot{\mathfrak{g}})}(\ell,0)$) (see Theorem 3.11).

Let $\mathfrak{g}$ be the untwisted affine Kac-Moody algebra
$\widetilde{\mathcal{L}}(\dot{\mathfrak{g}},\mathrm{id})$, and $\mu$ a
nontransitive diagram automorphism of $\mathfrak{g}$. We can also define the
twisted toroidal EALA $\widetilde{\mathfrak{g}}[\mu]$ similar to the
construction of twisted affine Kac-Moody algebra. Allison-Berman-Pianzola
proved in [ABP] that a nullity $2$ EALA of maximal type is either isomorphic
to a twisted toroidal EALA $\widetilde{\mathfrak{g}}[\mu]$, or to EALA
$\widetilde{\mathfrak{sl}}_{N}(\mathbb{C}_{q})$ of type $A_{N-1}$ coordinated
by an irrational quantum torus $\mathbb{C}_{q}$ (see Section 5 for details).
It was discovered by Billig in [B2] that an untwisted toroidal EALA is in
general not a vertex Lie algebra in the sense of [DLM]. Thus one cannot
associate the restricted $\widetilde{\mathfrak{g}}[\mu]$-modules, as the
affine Kac-Moody algebra case, to twisted modules of vertex algebras. It was
also pointed out in [Li3] that, since the generating functions of
$\widetilde{\mathfrak{sl}}_{N}(\mathbb{C}_{q})$ are not “local” in general,
the restricted modules of EALA $\widetilde{\mathfrak{sl}}_{N}(\mathbb{C}_{q})$
cannot be directly associated to twisted modules of vertex algebras.

As the main result of this paper, we prove that every nullity $2$ EALA of
maximal type can be associated with a vertex algebra through equivariant
$\phi$-coordinated quasi modules. More explicitly, for any nontransitive
diagram automorphism $\mu$ of an untwisted affine Kac-Moody algebra
$\mathfrak{g}$, we construct a vertex algebra
$V_{\widehat{\mathfrak{g}}}(\ell,0)$, a quotient vertex algebra
$L_{\widehat{\mathfrak{g}}}(\ell,0)$ of $V_{\widehat{\mathfrak{g}}}(\ell,0)$,
an automorphism group $G_{\mu}$ of $V_{\widehat{\mathfrak{g}}}(\ell,0)$ and
$L_{\widehat{\mathfrak{g}}}(\ell,0)$, and a linear character $\chi_{\omega}$
of $G_{\mu}$. And then we establish a module category isomorphism from the
category of restricted (resp. integrable restricted)
$\widetilde{\mathfrak{g}}[\mu]$-modules of level $\ell$ to the category of
$(G_{\mu},\chi_{\omega})$-equivariant $\phi$-coordinated quasi modules for
$V_{\widehat{\mathfrak{g}}}(\ell,0)$ (resp.
$L_{\widehat{\mathfrak{g}}}(\ell,0)$) (see Theorem 6.8). Meanwhile, for any
positive integer $N\geq 2$ and generic complex number $q$, we also construct a
universal (resp. simple) affine vertex algebras
$V_{\widehat{\mathcal{L}}\left(\mathfrak{sl}_{\infty}\right)}\left(\ell,0\right)$
(resp.
$L_{\widehat{\mathcal{L}}\left(\mathfrak{sl}_{\infty}\right)}\left(\ell,0\right)$)
associated to $\mathfrak{sl}_{\infty}$, an automorphism group $G_{N}$ of the
affine vertex algebras, and a linear character $\chi_{q}$ of $G_{N}$. And then
we prove that the category of restricted (resp. integrable restricted)
$\widetilde{\mathfrak{sl}}_{N}(\mathbb{C}_{q})$-modules of level $\ell$ is
canonically isomorphic to the category of $(G_{N},\chi_{q})$-equivariant
$\phi$-coordinated quasi modules for
$V_{\widehat{\mathcal{L}}\left(\mathfrak{sl}_{\infty}\right)}\left(\ell,0\right)$
(resp.
$L_{\widehat{\mathcal{L}}\left(\mathfrak{sl}_{\infty}\right)}\left(\ell,0\right)$)
(see Theorem 7.5).

The structure of the paper is given as follows. In Section 2 we recall the
notion of $(G,\chi)$-equivariant $\phi$-coordinated quasi module for a vertex
algebra introduced in [Li7], and consider the $(G,\chi)$-equivariant
$\phi$-coordinated quasi modules for the universal enveloping vertex algebra
of a conformal algebra. After giving natural connections among equivariant
$\phi$-coordinated quasi modules, equivariant quasi modules and twisted
modules for vertex operator algebras and (general) universal affine vertex
algebras, in Section 3 we prove the isomorphism between the categories of
restricted modules for affine Kac-Moody algebras and equivariant
$\phi$-coordinated quasi modules for affine vertex algebras. In Section 4, for
any diagram automorphism $\mu$ of an untwisted affine Kac-Moody algebra
$\mathfrak{g}$, we construct an automorphism $\widetilde{\mu}$ of the toroidal
EALA $\widetilde{\mathfrak{g}}$ associated to $\mathfrak{g}$ and study the Lie
subalgebra $\widetilde{\mathfrak{g}}[\mu]$ of $\widetilde{\mathfrak{g}}$ fixed
by the automorphism $\widetilde{\mu}$. We recall Allison-Berman-Pianzola’s
classification result of nullity 2 EALAs of maximal type in Section 5. And
then in Section 6, we define two vertex algebras
$V_{\widehat{\mathfrak{g}}}(\ell,0)$ and $L_{\widehat{\mathfrak{g}}}(\ell,0)$,
and associate restricted (resp. integrable restricted)
$\widetilde{\mathfrak{g}}[\mu]$-modules of level $\ell$ with equivariant
$\phi$-coordinated quasi modules for the vertex algebra
$V_{\widehat{\mathfrak{g}}}(\ell,0)$ (resp.
$L_{\widehat{\mathfrak{g}}}(\ell,0)$). Finally, in Section 7, we associate
restricted (resp. integrable restricted)
$\widetilde{\mathfrak{sl}}_{N}(\mathbb{C}_{q})$-modules of level $\ell$ with
equivariant $\phi$-coordinated quasi modules for universal (resp. simple)
affine vertex algebras associated to $\mathfrak{sl}_{\infty}$.

In this paper we denote by $\mathbb{Z}$, $\mathbb{Z}^{\ast}$, $\mathbb{N}$,
$\mathbb{C}$ and $\mathbb{C}^{\ast}$ respectively the sets of integers,
nonzero integers, nonnegative integers, complex numbers and nonzero complex
numbers. And if $\mathfrak{g}$ is a Lie algebra, we denote by
$\mathcal{U}(\mathfrak{g})$ the universal enveloping algebra.

##  2 Equivariant $\phi$-coordinated quasi modules for vertex algebras

In this section we recall the notion and some basics on equivariant
$\phi$-coordinated quasi modules for vertex algebras (cf. [Li8, JKLiT, CLTW]).

###  2.1 Definitions and basic properties

Throughout this paper, let $z,w,z_{0},z_{1},z_{2},\dots$ be mutually commuting
independent formal variables. We use the standard notations and conventions as
in [FHL, LLi]. For example, for a vector space $U$,
$U[[z_{1},z_{2},\dots,z_{r}]]$ is the space of formal (possibly doubly
infinite) power series in $z_{1},z_{2},\dots,z_{r}$ with coefficients in $U$,
and $U((z_{1},z_{2},\dots,z_{r}))$ is the space of lower truncated Laurent
power series in $z_{1},z_{2},\dots,z_{r}$ with coefficients in $U$. Denote a
vertex algebra by $V=(V,Y,\mathbf{1})$ , where ${\mathbf{1}}$ is the vacuum
vector and $Y(\cdot,z):V\rightarrow\mathrm{Hom}(V,V((z))),\
v\mapsto\sum_{n\in{\mathbb{Z}}}v_{n}z^{-n-1}$ is the vertex operator. And the
canonical derivation on $V$ defined by $v\mapsto v_{-2}\mathbf{1}$ for $v\in
V$ is denoted by $\mathcal{D}$.

For a subset $\Gamma$ of $\mathbb{C}^{\ast}$, denote by
$\mathbb{C}_{\Gamma}\left[z\right]$ the set of all polynomials in
$\mathbb{C}\left[z\right]$ whose roots are contained in $\Gamma$. Let $\phi$
be the formal power series $\phi\left(z_{2},z_{0}\right)=z_{2}e^{z_{0}}$,
which is a particular associate of the one-dimensional additive formal group
$F\left(z,w\right)=z+w$ as defined in [Li3]. Now we recall the notions of
equivariant $\phi$-coordinated quasi modules for a vertex algebra (see [Li8]).

######  Definition 2.1.

Let $(V,Y,\mathbf{1})$ be a vertex algebra, $G$ a group of automorphism on $V$
and $\chi:G\to\mathbb{C}^{\ast}$ a linear character of $G$. _A
$\left(G,\chi\right)$-equivariant $\phi$-coordinated quasi $V$-module
$\left(W,Y_{W}^{\phi}\right)$_ is a vector space $W$ equipped with a linear
map

$Y_{W}^{\phi}\left(\cdot,z\right):V\to\text{Hom}\left(W,W\left(\left(z\right)\right)\right)\subset\left(\text{End}W\right)\left[\left[z,z^{-1}\right]\right]$

satisfying the following three conditions:

  1. (i)

$Y_{W}^{\phi}\left(\textbf{1},z\right)=1_{W}$;

  2. (ii)

$Y_{W}^{\phi}\left(gv,z\right)=Y_{W}^{\phi}\left(v,\chi\left(g\right)z\right)$
for $g\in G,v\in V$;

  3. (iii)

For $u,v\in V,$ there exists
$f\left(z\right)\in\mathbb{C}_{\chi\left(G\right)}\left[z\right]$ such that

$\displaystyle
f\left(z_{1}/z_{2}\right)Y_{W}^{\phi}\left(u,z_{1}\right)Y_{W}^{\phi}\left(v,z_{2}\right)\in\text{Hom}\left(W,W\left(\left(z_{1},z_{2}\right)\right)\right),$
$\displaystyle
f\left(e^{z_{0}}\right)Y_{W}^{\phi}\left(Y\left(u,z_{0}\right)v,z_{2}\right)=\left(f\left(z_{1}/z_{2}\right)Y_{W}^{\phi}\left(u,z_{1}\right)Y_{W}^{\phi}\left(v,z_{2}\right)\right)|_{z_{1}=\phi(z_{2},z_{0})}.$

Furthermore, a $\left(G,\chi\right)$-equivariant $\phi$-coordinated quasi
$V$-module $\left(W,Y_{W}^{\phi},d\right)$ is a
$\left(G,\chi\right)$-equivariant $\phi$-coordinated quasi $V$-module
$\left(W,Y_{W}^{\phi}\right)$ equipped with an endomorphism $d$ of $W$ such
that
$[d,Y_{W}^{\phi}\left(v,z\right)]=Y_{W}^{\phi}\left(\mathcal{D}v,z\right)\
\text{for }\ v\in V$.

Now we fix a vertex algebra $V$, an automorphism group $G$ of $V$, and a
linear character $\chi$ of $G$. The following two results follow respectively
from [Li6, Lemma 3.7] and [CLTW, Proposition 5.2].

######  Lemma 2.2.

For a $\left(G,\chi\right)$-equivariant $\phi$-coordinated quasi $V$-module
$\left(W,Y_{W}^{\phi},d\right)$, we have

$\displaystyle\left[d,Y_{W}^{\phi}\left(v,z\right)\right]=Y_{W}^{\phi}\left(\mathcal{D}v,z\right)=z\frac{d}{dz}Y_{W}^{\phi}\left(v,z\right),\quad\forall\
v\in V.$

######  Proposition 2.3.

Let $\left(W,Y_{W}^{\phi}\right)$ be a $\left(G,\chi\right)$-equivariant
$\phi$-coordinated quasi $V$-module and $\psi:\chi\left(G\right)\to G$ be a
section of $\chi$. Then for $u,v\in V$,

$\displaystyle\left[Y_{W}^{\phi}\left(u,z_{1}\right),Y_{W}^{\phi}\left(v,z_{2}\right)\right]=\text{Res}_{z_{0}}\sum_{g\in\psi\left(\chi\left(G\right)\right)}Y_{W}^{\phi}\left(Y\left(gu,z_{0}\right)v,z_{2}\right)e^{z_{0}z_{2}\frac{\partial}{\partial
z_{2}}}\left(\chi\left(g\right)\delta\left(\frac{\chi\left(g\right)z_{2}}{z_{1}}\right)\right).$

Then, we have:

######  Lemma 2.4.

Let $\left(W,Y_{W}^{\phi}\right)$ be a $\left(G,\chi\right)$-equivariant
$\phi$-coordinated quasi $V$-module, and $u,v\in V$ be such that $u_{n}v=0$
for $n\geq 0$. Then there exists a polynomial
$q\left(z\right)\in\mathbb{C}_{\chi\left(G\right)\setminus\\{1\\}}\left[z\right]$
such that

$q\left(z_{1}/z_{2}\right)Y_{W}^{\phi}\left(u,z_{1}\right)Y_{W}^{\phi}\left(v,z_{2}\right)=q\left(z_{1}/z_{2}\right)Y_{W}^{\phi}\left(v,z_{2}\right)Y_{W}^{\phi}\left(u,z_{1}\right).$
(2.1)

Furthermore, for any such polynomial $q\left(z\right)$ we have

$Y_{W}^{\phi}\left(u_{-1}v,z_{2}\right)=\left(q\left(z_{1}/z_{2}\right)Y_{W}^{\phi}\left(u,z_{1}\right)Y_{W}^{\phi}\left(v,z_{2}\right)\right)|_{z_{1}=z_{2}}.$
(2.2)

###### Proof.

Since $u_{n}v=0$ for $n\geq 0$, from Proposition 2.3 it follows that there
exist (possibly same)
$\lambda_{1},\dots,\lambda_{r}\in\chi\left(G\right)\setminus\left\\{1\right\\}$
such that

$\left(z_{1}/z_{2}-\lambda_{1}\right)\cdots\left(z_{1}/z_{2}-\lambda_{r}\right)\left[Y_{W}^{\phi}\left(u,z_{1}\right),Y_{W}^{\phi}\left(v,z_{2}\right)\right]=0.$

This proves (2.1) with
$q\left(z\right)=\left(z-\lambda_{1}\right)\cdots\left(z-\lambda_{r}\right)$.
Then by definition we have

$q\left(e^{z_{0}}\right)Y_{W}^{\phi}\left(Y\left(u,z_{0}\right)v,z_{2}\right)=\left(q\left(z_{1}/z_{2}\right)Y_{W}^{\phi}\left(u,z_{1}\right)Y_{W}^{\phi}\left(v,z_{2}\right)\right)|_{z_{1}=z_{2}e^{z_{0}}}.$

Note that $Y\left(u,z_{0}\right)v\in V\left[\left[z_{0}\right]\right]$, one
can set $z_{0}=0$ in the above equality, which gives (2.2). ∎

Furthermore, we have the following results (cf. [Li1, Proposition 2.3.6],
[Li2, Proposition 2.10]).

######  Proposition 2.5.

Let $u\in V$ such that $u_{n}u=0$ for $n\geq 0$, $\ell$ a positive integer and
$\left(W,Y_{W}^{\phi}\right)$ a $\left(G,\chi\right)$-equivariant
$\phi$-coordinated quasi $V$-module. Then there exists a polynomial
$q\left(z\right)\in\mathbb{C}_{\chi\left(G\right)\setminus\\{1\\}}\left[z\right]$
such that

$\left(\prod_{1\leq
i<j\leq\ell+1}q\left(z_{i}/z_{j}\right)\right)Y_{W}^{\phi}\left(u,z_{1}\right)\cdots
Y_{W}^{\phi}\left(u,z_{\ell+1}\right)\in\text{Hom}\left(W,W\left(\left(z_{1},\cdots,z_{\ell+1}\right)\right)\right).$
(2.3)

Moreover, if $\left(u_{-1}\right)^{\ell+1}\boldsymbol{1}=0$, then

$\left(\prod_{1\leq
i<j\leq\ell+1}q\left(z_{i}/z_{j}\right)\right)Y_{W}^{\phi}\left(u,z_{1}\right)\cdots
Y_{W}^{\phi}\left(u,z_{\ell+1}\right)|_{z_{1}=\cdots=z_{\ell+1}}=0.$ (2.4)

And on the other hand, if $W$ is faithful and (2.4) holds, then
$\left(u_{-1}\right)^{\ell+1}\boldsymbol{1}=0.$

###### Proof.

Note that the first part of the proposition follows from (2.1). For the second
part, it suffices to prove the following identity

$\displaystyle
Y_{W}^{\phi}\left(\left(u_{-1}\right)^{\ell+1}\boldsymbol{1},z_{\ell+1}\right)=\left(\prod_{1\leq
i<j\leq\ell+1}q\left(z_{i}/z_{j}\right)\right)Y_{W}^{\phi}\left(u,z_{1}\right)\cdots
Y_{W}^{\phi}\left(u,z_{\ell+1}\right)|_{z_{1}=\cdots=z_{\ell+1}}.$ (2.5)

We prove this equation by induction on $\ell$. When $\ell=1$, the identity
(2.5) follows from (2.2). Now we assume that $\ell>1$ and set
$v=\left(u_{-1}\right){}^{\ell}\boldsymbol{1}$. Then by induction and (2.3),
we obtain

$\begin{split}&q\left(z_{1}/z_{\ell+1}\right)^{\ell}Y_{W}^{\phi}\left(u,z_{1}\right)Y_{W}^{\phi}\left(v,z_{\ell+1}\right)\\\
=&q\left(z_{1}/z_{\ell+1}\right)^{\ell}Y_{W}^{\phi}\left(u,z_{1}\right)\cdot\left(\prod_{2\leq
i<j\leq\ell+1}q\left(z_{i}/z_{j}\right)\right)Y_{W}^{\phi}\left(u,z_{2}\right)\cdots
Y_{W}^{\phi}\left(u,z_{\ell+1}\right)|_{z_{2}=\cdots=z_{\ell+1}}\\\
=&\left(\prod_{1\leq
i<j\leq\ell+1}q\left(z_{i}/z_{j}\right)\right)Y_{W}^{\phi}\left(u,z_{1}\right)\cdots
Y_{W}^{\phi}\left(u,z_{\ell+1}\right)|_{z_{2}=\cdots=z_{\ell+1}}\in\text{Hom}\left(W,W\left(\left(z_{1},z_{\ell+1}\right)\right)\right).\end{split}$

Since $u_{n}u=0$ for $n\geq 0$, we have $u_{n}v=0$ for $n\geq 0$. Then it
follows from (2.1) that

$\displaystyle
Y_{W}^{\phi}\left(\left(u_{-1}\right)^{\ell+1}\boldsymbol{1},z_{\ell+1}\right)=Y_{W}^{\phi}\left(u_{-1}v,z_{\ell+1}\right)=\left(q\left(z_{1}/z_{\ell+1}\right){}^{\ell}Y_{W}^{\phi}\left(u,z_{1}\right)Y_{W}^{\phi}\left(v,z_{\ell+1}\right)\right)|_{z_{1}=z_{\ell+1}}$
$\displaystyle=$ $\displaystyle\left(\prod_{1\leq
i<j\leq\ell+1}q\left(z_{i}/z_{j}\right)\right)Y_{W}^{\phi}\left(u,z_{1}\right)\cdots
Y_{W}^{\phi}\left(u,z_{\ell+1}\right)|_{z_{1}=\cdots=z_{\ell+1}},$

which proves the claim (2.5) and hence completes the proof of the proposition.
∎

###  2.2 $\left(G,\chi\right)$-equivariant $\phi$-coordinated quasi
$V_{\mathcal{C}}$-modules

In this subsection we first recall the notion of conformal algebra, then study
the equivariant $\phi$-coordinated quasi modules for the universal enveloping
vertex algebra constructed from a conformal algebra [CLTW].

A _conformal algebra_ , also known as a _vertex Lie algebra_ (see [P, DLM]),
is a vector space $\mathcal{C}$ equipped with a linear operator $\partial$ and
a linear map

$\displaystyle
Y^{-}:\mathcal{C}\rightarrow\text{Hom}\left(\mathcal{C},z^{-1}\mathcal{C}\left[z^{-1}\right]\right),\quad
u\mapsto Y^{-}\left(u,z\right)=\sum_{n\geq 0}u_{n}z^{-n-1}$ (2.6)

such that for any $u,v\in\mathcal{C}$,

$\displaystyle\left[\partial,Y^{-}\left(u,z\right)\right]$
$\displaystyle=Y^{-}\left(\partial
u,z\right)=\frac{d}{dz}Y^{-}\left(u,z\right),$ (2.7) $\displaystyle
Y^{-}\left(u,z\right)v$
$\displaystyle=\text{Sing}\left(e^{z\partial}Y^{-}\left(v,-z\right)u\right),$
$\displaystyle\left[Y^{-}\left(u,z\right),Y^{-}\left(v,w\right)\right]$
$\displaystyle=\text{Sing}\left(Y^{-}\left(Y^{-}\left(u,z-w\right)v,w\right)\right),$

where Sing stands for the singular part.

It was proved in [P, Remark 4.2] that a conformal algebra structure on a
vector space $\mathcal{C}$ amounts to a Lie algebra structure on the following
quotient space of $\mathbb{C}\left[t,t^{-1}\right]\otimes\mathcal{C}$:

$\widehat{\mathcal{C}}=\mathbb{C}\left[t,t^{-1}\right]\otimes\mathcal{C}/\left(1\otimes\partial+\frac{d}{dt}\otimes
1\right)\left(\mathbb{C}\left[t,t^{-1}\right]\otimes\mathcal{C}\right).$

######  Lemma 2.6.

Let $\mathcal{C}$ be a vector space equipped with a linear operator $\partial$
and a linear map $Y^{-}$ as given in (2.6) such that (2.7) holds. Then
$\mathcal{C}$ is a conformal algebra if and only if there is a Lie algebra
structure on $\widehat{\mathcal{C}}$ such that

$\displaystyle\left[u\left(m\right),v\left(n\right)\right]=\sum_{i\geq
0}{{m}\choose{i}}\left(u_{i}v\right)\left(m+n-i\right),$ (2.8)

for $u,v\in\mathcal{C}$, $m,n\in\mathbb{Z},$ where $u(m)$ stands for the image
of $t^{m}\otimes u$ in $\widehat{\mathcal{C}}$.

Let $\mathcal{C}$ be a conformal algebra. Set

$\displaystyle\widehat{\mathcal{C}}^{-}=\text{Span}\\{u(-m-1)\mid
u\in\mathcal{C},m\in\mathbb{N}\\}\ \text{\ and\ \
}\widehat{\mathcal{C}}^{+}=\text{Span}\\{u(m)\mid
u\in\mathcal{C},m\in\mathbb{N}\\}.$ (2.9)

Then
$\widehat{\mathcal{C}}=\widehat{\mathcal{C}}^{+}\oplus\widehat{\mathcal{C}}^{-}$
and both $\widehat{\mathcal{C}}^{+}$ and $\widehat{\mathcal{C}}^{-}$ are
subalgebras of the Lie algebra $\widehat{\mathcal{C}}$. Moreover, the map

$\displaystyle\mathcal{C}\to\widehat{\mathcal{C}}^{-},\quad u\mapsto
u\left(-1\right)$ (2.10)

is an isomorphism of vector spaces [P, Theorem 4.6]. Consider the induced
$\widehat{\mathcal{C}}$-module

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

where $\mathbb{C}$ is the one dimensional trivial
$\widehat{\mathcal{C}}^{+}$-module. Set $\boldsymbol{1}=1\otimes 1\in
V_{\mathcal{C}}.$ Identify $\mathcal{C}$ as a subspace of $V_{\mathcal{C}}$
through the linear map $u\mapsto u\left(-1\right)\boldsymbol{1}.$ It was
proved in [P] that there exists a unique vertex algebra structure on
$V_{\mathcal{C}}$, called _the universal enveloping vertex algebra of
$\mathcal{C}$_, with $\boldsymbol{1}$ as the vacuum vector and
$Y\left(u,z\right)=u\left(z\right)=\sum_{n\in\mathbb{Z}}u\left(n\right)z^{-n-1}$
for $u\in\mathcal{C}$. The map $-\frac{d}{dt}\otimes 1$ (or equivalently,
$1\otimes\partial$) on ${\mathbb{C}}[t,t^{-1}]\otimes\mathcal{C}$ induces a
derivation $\mathcal{D}$ on $\widehat{\mathcal{C}}$ such that

$\displaystyle\mathcal{D}\left(u\left(m\right)\right)=-mu\left(m-1\right),\quad\forall\
u\in\mathcal{C},\ m\in\mathbb{Z}.$ (2.11)

Note that $\mathcal{D}$ preserves the subalgebra $\widehat{\mathcal{C}}^{-}$.
Thus it can be uniquely extended to a derivation on
$\mathcal{U}\left(\widehat{\mathcal{C}}^{-}\right)\cong V_{\mathcal{C}}$,
which coincides with the canonical derivation on the vertex algebra
$V_{\mathcal{C}}$.

Recall that an _automorphism_ $\varphi$ of the conformal algebra $\mathcal{C}$
is a linear automorphism such that $\varphi\circ\partial=\partial\circ\varphi$
and
$\varphi\left(u_{i}v\right)=\varphi\left(u\right){}_{i}\varphi\left(v\right)$
for $u,v\in\mathcal{C}$ and $i\in\mathbb{N}$ (see [K2]). We have:

######  Lemma 2.7.

Let $\varphi$ be a linear map of the conformal algebra $\mathcal{C}$ such that
$\varphi\circ\partial=\partial\circ\varphi$. Then the linear map

$\displaystyle\hat{\varphi}:\widehat{\mathcal{C}}\to\widehat{\mathcal{C}},\quad
u\left(m\right)\mapsto\varphi\left(u\right)\left(m\right)\quad(u\in\mathcal{C},m\in\mathbb{Z})$
(2.12)

on the Lie algebra $\widehat{\mathcal{C}}$ is well-defined. Furthermore,
$\varphi$ is an automorphism of the conformal algebra $\mathcal{C}$ if and
only if $\hat{\varphi}$ is an automorphism of the Lie algebra
$\widehat{\mathcal{C}}.$

###### Proof.

The first assertion of the lemma is easy to see. For the second part, let
$u,v\in\mathcal{C}$ and $m,n\in\mathbb{Z}$. Then by (2.8) we have

$\displaystyle\hat{\varphi}\left(\left[u\left(m\right),v\left(n\right)\right]\right)=\sum_{i\geq
0}{{m}\choose{i}}\hat{\varphi}\left(\left(u_{i}v\right)\left(m+n-i\right)\right)=\sum_{i\geq
0}{{m}\choose{i}}\varphi\left(u_{i}v\right)\left(m+n-i\right),$ (2.13)

and

$\displaystyle\left[\hat{\varphi}\left(u\left(m\right)\right),\hat{\varphi}\left(v\left(n\right)\right)\right]=\left[\varphi(u)\left(m\right),\varphi(v)\left(n\right)\right]=\sum_{i\geq
0}{{m}\choose{i}}\left((\varphi u)_{i}(\varphi v)\right)\left(m+n-i\right).$
(2.14)

This immediately implies that if $\varphi$ is an automorphism of
$\mathcal{C}$, then $\hat{\varphi}$ is a Lie algebra automorphism of
$\widehat{\mathcal{C}}$. On the other hand, let $\hat{\varphi}$ be a Lie
algebra automorphism of $\widehat{\mathcal{C}}$. Assume that $\varphi$ is not
an automorphism of $\mathcal{C}$. Then there exist $u,v\in\mathcal{C}$ such
that $\varphi(u_{i}v)\neq(\varphi u)_{i}(\varphi v)$ for some
$i\in\mathbb{N}$. Let $i_{0}$ be the maximal one among such integers. Take
$m,n\in\mathbb{Z}$ such that $m+n=i_{0}-1$. Then we have

$\sum_{i=0}^{i_{0}-1}{{m}\choose{i}}\varphi\left(u_{i}v\right)\left(m+n-i\right),\
\sum_{i=0}^{i_{0}-1}{{m}\choose{i}}\left((\varphi u)_{i}(\varphi
v)\right)\left(m+n-i\right)\in\widehat{\mathcal{C}}^{+},$

and

$\sum_{i\geq
i_{0}}{{m}\choose{i}}\varphi\left(u_{i}v\right)\left(m+n-i\right),\
\sum_{i\geq i_{0}}{{m}\choose{i}}\left((\varphi u)_{i}(\varphi
v)\right)\left(m+n-i\right)\in\widehat{\mathcal{C}}^{-}.$

Recall that $\widehat{\mathcal{C}}^{+}\cap\widehat{\mathcal{C}}^{-}=\\{0\\}$
and $\hat{\varphi}$ is a Lie algebra automorphism. Then by (2.13), (2.14) and
the maximality of $i_{0}$ we obtain that $\varphi(u_{i_{0}}v)(-1)=((\varphi
u)_{i_{0}}(\varphi v))(-1)$. This together with (2.10) gives
$\varphi(u_{i_{0}}v)=(\varphi u)_{i_{0}}(\varphi v)$, a contradiction.
Therefore we have finished the proof of the lemma. ∎

We define a multiplication on the loop space
$\mathbb{C}\left[t,t^{-1}\right]\otimes\mathcal{C}$ by

$\left[f\left(t\right)\otimes u,g\left(t\right)\otimes v\right]=\sum_{i\geq
0}\left(\frac{1}{i!}\left(t\frac{d}{dt}\right)^{i}f\left(t\right)\right)g\left(t\right)\otimes\left(u_{i}v\right)\quad\left(f\left(t\right),g\left(t\right)\in\mathbb{C}\left[t,t^{-1}\right],\,u,v\in\mathcal{C}\right).$
(2.15)

By [K2, Remark 2.7d] this multiplication affords a Lie algebra structure on
the quotient space

$\overline{\mathcal{C}}=\left(\mathbb{C}\left[t,t^{-1}\right]\otimes\mathbb{\mathcal{C}}\right)/\text{Im}\left(1\otimes\partial+t\frac{d}{dt}\otimes
1\right).$

Denote by $\mathbf{d}$ the derivation on $\overline{\mathcal{C}}$ induced from
$-t\frac{d}{dt}\otimes
1\in\mathrm{End}\left(\mathbb{C}\left[t,t^{-1}\right]\otimes\mathcal{C}\right)$.
Form the semi-direct product Lie algebra

$\widetilde{\mathcal{C}}=\overline{\mathcal{C}}\rtimes{\mathbb{C}}\textbf{d}.$

For $u\in\mathcal{C}$ and $m\in\mathbb{Z}$, denote by $u\left[m\right]$ the
image of $t^{m}\otimes u$ in $\overline{\mathcal{C}}$. Then the Lie relations
on $\widetilde{\mathcal{C}}$ are given by

$\displaystyle\left[u\left[m\right],v\left[n\right]\right]=\sum_{i\geq
0}\frac{m^{i}}{i!}\left(u_{i}v\right)\left[m+n\right],\quad\left[\textbf{d},u\left[m\right]\right]=-mu\left[m\right]$
(2.16)

for $u,v\in\mathcal{C},$ $m,n\in\mathbb{Z}$. The following notion was first
introduced in [G-KK] (see also [Li3]).

######  Definition 2.8.

A _$G$ -conformal algebra_ is a conformal algebra $\mathcal{C}$ together with
an automorphism group $G$ of $\mathcal{C}$ such that for $u\in\mathcal{C}$,
$Y^{-}\left(gu,z\right)=0$ for all but finitely many $g\in G$.

Let $\mathcal{C}$ be a $G$-conformal algebra and $\chi:G\to\mathbb{C}^{\ast}$
be a linear character. For any $g\in G$, it is easy to check that (cf. [CLTW,
Lemma 5.8]) the linear map

$\displaystyle\bar{g}:\overline{\mathcal{C}}\to\overline{\mathcal{C}},\quad
u\left[m\right]\mapsto\chi\left(g\right)^{m}\left(gu\right)\left[m\right]$

defines an automorphism of the Lie algebra $\overline{\mathcal{C}}$.
Furthermore, $\overline{g}$ can be extended to be an automorphism $\tilde{g}$
of $\widetilde{\mathcal{C}}$ by

$\displaystyle\tilde{g}|_{\overline{\mathcal{C}}}=\bar{g}\quad\text{and}\quad\tilde{g}\left(\mathbf{d}\right)=\mathbf{d}.$
(2.17)

Following [Li6, Section 4], we define a new operation on $\bar{\mathcal{C}}$
by

$\left(a,b\right)\mapsto\sum_{g\in
G}\left[\overline{g}a,b\right]\quad(a,b\in\bar{\mathcal{C}}).$

It was proved therein that the quotient space

$\overline{\mathcal{C}}\left[G\right]=\bar{\mathcal{C}}/\text{Span$\left\\{\bar{g}a-a\mid
a\in\overline{\mathcal{C}},g\in G\right\\}$ }$ (2.18)

is a Lie algebra under the operation. For $u\in\mathcal{C}$ and
$n\in\mathbb{Z},$ we denote by $\overline{u\left[n\right]}$ the image of
$u\left[n\right]$ under the quotient map
$\overline{\mathcal{C}}\to\overline{\mathcal{C}}\left[G\right]$. Since
$g\circ\mathbf{d}=\mathbf{d}\circ g$ for $g\in G$, $\mathbf{d}$ descends to a
derivation on $\overline{\mathcal{C}}\left[G\right]$ and hence we have the
semi-direct product Lie algebra

$\displaystyle\widetilde{\mathcal{C}}\left[G\right]=\overline{\mathcal{C}}\left[G\right]\rtimes\mathbb{C}\mathbf{d}.$
(2.19)

For $u\in\mathcal{C}$, we form the generating function
$u\left[z\right]=\sum_{n\in\mathbb{Z}}\overline{u\left[n\right]}z^{-n}$, and
we say that a module $W$ for $\widetilde{\mathcal{C}}\left[G\right]$ or
$\overline{\mathcal{C}}\left[G\right]$ is _restricted_ if
$u\left[z\right]\in\text{Hom}\left(W,W\left(\left(z\right)\right)\right)$ for
$u\in\mathcal{C}$. It is known from [P] that an automorphism of $\mathcal{C}$
can be lifted uniquely to an automorphism of $V_{\mathcal{C}}$. In particular,
the automorphism group $G$ of $\mathcal{C}$ can be viewed as an automorphism
group of $V_{\mathcal{C}}$. Now we have the following result.

######  Proposition 2.9.

Let $\mathcal{C}$ be a $G$-conformal algebra and
$\chi:G\rightarrow\mathbb{C}^{\ast}$ be an injective linear character. Then
the $\left(G,\chi\right)$-equivariant $\phi$-coordinated quasi
$V_{\mathcal{C}}$-modules $\left(W,Y_{W}^{\phi},d\right)$ are exactly the
restricted $\widetilde{\mathcal{C}}\left[G\right]$-modules $W$ with

$\displaystyle d=\textbf{d}\quad\text{and}\quad
Y_{W}^{\phi}\left(u,z\right)=u\left[z\right]\quad\text{for}\ u\in\mathcal{C}.$

###### Proof.

It was proved in [CLTW, Theorem 5.12] that the
$\left(G,\chi\right)$-equivariant $\phi$-coordinated quasi
$V_{\mathcal{C}}$-modules $\left(W,Y_{W}^{\phi}\right)$ are exactly the
restricted $\overline{\mathcal{C}}\left[G\right]$-modules $W$ with
$Y_{W}^{\phi}\left(u,z\right)=u\left[z\right]$ for $u\in\mathcal{C}$. Note
that for $u\in\mathcal{C}$, we have

$\displaystyle\left[\textbf{d},u\left[z\right]\right]=\sum_{n\in\mathbb{Z}}\left[\textbf{d},\overline{u\left[n\right]}\right]z^{-n}=\sum_{n\in{\mathbb{Z}}}-n\overline{u\left[n\right]}z^{-n}=z\frac{d}{dz}u\left[z\right].$

This together with Lemma 2.2 proves the proposition. ∎

######  Remark 2.10.

Assume that $G=\left\langle g\right\rangle$ is a cyclic group of finite order
$T$, and the linear character $\chi$ is injective. Then it is straightforward
to see that $\overline{\mathcal{C}}\left[G\right]$ is isomorphic to the
subalgebra of $\overline{\mathcal{C}}$ fixed by $\bar{g}$. And the isomorphism
is given by

$\overline{u\left[n\right]}\mapsto\sum_{p=0}^{T-1}\left(\bar{g}\right)^{p}\left(u\left[n\right]\right)$

for $u\in\mathcal{C}$ and $n\in\mathbb{Z}.$ Furthermore, this isomorphism can
be extended to an isomorphism from $\widetilde{\mathcal{C}}\left[G\right]$ to
the subalgebra of $\widetilde{\mathcal{C}}$ fixed by $\tilde{g}$ such that
$\mathbf{d}\mapsto\mathbf{d}$.

##  3 Equivariant $\phi$-coordinated quasi modules for affine vertex algebras

In this section, we study the connection between equivariant
$\phi$-coordinated quasi modules for universal (resp. simple) affine vertex
algebras and restricted (resp. integrable restricted) modules for affine Kac-
Moody algebras.

###  3.1 Equivalence of module categories for vertex operator algebras

We first study connections among equivariant $\phi$-coordinated quasi modules,
equivariant quasi modules and twisted modules for vertex operator algebras
(see [FLM, FHL]). Recall that for any finite order automorphism $\sigma$ of a
vertex algebra $V$, there is (weak) _$\sigma$ -twisted $V$-module_
$\left(W,Y_{W}^{t}\right)$, where $W$ is a vector space and
$Y_{W}^{t}\in\mathrm{Hom}\left(W,W\left(\left(z^{1/N}\right)\right)\right)$
with $N$ the order of $\sigma$ (cf. [Li2, FLM]). We also recall that a
$\mathbb{Z}$-graded vertex algebra is a vertex algebra $V$ equipped with a
$\mathbb{Z}$-grading $V=\oplus_{n\in\mathbb{Z}}V_{(n)}$ such that ${\bf 1}\in
V_{(0)}$ and

$\displaystyle u_{m}V_{(n)}\subset V_{(n+k-m-1)}\ \ \mbox{ for }u\in V_{(k)},\
m,n,k\in\mathbb{Z}.$ (3.1)

Define the linear operator $L\left(0\right)$ on a $\mathbb{Z}$-graded vertex
algebra $V$ by $L\left(0\right)v=nv$ for $v\in V_{(n)}$ with
$n\in{\mathbb{Z}}$. Then we have the following definition (cf. [Li3, Li4]):

######  Definition 3.1.

Let $(V,Y,\mathbf{1})$ be a $\mathbb{Z}$-graded vertex algebra, $G$ an
automorphism group of $V$ preserving the $\mathbb{Z}$-grading of $V$ and
$\chi:G\rightarrow\mathbb{C}^{\ast}$ a linear character. _A
$\left(G,\chi\right)$-equivariant quasi $V$-module_ $\left(W,Y_{W}\right)$ is
a vector space $W$ equipped with a linear map
$Y_{W}\left(\cdot,z\right):V\to\text{Hom}\left(W,W\left(\left(z\right)\right)\right)$
satisfying the following conditions:

  1. (i)

$Y_{W}\left(\textbf{1},z\right)=1_{W}$;

  2. (ii)

$Y_{W}\left(\chi\left(g\right){}^{-L(0)}gv,z\right)=Y_{W}\left(v,\chi\left(g\right)z\right)$
for $g\in G,v\in V$;

  3. (iii)

For $u,v\in V,$ there exists
$f\left(z\right)\in\mathbb{C}_{\chi\left(G\right)}\left[z\right]$ such that

$\displaystyle
f\left(z_{1}/z_{2}\right)Y_{W}\left(u,z_{1}\right)Y_{W}\left(v,z_{2}\right)\in\text{Hom}\left(W,W\left(\left(z_{1},z_{2}\right)\right)\right),$
$\displaystyle
f\left((z_{2}+z_{0})/z_{2}\right)Y_{W}\left(Y\left(u,z_{0}\right)v,z_{2}\right)=\left(f\left(z_{1}/z_{2}\right)Y_{W}\left(u,z_{1}\right)Y_{W}\left(v,z_{2}\right)\right)|_{z_{1}=z_{2}+z_{0}}.$

######  Remark 3.2.

If the automorphism group $G=\left\\{1\right\\}$ is the trivial group in the
definition, one has the usual module for vertex algebra. And if replacing the
associate $\phi=z_{2}+z_{0}$ in (iii) by $\phi=z_{2}e^{z_{0}}$, one gets the
notion of a $\phi$-coordinated $V$-module defined in Definition 2.1 .

If $V=(V,Y(\cdot,z),{\mathbf{1}})$ is a $\mathbb{Z}$-graded vertex algebra, we
define a linear map

$\displaystyle
Y\left[\cdot,z\right]:V\rightarrow\mathrm{End}\left(V\right)\left[\left[z,z^{-1}\right]\right],\quad
v\mapsto Y\left(e^{zL(0)}v,e^{z}-1\right).$

Then $\left(V,Y\left[\cdot,z\right],{\mathbf{1}}\right)$ also carries a vertex
algebra structure [Z, Li7]. Note that if $G$ is an automorphism group of the
vertex algebra $V$ preserving the $\mathbb{Z}$-grading of $V$, then it is also
an automorphism group of $\left(V,Y\left[\cdot,v\right],\mathbf{1}\right)$.
The following result is a generalization of [Li7, Proposition 5.8].

######  Proposition 3.3.

Let $V=(V,Y(\cdot,z),{\mathbf{1}})$ be a $\mathbb{Z}$-graded vertex algebra,
$G$ an automorphism group of $V$ preserving the $\mathbb{Z}$-grading of $V$
and $\chi$ a linear character of $G$. Then the
$\left(G,\chi\right)$-equivariant quasi $V$-modules $\left(W,Y_{W}\right)$ are
exactly the $\left(G,\chi\right)$-equivariant $\phi$-coordinated quasi modules
$\left(W,Y_{W}^{\phi}\right)$ for the vertex algebra
$\left(V,Y\left[\cdot,z\right],\mathbf{1}\right)$ with

$\displaystyle
Y_{W}^{\phi}\left(v,z\right)=Y_{W}\left(z^{L(0)}v,z\right),\quad\forall\ v\in
V.$

###### Proof.

Assume first that $\left(W,Y_{W}\right)$ is a
$\left(G,\chi\right)$-equivariant quasi $V$-module. Then for $v\in V$ and
$g\in G$ we have

$\displaystyle
Y_{W}^{\phi}\left(v,\chi\left(g\right)z\right)=Y_{W}\left(\left(\chi(g)z\right){}^{L(0)}v,\chi(g)z\right)=Y_{W}\left(z^{L(0)}gv,z\right)=Y_{W}^{\phi}\left(gv,z\right).$

This together with [Li7, Proposition 5.8] shows that
$\left(W,Y_{W}^{\phi}\right)$ is a $\left(G,\chi\right)$-equivariant
$\phi$-coordinated quasi module for $\left(V,Y[\cdot,z],\mathbf{1}\right)$.
Conversely, let $\left(W,Y_{W}^{\phi}\right)$ be a
$\left(G,\chi\right)$-equivariant $\phi$-coordinated quasi module for
$\left(V,Y[\cdot,z],\mathbf{1}\right)$. Then we have
$Y_{W}\left(v,z\right)\in\mathrm{Hom}\left(W,W\left(\left(z\right)\right)\right)$
for $v\in V$,
$Y_{W}\left({\mathbf{1}},z\right)=Y_{W}^{\phi}\left(z^{-L(0)}{\mathbf{1}},z\right)=Y_{W}^{\phi}\left({\mathbf{1}},z\right)=1_{W}$
and

$\displaystyle
Y_{W}\left(\chi\left(g\right){}^{-L(0)}gv,z\right)=Y_{W}^{\phi}\left(\left(\chi\left(g\right)z\right){}^{-L(0)}gv,z\right)=Y_{W}^{\phi}\left(\left(\chi\left(g\right)z\right){}^{-L(0)}v,\chi\left(g\right)z\right)=Y_{W}\left(v,\chi\left(g\right)z\right),$

for $v\in V$ and $g\in G$. For $u,v\in V$, let $f\left(z\right)$ be a nonzero
polynomial in $\mathbb{C}_{\chi\left(G\right)}\left[z\right]$ such that
Definition 2.1 (iii) holds. Then we have

$\displaystyle
f\left(z_{1}/z_{2}\right)Y_{W}\left(u,z_{1}\right)Y_{W}\left(v,z_{2}\right)=f\left(z_{1}/z_{2}\right)Y_{W}^{\phi}\left(z_{1}^{-L(0)}u,z_{1}\right)Y_{W}^{\phi}\left(z_{2}^{-L(0)}v,z_{2}\right)\in\mathrm{Hom}\left(W,W\left(\left(z_{1},z_{2}\right)\right)\right).$

Furthermore, we have
$f\left(z_{1}/z_{2}\right)Y_{W}^{\phi}\left(\left(z_{2}+w_{0}\right){}^{-L(0)}u,z_{1}\right)Y_{W}\left(z_{2}^{-L(0)}v,z_{2}\right)\in\mathrm{Hom}\left(W,W\left(\left(z_{1},z_{2}\right)\right)\left[\left[w_{0}\right]\right]\right)$
and

$\displaystyle
f\left(z_{1}/z_{2}\right)Y_{W}^{\phi}\left(\left(z_{2}+w_{0}\right){}^{-L(0)}u,z_{1}\right)Y_{W}^{\phi}\left(z_{2}^{-L(0)}v,z_{2}\right)|_{z_{1}=z_{2}e^{z_{0}}}=f\left(e^{z_{0}}\right)Y_{W}^{\phi}\left(Y\left[\left(z_{2}+w_{0}\right)^{-L(0)}u,z_{0}\right]z_{2}^{-L(0)}v,z_{2}\right).$

By applying the substitution $z_{0}=\log\left(1+w_{0}/z_{2}\right)$ in the
previous equation, we obtain from the left hand side

$\displaystyle\left(f(z_{1}/z_{2})Y_{W}^{\phi}\left(\left(z_{2}+w_{0}\right){}^{-L(0)}u,z_{1}\right)Y_{W}^{\phi}\left(z_{2}^{-L(0)}v,z_{2}\right)|_{z_{1}=z_{2}e^{z_{0}}}\right)|_{z_{0}=\log(1+w_{0}/z_{2})}$
$\displaystyle=$ $\displaystyle
f\left(z_{1}/z_{2}\right)Y_{W}^{\phi}\left(z_{1}^{-L(0)}u,z_{1}\right)Y_{W}^{\phi}\left(z_{2}^{-L(0)}v,z_{2}\right)|_{z_{1}=z_{2}+w_{0}}$
$\displaystyle=$ $\displaystyle
f\left(z_{1}/z_{2}\right)Y_{W}\left(u,z_{1}\right)Y_{W}\left(v,z_{2}\right)|_{z_{1}=z_{2}+w_{0}},$

while by using the fact
$z_{2}^{L(0)}Y\left(u,z_{0}\right)z_{2}^{-L(0)}=Y\left(z_{2}^{L(0)}u,z_{0}z\right)$
(cf. [FHL]), we obtain from the right hand side

$\displaystyle\left(f\left(e^{z_{0}}\right)Y_{W}^{\phi}\left(Y\left[\left(z_{2}+w_{0}\right){}^{-L(0)}u,z_{0}\right]z_{2}^{-L(0)}v,z_{2}\right)\right)|_{z_{0}=\log(1+w_{0}/z_{2})}$
$\displaystyle=$
$\displaystyle\left(f\left(e^{z_{0}}\right)Y_{W}\left(z_{2}^{L(0)}Y\left(e^{z_{0}L(0)}\left(z_{2}+w_{0}\right){}^{-L(0)}u,e^{z_{0}}-1\right)z_{2}^{-L(0)}v,z_{2}\right)\right)|_{z_{0}=\log\left(1+w_{0}/z_{2}\right)}$
$\displaystyle=$
$\displaystyle\left(f\left(e^{z_{0}}\right)Y_{W}\left(Y\left(z_{2}^{L(0)}e^{z_{0}L(0)}\left(z_{2}+w_{0}\right){}^{-L(0)}u,\left(e^{z_{0}}-1\right)z_{2}\right)v,z_{2}\right)\right)|_{z_{0}=\log\left(1+w_{0}/z_{2}\right)}$
$\displaystyle=$ $\displaystyle
f\left(z_{2}+w_{0}/z_{2}\right)Y_{W}\left(Y\left(u,w_{0}\right)v,z_{2}\right).$

This proves the proposition. ∎

######  Proposition 3.4.

Assume that $V$ is a vertex operator algebra, $G$ is a group of automorphisms
of $V$ and $\chi$ a linear character of $G$. Then the following two categories
are isomorphic:

  1. (i)

the category of $\left(G,\chi\right)$-equivariant $\phi$-coordinated quasi
$V$-modules $\left(W,Y_{W}^{\phi}\right)$;

  2. (ii)

the category of $\left(G,\chi\right)$-equivariant quasi $V$-modules
$\left(W,Y_{W}\right)$.

Moreover, if $G=\langle\sigma\rangle$ is a cyclic group of finite order $N$,
and $\chi\left(\sigma\right)=e^{-\frac{2\pi\sqrt{-1}}{N}}$, then these two
categories are also isomorphic to the following category

  1. (iii)

the category of (weak) $\sigma$-twisted $V$-modules
$\left(W,Y_{W}^{t}\right)$.

And furthermore, if $v\in V$ is a primary vector, i.e., $L(n)v=0$ for all
$n\geq 1$, then we have the following identities

$\displaystyle
Y_{W}^{\phi}\left(z^{-L(0)}v,z\right)=Y_{W}\left(v,z\right)=Y_{W}^{t}\left(\left(Nz^{N-1}\right){}^{L(0)}v,z^{N}\right).$
(3.2)

###### Proof.

Let $\left(V,Y\left(\cdot,z\right),{\mathbf{1}},\omega\right)$ be a vertex
operator algebra of central charge $\ell$. Set
$\tilde{\omega}=\omega-\frac{\ell}{24}{\mathbf{1}}$. It was proved in [Z] that
$\left(V,Y\left[\cdot,z\right],{\mathbf{1}},\tilde{\omega}\right)$ carries a
vertex operator algebra structure. And, an explicit isomorphism $f$ from
$\left(V,Y\left(\cdot,z\right),{\mathbf{1}},\omega\right)$ to
$V\left(V,Y\left[\cdot,z\right],{\mathbf{1}},\tilde{\omega}\right)$ was
constructed therein. This together with Proposition 3.3 implies that the
$\left(G,\chi\right)$-equivariant $\phi$-coordinated quasi $V$-modules
$\left(W,Y_{W}^{\phi}\right)$ are exactly the
$\left(G,\chi\right)$-equivariant quasi $V$-modules $\left(W,Y_{W}\right)$
with

$\displaystyle
Y_{W}^{\phi}\left(v,z\right)=Y_{W}\left(z^{L(0)}f(v),z\right),\quad\text{for}\
v\in V.$

In particular, if $v$ is a primary vector, then from [Z] we have $f(v)=v$ and
hence $Y_{W}^{\phi}\left(z^{-L(0)}v,z\right)=Y_{W}\left(v,z\right)$.

Finally, if $G=\langle\sigma\rangle$ is a finite cyclic group of order $N$ and
$\chi(\sigma)=e^{-\frac{2\pi\sqrt{-1}}{N}}$, then it was proved in [Li5] that
the $\left(G,\chi\right)$-equivariant quasi $V$-modules $\left(W,Y_{W}\right)$
are exactly the (weak) $\sigma$-twisted $V$-modules $\left(W,Y_{W}^{t}\right)$
with

$\displaystyle
Y_{W}\left(v,z\right)=Y_{W}\left(\Phi\left(z\right)v,z^{N}\right),\quad\text{for}\
v\in V,$

where $\Phi\left(z\right)$ is an operator in
$\mathrm{Hom}\left(V,V\left(\left(z\right)\right)\right)$ such that
$\Phi\left(z\right)v=\left(Nz^{N-1}\right){}^{L(0)}v$ whenever $v$ is primary.
This completes the proof the proposition. ∎

###  3.2 Equivalence of module categories for universal affine vertex algebras

In this subsection, we study the equivalence of certain module categories for
universal affine vertex algebras. Let $\mathfrak{b}$ be a (possibly infinite
dimensional) Lie algebra equipped with a nondegenerate invariant symmetric
bilinear form $\left\langle\cdot,\cdot\right\rangle$. Then we have the affine
Lie algebra

$\displaystyle\widetilde{\mathcal{L}}\left(\mathfrak{b}\right)=\left(\mathbb{C}\left[t,t^{-1}\right]\otimes\mathfrak{b}\right)\oplus\mathbb{C}\text{k}\oplus\mathbb{C}\text{d},$

where k is a central element and

$\displaystyle\left[t^{m}\otimes a,t^{n}\otimes
b\right]=t^{m+n}\otimes\left[a,b\right]+\left\langle
a,b\right\rangle\delta_{m+n,0}\text{k},\quad\left[\text{d},t^{m}\otimes
a\right]=mt^{m}\otimes a,$ (3.3)

for $m,n\in\mathbb{Z}$ and $a,b\in\mathfrak{b}$. Equip
$\widetilde{\mathcal{L}}\left(\mathfrak{b}\right)$ with a $\mathbb{Z}$-grading
structure with respect to the adjoint action of $-\text{d}$, and form the
following $\mathbb{Z}$-graded Lie subalgebra of
$\widetilde{\mathcal{L}}\left(\mathfrak{b}\right)$:

$\widehat{\mathcal{L}}\left(\mathfrak{b}\right)=\left(\mathbb{C}\left[t,t^{-1}\right]\otimes\mathfrak{b}\right)\oplus\mathbb{C}\text{k}.$

Let $\ell$ be a fixed complex number. View $\mathbb{C}$ as a
$\left(\mathbb{C}\left[t\right]\otimes\mathfrak{b}\oplus\mathbb{C}\text{k}\right)$-module
with $\mathbb{C}\left[t\right]\otimes\mathfrak{b}$ acting trivially and k
acting as scalar $\ell$. Then we have the induced
$\widehat{\mathcal{L}}(\mathfrak{b})$-module

$\displaystyle
V_{\widehat{\mathcal{L}}\left(\mathfrak{b}\right)}\left(\ell,0\right)=\mathcal{U}\left(\widehat{\mathcal{L}}\left(\mathfrak{b}\right)\right)\otimes_{\mathcal{U}\left(\mathbb{C}[t]\otimes\mathfrak{b}\oplus\mathbb{C}\text{k}\right)}\mathbb{C},$
(3.4)

which is naturally $\mathbb{N}$-graded by defining $\deg\mathbb{C}=0$. Set
${\bf 1}=1\otimes 1\in
V_{\widehat{\mathcal{L}}\left(\mathfrak{b}\right)}\left(\ell,0\right)$ and
identify $\mathfrak{b}$ as the degree-one subspace of
$V_{\widehat{\mathcal{L}}(\mathfrak{b})}\left(\ell,0\right)$ through the
linear map

$a\mapsto\left(t^{-1}\otimes a\right){\bf 1}\in
V_{\widehat{\mathcal{L}}\left(\mathfrak{b}\right)}\left(\ell,0\right),$

for $a\in\mathfrak{b}$. It is known (cf. [FZ, Li1]) that there exists a vertex
algebra structure on
$V_{\widehat{\mathcal{L}}(\mathfrak{b})}\left(\ell,0\right)$, which is
uniquely determined by the condition that ${\bf 1}$ is the vacuum vector and

$Y\left(a,z\right)=a\left(z\right)=\sum_{m\in{\mathbb{Z}}}\left(t^{m}\otimes
a\right)z^{-m-1},$

for $a\in\mathfrak{b}$. The vertex algebra
$V_{\widehat{\mathcal{L}}(\mathfrak{b})}\left(\ell,0\right)$ is often called
_the universal affine vertex algebra_ associated to $\mathfrak{b}$. Denote by
$J_{\widehat{\mathcal{L}}(\mathfrak{b})}\left(\ell,0\right)$ the unique
maximal graded $\widehat{\mathcal{L}}\left(\mathfrak{b}\right)$-submodule of
$V_{\widehat{\mathcal{L}}\left(\mathfrak{b}\right)}\left(\ell,0\right)$. Then
$J_{\widehat{\mathcal{L}}\left(\mathfrak{b}\right)}\left(\ell,0\right)$ is an
ideal of the vertex algebra
$V_{\widehat{\mathcal{L}}\left(\mathfrak{b}\right)}\left(\ell,0\right)$.
Define

$\displaystyle
L_{\widehat{\mathcal{L}}\left(\mathfrak{b}\right)}\left(\ell,0\right)=V_{\widehat{\mathcal{L}}\left(\mathfrak{b}\right)}\left(\ell,0\right)/J_{\widehat{\mathcal{L}}\left(\mathfrak{b}\right)}\left(\ell,0\right),$

which is a simple ${\mathbb{Z}}$-graded vertex algebra.

Assume that $G$ is an automorphism group of $\mathfrak{b}$ preserving the
bilinear form. It is easy to see that $G$ can be uniquely lifted to an
automorphism group of
$V_{\widehat{\mathcal{L}}\left(\mathfrak{b}\right)}\left(\ell,0\right)$ that
preserving the $\mathbb{Z}$-grading. Moreover, as
$J_{\widehat{\mathcal{L}}\left(\mathfrak{b}\right)}\left(\ell,0\right)$ is
$G$-stable, $G$ also induces an automorphism group of
$L_{\widehat{\mathcal{L}}\left(\mathfrak{b}\right)}\left(\ell,0\right)$.
Assume further that for $a,b\in\mathfrak{b}$,

$\displaystyle\left[ga,b\right]=0,\quad\left\langle ga,b\right\rangle=0$ (3.5)

for all but finitely many $g\in G$. For any linear character
$\chi:G\rightarrow{\mathbb{C}}^{\ast}$, we define a quotient space

$\displaystyle\widehat{\mathcal{L}}\left(\mathfrak{b},G\right)=\widehat{\mathcal{L}}\left(\mathfrak{b}\right)/{\text{Span}}\left\\{\chi\left(g\right){}^{m}\left(t^{m}\otimes
ga\right)-t^{m}\otimes a\mid g\in G,m\in\mathbb{Z},a\in\mathfrak{b}\right\\}.$
(3.6)

It was proved in [Li4] that $\widehat{\mathcal{L}}\left(\mathfrak{b},G\right)$
carries a Lie algebra structure with Lie bracket given by

$\displaystyle\left[\overline{t^{m}\otimes a},\overline{t^{n}\otimes
b}\right]=\sum_{g\in
G}\chi\left(g\right){}^{m}\left(\overline{t^{m+n}\otimes\left[ga,b\right]}+\delta_{m+n,0}\left\langle
ga,b\right\rangle\bar{\text{k}}\right)$

for $a,b\in\mathfrak{b}$ and $m,n\in\mathbb{Z}$, where $\overline{t^{m}\otimes
a}$ and $\overline{\text{k}}$ stand for the images of $t^{m}\otimes a$ and k
in $\widehat{\mathcal{L}}\left(\mathfrak{b},G\right)$ respectively.

We say that an $\widehat{\mathcal{L}}\left(\mathfrak{b},G\right)$-module $W$
is _restricted_ if for any $a\in\mathfrak{b}$ and $w\in W$,
$\overline{t^{n}\otimes a}\cdot w=0$ for $n$ sufficiently large, and is _of
level $\ell$_ if $\overline{\text{k}}$ acts as the scalar $\ell$. From [CLTW,
Proposition 6.4] we have the following result.

######  Proposition 3.5.

Let $G,\chi$ be as above. The following categories are isomorphic to each
other:

  1. (i)

the category of $\left(G,\chi\right)$-equivariant $\phi$-coordinated quasi
$V_{\widehat{\mathcal{L}}\left(\mathfrak{b}\right)}\left(\ell,0\right)$-modules
$\left(W,Y_{W}^{\phi}\right)$;

  2. (ii)

the category of $\left(G,\chi\right)$-equivariant quasi
$V_{\widehat{\mathcal{L}}\left(\mathfrak{b}\right)}\left(\ell,0\right)$-modules
$\left(W,Y_{W}\right)$;

  3. (iii)

the category of restricted $\widehat{\mathcal{L}}(\mathfrak{b},G)$-modules $W$
of level $\ell$.

And, the isomorphisms are determined by

$\displaystyle
Y_{W}^{\phi}\left(a,z\right)=zY_{W}\left(a,z\right)=\sum_{n\in\mathbb{Z}}\left(\overline{t^{n}\otimes
a}\right)z^{-n},$

for $a\in\mathfrak{b}$.

We assume now that $\sigma$ is an automorphism of $\mathfrak{b}$ with finite
order $N$ and preserving the bilinear form, and set
$\omega_{N}=e^{\frac{2\pi\sqrt{-1}}{N}}$. For $a\in\mathfrak{b}$ and
$m\in\mathbb{Z}$, set
$a_{\left(m\right)}=\sum_{p=0}^{N-1}\omega_{N}^{-mp}\sigma^{p}(a)$ and
$\mathfrak{b}_{\left(m\right)}=\left\\{a_{\left(m\right)}\mid
a\in\mathfrak{b}\right\\}$ (the $\sigma$-eigenspace of $\mathfrak{b}$ with
eigenvalue $\omega_{N}^{m}$). Then we have the following Lie subalgebras of
$\widetilde{\mathcal{L}}(\mathfrak{b})$, called $\sigma$-twisted affine Lie
algebras (cf. [K1])

$\displaystyle\widetilde{\mathcal{L}}\left(\mathfrak{b},\sigma\right)=\left(\sum_{m\in\mathbb{Z}}\mathbb{C}t^{m}\otimes\mathfrak{b}_{(m)}\right)\oplus\mathbb{C}\text{k}\oplus\mathbb{C}\text{d},\quad\widehat{\mathcal{L}}\left(\mathfrak{b},\sigma\right)=\left(\sum_{m\in\mathbb{Z}}\mathbb{C}t^{m}\otimes\mathfrak{b}_{(m)}\right)\oplus\mathbb{C}\text{k}.$

Let $\chi_{\omega_{N}}$ be the linear character of the cyclic group
$\langle\sigma\rangle$ such that
$\chi_{\omega_{N}}\left(\sigma\right)=\omega_{N}^{-1}$. We take
$G=\left\langle\sigma\right\rangle$ and $\chi=\chi_{\omega_{N}}$ in (3.6).
Then it is easy to check that the linear map given by

$\displaystyle\overline{t^{m}\otimes a}\mapsto t^{m}\otimes
a_{(m)},\quad\overline{\text{k}}\mapsto N\text{k},\quad\text{for}\
a\in\mathfrak{b},m\in\mathbb{Z},$ (3.7)

is a Lie algebra isomorphism from
$\widehat{\mathcal{L}}\left(\mathfrak{b},\left\langle\sigma\right\rangle\right)$
to $\widehat{\mathcal{L}}\left(\mathfrak{b},\sigma\right)$.

######  Definition 3.6.

Let $W$ be a module of
$\widetilde{\mathcal{L}}\left(\mathfrak{b},\sigma\right)$ or
$\widehat{\mathcal{L}}\left(\mathfrak{b},\sigma\right)$. We say that $W$ is
_restricted_ if for any $a\in\mathfrak{b}$ and $w\in W$, $\left(t^{n}\otimes
a_{(n)}\right)\cdot w=0$ for $n$ sufficiently large, and is _of level $\ell$_
if k acts as scalar $\ell/N$.

From Proposition 3.5 and the isomorphism (3.7), one immediately obtains the
following result.

######  Proposition 3.7.

The following categories are isomorphic to each other:

  1. (i)

the category of
$\left(\langle\sigma\rangle,\chi_{\omega_{N}}\right)$-equivariant
$\phi$-coordinated quasi
$V_{\widehat{\mathcal{L}}\left(\mathfrak{b}\right)}\left(\ell,0\right)$-modules
$\left(W,Y_{W}^{\phi}\right)$;

  2. (ii)

the category of
$\left(\langle\sigma\rangle,\chi_{\omega_{N}}\right)$-equivariant quasi
$V_{\widehat{\mathcal{L}}\left(\mathfrak{b}\right)}\left(\ell,0\right)$-modules
$\left(W,Y_{W}\right)$;

  3. (iii)

the category of restricted
$\widehat{\mathcal{L}}\left(\mathfrak{b},\sigma\right)$-modules $W$ of level
$\ell$.

And, the isomorphisms are determined by

$\displaystyle
Y_{W}^{\phi}\left(a,z\right)=zY_{W}\left(a,z\right)=a\left[z\right]:=\sum_{n\in\mathbb{Z}}\left(t^{n}\otimes
a_{(m)}\right)z^{-n},$

for $a\in\mathfrak{b}$.

Furthermore, by Proposition 3.7, Lemma 2.2 and the fact that
$\left[\text{d},a\left[z\right]\right]=-z\frac{d}{dz}a\left[z\right]$ for
$a\in\mathfrak{b}$, we have

######  Proposition 3.8.

For any complex number $\ell$, the restricted
$\widetilde{\mathcal{L}}\left(\mathfrak{b},\sigma\right)$-modules $W$ of level
$\ell$ are exactly the
$\left(\langle\sigma\rangle,\chi_{\omega_{N}}\right)$-equivariant
$\phi$-coordinated quasi
$V_{\widehat{\mathcal{L}}\left(\mathfrak{b}\right)}\left(\ell,0\right)$-modules
$\left(W,Y_{W}^{\phi},d\right)$ with

$\displaystyle\text{d}=-d,\quad a\left[z\right]=Y_{W}^{\phi}\left(a,z\right),$

for $a\in\mathfrak{b}$.

######  Remark 3.9.

We also have the following variant of the twisted affine Lie algebra
$\widehat{\mathcal{L}}\left(\mathfrak{b},\sigma\right)$ (cf. [FLM]):

$\displaystyle\widehat{\mathcal{L}}\left[\mathfrak{b},\sigma\right]=\left(\sum_{m\in\frac{1}{N}\mathbb{Z}}\mathbb{C}t^{m}\otimes\mathfrak{b}_{(Nm)}\right)\oplus\mathbb{C}\text{k},$

where the Lie bracket is given by (3.3) with $m,n\in\frac{1}{N}\mathbb{Z}$.
Note that $\widehat{\mathcal{L}}\left(\mathfrak{b},\sigma\right)$ is
isomorphic to $\widehat{\mathcal{L}}\left[\mathfrak{b},\sigma\right]$ with the
mapping

$\displaystyle t^{m}\otimes a_{(m)}\mapsto t^{m/N}\otimes a_{(m)},\quad
N\text{k}\mapsto\text{k},$ (3.8)

for $a\in\mathfrak{b},m\in\mathbb{Z}.$ It is well-known that the
$\sigma$-twisted
$V_{\widehat{\mathcal{L}}\left(\mathfrak{b}\right)}\left(\ell,0\right)$-modules
$\left(W,Y_{W}^{t}\right)$ are exactly the restricted
$\widehat{\mathcal{L}}\left[\mathfrak{b},\sigma\right]$-modules $W$ of level
$\ell$ with
$Y_{W}^{t}\left(a,z\right)=\sum_{m\in\mathbb{Z}}\left(t^{\frac{m}{N}}\otimes\frac{a_{(m)}}{N}\right)z^{-\frac{m}{N}-1}$
for $a\in\mathfrak{b}$ (cf. [Li2]).

######  Remark 3.10.

One can also define the notion of a $\sigma$-twisted
$V_{\widehat{\mathcal{L}}\left(\mathfrak{b}\right)}\left(\ell,0\right)$-module
$\left(W,Y_{W}^{t},d\right)$ such that
$\left[d,Y_{W}^{t}\left(v,z\right)\right]=Y_{W}^{t}\left(\mathcal{D}v,z\right)$
for $v\in
V_{\widehat{\mathcal{L}}\left(\mathfrak{b}\right)}\left(\ell,0\right)$ (see
[Li2] for example). Then such $\sigma$-twisted
$V_{\widehat{\mathcal{L}}\left(\mathfrak{b}\right)}\left(\ell,0\right)$-modules
are precisely the restricted modules of level $\ell$ for the algebra
$\widehat{\mathcal{L}}\left[\mathfrak{b},\sigma\right]\rtimes\mathbb{C}\left(\frac{d}{dt}\otimes
1\right)$, as $[d,Y_{W}^{t}(v,z)]=\frac{d}{dz}Y_{W}^{t}(v,z)$ for $v\in
V_{\widehat{\mathcal{L}}\left(\mathfrak{b}\right)}\left(\ell,0\right)$ (cf.
[Li2]). And a similar result also holds for the equivariant quasi
$V_{\widehat{\mathcal{L}}\left(\mathfrak{b}\right)}\left(\ell,0\right)$-modules
(cf. [Li3]).

###  3.3 Associating affine Kac-Moody algebras with vertex algebras

In this subsection, we associate the nullity 1 EALAs (i.e. affine Kac-Moody
algebras with derivations) to vertex algebras via equivariant
$\phi$-coordinated quasi modules. As a by-product, we obtain a
characterization of integrable restricted modules for affine Kac-Moody
algebras, which will be used later to associate the nullity 2 EALAs with
vertex algebras.

Let $\dot{\mathfrak{g}}$ be a finite-dimensional simple Lie algebra over the
complex field, $\dot{\mathfrak{h}}$ a Cartan subalgebra of
$\dot{\mathfrak{g}}$, and $\dot{\Delta}$ the root system of
$\dot{\mathfrak{g}}$ with respect to $\dot{\mathfrak{h}}$. Let
$\left\langle\cdot,\cdot\right\rangle$ be the nondegenerate invariant
symmetric bilinear form on $\dot{\mathfrak{g}}$ which is normalized so that
the square length of long roots are equal to 2. For
$\dot{\alpha}\in\dot{\Delta}$, choose nonzero root vectors
$x_{\dot{\alpha}}\in\dot{\mathfrak{g}}_{\dot{\alpha}}$ such that
$\left\\{x_{\dot{\alpha}},\dot{\alpha}^{\vee},x_{-\dot{\alpha}}\right\\}$ form
an $\mathfrak{sl}_{2}$-triple, where
$\dot{\alpha}^{\vee}\in\dot{\mathfrak{h}}$ denotes the coroot of $\alpha$.

Fix a simple root system
$\dot{\Pi}=\left\\{\dot{\alpha}_{1},\dots,\dot{\alpha}_{l}\right\\}$ of
$\dot{\Delta}$, where $l=\dim\dot{\mathfrak{h}}$ is the rank of
$\dot{\mathfrak{g}}$. Let $\dot{\nu}$ be a diagram automorphism of
$\dot{\mathfrak{g}}$ of order $N$ ($N=1,2$ or $3$). By definition, there
exists a permutation $\dot{\nu}$ on the set
$\dot{I}=\left\\{1,2,\dots,l\right\\}$ such that
$\dot{\nu}\left(x_{\pm\dot{\alpha}_{i}}\right)=x_{\pm\dot{\alpha}_{\dot{\nu}(i)}}$
for $i\in\dot{I}$. Then the Lie algebra
$\widetilde{\mathcal{L}}\left(\dot{\mathfrak{g}},\dot{\nu}\right)$ as defined
in Section 3.2 is a Kac-Moody algebra of affine type, and any affine Kac-Moody
algebra has such a form [K1]. We say that an
$\widetilde{\mathcal{L}}\left(\dot{\mathfrak{g}},\dot{\nu}\right)$-module (or
an $\widehat{\mathcal{L}}\left(\dot{\mathfrak{g}},\dot{\nu}\right)$-module) is
_integrable_ if all real root vectors $t^{m}\otimes x_{\dot{\alpha}(m)}$ for
$\dot{\alpha}\in\dot{\Delta},m\in\mathbb{Z}$ act locally nilpotent on it [K1].
Now we give the main result of this section.

######  Theorem 3.11.

Let $\ell$ be any complex number. For any restricted
$\widetilde{\mathcal{L}}\left(\dot{\mathfrak{g}},\dot{\nu}\right)$-module $W$
of level $\ell$, there is a
$\left(\langle\dot{\nu}\rangle,\chi_{\omega_{N}}\right)$-equivariant
$\phi$-coordinated quasi
$V_{\widehat{\mathcal{L}}\left(\dot{\mathfrak{g}}\right)}\left(\ell,0\right)$-module
structure $(Y_{W}^{\phi},d)$ on $W$, which is uniquely determined by

$\displaystyle d=-\text{d},\quad
Y_{W}^{\phi}\left(a,z\right)=a\left[z\right]\left(=\sum_{n\in\mathbb{Z}}\left(t^{m}\otimes
a_{(m)}\right)z^{-m}\right),$

for $a\in\dot{\mathfrak{g}}$. On the other hand, for any
$\left(\langle\dot{\nu}\rangle,\chi_{\omega_{N}}\right)$-equivariant
$\phi$-coordinated quasi
$V_{\widehat{\mathcal{L}}\left(\dot{\mathfrak{g}}\right)}\left(\ell,0\right)$-module
$(W,Y_{W}^{\phi},d)$, $W$ is a restricted
$\widetilde{\mathcal{L}}\left(\dot{\mathfrak{g}},\dot{\nu}\right)$-module of
level $\ell$ with action given by

$\displaystyle\text{d}=-d,\quad a\left[z\right]=Y_{W}^{\phi}\left(a,z\right),$

for $a\in\dot{\mathfrak{g}}$. Furthermore, when $\ell$ is a nonnegative
integer, the integrable restricted
$\widetilde{\mathcal{L}}\left(\dot{\mathfrak{g}},\dot{\nu}\right)$-modules of
level $\ell$ exactly correspond to the
$\left(\langle\dot{\nu}\rangle,\chi_{\omega_{N}}\right)$-equivariant
$\phi$-coordinated quasi
$L_{\widehat{\mathcal{L}}\left(\dot{\mathfrak{g}}\right)}\left(\ell,0\right)$-modules.

###### Proof.

The one-to-one correspondence between the category of restricted
$\widetilde{\mathcal{L}}\left(\dot{\mathfrak{g}},\dot{\nu}\right)$-modules of
level $\ell$ and the category of
$\left(\langle\dot{\nu}\rangle,\chi_{\omega_{N}}\right)$-equivariant
$\phi$-coordinated quasi
$V_{\widehat{\mathcal{L}}\left(\dot{\mathfrak{g}}\right)}\left(\ell,0\right)$-modules
follows from Proposition 3.8.

For the last assertion of the theorem, we note that
$L_{\widehat{\mathcal{L}}\left(\dot{\mathfrak{g}}\right)}\left(\ell,0\right)$
is a rational vertex operator algebra, and each $a\in\dot{\mathfrak{g}}$ is a
primary vector of weight $1$ in
$L_{\widehat{\mathcal{L}}\left(\dot{\mathfrak{g}}\right)}\left(\ell,0\right)$.
Thus, by Proposition 3.4, it is easy to see that the
$\left(\langle\dot{\nu}\rangle,\chi_{\omega_{N}}\right)$-equivariant
$\phi$-coordinated quasi
$L_{\widehat{\mathcal{L}}\left(\dot{\mathfrak{g}}\right)}\left(\ell,0\right)$-modules
$\left(W,Y_{W}^{\phi}\right)$ are exactly the $\dot{\nu}$-twisted
$L_{\widehat{\mathcal{L}}(\dot{\mathfrak{g}})}\left(\ell,0\right)$-modules
$\left(W,Y_{W}^{t}\right)$ with

$\displaystyle Y_{W}^{\phi}\left(a,z\right)=Nz^{N}Y_{W}^{t}\left(a,z\right),$

for $a\in\dot{\mathfrak{g}}$. Recall the algebra
$\widehat{\mathcal{L}}\left[\dot{\mathfrak{g}},\dot{\nu}\right]$ defined in
Remark 3.9. It is known (cf. [Li2]) that the integrable restricted
$\widehat{\mathcal{L}}\left[\dot{\mathfrak{g}},\dot{\nu}\right]$-modules of
level $\ell$ are exactly the $\dot{\nu}$-twisted
$L_{\widehat{\mathcal{L}}\left(\dot{\mathfrak{g}}\right)}\left(\ell,0\right)$-modules
with

$\displaystyle
Y_{W}^{t}\left(a,z\right)=\sum_{m\in\mathbb{Z}}\left(t^{\frac{m}{N}}\otimes\frac{a_{(m)}}{N}\right)z^{-\frac{m}{N}-1},$

for $a\in\dot{\mathfrak{g}}$. Via the isomorphism (3.8), we obtain that the
integrable restricted
$\widehat{\mathcal{L}}\left(\dot{\mathfrak{g}},\dot{\nu}\right)$-modules of
level $\ell$ are exactly the
$\left(\langle\dot{\nu}\rangle,\chi_{\omega_{N}}\right)$-equivariant
$\phi$-coordinated quasi
$L_{\widehat{\mathcal{L}}\left(\dot{\mathfrak{g}}\right)}\left(\ell,0\right)$-modules
$\left(W,Y_{W}^{\phi}\right)$ with

$\displaystyle
Y_{W}^{\phi}\left(a,z\right)=Nz^{N}Y_{W}^{t}\left(a,z\right)=\sum_{m\in\mathbb{Z}}\left(t^{\frac{m}{N}}\otimes
a_{(m)}\right)z^{-m}=\sum_{m\in\mathbb{Z}}\left(t^{m}\otimes
a_{(m)}\right)z^{-m}=a\left[z\right],$

for $a\in\dot{\mathfrak{g}}$. Thus, the last assertion of the theorem follows
from this and Lemma 2.2. ∎

We write $\dot{\Delta}=\dot{\Delta}_{1}\cup\dot{\Delta}_{2}$, where
$\dot{\Delta}_{2}=\left\\{\dot{\alpha}\in\dot{\Delta}\mid\langle\dot{\alpha},\dot{\nu}\left(\dot{\alpha}\right)\rangle=-1\right\\}$,
and $\dot{\Delta}_{1}=\dot{\Delta}\setminus\dot{\Delta}_{2}$. For each
$\dot{\alpha}\in\dot{\Delta}$, we set
$\epsilon_{\dot{\alpha}}=\frac{2}{\langle\dot{\alpha},\dot{\alpha}\rangle}\
(=1,2\,\text{or}\,3)$, and
$p_{\dot{\alpha}}\left(z\right)=\frac{1-z^{s}}{1-z}$ if
$\dot{\alpha}\in\dot{\Delta}_{s}$ for $s=1,2.$

Note that if $\dot{\alpha}\in\dot{\Delta}_{2}$ (resp. $\dot{\Delta}_{1}$),
then the Dynkin diagram associated to the $\dot{\nu}$-orbit
$\left\\{\dot{\nu}^{p}\left(\dot{\alpha}\right)\mid p=0,\dots,N-1\right\\}$ of
$\dot{\alpha}$ is of type $A_{2}$ (resp. a direct sum of type $A_{1}$). Using
this, one can check that

$\displaystyle
p_{\dot{\alpha}}\left(z/w\right)\cdot\left[x_{\dot{\alpha}}\left[z\right],x_{\dot{\alpha}}\left[w\right]\right]=0,$
(3.9)

for $\dot{\alpha}\in\dot{\Delta}$. As a by-product of Theorem 3.11, we obtain
a characterization of the integrable restricted
$\widetilde{\mathcal{L}}\left(\dot{\mathfrak{g}},\dot{\nu}\right)$-modules. We
note that when $\dot{\nu}=\mathrm{Id}$, such a characterization is known (cf.
[Li1, Proposition 5.2.3]).

######  Proposition 3.12.

Let $W$ be a restricted
$\widetilde{\mathcal{L}}\left(\dot{\mathfrak{g}},\dot{\nu}\right)$-module of
level $\ell$. Then $W$ is integrable if and only if $\ell$ is a nonnegative
integer, and for each $\dot{\alpha}\in\dot{\Delta}$,

$\displaystyle\left(\prod_{1\leq
i<j\leq\epsilon_{\dot{\alpha}}\ell+1}p_{\dot{\alpha}}\left(z_{i}/z_{j}\right)\right)x_{\dot{\alpha}}\left[z_{1}\right]x_{\dot{\alpha}}\left[z_{2}\right]\cdots
x_{\dot{\alpha}}\left[z_{\epsilon_{\dot{\alpha}}\ell+1}\right]|_{z_{1}=z_{2}\cdots=z_{\epsilon_{\dot{\alpha}}\ell+1}}=0\quad\text{on}\
W.$ (3.10)

###### Proof.

Note that if $W$ is integrable, then $\ell$ is a nonnegative integer and the
ideal $J_{\widehat{\mathcal{L}}(\mathfrak{b})}\left(\ell,0\right)$ is
generated by the elements
$\left(\left(x_{\dot{\alpha}}\right){}_{-1}\right){}^{\epsilon_{\dot{\alpha}}\ell+1}{\mathbf{1}}$
for $\dot{\alpha}\in\dot{\Delta}$ [K1]. Let $W$ be a restricted
$\widetilde{\mathcal{L}}\left(\dot{\mathfrak{g}},\dot{\nu}\right)$-module of
level $\ell$, and hence by Theorem 3.11 there is a
$\left(\langle\dot{\nu}\rangle,\chi_{\omega_{N}}\right)$-equivariant
$\phi$-coordinated quasi
$V_{\widehat{\mathcal{L}}(\dot{\mathfrak{g}})}\left(\ell,0\right)$-module
$(W,Y_{W}^{\phi},d)$. Furthermore, if $W$ is also integrable, then
$(W,Y_{W}^{\phi},d)$ becomes a
$\left(\langle\dot{\nu}\rangle,\chi_{\omega_{N}}\right)$-equivariant
$\phi$-coordinated quasi
$L_{\widehat{\mathcal{L}}\left(\dot{\mathfrak{g}}\right)}\left(\ell,0\right)$-module.
This then implies that
$\left(\left(x_{\dot{\alpha}}\right){}_{-1}\right){}^{\epsilon_{\dot{\alpha}}\ell+1}{\mathbf{1}}=0$
on $W$. Note that
$\left(x_{\dot{\alpha}}\right){}_{n}\left(x_{\dot{\alpha}}\right)=0$ for
$n\geq 0$ in
$V_{\widehat{\mathcal{L}}(\dot{\mathfrak{g}})}\left(\ell,0\right)$. Thus by
Proposition 2.5 and (3.9), we see that (3.10) holds.

Conversely, if (3.10) holds, again by Proposition 2.5 and (3.9), we see that
$\left(\left(x_{\dot{\alpha}}\right){}_{-1}\right){}^{\epsilon_{\dot{\alpha}}\ell+1}{\mathbf{1}}=0$
on $W$, where $W$ is now viewed as a faithful
$\left(\langle\dot{\nu}\rangle,\chi_{\omega_{N}}\right)$-equivariant
$\phi$-coordinated quasi module for
$V_{\widehat{\mathcal{L}}(\dot{\mathfrak{g}})}\left(\ell,0\right)/\ker
Y_{W}^{\phi}$. This implies that
$J_{\widehat{\mathcal{L}}\left(\mathfrak{b}\right)}\left(\ell,0\right)\subset\ker
Y_{W}^{\phi}$, and hence $(W,Y_{W}^{\phi},d)$ becomes a
$\left(\langle\dot{\nu}\rangle,\chi_{\omega_{N}}\right)$-equivariant
$\phi$-coordinated quasi
$L_{\widehat{\mathcal{L}}(\dot{\mathfrak{g}})}\left(\ell,0\right)$-module.
Finally, the integrability of $W$ follows from Theorem 3.11. ∎

##  4 Twisted toroidal extended affine Lie algebras

In this section, we recall the construction of the nullity $2$ twisted
toroidal extended affine Lie algebras arising from diagram automorphisms of
affine Kac-Moody algebras.

###  4.1 Toroidal EALA $\widetilde{\mathfrak{g}}$

We first recall the definition of nullity $2$ toroidal extended affine Lie
algebras in this subsection [BGK, B2]. Let
$\mathcal{R}=\mathbb{C}\left[t_{0},t_{0}^{-1},t_{1},t_{1}^{-1}\right]$ be
Laurent polynomial ring in commuting variables $t_{0}$ and $t_{1}.$ Denote by
$\Omega_{\mathcal{R}}^{1}=\mathcal{R}dt_{0}\oplus\mathcal{R}dt_{1}$ the space
of K$\ddot{a}$hler differentials on $\mathcal{R}.$ Define the $1$-forms

$\text{k}_{0}=t_{0}^{-1}dt_{0,}\qquad\text{k}_{1}=t_{1}^{-1}dt_{1}.$

Then $\left\\{\text{k}_{0},\text{k}_{1}\right\\}$ form a $\mathcal{R}$-basis
of $\Omega_{\mathcal{R}}^{1}.$ Let

$d\left(\mathcal{R}\right)=\left\\{df=\frac{\partial f}{\partial
t_{0}}dt_{0}+\frac{\partial f}{\partial t_{1}}dt_{1}\mid
f\in\mathcal{R}\right\\}$

be the space of exact $1$-forms in $\Omega_{\mathcal{R}}^{1}$, and set

$\mathcal{K}=\Omega_{\mathcal{R}}^{1}/d\left(\mathcal{R}\right).$

and

$\text{k}_{m_{0},m_{1}}=\begin{cases}\frac{1}{m_{1}}t_{0}^{m_{0}}t_{1}^{m_{1}}\text{k}_{0}\
&\text{if}\ m_{1}\neq 0,\\\ -\frac{1}{m_{0}}t_{0}^{m_{0}}\text{k}_{1}\
&\text{if}\ m_{0}\neq 0,m_{1}=0,\\\ 0\ &\text{if}\ m_{0}\neq
0,m_{1}=0.\end{cases}$ (4.1)

for $m_{0},m_{1}\in\mathbb{Z}$. Then the set

$\displaystyle\mathbb{B}_{\mathcal{K}}=\left\\{\text{k}_{0},\text{k}_{1}\right\\}\cup\left\\{\text{k}_{m_{0},m_{1}}\mid\left(m_{0},m_{1}\right)\in\mathbb{Z}^{2}\setminus\left\\{\left(0,0\right)\right\\}\right\\}$
(4.2)

forms a basis of $\mathcal{K}$. The set $\mathbb{B}_{\mathcal{K}}$ can also be
expressed as follows

$\mathbb{B}_{\mathcal{K}}=\left\\{\text{k}_{0}\right\\}\cup\left\\{t_{0}^{m_{0}}\text{k}_{1},\text{k}_{m_{0},m_{1}}\mid
m_{0}\in\mathbb{Z},m_{1}\in\mathbb{Z}^{\ast}\right\\},$ (4.3)

which will be used later on. Recall that $\dot{\mathfrak{g}}$ is a finite-
dimensional simple Lie algebra and $\langle\cdot,\cdot\rangle$ is the
normalized bilinear form on $\dot{\mathfrak{g}}$. Let

$\mathfrak{t}\left(\dot{\mathfrak{g}}\right)=\left(\mathcal{R}\otimes\dot{\mathfrak{g}}\right)\oplus\mathcal{K}$

be a central extension of the double-loop algebra
$\mathcal{R}\otimes\dot{\mathfrak{g}}$ by $\mathcal{K}$ with Lie product

$\left[t_{0}^{m_{0}}t_{1}^{m_{1}}\otimes x,t_{0}^{n_{0}}t_{1}^{n_{1}}\otimes
y\right]=t_{0}^{m_{0}+n_{0}}t_{1}^{m_{1}+n_{1}}\otimes\left[x,y\right]+\left\langle
x,y\right\rangle\sum_{i=0,1}m_{i}t_{0}^{m_{0}+n_{0}}t_{1}^{m_{1}+n_{1}}\text{k}_{i},$
(4.4)

for $x,y\in\dot{\mathfrak{g}}$ and $m_{0},m_{1},n_{0},n_{1}\in\mathbb{Z}.$ It
is proved in [MRY] that the Lie algebra
$\mathfrak{t}\left(\dot{\mathfrak{g}}\right)$ is the universal central
extension of $\mathcal{R}\otimes\dot{\mathfrak{g}}$, and it is often called
_the nullity 2 toroidal Lie algebra_.

Let

$\text{Der}\left(\mathcal{R}\right)=\mathcal{R}\frac{\partial}{\partial
t_{0}}\oplus\mathcal{R}\frac{\partial}{\partial t_{1}}$

be the space of derivations of $\mathcal{R}.$ Set

$\text{d}_{0}=t_{0}\frac{\partial}{\partial
t_{0}},\quad\text{d}_{1}=t_{1}\frac{\partial}{\partial t_{1}}.$

Then $\left\\{\text{d}_{0},\text{d}_{1}\right\\}$ form a $\mathcal{R}$-basis
of $\text{Der}\left(\mathcal{R}\right)$, and the derivation Lie algebra
$\text{Der}\left(\mathcal{R}\right)$ also acts on
$\mathcal{R}\otimes\dot{\mathfrak{g}}$ with

$\psi\left(f\otimes x\right)=\psi\left(f\right)\otimes x,$

for
$\psi\in\text{Der}\left(\mathcal{R}\right),f\in\mathcal{R},x\in\dot{\mathfrak{g}}$.
One notes that the $\text{Der}\left(\mathcal{R}\right)$-action on
$\mathcal{R}\otimes\dot{\mathfrak{g}}$ can be uniquely extended to an action
on the center $\mathcal{K}$ of the toroidal Lie algebra
$\mathfrak{t}\left(\dot{\mathfrak{g}}\right)$ with

$\psi\left(f\text{d}g\right)=\psi\left(f\right)\text{d}g+f\text{d}\psi\left(g\right),$

for $\psi\in\text{Der}\left(\mathcal{R}\right),g\in\mathcal{R}$. Now, we form
the semi-direct product Lie algebra

$\mathcal{T}\left(\dot{\mathfrak{g}}\right)=\mathfrak{t}\left(\dot{\mathfrak{g}}\right)\rtimes\text{Der}\left(\mathcal{R}\right)=\mathcal{R\otimes\dot{\mathfrak{g}}}\oplus\mathcal{K}\oplus\text{Der}\left(\mathcal{R}\right),$

which is often called _the full toroidal Lie algebra_ [B1]. Note that in
$\mathcal{T}\left(\dot{\mathfrak{g}}\right)$ we have

$\displaystyle\left[t_{0}^{m_{0}}t_{1}^{m_{1}}\text{d}_{i},t_{0}^{n_{0}}t_{1}^{n_{1}}\otimes
x\right]$
$\displaystyle=n_{i}\left(t_{0}^{m_{0}+n_{0}}t_{1}^{m_{1}+n_{1}}\otimes
x\right),$ (4.5)
$\displaystyle\left[t_{0}^{m_{0}}t_{1}^{m_{1}}\text{d}_{i},t_{0}^{n_{0}}t_{1}^{n_{1}}\text{k}_{j}\right]$
$\displaystyle=n_{i}t_{0}^{m_{0}+n_{0}}t_{1}^{m_{1}+n_{1}}\text{k}_{j}+\delta_{i,j}\sum_{r=0,1}m_{r}t_{0}^{m_{0}+n_{0}}t_{1}^{m_{1}+n_{1}}\text{k}_{r},$
(4.6)
$\displaystyle\left[t_{0}^{m_{0}}t_{1}^{m_{1}}\text{d}_{i},t_{0}^{n_{0}}t_{1}^{n_{1}}\text{d}_{j}\right]$
$\displaystyle=n_{i}t_{0}^{m_{0}+n_{0}}t_{1}^{m_{1}+n_{1}}\text{d}_{j}-m_{j}t_{0}^{m_{0}+n_{0}}t_{1}^{m_{1}+n_{1}}\text{d}_{i},$
(4.7)

for $x\in\mathfrak{g},$ $m_{0},n_{0},m_{1},n_{1}\in\mathbb{Z}$, and
$i,j\in\left\\{0,1\right\\}.$

We define a Lie subalgebra $\mathcal{S}$ of
$\text{Der}\left(\mathcal{R}\right)$ as follows

$\mathcal{S}=\left\\{f_{0}\text{d}_{0}+f_{1}\text{d}_{1}\in\text{Der}\left(\mathcal{R}\right)\mid
f_{0},f_{1}\in\mathcal{R},\text{d}_{0}\left(f_{0}\right)+\text{d}_{1}\left(f_{1}\right)=0\right\\}.$

The elements in $\mathcal{S}$ is often called _skew derivations_ over
$\mathcal{R}$ (cf. [BGK, N]), and also known as _divergence-zero derivations_
(cf. [B2]). It is easy to see that, for $m_{0},m_{1}\in\mathbb{Z}$

$\widetilde{\text{d}}_{m_{0},m_{1}}=m_{0}t_{0}^{m_{0}}t_{1}^{m_{1}}\text{d}_{1}-m_{1}t_{0}^{m_{0}}t_{1}^{m_{1}}\text{d}_{0},$

are elements of $\mathcal{S},$ and the set

$\mathbb{B}_{\mathcal{S}}=\left\\{\text{d}_{0},\text{d}_{1}\right\\}\cup\left\\{\widetilde{\text{d}}_{m_{0},m_{1}}\mid\left(m_{0},m_{1}\right)\in\mathbb{Z}\times\mathbb{Z}\setminus\left\\{\left(0,0\right)\right\\}\right\\}$

forms a $\mathbb{C}$-basis of $\mathcal{S}$. Note that
$\widetilde{\text{d}}_{m,0}=mt_{0}^{m}\text{d}_{1}$ for $m\in\mathbb{Z}$, we
can rewrite $\mathbb{B}_{\mathcal{S}}$ as follows

$\displaystyle\mathbb{B}_{\mathcal{S}}=\left\\{\text{d}_{0}\right\\}\cup\left\\{t_{0}^{m_{0}}\text{d}_{1},\widetilde{\text{d}}_{m_{0},m_{1}}\mid
m_{0}\in\mathbb{Z},m_{1}\in\mathbb{Z}^{\ast}\right\\},$ (4.8)

and the Lie product relations are given as follows

$\displaystyle\left[\text{d}_{i},\widetilde{\text{d}}_{m_{0},m_{1}}\right]=m_{i}\widetilde{\text{d}}_{m_{0},m_{1}},\quad\left[\widetilde{\text{d}}_{m_{0},m_{1}},\widetilde{\text{d}}_{n_{0},n_{1}}\right]=\left(m_{0}n_{1}-m_{1}n_{0}\right)\widetilde{\text{d}}_{m_{0}+n_{0},m_{1}+n_{1}}$
(4.9)

for $i\in\left\\{0,1\right\\}$, and $m_{0},m_{1},n_{0},n_{1}\in\mathbb{Z}.$

Form the following subalgebra of $\mathcal{T}\left(\dot{\mathfrak{g}}\right)$:

$\displaystyle\widetilde{\mathfrak{g}}=\mathfrak{t}\left(\dot{\mathfrak{g}}\right)\rtimes\mathcal{S}=\mathcal{R}\otimes\dot{\mathfrak{g}}\oplus\mathcal{K}\oplus\mathcal{S},$
(4.10)

which is often called _the nullity 2 toroidal extended affine Lie algebra_
[BGK, B2].

From (4.5) and (4.6), one has the following Lie product relations in
$\widetilde{\mathfrak{g}}$

$\displaystyle\left[\widetilde{\text{d}}_{m_{0},m_{1}},t_{0}^{n_{0}}t_{1}^{n_{1}}\otimes
x\right]$
$\displaystyle=\left(m_{0}n_{1}-m_{1}n_{0}\right)t_{0}^{m_{0}+n_{0}}t_{1}^{m_{1}+n_{1}}\otimes
x,$ (4.11)
$\displaystyle\left[\text{$\widetilde{\text{d}}$}_{m_{0},m_{1}},\text{k}_{n_{0},n_{1}}\right]$
$\displaystyle=\left(m_{0}n_{1}-m_{1}n_{0}\right)\text{k}_{m_{0}+n_{0},m_{1}+n_{1}}+\delta_{m_{0}+n_{0},0}\delta_{m_{1}+n_{1},0}\left(m_{0}\text{k}_{0}+m_{1}\text{k}_{0}\right),$
(4.12)

for $m_{0},n_{0},m_{1},n_{1}\in\mathbb{Z}$, and $x\in\dot{\mathfrak{g}}$. And
the Lie subalgebra

$\mathfrak{g}:=\left(\mathbb{C}\left[t_{1},t_{1}^{-1}\right]\otimes\dot{\mathfrak{g}}\right)\oplus\mathbb{C}\text{k}_{1}\oplus\mathbb{C}\text{d}_{1}$

of $\widetilde{\mathfrak{g}}$ is isomorphic to the affine Kac-Moody algebra
$\widetilde{\mathcal{L}}\left(\dot{\mathfrak{g}}\right)$. We extend the
normalized bilinear form $\left\langle\cdot,\cdot\right\rangle$ of
$\dot{\mathfrak{g}}$ to a nondegenerate invariant symmetric bilinear form on
$\mathfrak{g}$ by defining

$\left\langle t_{1}^{m}\otimes x+a\text{k}_{1}+b\text{d}_{1},t_{1}^{n}\otimes
y+a^{\prime}\text{k}_{1}+b^{\prime}\text{d}_{1}\right\rangle=\delta_{m+n,0}\left\langle
x,y\right\rangle+ab^{\prime}+ba^{\prime},$

where $m,n\in\mathbb{Z},$ $x,y\in\dot{\mathfrak{g}}$, and
$a,b,a^{\prime},b^{\prime}\in\mathbb{C}.$ It is easy to see from (4.3) and
(4.8) that the Lie algebra $\widetilde{\mathfrak{g}}$ is linearly spanned by
the set

$\displaystyle\left\\{t_{0}^{m}u,\ \text{k}_{m,n},\
\widetilde{\text{d}}_{m,n},\ \text{d}_{0},\ \text{k}_{0}\mid
u\in\mathfrak{g},\ m\in\mathbb{Z},\ n\in\mathbb{Z}^{\ast}\right\\},$ (4.13)

with a nondegenerate and invariant symmetric bilinear form
$\left\langle\cdot,\cdot\right\rangle$ defined by

$\displaystyle\left\langle t_{0}^{m}x,t_{0}^{-m}y\right\rangle=\left\langle
x,y\right\rangle,\quad\left\langle\text{k}_{0},\text{d}_{0}\right\rangle=\left\langle\widetilde{\text{d}}_{m,n},\text{k}_{-m,-n}\right\rangle=1,$
(4.14)

for $x,y\in\mathfrak{g}$, $m\in\mathbb{Z}$, $n\in\mathbb{Z}^{\ast}$, and a
self-centralizing ad-diagonalizable subalgebra

$\widetilde{\mathfrak{h}}=\dot{\mathfrak{h}}\oplus\mathbb{C}\text{k}_{0}\oplus\mathbb{C}\text{k}_{1}\oplus\mathbb{C}\text{d}_{0}\oplus\mathbb{C}\text{d}_{1}.$

###  4.2 Diagram automorphisms of affine Kac-Moody algebras

For any diagram automorphism $\mu$ of the affine Kac-Moody algebra
$\mathfrak{g}$, we define an automorphism $\hat{\mu}$ for the full toroidal
Lie algebra $\mathcal{T}\left(\dot{\mathfrak{g}}\right)$, which will be used
to construct twisted toroidal extended affine Lie algebras in the next
subsection.

Denote by

$\mathfrak{h}=\dot{\mathfrak{h}}\oplus\mathbb{C}\text{k}_{1}\oplus\mathbb{C}\text{d}_{1}$

the Cartan subalgebra of the affine Kac-Moody algebra $\mathfrak{g},$ and
$\mathfrak{h}^{\ast}$ the dual space of $\mathfrak{h}$. We identify
$\dot{\mathfrak{h}}^{\ast}$ as a subspace of $\mathfrak{h}^{\ast}$ such that
$\dot{\alpha}\left(\text{k}_{1}\right)=\dot{\alpha}\left(\text{d}_{1}\right)=0$
for $\dot{\alpha}\in\dot{\mathfrak{h}}^{\ast},$ and define the null root
$\delta_{1}\in\mathfrak{h}^{\ast}$ by setting

$\delta_{1}\left(\dot{\mathfrak{h}}\right)=\delta\left(\text{k}_{1}\right)=0,\quad\delta_{1}\left(\text{d}_{1}\right)=1.$

Then we have the root space decomposition
$\mathfrak{g}=\mathfrak{h}\oplus\sum_{\alpha\in\Delta}\mathfrak{g}_{\alpha},$
where

$\Delta=\left\\{\dot{\alpha}+m\delta_{1},n\delta_{1}\mid\dot{\alpha}\in{\dot{\Delta}},m\in\mathbb{Z},n\in\mathbb{Z}^{\ast}\right\\}.$

Denote by
$\Delta^{\times}=\left\\{\dot{\alpha}+m\delta_{1}\mid\dot{\alpha}\in\dot{\Delta},m\in\mathbb{Z}\right\\}$
the set of real roots in $\Delta.$ Recall that for each
$\dot{\alpha}\in\dot{\Delta}$ and
$\alpha=\dot{\alpha}+m\delta_{1}\in\Delta^{\times}$, there are
$\mathfrak{sl}_{2}$-triples
$\left\\{x_{\dot{\alpha}},\dot{\alpha}^{\vee},x_{-\dot{\alpha}}\right\\}$ in
$\dot{\mathfrak{g}}$, and
$\left\\{x_{\alpha},\alpha^{\vee},x_{-\alpha}\right\\}$ in $\mathfrak{g}$,
where

$\alpha^{\vee}=\dot{\alpha}^{\vee}+\frac{2m}{\left\langle\dot{\alpha},\dot{\alpha}\right\rangle}\text{k}_{1}\,\quad
x_{\alpha}=t_{1}^{m}\otimes x_{\dot{\alpha}}\in\mathfrak{g}_{\alpha}.$

Let $\dot{\theta}$ be the highest root in $\dot{{\Delta}}$ with respect to the
simple root system
$\dot{\Pi}$=$\left\\{\dot{\alpha}_{1},\cdots,\dot{\alpha}_{l}\right\\}$ of
$\dot{\Delta}$. Then $\Pi=\left\\{\alpha_{0},\cdots,\alpha_{l}\right\\}$ forms
a simple root system of $\Delta$ with $\alpha_{0}=\delta_{1}-\dot{\theta}$ and
$\alpha_{i}=\dot{\alpha}_{i}$ for $1\leq i\leq l$. Denote by

$A=\left(a_{ij}\right)_{i,j\in
I}=\left(2\frac{\left\langle\alpha_{i},\alpha_{j}\right\rangle}{\left\langle\alpha_{i},\alpha_{i}\right\rangle}\right)_{i,j\in
I}$

the generalized Cartan matrix of the affine Kac-Moody algebra $\mathfrak{g},$
where $I=\left\\{0,1,\cdots,l\right\\}.$ Let $\mu$ be an automorphism of the
generalized Cartan matrix $A$, which by definition is a permutation of $I$
such that $a_{ij}=a_{\mu\left(i\right)\mu\left(j\right)}$ for $i,j\in I$. It
is known (cf. [KW], [FSS]) that there is a unique automorphism of
$\mathfrak{g}$, still denoted by $\mu$, such that

$\mu\left(x_{\pm\alpha_{i}}\right)=x_{\pm\alpha_{\mu\left(i\right)}},\ \
\left\langle\mu\left(x\right),\mu\left(y\right)\right\rangle=\left\langle
x,y\right\rangle$ (4.15)

for $i\in I$ and $x,y\in\mathfrak{g}$. Moreover, $\mu$ stabilizes
$\mathfrak{h}$ and has the same order as it is viewed as a permutation of $I$.

######  Remark 4.1.

The automorphism $\mu$ of the affine Kac-Moody algebra $\mathfrak{g}$
determined by (4.15) is called a _diagram automorphism_ associated to an
automorphism $\mu$ of the generalized Cartan matrix $A$.

From now on, we fix a diagram automorphism $\mu$ of $\mathfrak{g}$ with order
$T$. Recall that $\mu\left(\mathfrak{h}\right)=\mathfrak{h}$. Then there is an
action of $\mu$ on $\mathfrak{h}^{\ast}$ defined by

$\mu\left(\alpha\right)\left(h\right)=\alpha\left(\mu^{-1}\left(h\right)\right)$

for $\alpha\in\mathfrak{h}^{\ast},$ $h\in\mathfrak{h}$. Let
$Q=\mathbb{Z}\Pi=\mathbb{Z}\alpha_{0}\oplus\cdots\oplus\mathbb{Z}\alpha_{l}$
be the root lattice of $\mathfrak{g}$, and $\dot{Q}=\mathbb{Z}\dot{\Pi}$ be
the root lattice of $\dot{\mathfrak{g}}$. Note that
$\mu\left(\alpha_{i}\right)=\alpha_{\mu\left(i\right)}$ for $i\in I$ and hence
$\mu(Q)=Q$. Furthermore, it is known that
$\mu\left(\delta_{1}\right)=\delta_{1}$ (cf. [FSS]). Since
$Q=\dot{Q}\oplus\mathbb{Z}\delta_{1}$, for each $\dot{\alpha}\in\dot{Q},$ we
can write

$\mu\left(\dot{\alpha}\right)=\dot{\mu}\left(\dot{\alpha}\right)+\rho_{\mu}\left(\dot{\alpha}\right)\delta_{1},$

where $\dot{\mu}\left(\dot{\alpha}\right)\in\dot{Q}$ and
$\rho_{\mu}\left(\dot{\alpha}\right)\in\mathbb{Z}$. One can easily check that

$\displaystyle\dot{\mu}:\dot{Q}\to\dot{Q},\quad\dot{\alpha}\mapsto\dot{\mu}\left(\dot{\alpha}\right)$

defines an automorphism of $\dot{Q}$, and the map

$\displaystyle\rho_{\mu}:\dot{Q}\to\mathbb{Z},\quad\dot{\alpha}\mapsto\rho_{\mu}\left(\dot{\alpha}\right)$

is a homomorphism of abelian groups.

Now identity $\dot{\mathfrak{h}}$ with $\dot{\mathfrak{h}}^{\ast}$ via
$\left\langle\cdot,\cdot\right\rangle$ so that the coroot
$\dot{\alpha}^{\vee}=\frac{2\dot{\alpha}}{\left\langle\dot{\alpha},\dot{\alpha}\right\rangle}$
for $\dot{\alpha}\in\dot{\Delta}.$ Then by $\mathbb{C}$-linearity we obtain
two the linear maps $\dot{\mu}:\dot{\mathfrak{h}}\to\dot{\mathfrak{h}}$ and
$\rho_{\mu}:\dot{\mathfrak{h}}\to\mathbb{C}$ such that

$\displaystyle\mu(h)=\dot{\mu}(h)+\rho_{\mu}(h)\text{k}_{1}$ (4.16)

for $h\in\dot{\mathfrak{h}}$. Furthermore, we can extend $\dot{\mu}$ to an
automorphism of the Lie algebra $\dot{\mathfrak{g}}$ by the following rule
(cf. [CJKT]):

$\displaystyle\mu\left(x_{\dot{\alpha}}\right)=t_{1}^{\rho_{\mu}(\dot{\alpha})}\otimes\dot{\mu}\left(x_{\dot{\alpha}}\right)\in\mathfrak{g}_{\dot{\mu}(\dot{\alpha})+\rho_{\mu}(\dot{\alpha})\delta_{1}}{\color[rgb]{1,0,0}}$
(4.17)

for $\dot{\alpha}\in\dot{\Delta}$ (recall that
$\mu\left(\dot{\alpha}\right)=\dot{\mu}\left(\dot{\alpha}\right)+\rho_{\mu}\left(\dot{\alpha}\right)\delta_{1}$
). Then we have the following results:

######  Lemma 4.2.

For $\dot{\alpha}\in\dot{\Delta}$ and $n\in\mathbb{Z}$, we have

$\displaystyle\mu\left(t_{1}^{n}\otimes x_{\dot{\alpha}}\right)$
$\displaystyle=t_{1}^{\rho_{\mu}\left(\dot{\alpha}\right)+n}\otimes\dot{\mu}\left(x_{\dot{\alpha}}\right),\quad\mu\left(\text{k}_{1}\right)=\text{k}_{1},$
(4.18) $\displaystyle\mu\left(t_{1}^{n}\otimes\dot{\alpha}^{\vee}\right)$
$\displaystyle=t_{1}^{n}\otimes\dot{\mu}\left(\dot{\alpha}^{\vee}\right)+\delta_{n,0}\rho_{\mu}\left(\dot{\alpha}^{\vee}\right)\text{k}_{1},,$
(4.19) $\displaystyle\mu\left(\text{d}_{1}\right)$
$\displaystyle=\text{d}_{1}+\boldsymbol{h}-\frac{\left\langle\boldsymbol{h},\boldsymbol{h}\right\rangle}{2}\text{k}_{1},$
(4.20)

where $\boldsymbol{h}\in\dot{\mathfrak{h}}$ is determined by

$\dot{\mu}\left(\dot{\gamma}\right)\left(\boldsymbol{h}\right)=-\rho_{\mu}\left(\dot{\gamma}\right)\quad\text{for}\
\dot{\gamma}\in\dot{Q}.$ (4.21)

Furthermore, for any $h\in\dot{\mathfrak{h}}$ we have

$\displaystyle\dot{\mu}^{T}\left(h\right)=h,\quad\sum_{p=0}^{T-1}\rho_{\mu}\left(\dot{\mu}^{p}\left(h\right)\right)=0,\quad\sum_{p=0}^{T-1}\dot{\mu}^{p}\left(\boldsymbol{h}\right)=0=\sum_{p=1}^{T-1}\left(T-p\right)\left\langle\dot{\mu}^{p}(\boldsymbol{h}),\boldsymbol{h}\right\rangle+\frac{T\left\langle\boldsymbol{h},\boldsymbol{h}\right\rangle}{2}.$
(4.22)

###### Proof.

The equalities (4.18) and (4.19) were proved in [CJKT, Proposition 2.2].
Assume that
$\mu\left(\text{d}_{1}\right)=\boldsymbol{h}+a\text{k}_{1}+b\text{d}_{1}$ with
$\boldsymbol{h}\in\dot{\mathfrak{h}}$ and $a,b\in\mathbb{C}.$ Since

$1=\left\langle\text{d}_{1},\text{k}_{1}\right\rangle=\left\langle\mu\left(\text{d}_{1}\right),\
\
\mu\left(\text{k}_{1}\right)\right\rangle=\left\langle\boldsymbol{h}+a\text{k}_{1}+b\text{d}_{1},\text{k}_{1}\right\rangle=b,$

we obtain $b=1$. Then for any $\dot{\alpha}\in\dot{Q}$,

$0=\left\langle\mu\left(\text{d}_{1}\right),\ \
\mu\left(\dot{\alpha}\right)\right\rangle=\left\langle\boldsymbol{h}+a\text{k}_{1}+\text{d}_{1},\
\
\dot{\mu}\left(\dot{\alpha}\right)+\rho_{\mu}\left(\dot{\alpha}\right)\text{k}_{1}\right\rangle=\left\langle\boldsymbol{h},\dot{\mu}\left(\dot{\alpha}\right)\right\rangle+\rho_{\mu}\left(\dot{\alpha}\right),$

and hence $\boldsymbol{h}$ is determined by the condition (4.21). And by using
the fact that
$\left\langle\mu\left(\text{d}_{1}\right),\mu\left(\text{d}_{1}\right)\right\rangle=0$,
one obtains
$a=-\frac{\left\langle\boldsymbol{h},\boldsymbol{h}\right\rangle}{2}$, which
implies (4.20). Finally, (4.22) follows from the equalities (4.19), (4.20) and
the fact that $\mu$ has order $T$. ∎

Now we are ready to define an automorphism of the full toroidal Lie algebra
$\mathcal{T}\left(\dot{\mathfrak{g}}\right)$ from the diagram automorphism
$\mu$ of $\mathfrak{g}$.

######  Proposition 4.3.

The following assignment, for $\dot{\alpha}\in\dot{\Delta}$,
$m_{0},m_{1}\in\mathbb{Z}$ and $i=0,1,$

$\displaystyle t_{0}^{m_{0}}t_{1}^{m_{1}}\otimes x_{\dot{\alpha}}$
$\displaystyle\mapsto$ $\displaystyle
t_{0}^{m_{0}}t_{1}^{m_{1}+\rho_{\mu}\left(\dot{\alpha}\right)}\otimes\dot{\mu}\left(x_{\dot{\alpha}}\right),$
$\displaystyle t_{0}^{m_{0}}t_{1}^{m_{1}}\otimes\dot{\alpha}^{\vee}$
$\displaystyle\mapsto$ $\displaystyle
t_{0}^{m_{0}}t_{1}^{m_{1}}\otimes\dot{\mu}\left(\dot{\alpha}^{\vee}\right)+\rho_{\mu}\left(\dot{\alpha}^{\vee}\right)t_{0}^{m_{0}}t_{1}^{m_{1}}\text{k}_{1},$
$\displaystyle t_{0}^{m_{0}}t_{1}^{m_{1}}\text{k}_{i}$ $\displaystyle\mapsto$
$\displaystyle t_{0}^{m_{0}}t_{1}^{m_{1}}\text{k}_{i},\quad
t_{0}^{m_{0}}t_{1}^{m_{1}}\text{d}_{0}\mapsto
t_{0}^{m_{0}}t_{1}^{m_{1}}\text{d}_{0},$ $\displaystyle
t_{0}^{m_{0}}t_{1}^{m_{1}}\text{d}_{1}$ $\displaystyle\mapsto$ $\displaystyle
t_{0}^{m_{0}}t_{1}^{m_{1}}\text{d}_{1}+t_{0}^{m_{0}}t_{1}^{m_{1}}\otimes\boldsymbol{h}-\frac{\left\langle\boldsymbol{h},\boldsymbol{h}\right\rangle}{2}t_{0}^{m_{0}}t_{1}^{m_{1}}\text{k}_{1},$

defines an automorphism, denoted by $\hat{\mu}$, of the full toroidal Lie
algebra $\mathcal{T}\left(\dot{\mathfrak{g}}\right)$ with order $T$.

###### Proof.

For $m_{0},m_{1},n_{0},n_{1}\in\mathbb{Z}$, by applying (4.4)-(4.7), we have

$\displaystyle\left[\hat{\mu}\left(t_{0}^{m_{0}}t_{1}^{m_{1}}\text{d}_{1}\right),\hat{\mu}\left(t_{0}^{n_{0}}t_{1}^{n_{1}}\text{d}_{1}\right)\right]$
$\displaystyle=\ $
$\displaystyle\left[t_{0}^{m_{0}}t_{1}^{m_{1}}\text{d}_{1},t_{0}^{n_{0}}t_{1}^{n_{1}}\text{d}_{1}\right]-\frac{\left\langle\boldsymbol{h},\boldsymbol{h}\right\rangle}{2}\left(\left[t_{0}^{m_{0}}t_{1}^{m_{1}}\text{d}_{1},t_{0}^{n_{0}}t_{1}^{n_{1}}\text{k}_{1}\right]+\left[t_{0}^{m_{0}}t_{1}^{m_{1}}\text{k}_{1},t_{0}^{n_{0}}t_{1}^{n_{1}}\text{d}_{1}\right]\right)$
$\displaystyle+\left[t_{0}^{m_{0}}t_{1}^{m_{1}}\text{d}_{1},t_{0}^{n_{0}}t_{1}^{n_{1}}\otimes\boldsymbol{h}\right]+\left[t_{0}^{m_{0}}t_{1}^{m_{1}}\otimes\boldsymbol{h},t_{0}^{n_{0}}t_{1}^{n_{1}}\text{d}_{1}\right]+\left[t_{0}^{m_{0}}t_{1}^{m_{1}}\otimes\boldsymbol{h},t_{0}^{n_{0}}t_{1}^{n_{1}}\otimes\boldsymbol{h}\right]$
$\displaystyle=\ $
$\displaystyle(n_{1}-m_{1})\left(t_{0}^{m_{0}+n_{0}}t_{1}^{m_{1}+n_{1}}\text{d}_{1}+t_{0}^{m_{0}+n_{0}}t_{1}^{m_{1}+n_{1}}\otimes\boldsymbol{h}-\frac{\left\langle\boldsymbol{h},\boldsymbol{h}\right\rangle}{2}t_{0}^{m_{0}+n_{0}}t_{1}^{m_{1}+n_{1}}\text{k}_{1}\right)$
$\displaystyle-\frac{\left\langle\boldsymbol{h},\boldsymbol{h}\right\rangle}{2}\left(\sum_{i=1,2}m_{i}t_{0}^{m_{0}+n_{0}}t_{1}^{m_{1}+n_{1}}\text{k}_{i}-\sum_{i=1,2}n_{i}t_{0}^{m_{0}+n_{0}}t_{1}^{m_{1}+n_{1}}\text{k}_{i}\right)+\left\langle\boldsymbol{h},\boldsymbol{h}\right\rangle\sum_{i=1,2}m_{i}t_{0}^{m_{0}+n_{0}}t_{1}^{m_{1}+n_{1}}\text{k}_{i}$
$\displaystyle=\ $
$\displaystyle\hat{\mu}\left(\left[t_{0}^{m_{0}}t_{1}^{m_{1}}\text{d}_{1},t_{0}^{n_{0}}t_{1}^{n_{1}}\text{d}_{1}\right]\right).$

One can check $\hat{\mu}$ preserves all other relations by a similar argument
as above. And it follows from (4.22) that the order of the automorphism
$\hat{\mu}$ is equal to $T$. ∎

###  4.3 Subalgebra of $\widetilde{\mathfrak{g}}$ fixed by the automorphism
$\tilde{\mu}$

From the definition of the automorphism $\hat{\mu}$ of the full toroidal Lie
algebra $\mathcal{T}\left(\dot{\mathfrak{g}}\right)$ given in the previous
subsection, it is easy to see that
$\hat{\mu}\left(\widetilde{\mathfrak{g}}\right)=\widetilde{\mathfrak{g}}$. Fix
a $\mathbb{Z}$-grading
$\widetilde{\mathfrak{g}}=\oplus_{m\in\mathbb{Z}}\widetilde{\mathfrak{g}}_{\left(m\right)}$
by the adjoint action of $-\text{d}_{0}$, i.e.,
$\widetilde{\mathfrak{g}}_{\left(m\right)}$=$\left\\{x\in\widetilde{\mathfrak{g}}\mid\left[\text{d}_{0},x\right]=-mx\right\\}$.
Set $\omega=\omega_{T}=e^{2\pi\sqrt{-1}/T}$ and let $\omega^{-\text{d}_{0}}$
be the automorphism of $\widetilde{\mathfrak{g}}$ defined by
$\omega^{-\text{d}_{0}}\left(a\right)=\omega^{m}a$ for
$a\in\widetilde{\mathfrak{g}}_{\left(m\right)}$. Therefore,

$\tilde{\mu}=\omega^{-\text{d}_{0}}\circ\hat{\mu}|_{\widetilde{\mathfrak{g}}}$

defines an automorphism for the toroidal EALA $\widetilde{\mathfrak{g}}$. Now
we investigate the Lie subalgebra $\widetilde{\mathfrak{g}}\left[\mu\right]$
of $\widetilde{\mathfrak{g}}$ fixed by the automorphism $\tilde{\mu}$.

Firstly, for $x\in\mathfrak{g}_{\alpha}$ with
$\alpha\in\Delta^{\times}\cup\left\\{0\right\\}$, $h\in\dot{\mathfrak{h}}$,
$m\in\mathbb{Z}$ and $n\in\mathbb{Z}^{\ast}$, we have

$\displaystyle\tilde{\mu}\left(t_{0}^{m}x\right)=\omega^{-m}t_{0}^{m}\mu\left(x\right),\quad\tilde{\mu}\left(t_{0}^{m}t_{1}^{n}\otimes
h\right)=\omega^{-m}\left(t_{0}^{m}t_{1}^{n}\otimes\dot{\mu}\left(h\right)-\rho_{\mu}\left(h\right)m\text{k}_{m,n}\right),\quad\tilde{\mu}\left(\text{k}_{0}\right)=\text{k}_{0},$
$\displaystyle\tilde{\mu}\left(\text{d}_{0}\right)=\text{d}_{0},\quad\tilde{\mu}\left(\text{k}_{m,n}\right)=\omega^{-m}\text{k}_{m,n},\quad\tilde{\mu}\left(\widetilde{\text{d}}_{m,n}\right)=\omega^{-m}\left(\widetilde{\text{d}}_{m,n}+mt_{0}^{m}t_{1}^{n}\otimes\boldsymbol{h}+\frac{\langle\boldsymbol{h},\boldsymbol{h}\rangle}{2}m^{2}\text{k}_{m,n}\right).$
(4.23)

Moreover, we recall that the diagram automorphism $\mu$ preserves the Cartan
subalgebra $\mathfrak{h}$ and the bilinear form
$\left\langle\cdot,\cdot\right\rangle$. Therefore, by (4.14), (4.21) and
(4.23), we have the following results:

######  Lemma 4.4.

$\tilde{\mu}\left(\widetilde{\mathfrak{h}}\right)=\widetilde{\mathfrak{h}}$,
and
$\left\langle\tilde{\mu}\left(x\right),\tilde{\mu}\left(y\right)\right\rangle=\left\langle
x,y\right\rangle$ for $x,y\in\widetilde{\mathfrak{g}}$.

Denoted by $\widetilde{\mathfrak{g}}\left[\mu\right]$ and
$\widetilde{\mathfrak{h}}\left[\mu\right]$ the Lie subalgebras of
$\widetilde{\mathfrak{g}}$ and $\mathfrak{\widetilde{\mathfrak{h}}}$
respectively fixed by $\tilde{\mu}$. We consider the root space decomposition
of $\widetilde{\mathfrak{g}}\left[\mu\right]$ with respect to its abelian
subalgebra $\widetilde{\mathfrak{h}}\left[\mu\right]$. Denote by
$\left(\mathfrak{h}^{\ast}\right)^{\mu}$ the subspace of $\mathfrak{h}^{\ast}$
that is fixed by $\mu$ and denote by

$\displaystyle\pi_{\mu}:\mathfrak{h}^{\ast}\to\left(\mathfrak{h}^{\ast}\right)^{\mu},\quad\alpha\mapsto\check{\alpha}=\frac{1}{T}\sum_{p=0}^{T-1}\mu^{p}\left(\alpha\right)$

the canonical projection. Since
$\check{\alpha}\left(h-\mu\left(h\right)\right)=0$ for
$\alpha\in\mathfrak{h}^{\ast}$ and $h\in\mathfrak{h}$, we may identify
$\left(\mathfrak{h}^{\mu}\right)^{\ast}$ with
$\left(\mathfrak{h}^{\ast}\right)^{\mu},$ where $\mathfrak{h}^{\mu}$ is the
subspace of $\mathfrak{h}$ fixed by $\mu$. Note that
$\widetilde{\mathfrak{h}}\left[\mu\right]=\mathfrak{h}^{\mu}\oplus\mathbb{C}\text{k}_{0}\oplus\mathbb{C}\text{d}_{0}$.
We view $\left(\mathfrak{h}^{\mu}\right)^{\ast}$ as a subspace of
$\widetilde{\mathfrak{h}}\left[\mu\right]^{\ast}$ such that
$\alpha\left(\text{k}_{0}\right)=\alpha\left(\text{d}_{0}\right)=0$ for
$\alpha\in\left(\mathfrak{h}^{\mu}\right)^{\ast}$, and define
$\delta_{0}\in\widetilde{\mathfrak{h}}\left[\mu\right]^{\ast}$ by

$\delta_{0}\left(\mathfrak{h}^{\mu}\right)=\delta_{0}\left(\text{k}_{0}\right)=0,\
\ \delta_{0}\left(\text{d}_{0}\right)=1.$ (4.24)

In particular, for each $\alpha\in\Delta$,
$\check{\alpha}\in\left(\mathfrak{h}^{\ast}\right)^{\mu}=\left(\mathfrak{h}^{\mu}\right)^{\ast}$
is an element in $\widetilde{\mathfrak{h}}\left[\mu\right]^{\ast}$.

For $\alpha\in\widetilde{\mathfrak{h}}\left[\mu\right]^{\ast},$ set

$\widetilde{\mathfrak{g}}\left[\mu\right]_{\alpha}=\left\\{x\in\widetilde{\mathfrak{g}}\left[\mu\right]\mid\left[h,x\right]=\alpha\left(h\right)x,\
h\in\widetilde{\mathfrak{h}}\left[\mu\right]\right\\}.$

Then we have the following root space decomposition

$\widetilde{\mathfrak{g}}\left[\mu\right]=\widetilde{\mathfrak{g}}\left[\mu\right]_{0}\oplus\sum_{\alpha\in\tilde{\Delta}_{\mu}}\widetilde{\mathfrak{g}}\left[\mu\right]_{\alpha},$

where
$\tilde{\Delta}_{\mu}=\left\\{\alpha\in\tilde{\mathfrak{h}}\left[\mu\right]^{\ast}\backslash\left\\{0\right\\}\mid\widetilde{\mathfrak{g}}\left[\mu\right]_{\alpha}\not=0\right\\}.$
Let

$\displaystyle\eta_{\mu}:\widetilde{\mathfrak{g}}\to\widetilde{\mathfrak{g}}\left[\mu\right],\quad
x\mapsto\sum_{p=0}^{T-1}\tilde{\mu}^{p}\left(x\right)$ (4.25)

be a projection from $\widetilde{\mathfrak{g}}$ to
$\widetilde{\mathfrak{g}}\left[\mu\right]$. It can be readily seen that

$\displaystyle\eta_{\mu}\left(t_{0}^{m}u\right)\in\widetilde{\mathfrak{g}}\left[\mu\right]_{\check{\alpha}+m\delta_{0}},\quad\eta_{\mu}\left(\text{k}_{m,n}\right),\eta_{\mu}\left(\tilde{\text{d}}_{m,n}\right)\in\widetilde{\mathfrak{g}}\left[\mu\right]_{m\delta_{0}+n\delta_{1}},\quad\text{k}_{0},\text{d}_{0}\in\widetilde{\mathfrak{g}}\left[\mu\right]_{0}$
(4.26)

for $u\in\mathfrak{g}_{\alpha}$ with $\alpha\in\Delta\cup\left\\{0\right\\}$,
$m\in\mathbb{Z}$ and $n\in\mathbb{Z}^{\ast}$. Then we have

######  Lemma 4.5.

The root system
$\tilde{\Delta}_{\mu}\subset\left(\pi_{\mu}\left(\Delta\right)+\mathbb{Z}\delta_{1}\right)\cup\mathbb{Z}\delta_{1}$
and
$\mathfrak{\widetilde{\mathfrak{g}}}\left[\mu\right]_{0}=\mathfrak{\widetilde{h}}\left[\mu\right]$.

###### Proof.

The first assertion follows from (4.13) and (4.26). For the second one, note
that if $\alpha=\sum_{i\in I}a_{i}\alpha_{i}$ is a root in $\Delta$, then
$\check{\alpha}=\frac{1}{T}\sum_{i\in
I}\sum_{p=0}^{T-1}a_{i}\alpha_{\mu^{p}(i)}$ is clearly nonzero. This together
with (4.26) implies that

$\widetilde{\mathfrak{g}}\left[\mu\right]_{0}=\eta_{\mu}\left(\mathfrak{h}\right)\oplus\mathbb{C}\text{k}_{0}\oplus\mathbb{C}\text{d}_{0}=\widetilde{\mathfrak{h}}\left[\mu\right].$

∎

Denote by $\mathfrak{t}\left(\dot{\mathfrak{g}},\mu\right)$ the subalgebra of
$\mathfrak{t}\left(\dot{\mathfrak{g}}\right)$ fixed by $\tilde{\mu}$ (noting
that
$\tilde{\mu}\left(\mathfrak{t}\left(\dot{\mathfrak{g}}\right)\right)=\mathfrak{t}\left(\dot{\mathfrak{g}}\right)$).
We have the following result from [CJKT].

######  Proposition 4.6.

The Lie algebra $\mathfrak{t}\left(\dot{\mathfrak{g}},\mu\right)$ is centrally
closed.

In the rest of this subsection, we recall a characterization for the subset of
the root system $\tilde{\Delta}_{\mu}$ given in [CJKT]:

$\tilde{\Delta}_{\mu}^{\times}=\left\\{\check{\alpha}+m\delta_{0}\in\tilde{\Delta}_{\mu}\mid\alpha\in\Delta,m\in\mathbb{Z}\
\text{and}\
\left\langle\check{\alpha},\check{\alpha}\right\rangle\not=0\right\\}.$

For every $i\in I=\left\\{0,1,\cdots,l\right\\}$, denote by
$\mathcal{O}\left(i\right)$ the orbit containing $i$ under the action of the
group $\left\langle\mu\right\rangle$. We say $\mu$ is _transitive_ if
$\mathcal{O}\left(i\right)=I$ for each $i\in I$. Observe that a diagram
automorphism on $\mathfrak{g}$ is transitive if and only if $\mathfrak{g}$ is
of type $A_{l}^{\left(1\right)}$, and it has order $l+1$. Note that in this
case we have $\pi_{\mu}\left(\alpha_{i}\right)=\delta_{1}$ for any $i\in I$,
and hence $\tilde{\Delta}_{\mu}^{\times}=\emptyset.$

From now on we assume that $\mu$ is nontransitive, and in this case, it is
known that the folded matrix
$\check{A}=\left(2\frac{\left\langle\check{\alpha}_{i},\check{\alpha}_{j}\right\rangle}{\left\langle\check{\alpha}_{i},\check{\alpha}_{i}\right\rangle}\right)_{i,j\in\check{I}}$
of $A$ associated to $\mu$ is also an affine generalized Cartan matrix (cf.
[FSS, ABP]), where $\check{I}=\left\\{i\in I\mid\mu^{k}\left(i\right)\geq i\
\text{for\ }k\in\mathbb{Z}\right\\}$ is a set of representative elements in
$I$. Denote by $\check{\Delta}$ and $\check{W}$ respectively the root system
and the Weyl group associated to the folded matrix $\check{A}$. Then
$\left\\{\check{\alpha}_{i}\right\\}_{i\in\check{I}}$ is a simple root base of
$\check{\Delta}$. Furthermore, we have the following result from [ABP, Lemma
12.15].

######  Lemma 4.7.

For each $i\in I$, one and only one of the following statement holds:

(a) The elements $\alpha_{p},$ for $p\in\mathcal{O}\left(i\right)$, are
pairwise orthogonal;

(b) $\mathcal{O}\left(i\right)=\left\\{i,\mu\left(i\right)\right\\}$, and
$a_{i\mu\left(i\right)}=-1=a_{\mu\left(i\right)i}.$

Recall that $T$ is the order of the diagram automorphism $\mu$. For every
$i\in I$, we set

$\displaystyle T_{i}=T/\left|\mathcal{O}\left(i\right)\right|,\ \text{and}\
s_{i}=\begin{cases}1,&\text{if}\ \text{(a)\ holds\ in Lemma
\ref{lem:defsi};}\\\ 2,&\text{if\ (b)\ holds\ in Lemma \ref{lem:defsi}.
}\end{cases}$ (4.27)

The following result was proved in [CJKT, Proposition 5.1].

######  Proposition 4.8.

If $\mu$ is nontransitive, then

$\displaystyle\tilde{\Delta}_{\mu}^{\times}=\left\\{\check{w}\check{\alpha}_{i}+T_{i}m\delta_{0}\mid\check{w}\in\check{W},i\in\check{I},m\in\mathbb{Z}\right\\}\cup\left\\{2\check{w}\check{\alpha}_{i}+\left(\frac{T}{2}+mT\right)\delta_{0}\mid\check{w}\in\check{W},i\in\check{I}\
\text{with\ }s_{i}=2,m\in\mathbb{Z}\right\\}.$

######  Remark 4.9.

If $\mu$ is nontransitive, then $\tilde{\Delta}_{\mu}^{\times}$ is a nullity 2
reduced extended affine root system introduced by Saito (cf. [Sa]).
Conversely, any nullity 2 reduced extended affine root system is of such a
form (cf. [ABP]). Furthermore, we will see in the next section that the triple
$\left(\widetilde{\mathfrak{g}}\left[\mu\right],\widetilde{\mathfrak{h}}\left[\mu\right],\langle\cdot,\cdot\rangle\right)$
is a nullity $2$ EALA, and which we call a _nullity $2$ twisted toroidal
extended affine Lie algebra_.

##  5 Nullity 2 EALA of maximal type

In this section, we first recall the definition of extended affine Lie algebra
(cf. [AABGP, N]), and the classification of nullity $2$ extended affine Lie
algebras of maximal type based on Allison-Berman-Pianzola’s work [ABP].

Let $\mathcal{E}$ be a Lie algebra equipped with a nontrivial finite-
dimensional self-centralizing ad-diagonalizable subalgebra $\mathcal{H}$ and a
nondegenerate invariant symmetric bilinear form $\left(\cdot|\cdot\right).$
Let $\mathcal{E}=\mathcal{H}\oplus\sum_{\alpha\in\Phi}\mathcal{E}_{\alpha}$ be
the root space decomposition with respect to $\mathcal{H},$ where $\Phi$ is
the corresponding root system. The form $\left(\cdot|\cdot\right)$ is also
nondegenerate when it is restricted to $\mathcal{H}=\mathcal{E}_{0}$. Hence it
induces a nondegenerate symmetric bilinear form on $\mathcal{H}^{\ast}$. Set

$\Phi^{\times}=\left\\{\alpha\in\Phi\mid\left(\alpha|\alpha\right)\not=0\right\\},\quad\Phi^{0}=\left\\{\alpha\in\Phi\mid\left(\alpha|\alpha\right)=0\right\\}.$

Let $\mathcal{E}_{c}$ be the subalgebra of $\mathcal{E}$ generated by the root
spaces $\mathcal{E}_{\alpha},\alpha\in\Phi^{\times}$, which is called the
_core_ of $\mathcal{E}$.

######  Definition 5.1.

$\mathcal{E}:=\left(\mathcal{E},\mathcal{H},(\cdot|\cdot)\right)$ is called an
_extended affine Lie algebra_ (EALA for short) if

(1) $\text{ad}\left(x\right)$ is locally nilpotent for
$x\in\mathcal{E}_{\alpha},\alpha\in\Phi^{\times};$

(2) $\Phi^{\times}$ cannot be decomposed as a union of two orthogonal nonempty
subsets;

(3) The centralizer of $\mathcal{E}_{c}$ in $\mathcal{E}$ is contained in
$\mathcal{E}_{c}$;

(4) $\Phi$ is a discrete subset of $\mathcal{H}^{\ast}$ with respect to its
Euclidean topology.

The axiom (4) implies that the subgroup $\left\langle\Phi^{0}\right\rangle$ of
$\mathcal{H}^{\ast}$ generated by $\Phi^{0}$ is a free abelian group of finite
rank, and this rank is called the _nullity_ of $\mathcal{E}$. Indeed, the
nullity $0$ EALAs are exactly the finite dimensional simple Lie algebras,
while nullity $1$ EALAs are exactly the affine Kac-Moody algebras (cf.
[ABGP]). For the purpose of classifying EALAs, the following notion was
introduced in [BGK].

######  Definition 5.2.

An EALA is of _maximal type_ if its core is centrally closed.

Note that the nullity $0$ and nullity $1$ EALAs are all of maximal type. In
general, an EALA may not be of maximal type. However, the maximal EALAs appear
to be the most interesting ones as from them one can know the structure of
other EALAs (see [BGK, Remark 3.73] and [N] for details). In what follows, we
give two classes of nullity $2$ EALAs of maximal type.

######  Proposition 5.3.

Let $\mu$ be a nontransitive diagram automorphism of an untwisted affine Kac-
Moody algebra $\mathfrak{g}$. Then the triple
$\left(\widetilde{\mathfrak{g}}\left[\mu\right],\widetilde{\mathfrak{h}}\left[\mu\right],\left\langle\cdot,\cdot\right\rangle\right)$,
defined in the previous section, is a nullity 2 EALA of maximal type, and
$\mathfrak{t}\left(\dot{\mathfrak{g}},\mu\right)$ is the core of
$\widetilde{\mathfrak{g}}\left[\mu\right]$.

###### Proof.

Recall from Lemma 4.5 that the ad-diagonalizable subalgebra
$\widetilde{\mathfrak{h}}\left[\mu\right]$ of
$\widetilde{\mathfrak{g}}\left[\mu\right]$ is self-centralized, and it follows
from Lemma 4.4 that the invariant form $\left\langle\cdot,\cdot\right\rangle$
restricted to $\widetilde{\mathfrak{g}}\left[\mu\right]$ is still
nondegenerate. One can easily check that
$\mathfrak{t}\left(\dot{\mathfrak{g}},\mu\right)$ is generated by the elements
$\eta_{\mu}\left(t_{0}^{m}x_{\pm\alpha_{i}}\right)$ for $m\in\mathbb{Z}$ and
$i\in I$. This together with Proposition 4.8 implies that the core of
$\widetilde{\mathfrak{g}}\left[\mu\right]$ is
$\mathfrak{t}\left(\dot{\mathfrak{g}},\mu\right)$. Now we check the axioms
(1)-(4) in Definition 5.1. The axiom (1) follows from Proposition 4.8, while
the axiom (2) is implied by Remark 4.9 as the root system defined by Saito is
irreducible (see [Sa, Definition 1]). The axioms (3) and (4) are obvious.
Finally, Proposition 4.6 implies the maximality of
$\widetilde{\mathfrak{g}}\left[\mu\right]$. ∎

For the second class of nullity $2$ EALAs of maximal type. We let
$q\in\mathbb{C}^{\ast}$ be generic, i.e, $q$ is not a root of unity, and
$\mathbb{C}_{q}:=\mathbb{C}_{q}\left[t_{0}^{\pm 1},t_{1}^{\pm 1}\right]$ the
quantum torus associated to $q$ such that $t_{0}t_{1}=qt_{1}t_{0}$. For any
positive integer $N\geq 2$, denoted by
$\mathfrak{gl}_{N}\left(\mathbb{C}_{q}\right)$ the general linear Lie algebra
over $\mathbb{C}_{q}$, and $\mathfrak{sl}_{N}\left(\mathbb{C}_{q}\right)$ the
derived subalgebra of $\mathfrak{gl}_{N}\left(\mathbb{C}_{q}\right)$. For
$1\leq i,j\leq N$, $a\in\mathbb{C}_{q}$, we write $E_{i,j}a$ the matrix whose
only possible nonzero entry is the $\left(i,j\right)$-entry which is $a$. We
consider the central extension of the Lie algebra
$\mathfrak{gl}_{N}(\mathbb{C}_{q})$:

$\displaystyle\mathfrak{\widehat{gl}}_{N}\left(\mathbb{C}_{q}\right):=\mathfrak{gl}_{N}\left(\mathbb{C}_{q}\right)\oplus\mathbb{C}\text{k}_{0}\oplus\mathbb{C}\text{k}_{1},$

where $\text{k}_{0},\text{k}_{1}$ are central elements, and

$\begin{split}\left[E_{i,j}t_{0}^{m_{0}}t_{1}^{m_{1}},E_{k,l}t_{0}^{n_{0}}t_{1}^{n_{1}}\right]=\
&\delta_{j,k}q^{m_{1}n_{0}}E_{i,l}t_{0}^{m_{0}+n_{0}}t_{1}^{m_{1}+n_{1}}-\delta_{i,l}q^{n_{1}m_{0}}E_{k,j}t_{0}^{m_{0}+n_{0}}t_{1}^{m_{1}+n_{1}}\\\
&+\delta_{j,k}\delta_{i,l}\delta_{m_{0}+n_{0},0}\delta_{m_{1}+n_{1},0}q^{m_{1}n_{0}}\left(m_{0}\text{k}_{0}+m_{1}\text{k}_{1}\right),\end{split}$
(5.1)

for $1\leq i,j,k,l\leq N$ and $m_{0},m_{1},n_{0},n_{1}\in\mathbb{Z}$.
Moreover, we define two derivations $\text{d}_{0},\text{d}_{1}$ acting on
$\mathfrak{\widehat{gl}}_{N}\left(\mathbb{C}_{q}\right)$ by

$\displaystyle[\text{d}_{r},E_{i,j}t_{0}^{m_{0}}t_{1}^{m_{1}}]=m_{r}E_{i,j}t_{0}^{m_{0}}t_{1}^{m_{1}},\quad[\text{d}_{r},\text{k}_{s}]=0=[\text{d}_{r},\text{d}_{s}],$

for $r,s\in\\{0,1\\}$ and for $1\leq i,j\leq N,m_{0},m_{1}\in\mathbb{Z}$.
Therefore, we obtain a Lie algebra

$\displaystyle\mathfrak{\widetilde{gl}}_{N}\left(\mathbb{C}_{q}\right):=\mathfrak{\widehat{gl}}_{N}\left(\mathbb{C}_{q}\right)\oplus\mathbb{C}\text{d}_{0}\oplus\mathbb{C}\text{d}_{1}.$

Set

$\displaystyle\widehat{\mathfrak{sl}}_{N}\left(\mathbb{C}_{q}\right)=\left[\mathfrak{\widehat{gl}}_{N}\left(\mathbb{C}_{q}\right),\mathfrak{\widehat{gl}}_{N}\left(\mathbb{C}_{q}\right)\right]=\mathfrak{sl}_{N}\left(\mathbb{C}_{q}\right)\oplus\mathbb{C}\text{k}_{0}\oplus\mathbb{C}\text{k}_{1},$

the derived subalgebra of
$\mathfrak{\widehat{gl}}_{N}\left(\mathbb{C}_{q}\right)$. It is known that
$\widehat{\mathfrak{sl}}_{N}\left(\mathbb{C}_{q}\right)$ is central closed and
$\mathfrak{\widehat{gl}}_{N}\left(\mathbb{C}_{q}\right)=\widehat{\mathfrak{sl}}_{N}\left(\mathbb{C}_{q}\right)\oplus\mathbb{C}I_{N}$
(cf. [BGK]), where $I_{N}$ is the identity matrix. Furthermore, we define

$\displaystyle\widetilde{\mathfrak{sl}}_{N}\left(\mathbb{C}_{q}\right):=\widehat{\mathfrak{sl}}_{N}\left(\mathbb{C}_{q}\right)\oplus\mathbb{C}\text{d}_{0}\oplus\mathbb{C}\text{d}_{1},\quad\widetilde{\mathcal{H}}:=\sum_{i=1}^{N-1}\mathbb{C}\left(E_{i,i}-E_{i+1,i+1}\right)\oplus\mathbb{C}\text{k}_{0}\oplus\mathbb{C}\text{k}_{1}\oplus\mathbb{C}\text{d}_{0}\oplus\mathbb{C}\text{d}_{1},$

and a bilinear form $\left\langle\cdot,\cdot\right\rangle$ of
$\widetilde{\mathfrak{sl}}_{N}\left(\mathbb{C}_{q}\right)$ such that

$\displaystyle\left\langle
E_{i,j}t_{0}^{m_{0}}t_{1}^{m_{1}},E_{j,i}t_{0}^{-m_{0}}t_{1}^{-m_{1}}\right\rangle=1=\left\langle\text{d}_{r},\text{k}_{r}\right\rangle,$

for $1\leq i,j\leq N,m_{0},m_{1}\in\mathbb{Z}$, $r\in\\{0,1\\}$, and trivial
for others. The following result is from [BGK]:

######  Proposition 5.4.

Let $N\geq 2$ be a positive integer and $q\in\mathbb{C}^{\ast}$ be generic.
Then the triple
$\left(\widetilde{\mathfrak{sl}}_{N}\left(\mathbb{C}_{q}\right),\widetilde{\mathcal{H}},\left\langle\cdot,\cdot\right\rangle\right)$
is a nullity 2 EALA of maximal type, and
$\widehat{\mathfrak{sl}}_{N}\left(\mathbb{C}_{q}\right)$ is the core of
$\mathfrak{\widetilde{sl}}_{N}(\mathbb{C}_{q})$.

For any given EALA $\mathcal{E}$, we call
$\mathcal{E}_{cc}:=\mathcal{E}_{c}/{\text{Z}}\left(\mathcal{E}_{c}\right)$
_the centerless core_ of $\mathcal{E}$, where
${\text{Z}}\left(\mathcal{E}_{c}\right)$ is the center of the core
$\mathcal{E}_{c}$. One notes that the centerless core of the affine Kac-Moody
algebra $\mathfrak{g}$ is isomorphic to the loop algebra
$\overline{\mathfrak{g}}=\mathbb{C}\left[t_{1},t_{1}^{-1}\right]\otimes\dot{\mathfrak{g}}.$
Denote by $\bar{\mu}$ the automorphism of $\overline{\mathfrak{g}}$ induced by
$\mu$. Then the centerless core of $\widetilde{\mathfrak{g}}\left[\mu\right]$
is isomorphic to the $\bar{\mu}$-twisted loop algebra

$\mathcal{L}\left(\overline{\mathfrak{g}},\bar{\mu}\right)=\sum_{n\in\mathbb{Z}}t_{0}^{n}\otimes\overline{\mathfrak{g}}_{\left(m\right)}\subset\mathbb{C}\left[t_{0},t_{0}^{-1}\right]\otimes\overline{\mathfrak{g}}=\mathcal{R}\otimes\dot{\mathfrak{g}},$

where
$\overline{\mathfrak{g}}_{\left(m\right)}=\left\\{x\in\overline{\mathfrak{g}}\mid\bar{\mu}\left(x\right)=\omega^{m}x\right\\}$.

It is clear that the centerless core of
$\widetilde{\mathfrak{sl}}_{N}\left(\mathbb{C}_{q}\right)$ with $q$ generic is
isomorphic to $\mathfrak{sl}_{N}\left(\mathbb{C}_{q}\right)$. The following
classification of centerless cores of nullity 2 EALAs was given in [ABP]:

######  Proposition 5.5.

The centerless core of a nullity 2 EALA is either isomorphic to
$\mathfrak{sl}_{N}\left(\mathbb{C}_{q}\right)$ for some positive integer
$N\geq 2$ and generic $q\in\mathbb{C}^{\ast}$, or to
$\mathcal{L}\left(\overline{\mathfrak{g}},\bar{\mu}\right)$ for some
nontransitive diagram automorphism $\mu$ of an untwisted affine Kac-Moody
algebra $\mathfrak{g}$.

Two EALAs are said to be _equivalent_ if their cores are isomorphic. Then we
have the following result.

######  Theorem 5.6.

Up to equivalence, a nullity $2$ extended affine Lie algebra of maximal type
either has the form $\widetilde{\mathfrak{sl}}_{N}\left(\mathbb{C}_{q}\right)$
for some positive integer $N\geq 2$ and generic $q\in\mathbb{C}^{\ast}$, or
has the form $\widetilde{\mathfrak{g}}\left[\mu\right]$ for some nontransitive
diagram automorphism $\mu$ of an untwisted affine Kac-Moody algebra
$\mathfrak{g}$.

###### Proof.

Let $\mathcal{E}$ be a nullity $2$ EALA of maximal type. Then by Proposition
5.5, either
$\mathcal{E}_{cc}\cong\mathfrak{sl}_{N}\left(\mathbb{C}_{q}\right)$, or
$\mathcal{E}_{cc}\cong\mathcal{L}\left(\overline{\mathfrak{g}},\bar{\mu}\right)$.
Then the maximality of $\mathcal{E}$ forces that either
$\mathcal{E}_{c}\cong\widehat{\mathfrak{sl}}_{N}\left(\mathbb{C}_{q}\right)$
or $\mathcal{E}_{c}\cong\mathfrak{t}\left(\dot{\mathfrak{g}},\mu\right)$. Thus
it follows from Propositions 5.3 and 5.4 that $\mathcal{E}$ is either
equivalent to $\widetilde{\mathfrak{sl}}_{N}\left(\mathbb{C}_{q}\right)$ or to
$\widetilde{\mathfrak{g}}\left[\mu\right]$. ∎

######  Remark 5.7.

Following [BGK, N], a _$\mathcal{K}$ -valued affine cocycle $\tau$ on
$\mathcal{S}$_ is an abelian $2$-cocycle
$\tau:\mathcal{S}\times\mathcal{S}\rightarrow\mathcal{K}$ such that
$\tau\left(\mathcal{S},\text{d}_{i}\right)=0$ for $i=0,1$, and
$\left\langle\tau\left(s_{1},s_{2}\right),s_{3}\right\rangle=\left\langle
s_{1},\tau\left(s_{2},s_{3}\right)\right\rangle$ for
$s_{1},s_{2},s_{3}\in\mathcal{S}$. For any affine cocycle $\tau$, one can
define a new Lie multiplication $[\cdot,\cdot]_{\tau}$ on
$\widetilde{\mathfrak{g}}$ defined by

$\displaystyle\left[x_{1}+s_{1},x_{2}+s_{2}\right]{}_{\tau}=\left[x_{1}+s_{1},x_{2}+s_{2}\right]+\tau\left(s_{1},s_{2}\right)$

for $x_{1},x_{2}\in\mathfrak{t}\left(\dot{\mathfrak{g}}\right)$ and
$s_{1},s_{2}\in\mathcal{S}$. We denote the resulting Lie algebra by
$\widetilde{\mathfrak{g}}_{\tau}$. It is easy to see that the action (4.23)
also defines an automorphism $\tilde{\mu}$ of
$\widetilde{\mathfrak{g}}_{\tau}$. Denoted by
$\widetilde{\mathfrak{g}}\left[\mu\right]{}_{\tau}$ the subalgebra of
$\widetilde{\mathfrak{g}}_{\tau}$ fixed by $\tilde{\mu}$. Similar to the proof
of Proposition 5.3, one can check that
$\left(\widetilde{\mathfrak{g}}\left[\mu\right]_{\tau},\widetilde{\mathfrak{h}}\left[\mu\right],\left\langle\cdot,\cdot\right\rangle\right)$
is a nullity 2 EALA of maximal type if $\mu$ is nontransitive. By the explicit
construction of EALAs (of maximal type) given by Neher in [N], one can prove
that if an EALA is equivalent to $\widetilde{\mathfrak{g}}\left[\mu\right]$,
then it is isomorphic to $\widetilde{\mathfrak{g}}\left[\mu\right]_{\tau}$ for
some affine cocycle $\tau$. And, if an EALA is equivalent to
$\widetilde{\mathfrak{sl}}_{N}\left(\mathbb{C}_{q}\right)$, then it must be
isomorphic to $\widetilde{\mathfrak{sl}}_{N}\left(\mathbb{C}_{q}\right)$.

######  Remark 5.8.

It was shown in [MRY] that there exist nontrivial affine cocycles. For
example, for any complex number $a$, the bilinear map
$\tau_{a}:\mathcal{S}\times\mathcal{S}\rightarrow\mathcal{K}$ defined by

$\tau_{a}\left(\mathcal{S},\text{d}_{r}\right)=0,\quad\tau_{a}\left(\text{d}_{m_{0},m_{1}},\text{d}_{n_{0},n_{1}}\right)=a\left(m_{0}n_{1}-n_{0}m_{1}\right){}^{3}\text{k}_{m_{0}+n_{0},m_{1}+n_{1}}$

for $m_{0},m_{1},n_{0},n_{1}\in\mathbb{Z}$ and $r=0,1$, is an affine cocycle.
Then we have the Lie algebras
$\widetilde{\mathfrak{g}}\left[\mu\right]_{\tau_{a}}$ with
$\widetilde{\mathfrak{g}}\left[\mu\right]_{\tau_{0}}=\widetilde{\mathfrak{g}}\left[\mu\right]$.
It is conjectured that any $\mathcal{K}$-valued affine cocycle on
$\mathcal{S}$ has the form $\tau_{a}$ for some $a\in\mathbb{C}$. This will
imply that $\widetilde{\mathfrak{g}}\left[\mu\right]_{\tau_{a}}$ and
$\widetilde{\mathfrak{sl}}_{N}\left(\mathbb{C}_{q}\right)$ exhaust all nullity
$2$ EALAs of maximal type up to isomorphism.

##  6 Associating $\tilde{\mathfrak{g}}\left[\mu\right]$ with vertex algebras

Let $\mu$ be a fixed diagram automorphism of the untwisted affine Kac-Moody
algebra $\mathfrak{g}$. In this section, we associate the twisted toroidal
EALA $\widetilde{\mathfrak{g}}\left[\mu\right]$ with vertex algebras through
equivariant $\phi$-coordinated quasi modules.

###  6.1 Vertex algebras $V_{\widehat{\mathfrak{g}}}\left(\ell,0\right)$ and
$L_{\widehat{\mathfrak{g}}}\left(\ell,0\right)$

We recall the variant of the skew derivation algebra $\mathcal{S}$ introduced
in [CLiT]:

$\widehat{\mathcal{S}}=\left\\{f_{0}\frac{\partial}{\partial
t_{0}}+f_{1}\text{d}_{1}\mid
f_{0},f_{1}\in\mathcal{R},\frac{\partial}{\partial
t_{0}}\left(f_{0}\right)+\text{d}_{1}\left(f_{1}\right)=0\right\\}\subset\mathrm{Der}(\mathcal{R}).$

For $m,n\in\mathbb{Z}$, set

$\widehat{\text{d}}_{n,m}=\left(n+1\right)t_{0}^{n}t_{1}^{m}\text{d}_{1}-mt_{0}^{n}t_{1}^{m}\text{d}_{0},$

It is easy to see that the set

$\mathbb{B}_{\widehat{\mathcal{S}}}=\left\\{t_{0}^{-1}\text{d}_{0},t_{0}^{-1}\text{d}_{1}\right\\}\cup\left\\{\widehat{\text{d}}_{n,m}\mid\left(n,m\right)\in\mathbb{Z}\times\mathbb{Z}\setminus\left\\{\left(-1,0\right)\right\\}\right\\}$
$=\left\\{t_{0}^{-1}\text{d}_{0}\right\\}\cup\left\\{t_{0}^{n}\text{d}_{1},\
\widehat{\text{d}}_{n,m}\mid n\in\mathbb{Z},m\in\mathbb{Z}^{\ast}\right\\}$

forms a $\mathbb{C}$-basis of $\widehat{\mathcal{S}}$, and subject to the
following relations

$\displaystyle\left[t_{0}^{-1}\text{d}_{0},\widehat{\text{d}}{}_{n,m}\right]=\left(n+1\right)\widehat{\text{d}}_{n-1,m},\quad\left[t_{0}^{-1}\text{d}_{1},\widehat{\text{d}}_{n,m}\right]=m\widehat{\text{d}}_{n-1,m},$
(6.1)
$\displaystyle\left[\widehat{\text{d}}_{n,m},\widehat{\text{d}}_{n_{1},m_{1}}\right]=\left(\left(n+1\right)m_{1}-m\left(n_{1}+1\right)\right)\widehat{\text{d}}_{n+n_{1},m+m_{1}},$
(6.2)

for $n,m,n_{1},m_{1}\in\mathbb{Z}$. In view of this, we have the following
subalgebra of the full toroidal Lie algebra $\mathcal{T}(\dot{\mathfrak{g}})$:

$\displaystyle\widehat{\mathfrak{g}}=\mathfrak{t}\left(\dot{\mathfrak{g}}\right)+\left[\widehat{\mathcal{S}},\widehat{\mathcal{S}}\right]+\mathbb{C}t_{0}^{-1}\text{d}_{1}.$

Note that $\widehat{\mathfrak{g}}$ is linearly spanned by the set

$\displaystyle\left\\{t_{0}^{n}u,\ \text{k}_{n,m},\ \text{k}_{0},\
\widehat{\text{d}}_{n,m}\mid u\in\mathfrak{g},\ n\in\mathbb{Z},\
m\in\mathbb{Z}^{\ast}\right\\}.$ (6.3)

From (4.5) and (4.6), we have

$\displaystyle\left[\widehat{\text{d}}_{i,m},t_{0}^{j}t_{1}^{n}\otimes
x\right]=\left(\left(i+1\right)n-mj\right)t_{0}^{i+j}t_{1}^{m+n}\otimes x,$
(6.4)
$\displaystyle\left[\widehat{\text{d}}_{i,m},\text{k}_{j,n}\right]=\left(\left(i+1\right)\left(m+n\right)-m\left(i+j\right)\right)\text{k}_{i+j,m+n}+\delta_{m+n,0}\delta_{i+j,0}\left(\left(i+1\right)\text{k}_{0}+m\text{k}_{1}\right),$
(6.5)

for $i,j,m,n\in\mathbb{Z}$ and $x\in\dot{\mathfrak{g}}$. Set

$\displaystyle\widehat{\mathfrak{g}}_{+}$
$\displaystyle=\text{Span}\left\\{t_{0}^{n}u,\text{k}_{n+1,m},\widehat{\text{d}}_{n-1,m}\mid
u\in\mathfrak{g},\ n\in\mathbb{N},\ m\in\mathbb{Z}\right\\},$ (6.6)
$\displaystyle\widehat{\mathfrak{g}}_{-}$
$\displaystyle=\text{Span}\left\\{t_{0}^{-n-1}u,\text{k}_{-n,m},\widehat{\text{d}}_{-n-2,m}\mid
u\in\mathfrak{g},\ n\in\mathbb{N},\ m\in\mathbb{Z}\right\\}.$ (6.7)

Then both $\widehat{\mathfrak{g}}_{+}$ and $\widehat{\mathfrak{g}}_{-}$ are
subalgebras of $\widehat{\mathfrak{g}}$. Furthermore, we have the
decomposition:

$\displaystyle\widehat{\mathfrak{g}}=\widehat{\mathfrak{g}}_{+}\oplus\mathbb{C}\text{k}_{0}\oplus\widehat{\mathfrak{g}}_{-}.$
(6.8)

Let $\ell$ be a complex number. View $\mathbb{C}$ as a
$\left(\widehat{\mathfrak{g}}_{+}+\mathbb{C}\text{k}_{0}\right)$-module with
$\widehat{\mathfrak{g}}_{+}$ acting trivially and with $\text{k}_{0}$ acting
as scalar $\ell$. We form the induced $\widehat{\mathfrak{g}}$-module

$\displaystyle
V_{\widehat{\mathfrak{g}}}\left(\ell,0\right)=\mathcal{U}\left(\widehat{\mathfrak{g}}\right)\otimes_{\mathcal{U}\left(\widehat{\mathfrak{g}}_{+}+\mathbb{C}\text{k}_{0}\right)}\mathbb{C}.$
(6.9)

Let $\mathcal{A}$ be a vector space with a basis $\left\\{K_{n},D_{n}\mid
n\in\mathbb{Z}^{\ast}\right\\}$, and set

$\mathcal{A}_{\mathfrak{g}}=\mathfrak{g}\oplus\mathcal{A}.$

Form the generating functions $a\left(z\right)$,
$a\in\mathcal{A}_{\mathfrak{g}}$ in
$\widehat{\mathfrak{g}}\left[\left[z,z^{-1}\right]\right]$ as follows:

$\displaystyle
u\left(z\right)=\sum_{n\in\mathbb{Z}}\left(t_{0}^{n}u\right)z^{-n-1},\quad
D_{m}\left(z\right)=\sum_{n\in\mathbb{Z}}\widehat{\text{d}}_{n,m}z^{-n-2},\quad
K_{m}\left(z\right)=\sum_{n\in\mathbb{Z}}\text{k}_{n,m}z^{-n},$

for $u\in\mathfrak{g}$ and $m\in\text{$\mathbb{Z}^{\ast}.$}$ Set
$\mathbf{1}=1\otimes 1\in V_{\widehat{\mathfrak{g}}}\left(\ell,0\right)$.
Identify $\mathcal{A}_{\mathfrak{g}}$ as a subspace of
$V_{\widehat{\mathfrak{g}}}\left(\ell,0\right)$ through the linear map

$\displaystyle u\mapsto\left(t_{0}^{-1}u\right)\mathbf{1},\quad
K_{n}\mapsto\text{k}_{0,n}\mathbf{1},\quad
D_{n}\mapsto\hat{\text{d}}_{-2,n}\mathbf{1},$ (6.10)

for $u\in\mathfrak{g},\,n\in\mathbb{Z}^{\ast}$. It is proved in [CLiT] that
there is a unique vertex algebra structure on
$V_{\widehat{\mathfrak{g}}}\left(\ell,0\right)$ with
$Y\left(a,z\right)=a\left(z\right)$ for $a\in\mathcal{A}_{\mathfrak{g}}$, and
$\mathbf{1}$ the vacuum vector.

######  Remark 6.1.

Note that
$\mathfrak{t}\left(\dot{\mathfrak{g}}\right)\oplus\widehat{\mathcal{S}}=\widehat{\mathfrak{g}}\oplus\mathbb{C}t_{0}^{-1}\text{d}_{0}$
with

$\left[-t_{0}^{-1}\text{d}_{0},a(z)\right]=\frac{d}{dz}a\left(z\right),\quad
a\in\mathcal{A}_{\mathfrak{g}}.$

This implies that $V_{\widehat{\mathfrak{g}}}\left(\ell\right)$ is a
$\mathfrak{t}\left(\dot{\mathfrak{g}}\right)\oplus\widehat{\mathcal{S}}$-module
with $-t_{0}^{-1}\text{d}_{0}$ acts as the canonical derivation $\mathcal{D}$.
In particular, from [LLi] it follows that an ideal of vertex algebra
$V_{\widehat{\mathfrak{g}}}\left(\ell,0\right)$ is the same as a
$\mathfrak{t}\left(\dot{\mathfrak{g}}\right)\oplus\widehat{\mathcal{S}}$-submodule.

When $\ell$ is a nonnegative integer, denoted by
$J_{\widehat{\mathfrak{g}}}\left(\ell,0\right)$ the
$\widehat{\mathfrak{g}}$-submodule of
$V_{\widehat{\mathfrak{g}}}\left(\ell,0\right)$ generated by the vectors

$\displaystyle\left(t_{0}^{-1}x_{\pm\alpha_{i}}\right){}^{\epsilon_{i}\ell+1}\mathbf{1},\quad
i\in I,$ (6.11)

where $\epsilon_{i}=\frac{2}{\langle\alpha_{i},\alpha_{i}\rangle}$. It is
straightforward to check that $J_{\widehat{\mathfrak{g}}}\left(\ell,0\right)$
is $t_{0}^{-1}\text{d}_{0}$-stable (cf. [CLiT, Lemma 3.13]), and hence by
Remark 6.1 it is an ideal of the vertex algebra
$V_{\widehat{\mathfrak{g}}}\left(\ell,0\right)$. Let

$\displaystyle
L_{\widehat{\mathfrak{g}}}\left(\ell,0\right)=V_{\widehat{\mathfrak{g}}}\left(\ell,0\right)/J_{\widehat{\mathfrak{g}}}\left(\ell,0\right)$
(6.12)

be a quotient vertex algebra of
$V_{\widehat{\mathfrak{g}}}\left(\ell,0\right)$.

###  6.2 Conformal algebra $\mathcal{C}_{\mathfrak{g}}$

In order to associate the twisted toroidal EALA
$\widetilde{\mathfrak{g}}[\mu]$ with the vertex algebras
$V_{\widehat{\mathfrak{g}}}\left(\ell,0\right)$ and
$L_{\widehat{\mathfrak{g}}}\left(\ell,0\right)$, we define a
$G_{\mu}$-conformal algebra $\mathcal{C}_{\mathfrak{g}}$ such that
$\widehat{\mathcal{C}}_{\mathfrak{g}}\cong\widehat{\mathfrak{g}}$, and
$\widetilde{\mathcal{C}}_{\mathfrak{g}}\left[G_{\mu}\right]\cong\widetilde{\mathfrak{g}}\left[\mu\right]$.
As a vector space, we set

$\mathcal{C}_{\mathfrak{g}}=\left(\mathbb{C}\left[\partial\right]\otimes\mathcal{A}_{\mathfrak{g}}\right)\oplus\mathbb{C}\text{k}_{0},$

and define $\partial$ to be a linear transformation on
$\mathcal{C}_{\mathfrak{g}}$ such that

$\partial\left(\partial^{m}\otimes x\right)=\partial^{m+1}\otimes
x,\quad\partial\left(\text{k}_{0}\right)=0$

for $m\in\mathbb{N},$ $x\in\mathcal{A}_{\mathfrak{g}}.$ Let

$\displaystyle
Y^{-}:\mathcal{C}_{\mathfrak{g}}\to\text{Hom}\left(\mathcal{C}_{\mathfrak{g}},z^{-1}\mathcal{C}_{\mathfrak{g}}\left[z^{-1}\right]\right),\quad
a\mapsto\sum_{i\in\mathbb{N}}a_{i}z^{-i-1}$

be the unique linear map such that the property (2.7) holds, and the
nontrivial $i$-products on
$\mathcal{A}_{\mathfrak{g}}\oplus{\mathbb{C}}\text{k}_{0}$ are as follows:

$\displaystyle\left(t_{1}^{m}\otimes u\right)_{0}\left(t_{1}^{n}\otimes
v\right)=t_{1}^{m+n}\otimes\left[u,v\right]+\left\langle u,v\right\rangle
m\left(\partial\otimes K_{m+n}\right)+\delta_{m+n,0}\left\langle
u,v\right\rangle m\text{k}_{1},$ $\displaystyle\left(t_{1}^{m}\otimes
u\right)_{1}\left(t_{1}^{n}\otimes v\right)=\left(m+n\right)\left\langle
u,v\right\rangle K_{m+n}+\delta_{m+n,0}\left\langle
u,v\right\rangle\text{k}_{0},$
$\displaystyle\left(D_{r}\right)_{0}\left(t_{1}^{m}\otimes
u\right)=m\left(\partial\otimes\left(t_{1}^{r+m}\otimes
u\right)\right),\quad\left(t_{1}^{m}\otimes
u\right)_{0}\left(D_{r}\right)=r\left(\partial\otimes\left(t_{1}^{r+m}\otimes
u\right)\right),$ $\displaystyle\left(D_{r}\right)_{1}\left(t_{1}^{m}\otimes
u\right)=\left(t_{1}^{m}\otimes
u\right)_{1}\left(D_{r}\right)=\left(r+m\right)t_{1}^{r+m}\otimes u,$
$\displaystyle\left(D_{r}\right)_{0}\left(K_{s}\right)=r\left(\partial\otimes
K_{r+s}\right)+\delta_{r+s,0}r\text{k}_{1},$
$\displaystyle\left(K_{s}\right)_{0}\left(D_{r}\right)=s\left(\partial\otimes
K_{r+s}\right)+\delta_{r+s,0}\left(-r\text{k}_{1}+\partial\otimes\text{k}_{0}\right),$
$\displaystyle\left(D_{r}\right)_{1}\left(K_{s}\right)=\left(K_{s}\right)_{1}\left(D_{r}\right)=\left(r+s\right)K_{r+s}+\delta_{r+s,0}\text{k}_{0},$
$\displaystyle\left(\text{d}_{1}\right)_{0}\left(t_{1}^{m}\otimes
u\right)=-\left(t_{1}^{m}\otimes
u\right)_{0}\left(\text{d}_{1}\right)=mt_{1}^{m}\otimes
u,\quad\left(\text{d}_{1}\right)_{1}\left(\text{k}_{1}\right)=\left(\text{k}_{1}\right)_{1}\left(\text{d}_{1}\right)=\text{k}_{0},$
$\displaystyle\left(\text{d}_{1}\right)_{0}\left(K_{r}\right)=-\left(K_{r}\right)_{0}\left(\text{d}_{1}\right)=rK_{r},\quad\left(\text{d}_{1}\right)_{0}\left(D_{r}\right)=-\left(D_{r}\right)_{0}\left(\text{d}_{1}\right)=D_{r},$
$\displaystyle\left(D_{r}\right)_{0}\left(D_{s}\right)=r\partial\otimes
D_{r+s}+\delta_{r+s,0}\left(-r\partial^{2}\otimes\text{d}_{1}\right),\quad\left(D_{r}\right)_{1}\left(D_{s}\right)=\left(r+s\right)D_{r+s},$

where $u,v\in\dot{\mathfrak{g}},$ $m,n\in\mathbb{Z},$
$r,s\in\mathbb{Z}^{\ast}$, and we have used the convention $K_{0}=0=D_{0}$.

######  Proposition 6.2.

The vector space $\mathcal{C}_{\mathfrak{g}}$, together with the linear maps
$\partial$ and $Y^{-},$ as defined above, is a conformal algebra, and the
linear map
$\hat{i}_{\mathfrak{g}}:\widehat{\mathcal{C}}_{\mathfrak{g}}\to\widehat{\mathfrak{g}}$
defined by

$\displaystyle u\left(m\right)\mapsto t_{0}^{m}u,\
K_{n}\left(m\right)\mapsto\text{k}_{m+1,n},\
D_{n}\left(m\right)\mapsto\widehat{\text{d}}_{m-1,n},\
\text{k}_{0}\left(m\right)\mapsto\delta_{m,-1}\text{k}_{0},$

for $u\in\mathfrak{g},$ $m\in\mathbb{Z}$ and $n\in\mathbb{Z}^{\ast}$, is an
isomorphism of Lie algebras. Furthermore, the linear map
$\tilde{i}_{\mathfrak{g}}:\widetilde{\mathcal{C}}_{\mathfrak{g}}\to\widetilde{\mathfrak{g}}$
defined by

$\displaystyle u\left[m\right]\mapsto t_{0}^{m}u,\
K_{n}\left[m\right]\mapsto\text{k}_{m,n},\
D_{n}\left[m\right]\mapsto\widetilde{\text{d}}_{m,n},\
\text{k}_{0}\left[m\right]\mapsto\delta_{m,0}\text{k}_{0},\
\mathbf{d}\mapsto-\text{d}_{0}$ (6.13)

for $u\in\mathfrak{g},$ $m\in\mathbb{Z}$ and $n\in\mathbb{Z}^{\ast}$, is also
an isomorphism of Lie algebras.

###### Proof.

By definition, $\widehat{\mathcal{C}}_{\mathfrak{g}}$ is linearly spanned by
the elements $a\left(m\right),\text{k}_{0}\left(m\right)$ for
$a\in\mathcal{A}_{\mathfrak{g}}$, $m\in{\mathbb{Z}}$. Note that
$\text{k}_{0}\left(m\right)=0$ if $m\neq-1$. Using these with (6.3), it is
easy to see that the map $\hat{i}_{\mathfrak{g}}$ is an isomorphism of vector
spaces. Then $\widehat{\mathcal{C}}_{\mathfrak{g}}$ admits a Lie algebra
structure transferring from $\widehat{\mathfrak{g}}$ via the linear
isomorphism $\hat{i}_{\mathfrak{g}}$. By using the relations (4.4), (6.1),
(6.2), (6.4), (6.5) and the facts that
$\widehat{\text{d}}_{n,0}=\left(n+1\right)t_{0}^{n}\text{d}_{1}$ and
$\text{k}_{m,0}=-\frac{1}{m}t_{0}^{m}\text{k}_{1}$ for $n\in\mathbb{Z}$ and
$m\in\mathbb{Z}^{\ast}$, one can check that the Lie brackets on
$\widehat{\mathcal{C}}_{\mathfrak{g}}$ coincide with that in (2.8) with the
$i$-products defined above. Thus, by Lemma 2.6,
$\left(\mathcal{C}_{\mathfrak{g}},\partial,Y_{-}\right)$ is a conformal
algebra with $\hat{i}_{\mathfrak{g}}$ a Lie algebra isomorphism.

For the second assertion of the proposition, we note that
$\widetilde{\mathcal{C}}_{\mathfrak{g}}$ is linearly spanned by the elements
$a\left[m\right],\text{k}_{0}\left[m\right]$, $\mathbf{d}$ for
$a\in\mathcal{A}_{\mathfrak{g}}$ and $m\in{\mathbb{Z}}$. Also note that
$\text{k}_{0}\left[n\right]=0$ if $n\neq 0$. These together with (4.13) give
that $\tilde{i}_{\mathfrak{g}}$ is a linear isomorphism. Furthermore, by
comparing the Lie relation (2.16) in $\widetilde{\mathcal{C}}_{\mathfrak{g}}$
and the Lie relations (4.4), (4.9), (4.11), and (4.12) in
$\widetilde{\mathfrak{g}}$, it is straightforward to check that
$\tilde{i}_{\mathfrak{g}}$ is a Lie algebra homomorphism, as required. ∎

Now we define an automorphism group $G_{\mu}$ on $\mathcal{C}_{\mathfrak{g}}$
so that
$\widetilde{\mathcal{C}}_{\mathfrak{g}}\left[G_{\mu}\right]\cong\widetilde{\mathfrak{g}}\left[\mu\right]$.
We first define a linear transformation $R_{\mu}$ on
$\mathcal{C}_{\mathfrak{g}}$ by

$\displaystyle R_{\mu}\left(\partial^{m}\otimes
x\right)=\partial^{m}\otimes\mu\left(x\right),\
R_{\mu}\left(\text{k}_{0}\right)=\text{k}_{0},\
R_{\mu}\left(\partial^{m}\otimes K_{n}\right)=\partial^{m}\otimes K_{n},$
$\displaystyle R_{\mu}\left(\partial^{m}\otimes\left(t_{1}^{n}\otimes
h\right)\right)=\partial^{m}\otimes\left(t_{1}^{n}\otimes\dot{\mu}\left(h\right)\right)+\rho_{\mu}\left(h\right)\partial^{m+1}\otimes
K_{n},$ $\displaystyle R_{\mu}\left(\partial^{m}\otimes
D_{n}\right)=\partial^{m}\otimes
D_{n}-\partial^{m+1}\otimes\left(t_{1}^{n}\otimes\boldsymbol{h}\right)+\frac{\left\langle\boldsymbol{h},\boldsymbol{h}\right\rangle}{2}\partial^{m+2}\otimes
K_{n},$

for $x\in\mathfrak{g}_{\alpha}$ with
$\alpha\in\Delta^{\times}\cup\left\\{0\right\\}$, $h\in\dot{\mathfrak{h}},$
$m\in\mathbb{Z}$ and $n\in\mathbb{Z}^{\ast}$.

######  Lemma 6.3.

The linear transformation $R_{\mu}$, as defined above, is an automorphism of
$\mathcal{C}_{\mathfrak{g}}$ with order $T$.

###### Proof.

Note that $R_{\mu}\circ\partial=\partial\circ R_{\mu}$ on
$\mathcal{C}_{\mathfrak{g}}$. Then we have a linear map
$\widehat{R}_{\mu}:\widehat{\mathcal{C}}_{\mathfrak{g}}\rightarrow\widehat{\mathcal{C}}_{\mathfrak{g}}$
determined by (2.12). Via the isomorphism
$\hat{i}_{\mathfrak{g}}:\widehat{\mathcal{C}}_{\mathfrak{g}}\rightarrow\widehat{\mathfrak{g}}$
given in Proposition 6.2, $\widehat{R}_{\mu}$ induces a linear map on
$\widehat{\mathfrak{g}}$ determined by

$\begin{split}\widehat{R}_{\mu}\left(t_{0}^{m}x\right)=t_{0}^{m}\mu\left(x\right),\quad\widehat{R}_{\mu}\left(t_{0}^{m}t_{1}^{m}\otimes