# Simple modules for twisted Hamiltonian extended affine Lie algebras

Santanu Tantubay  Santanu Tantubay: Harish-Chandra Research Institute, A CI of
Homi Bhaba National Institute, Chhatnag Road, Jhunsi, Allahabad 211 019,
India<EMAIL_ADDRESS><EMAIL_ADDRESS>,  Priyanshu
Chakraborty  Priyanshu Chakraborty: Inistitute of Mathematical Sciences, A CI
of Homi Bhaba National Institute, Chennai, 600113, India,
<EMAIL_ADDRESS><EMAIL_ADDRESS>and  Punita Batra  Punita
Batra: Harish-Chandra Research Institute, A CI of Homi Bhaba National
Institute, Chhatnag Road, Jhunsi, Allahabad 211 019, India<EMAIL_ADDRESS>
###### Abstract.

In this paper we consider the twisted Hamiltonian extended affine Lie algebra
(THEALA). We classify the irreducible integrable modules for these Lie
algebras with finite dimensional weight spaces when the finite dimensional
center acts non-trivially. This Lie algebra has a triangular decomposition,
which is different from the natural triangular decomposition of twisted full
toroidal Lie algebra. Any irreducible integrable module of it is a highest
weight module with respect to the given triangular decomposition. In this
paper we describe the highest weight space in detail.

###### Key words and phrases:

Extended affine Lie algebras, Classical Hamiltonian Lie algebras

###### 2010 Mathematics Subject Classification:

17B67, 17B66

##  1\. Introduction

Extended affine Lie algebras (EALAs) form a category of important Lie algebras
consisting of finite dimensional simple Lie algebras, affine Lie algebras and
class of some other Lie algebras. Twisted toroidal extended affine Lie
algebras are examples of EALAs. The structure theory of EALAs have been
developed by several mathematicians like Allison, Azam, Berman, Gao, Neher,
Pianzola and Yoshii (see [2], [16], [17] and references therein). In 2004, Rao
classified all the irreducible integrable modules for toroidal Lie algebras
with finite dimensional weight spaces. In 2005, Rao and Jiang considered the
full toroidal Lie algebra and they identified the highest weight space with
Jet modules of derivations of Laurent polynomial rings. Later in 2018, Rao and
Batra considered the more general Lie algebra, called twisted full toroidal
Lie algebra and they classified all the irreducible integrable weight modules
for this Lie algebra. Full toroidal or twisted full toroidal Lie algerbas are
not examples of extended affine Lie algebras. So instead of adding full
derivations one adds a subalgebra of derivations space in order to make it an
extended affine Lie algebra. Twisted toroidal extended affine algebra is such
an example. In 2020, Rao, Batra, Sharma classified all the irreducible
integrable modules with finite dimensional weight spaces for twisted toridal
extended affine Lie algebra for nonzero level. In [22], Yuly Billig and John
Talboom gave a classification result for the category of jet modules for
divergence zero vector fields on torus, which plays a pivotal role in the
classification result of twisted toridal extended affine Lie algebras. John
Talboom gave a similar result for Hamiltonian vector fields on torus in [11].
In [1], Rao studied structure theory of Hamiltonian extended affine Lie
algebra and its integrable simple weight modules. In this article we consider
the twisted form of Hamiltonian extended affine Lie algebra and study its
integrable simple weight modules. In order to classify all these modules we
use a result of [11].

Let $\mathfrak{g}$ be a finite dimensional simple Lie algebra over
$\mathbb{C}$ and $A_{n}$ be the Laurent polynomial ring in $n\geq 2$ commuting
variables $t_{1},\dots,t_{n}$. Let $L(\mathfrak{g})=\mathfrak{g}\otimes A_{n}$
be a loop algebra. Let $L(\mathfrak{g})\oplus\Omega_{A_{n}}/dA_{n}$ be the
universal central extension of $L(\mathfrak{g})$. This is called the toroidal
Lie algebra (TLA). Let $Der(A_{n})$ be the Lie algebra of all the derivations
of $A_{n}$. Then $L(\mathfrak{g})\oplus\Omega_{A_{n}}/dA_{n}\oplus Der(A_{n})$
is called the full toroidal Lie algebra (FTLA). TLA and FTLA are not EALA, as
they do not admit any non-degenerate symmetric invariant bilinear form. So
instead of $Der(A_{n})$, one takes ${S}_{n}$ the subalgebra of $Der(A_{n})$,
consisting of divergence zero vector fields. The Lie algebra
$L(\mathfrak{g})\oplus\Omega_{A_{n}}/dA_{n}\oplus S_{n}$ is called a toroidal
extended affine Lie algebra which admits a non-degenerate symmetric bilinear
form and hence is an example of EALA. Now we take $n=2k$, a positive even
integer and let $H_{n}$ be the classical Hamiltonian Lie subalgebra of
Der($A_{n}$), then a quotient algebra of
$L(\mathfrak{g})\oplus\Omega_{A_{n}}/dA_{n}\oplus H_{n}$ becomes an extended
affine Lie algebra, called as Hamiltonian extended affine algebra.

Here we consider more general EALAs. We take $\sigma_{1},\dots,\sigma_{n}$ to
be finite order commuting automorphisms of $\mathfrak{g}$ and consider the
multi-loop algebra
$L(\mathfrak{g},\sigma)=\oplus_{r\in\mathbb{Z}^{n}}\mathfrak{g}(\bar{r})\otimes
t^{k}$. Corresponding to a diagram automorphism, Batra studied finite
dimensional modules of twisted multiloop algebras in [23]. The finite
dimensional representations of multiloop algebras (both untwisted and twisted)
have been studied by Lau in [18]. We assume that this multi-loop algebra is a
Lie torus. Now we consider the universal central extension of multi-loop
algebra and add $H_{n}$, the classical Hamiltonian Lie algebra. We quotient
this resulatant Lie algebra by $K(m)$ (See definition of $K(m)$ above
Proposition 2.1). We denote it by $\tau$. In this paper we will classify
irreducible integrable modules with finite dimensional weight spaces for
$\tau$, where the zero degree central operators act non-trivially for any
$n\geq 2$. We make use of some important results from [1] and [11] in order to
classify our modules.

The paper has been organized as follows. In Section 2, we recall the
definition of multiloop algebra, Lie torus and classical Hamiltonian Lie
algebra and then construct the twisted Hamiltonian extended affine Lie
algebra. Towards the end of Section 2, we recall a Proposition about the
dimension of graded component of central extension part. In Section 3, we fix
a Cartan subalgebra and with respect to this Cartan subalgebra we give a root
space decomposition of $\tau$. We define integrable modules for this Lie
algebra and give a triangular decomposition of $\tau$. Now we are taking a
different triangular decomposition of $\tau$ from the triangular decomposition
of toroidal or full toroidal Lie algebra for the level zero or non zero case.
With respect to this triangular decomposition we prove that any irreducible
integrable weight module is a highest weight module. Then the highest weight
space becomes an irreducible module for the zeroth part of triangular
decomposition of $\tau$. Infact it becomes $Z^{n-2}$-graded. We also show that
the whole central extension part of zeroth part acts trivially on the highest
weight space. This was not the case for twisted full toridal Lie algebra. In
Section 4, we study the highest weight space in detail. We take subalgbera $L$
of $\tau^{0}$ and a subspace $W$ of $M$, and we prove that $W$ is an $L$
submodule of $M$. We also prove that $M/W$ is completely reducible module for
$L$ and we identify each irreducible component of $M/W$ with tensor product of
irreducible module for a semisimple Lie algebra and symplectic algebra.
Finally we prove our main Theorem 4.3.

##  2\. Notation and Preliminaries

Let $n=2k$ be a positive integer and consider $A_{n}=C[t_{1}^{\pm
1},t_{2}^{\pm 1}\dots,t_{n}^{\pm 1}]$ be a Laurent polynomial ring in $n$
commuting variables $t_{1},\dots,t_{n}$ . We assume $\mathfrak{g}$ a finite
dimensional simple Lie algebra over $\mathbb{C}$ with a Cartan subalgebra
$\mathfrak{h}$ and let $\sigma_{1},\sigma_{2},\dots,\sigma_{n}$ be the
commuting automorphisms of $\mathfrak{g}$ of order $m_{1},m_{2},\dots,m_{n}$
respectively. For $m=(m_{1},m_{2},\dots,m_{n})\in\mathbb{Z}^{n}$ we define
$\Gamma=m_{1}\mathbb{Z}\oplus\dots\oplus m_{k-1}\mathbb{Z}\oplus
m_{k+1}\mathbb{Z}\oplus\dots\oplus m_{2k-1}\mathbb{Z}$,
$\Gamma_{0}=m_{k}\mathbb{Z}\oplus m_{2k}\mathbb{Z}$ and
$\Lambda:=\mathbb{Z}^{n-2}/\Gamma$, then
$m^{\prime}=(m_{1},\dots,m_{k-1},m_{k+1},\dots,m_{2k-1})\in\Gamma$. Set
$\bar{\Gamma}=m_{1}\mathbb{Z}\oplus\dots\oplus m_{2k}\mathbb{Z}$. For
convenience we write $\Gamma\oplus\Gamma_{0}=$ $\bar{\Gamma}$ throughout the
paper. Then we have
$\mathfrak{g}=\displaystyle{\bigoplus_{\underline{r}\in{\mathbb{Z}^{n}}/{\bar{\Gamma}}}}\mathfrak{g}(\underline{r})$,
where
$\mathfrak{g}(\underline{r})=\\{X\in\mathfrak{g}|\sigma_{i}(X)=\zeta_{i}^{r_{i}}X,1\leq
i\leq n\\}$, where $\zeta_{i}$ are $m_{i}$-th primitive root of unity for
$i=1,\dots,n$.
Define the multiloop algebra
$L(\mathfrak{g},\sigma)=\displaystyle{\bigoplus_{{r}\in{\mathbb{Z}^{n}}}}\mathfrak{g}(\underline{r})\otimes
t^{r}$ with the Lie brackets

$[x\otimes t^{a},y\otimes t^{b}]=[x,y]\otimes t^{a+b},$

for $x\in\mathfrak{g}(\underline{a}),y\in\mathfrak{g}(\underline{b})$ and
$a,b\in\mathbb{Z}^{n}$.
Now let $\mathfrak{g}_{1}$ be any arbitrary finite dimensional simple Lie
algebra over $\mathbb{C}$ with a cartan subalgebra $\mathfrak{h}_{1}$. Let
$\Delta(\mathfrak{g}_{1},\mathfrak{h}_{1})=supp_{\mathfrak{h}_{1}}(\mathfrak{g}_{1})$.
Then
$\Delta_{1}^{\times}=\Delta^{\times}(\mathfrak{g}_{1},\mathfrak{h}_{1})=\Delta(\mathfrak{g}_{1},\mathfrak{h}_{1})-\\{0\\}$
is an irreducible reduced finite root system with atmost two root lengths.
Define

$\Delta_{1,en}^{\times}=\begin{cases}\Delta_{1}^{\times}\cup
2\Delta_{1,sh}^{\times}&\text{if $\Delta_{1}^{\times}=B_{l}$ types}\\\
\Delta_{1}^{\times}&\text{ otherwise}.\end{cases}$

######  Definition 2.1.

A finite dimensional $\mathfrak{g}_{1}$-module $V$ is said to satisfy
condition $(M)$ if $V$ is irreducible with dimension greater than 1 and
weights of $V$ relative to $\mathfrak{h}_{1}$ are contained in
$\Delta_{1,en}^{\times}$.

######  Definition 2.2.

A multiloop algebra $L(\mathfrak{g},\sigma)$ is called a Lie torus ($LT$) if

  * (1)

$\mathfrak{g}(\bar{0})$ is a finite dimensional simple Lie algebra.

  * (2)

For $\underline{r}\neq 0$ and $\mathfrak{g}(\underline{r})\neq 0$,
$\mathfrak{g}(\underline{r})\cong U(\underline{r})\oplus W(\underline{r})$,
where $U(\underline{r})$ is trivial as $\mathfrak{g}(\underline{0})$-module
and either $W(\underline{r})$ is zero or satisfy condition (M).

  * (3)

The order of the group generated by $\sigma_{i},1\leq i\leq n$ is equal to the
product of the orders of each $\sigma_{i}$, for $1\leq i\leq n$.

Let $A_{n}(m)=\mathbb{C}[t_{1}^{\pm m_{1}},\dots,t_{n}^{\pm m_{n}}]$ and
$\Omega_{A_{n}(m)}$ be a vector space spanned by the symbols $t^{r}K_{i},1\leq
i\leq n,r\in\bar{\Gamma}$. Also assume that $dA_{n}(m)$ be the subspace of
$\Omega_{A_{n}(m)}$ spanned by $\sum_{i=1}^{n}r_{i}t^{r}K_{i},$
$r\in\bar{\Gamma}$. Set $Z=\Omega_{A_{n}(m)}/dA_{n}(m)$, a typical element of
$Z$ is given by $K(u,r)=\displaystyle{\sum_{i=1}^{n}}u_{i}t^{r}K_{i}$, for
$u=(u_{1},\dots,u_{n})\in\mathbb{C}^{n}$ and $r\in\bar{\Gamma}$. It is well
known that

$\overline{LT}=LT\oplus\Omega_{A_{n}(m)}/dA_{n}(m)$

is the universal central extension of ${LT}$ with the following brackets:

$\displaystyle[x(p),y(q)]=[x,y](p+q)+(x|y)K(p,p+q),$ (2.1)

where $x(p)=x\otimes t^{p}$, $p,q\in\mathbb{Z}^{n}$.

Let $Der(A_{n}(m))$ be the space of all derivations of $A_{n}(m)$. A basis for
$Der(A_{n}(m))$ is given by {$d_{i},t^{r}d_{i}|1\leq i\leq n,0\neq
r\in\bar{\Gamma}$}. For
$u=(u_{1},\dots,u_{n})\in\mathbb{C}^{n},r\in\bar{\Gamma}$, set
$D(u,r)=\displaystyle{\sum_{i=1}^{n}}u_{i}t^{r}d_{i}$. $Der(A_{n}(m))$ forms a
Lie algebra with the Lie brackets

$[D(u,r),D(v,s)]=D(w,r+s),$

where $u,v\in\mathbb{C}^{n},r,s\in\bar{\Gamma}$ and $w=(u,s)v-(v,r)u$.
Now $Der(A_{n}(m))$ acts on $\Omega_{A_{n}(m)}/dA_{n}(m)$ by

$D(u,r).K(v,s)=(u,s)K(v,r+s)+(u,v)K(r,r+s).$

It is well known that $Der(A_{n}(m))$ admits an abelian extension of
$\Omega_{A_{n}(m)}/dA_{n}(m)$ with the Lie brackets

$\displaystyle[D(u,r),K(v,s)]=(u,s)K(v,r+s)+(u,v)K(r,r+s),$ (2.2)
$\displaystyle[D(u,r),D(v,s)]=D(w,r+s)+(u,s)(v,r)K(r,r+s),$ (2.3)

where $u,v\in\mathbb{C}^{n},r,s\in\bar{\Gamma}$ and $w=(u,s)v-(v,r)u$.
For $s=(s_{1}\dots,s_{n})\in\bar{\Gamma}$ we define
$\bar{s}=(s_{k+1},\dots,s_{2k},-s_{1},\dots,-s_{k}).$ Now consider a Lie
subalgebra of $Der(A_{n}(m))$ given by

$H_{n}(m)=span\\{d_{i},h_{r}=D(\bar{r},r)|1\leq i\leq n,r\in\bar{\Gamma}\\}.$

It is easy to see that $[h_{r},h_{s}]=(\bar{r},s)h_{r+s}$. This Lie algebra is
known as Hamiltonian Lie algebra (See [1],[11]). Clearly $H_{n}(m)$ induces an
action on $Z$. Let $\bar{\tau}=\overline{LT}\oplus H_{n}(m)$, this becomes a
Lie algebra with the brackets (2.1), (2.2), (2.3) and

$[h_{r},x\otimes t^{s}]=(\bar{r},s)x\otimes t^{r+s},$

for all $x\in\mathfrak{g}(\underline{s}),r\in\bar{\Gamma},s\in\mathbb{Z}^{n}$.
Define
$K(m)=span\\{K(u,r)|u\in\mathbb{C}^{n},r\in\bar{\Gamma}\setminus\\{0\\},(u,\bar{r})=0\\}$.
It is easy to see that $K(m)$ is an ideal of $\bar{\tau}$.

######  Proposition 2.1.

([1], Proposition 3.1)

  1. (1)

$Z/K(m)$ is $\bar{\Gamma}$-graded with dim$(Z/K(m))_{r}=1$ having basis
element $K(\bar{r},r),$ for $r\neq 0.$

  2. (2)

dim$(Z/K(m))_{0}=n$ with basis $K_{i},1\leq i\leq n$.

∎

Define $\tau_{n}=\tau=LT\oplus Z/K(m)\oplus H_{n}(m)$ and construct a bilinear
form on $\tau$ by

$(x(r)|y(s))=\delta_{r,-s}(x|y),$ $\forall
x\in\mathfrak{g}(\underline{r}),y\in\mathfrak{g}(\underline{s}),r,s\in\mathbb{Z}^{n};$

$(h_{r}|K(\bar{s},s))=\delta_{r,-s}(\bar{r},\bar{s}),$ where
$r,s\in\bar{\Gamma}\setminus\\{0\\}$.

$(d_{i},K_{j})=\delta_{i,j}$ for $1\leq i,j\leq n$.

All other brackets of bilinear form are zero. We see that $\tau$ becomes an
extended affine Lie algebra and is called a twisted Hamiltonian extended
affine Lie algebra (Twisted HEALA).

Let $\mathfrak{h}(\underline{0})$ denote a Cartan subalgebra of
$\mathfrak{g}(\underline{0})$. Then by [13] Lemma 3.1.3,
$\mathfrak{h}(\underline{0})$ is ad-diagonalizable on $\mathfrak{g}$ and
$\bigtriangleup^{\times}=\bigtriangleup^{\times}(\mathfrak{g},\mathfrak{h}(\underline{0}))$
is an irreducible finite root system in $\mathfrak{h}(\underline{0})$
(Proposition 3.3.5,[13]). Let
$\bigtriangleup_{0}:=\bigtriangleup(\mathfrak{g}(\underline{0}),\mathfrak{h}(\underline{0}))$.
One of the main properties of Lie tori is that
$\bigtriangleup:=\bigtriangleup_{0,en}$ (Proposition 3.2.5,[2]). Let
$\pi=\\{\alpha_{1}.\alpha_{2},...,\alpha_{d}\\}$ be the simple roots of
$\bigtriangleup_{0}.$ Let $Q$ be the root system of $\bigtriangleup_{0}$ and
$Q^{+}=\displaystyle{\bigoplus_{i=1}^{d}}\mathbb{Z}_{+}\alpha_{i}$. Here
$\mathbb{Z}_{+}$ denote the set of non-negative integers.

######  Remark 1.

$\overline{LT}$ is a Lie $\mathbb{Z}^{n}$ torus of type
$\bigtriangleup_{0,en}.$ Then by [20], Definition 4.2, $\overline{LT}$ is
generated as Lie algebra by
$(\overline{LT})_{\alpha}=\displaystyle{\bigoplus_{r\in\mathbb{Z}^{n}}}\mathfrak{g}(\underline{r},\alpha)\otimes
t^{r}$, $\alpha\in(\bigtriangleup_{0,en})^{\times}$.

##  3\. Existence of Highest weight space

In this section we will give a root space decomposition of $\tau$. Let
${H}=\mathfrak{h}(\underline{0})\displaystyle{\bigoplus_{i=1}^{n}}\mathbb{C}K_{i}\bigoplus_{i=1}^{n}\mathbb{C}d_{i}$
be our Cartan subalgebra for the root space decomposition of $\tau$. Define
$\delta_{i},w_{i}\in H^{*}$ by setting

$\delta_{i}(\mathfrak{h}(\underline{0}))=0,\delta_{i}(K_{j})=0$ and
$\delta_{i}(d_{j})=\delta_{ij}$, and

$w_{i}(\mathfrak{h}(\underline{0}))=0$,$w_{i}(K_{j})=\delta_{ij}$ and
$w_{i}(d_{j})=0$.

Take $\delta_{\beta}=\sum_{i=1}^{n}\beta_{i}\delta_{i}$ for
$\beta\in\mathbb{C}^{n}$. For $r\in\mathbb{Z}^{n}$, we shall refer to the
vector $\delta_{r+\gamma}$ as the translate of $\delta_{r}$ by the vector
$\delta_{\gamma}$, where $\gamma\in\mathbb{C}^{n}$. Define
$\mathfrak{g}(\underline{r},\alpha):=\\{x\in\mathfrak{g}(\underline{r})|[h,x]=\alpha(h)x,\forall
h\in\mathfrak{h}(\underline{0})\\}$. By the properties of Lie torus
$\mathfrak{g}(\underline{r})=\displaystyle{\bigoplus_{\alpha\in\mathfrak{h}(\underline{0})^{*}}}\mathfrak{g}(\underline{r},\alpha)$.
Then we have
$\tau=\displaystyle{\bigoplus_{\beta\in\bigtriangleup}}\tau_{\beta}$, where
$\bigtriangleup\subseteq\\{\alpha+\delta_{r}|\alpha\in\bigtriangleup_{0,en},r\in\mathbb{Z}^{n}\\}$,

$\tau_{\alpha+\delta_{r}}=\mathfrak{g}(\underline{r},\alpha)\otimes t^{r}$ for
$\alpha\neq 0$,

$\tau_{\delta_{r}}=\mathfrak{g}(\underline{r},0)\otimes
t^{r}\oplus\mathbb{C}K(\bar{r},r)\oplus\mathbb{C}h_{r}$ for $0\neq
r\in\bar{\Gamma}$,

$\tau_{\delta_{r}}=\mathfrak{g}(\underline{r},0)\otimes t^{r}$ for
$r\in\mathbb{Z}^{n}\setminus\bar{\Gamma}$ and $\tau_{0}=H$.

We call elements of $\bigtriangleup$ roots of $\tau$. A root
$\beta=\alpha+\delta_{r}$ is called a real root if $\alpha\neq 0$. Let
$\bigtriangleup^{re}$ denote the set of all real roots and
$\beta^{\vee}=\alpha^{\vee}+\frac{2}{(\alpha|\alpha)}\displaystyle{\sum_{i=1}^{n}}r_{i}K_{i}$
be co-root of $\beta$, where $\alpha^{\vee}$ is the co-root of
$\alpha\in\bigtriangleup_{0,en}$. For $\gamma\in\bigtriangleup^{re}$, define
$r_{\gamma}(\lambda)=\lambda-\lambda(\gamma^{\vee})\gamma$ for $\lambda\in
H^{*}$. Let ${W}$ be the Weyl group of $\tau$ generated by
$r_{\gamma},\forall\gamma\in\bigtriangleup^{re}$.

######  Definition 3.1.

A $\tau$ -module $V$ is called integrable if

  * (1)

$V=\bigoplus_{\lambda\in H^{*}}V_{\lambda}$, where $V_{\lambda}=\\{v\in
V|h.v=\lambda(h)v,$ $\forall\,h\in H\\}$ and $dim(V_{\lambda})<\infty$.

  * (2)

All the real root vectors act locally nilpotently on $V,$ i.e.,
$\mathfrak{g}(\underline{r},\alpha)\otimes t^{r}$ acts locally nilpotently on
$V$ for all $0\neq\alpha\in\bigtriangleup_{0,en}$.

We have the following Proposiotion as [5].

######  Proposition 3.1.

([5]) Let $V$ be an irreducible integrable module for $\tau$. Then

  * (1)

$P(V)=\\{\gamma\in H^{*}|V_{\gamma}\neq 0\\}$ is $W$\- invariant.

  * (2)

$dim(V_{\gamma})=dim(V_{w\gamma}),\forall w\in W$.

  * (3)

If $\lambda\in P(V)$ and $\gamma\in\bigtriangleup^{re}$, then
$\lambda(\gamma^{\vee})\in\mathbb{Z}$.

  * (4)

If $\lambda\in P(V)$ and $\gamma\in\bigtriangleup^{re}$, and
$\lambda(\gamma^{\vee})>0$, then $\lambda-\gamma\in P(V)$.

Now consider the triangular decomposition of $\tau$

$\tau_{++}=span\\{X(r),h_{s},K(\bar{s},s):r\in\mathbb{Z}^{n},s\in\bar{\Gamma},X\in\mathfrak{g}(\underline{r}),r_{k}>r_{2k},s_{k}>s_{2k}\\},$

$\tau_{+}=span\\{X_{\alpha}(r),h_{s},K(\bar{s},s):r\in\mathbb{Z}^{n},s\in\bar{\Gamma},X_{\alpha}\in\mathfrak{g}(\underline{r},\alpha),r_{k}=r_{2k}>0$

$\;or\;r_{k}=r_{2k}=0,\alpha>0,\;s_{k}=s_{2k}>0\\},$

$\tau^{0}=span\\{X(r),h_{s},K(\bar{s},s):r\in\mathbb{Z}^{n},s\in\bar{\Gamma},X\in\mathfrak{g}(\underline{r},0),r_{k}=r_{2k}=0,s_{k}=s_{2k}=0\\}$,

$\tau_{--}=span\\{X(r),h_{s},K(\bar{s},s):r\in\mathbb{Z}^{n},s\in\bar{\Gamma},X\in\mathfrak{g}(\underline{r}),r_{k}<r_{2k},s_{k}<s_{2k}\\},$

$\tau_{-}=span\\{X_{\alpha}(r),h_{s},K(\bar{s},s):r\in\mathbb{Z}^{n},s\in\bar{\Gamma},X_{\alpha}\in\mathfrak{g}(\underline{r},\alpha),r_{k}=r_{2k}<0$

$\;or\;r_{k}=r_{2k}=0,\alpha<0,\;s_{k}=s_{2k}<0\\}.$

Now we define $\tau^{+}=\tau_{++}\oplus\tau_{+}$ and
$\tau^{-}=\tau_{--}\oplus\tau_{-}$, therefor
$\tau=\tau^{-}\oplus\tau^{0}\oplus\tau^{+}$ is the triangular decomposition of
$\tau$. Note that
$[\tau_{++},\tau_{-}\oplus\tau_{+}]\subset\tau_{++},\;[\tau_{--},\tau_{-}\oplus\tau_{+}]\subset\tau_{--}$.

######  Remark 2.

We see that
$LT_{n-2}=span\\{X_{\alpha}(r),X(r),K(\bar{s},s):\alpha\in\bigtriangleup_{0,en},r_{k}=r_{2k}=s_{k}=s_{2k}=0\\}$
forms a Lie torus from the Lie algebra $\mathfrak{g}$ with automorphisms
$\sigma_{1},\dots,\sigma_{k-1},\sigma_{k+1},\dots,\sigma_{2k-1}$.

We shall now define automorphism of full toroidal Lie algebra
$FT=\mathfrak{g}\otimes A_{n}\oplus\Omega A_{n}/dA_{n}\oplus Der(A_{n})$,
where $\Omega A_{n}/dA_{n}$ can be defined as quotient space of $\Omega
A_{n}=span\\{t^{r}K_{i}:r\in\mathbb{Z}^{n},1\leq i\leq n\\}$ by
$d{A_{n}}=span\\{\sum_{i=1}^{n}r_{i}t^{r}K_{i}\\}$. Let $GL(n,\mathbb{Z})$ be
the group of invertible matrices with integer coefficients. Let $B\in
GL(n,\mathbb{Z})$ which acts on $\mathbb{Z}^{n}$, denote this action by
$Bu,\;u\in\mathbb{C}^{n}$. Now we define an automorphism of $FT$, again denote
it by $B$ by,

$B.X(r)=X(Br),\;B.K(u,r)=K(Bu,Br),\;B.D(u,r)=D(Fu,Br),$

where $F=(B^{T})^{-1}$.
Define

$K_{B}=span\\{K(Bu,Br)|(u,\bar{r})=0,u\in\mathbb{C}^{n},r\in\bar{\Gamma}\\}$
$H_{B}=span\\{d_{i},D(F\bar{r},Br)|1\leq i\leq
n,\;r\in\bar{\Gamma}\setminus\\{0\\}\\}$

We see that $[H_{B},K_{B}]\subseteq K_{B}$. Now take $\tau_{B}=LT_{B}\oplus
Z/K_{B}\oplus H_{B}$, where $LT_{B}$ is the image of $LT$ under the
automorphism $B$ of $FT$. Now under this automorphism $\tau\cong\tau_{B}$. In
order to avoid notational complexity instead of working with $\tau_{B}$, we
will work with $\tau$ after applying $B$ also.
Let $V$ be an irreducible integrable module with finite dimensional weight
spaces with respect to the Cartan subalgebra $H$, on which not all $K_{i}$ act
trivially. Then after twisting the module by a suitable isomorphism of above
type, we can assume that only $K_{k}$ acts non-trivially on the module and
$K_{i}$ acts trivially for $1\leq i\leq n$ and $i\neq k$.

######  Proposition 3.2.

Suppose $V$ is an irreducible integrable module for $\tau$ with finite
dimensional weight spaces with respect to the Cartan $H$. If the central
element $K_{k}$ acts as positive integer and $K_{i}$ acts trivially for $1\leq
i\leq n$ and $i\neq k$, then there exists a weight $\lambda$ such that

  1. (1)

$X_{\alpha}(r).V_{\lambda}=0$, where
$r\in\mathbb{Z}^{n},\;X_{\alpha}\in\mathfrak{g}(\underline{r},\alpha),\;r_{k}>0$.

  2. (2)

$h_{s}.V_{\lambda}=K(\bar{s},s).V_{\lambda}=0$, where
$s\in\bar{\Gamma},\;s_{k}>0$

###### Proof.

We can prove this Proposition similarly as Theorem 5.2 of [4]. Note that
instead of zero-th coordinte we need to work out with $k$-th coordinate here.
∎

Let

$B_{n,n}=\begin{pmatrix}1&0&\cdots&0&\cdots&0\\\ 0&1&\cdots&0&\cdots&0\\\
\vdots&\vdots&\ddots&\vdots&\ddots&\vdots\\\
0&0&\cdots&b_{k,k}&\cdots&b_{k,2k}\\\
\vdots&\vdots&\ddots&\vdots&\ddots&\vdots\\\
0&0&\cdots&b_{2k,k}&\cdots&b_{2k,2k}\end{pmatrix}$

and

$\left(\begin{array}[]{ccc}b_{k,k}&b_{k,2k}\\\
b_{2k,k}&b_{2k,2k}\end{array}\right)=\left(\begin{array}[]{ccc}a&1\\\
a-1&1\end{array}\right)$

with $2a-1>0$, all diagonal entries of $B_{n,n}$ are 1, rest of all other
entries are zero except $b_{k,k},b_{k,2k},b_{2k,k},b_{2k,2k}$. Now we twist
the module by an isomorphism $B\in GL(n,\mathbb{Z})$. Note that after twisting
the module by $B_{n,n}$, we have a weight space $V_{\lambda}$ of $V$ such that
$V_{\lambda+\eta+\delta_{s}}=0$, where $\eta\in Q$, $s_{k}-s_{2k}>0$, by
Proposition 3.2.
Define an ordering on the $H^{*}$ by $\lambda\leq\mu$ in $H^{*}$ if and only
if
$\mu-\lambda=\displaystyle{\sum_{i=1}^{d}n_{i}\alpha_{i}}+n_{d+k}\delta_{k}+n_{d+2k}\delta_{2k}$
where $n_{i}\in\mathbb{Z}$, either $(i)$ $n_{d+k}-n_{d+2k}>0$, or $(ii)$
$n_{d+k}=n_{d+2k}>0$ or $(iii)$ $n_{d+k}=n_{d+2k}=0$ and
$n_{i}\in\mathbb{Z}_{+}$, $1\leq i\leq d$. By an ordering of
$\mathfrak{h}(\underline{0})^{*}$ we mean that the oredering ”$\leq$”
restricted to $\mathfrak{h}(\underline{0})^{*}$.

######  Theorem 3.1.

Let $V$ be an irreducible integrable module for $\tau$ with finite dimensional
weight spaces with respect to the Cartan subalgebra $H$, then there exists a
weight space $V_{\mu}$ of $V$ such that $\tau^{+}.V_{\mu}=0$.

###### Proof.

We consider the Lie algebra
$\tilde{g}=span\\{X(r),K(\bar{s},s),h_{s},K(u,0),D(u,0):X\in\mathfrak{g}(\underline{r}),r\in\mathbb{Z}^{n},\;s\in\bar{\Gamma},r_{k}=r_{2k},s_{k}=s_{2k},u\in\mathbb{C}^{n}\\}$.
Consider the subspace

$W=\displaystyle{\bigoplus_{\eta\in
Q,\;l\in\mathbb{Z}^{n},\;l_{k}=l_{2k}}}V_{\lambda+\eta+\delta_{l}}.$

We can see that $W$ is an integrable (may not be irreducible)
$\tilde{g}$-module with respect to the same Cartan $H$.
Fix a weight $\lambda_{1}=\lambda+\eta+\delta_{l}$ of $W$. Let
$g_{i}=\lambda_{1}(d_{i})$ for $1\leq i\leq n$ and set
$g=(g_{1},g_{2},..,g_{n})$. Consider $P_{g}(W)=\\{v\in
W:d_{i}.v=g_{i}v,\;1\leq i\leq n\\}$. It is easy to see that $P_{g}(V)$ is an
integrable module for $\mathfrak{g}(\underline{0})$ with finite dimensional
weight spaces. Now by Lemma 3.5 of ([19]), we have $P_{g}(V)$ a finite
dimensional module for $\mathfrak{g}(\underline{0})$. Hence $P_{g}(V)$ has
only finitely many weights and consider the maximal weight $\lambda_{2}$
corresponding to $\lambda_{1}$ with respect to the ordering $"\leq"$. So
$\lambda_{1}\leq\lambda_{2}$ and $\lambda_{2}+\eta$ is not a weight of $W$,
for all $\eta>0$ and $\eta\in\bigtriangleup_{0,en}.$ Now we can use the method
used in Theorem 5.2 of [4] to prove that there exists a weight
$\mu=\lambda+\eta^{\prime}+\delta_{r}$ of $W$, where $\eta^{\prime}\in
Q,r\in\mathbb{Z}^{n},r_{k}=r_{2k}$ such that $V_{\mu+\eta+\delta_{s}}=0$ for
$\eta\in Q,s\in\mathbb{Z}^{n},s_{k}=s_{2k}>0$ or $s_{k}=s_{2k}=0$ but $\eta\in
Q^{+}$. Now we can prove $\tau^{+}.V_{\mu}=0$ with the help of similar method
used in Theorem 5.3 of [1]. ∎

The following Proposition is standard.

######  Proposition 3.3.

  1. (1)

$M=\\{v\in V:\tau^{+}.v=0\\}$ is a non-zero irreducible $\tau^{0}$ module.

  2. (2)

The weights of $M$ lies in the set
$\\{\mu+\delta_{r}:r_{k}=r_{2k}=0,r\in\mathbb{Z}^{n}\\},$ for a fixed weight
$\mu$ of $M$.

∎

######  Proposition 3.4.

If $\mu(h)=0,\;\forall\;h\in\mathfrak{h}(\underline{0})$ and for some weight
$\mu$ of $M$, then $M$ becomes an $H_{n-2}(m^{\prime})$ irreducible module.

###### Proof.

Suppose $\alpha$ be a positive root of $\bigtriangleup_{0,en}$ such that
$\mathfrak{g}(\underline{r},-\alpha)\neq 0$. Let $Y_{\alpha}\otimes
t^{r}\in\mathfrak{g}(\underline{r},-\alpha)\otimes t^{r}$ acting non-trivially
on $v_{\mu}$ for some weight vector $v_{\mu}$ of $M$. Then
$\mu-\alpha+\delta_{r}$ is a weight of $V$. Hence by Proposition 3.1(3), we
have
$r_{\alpha}(\mu-\alpha+\delta_{r})=\mu-\alpha+\delta_{r}-(\mu-\alpha+\delta_{r})(\alpha^{\vee})\alpha=\mu+\alpha+\delta_{r}$
is a weight of $V$, a contradiction. By Remark 1 and Remark 2, $(LT\oplus
Z/K(m))\cap\tau^{0}$ acting trivially on $v_{\mu}$. Consider the non-zero
subspace $W=\\{v\in M:(LT\oplus Z/K(m))\cap\tau^{0}.v=0\\}$ of $M$. We see
that $W$ is a $\tau^{0}$-module, hence by irreducibility $W=M$. Note that
$d_{k},d_{2k}$ are central in $\tau^{0}$ and hence acts as scalar on $M$.
Therefore $M$ is irreducible module for $H_{n-2}(m^{\prime})$. ∎

Classification of irreducible weight modules for $H_{n}$ is still unknown, So
we can not comment on irreducible weight modules of $H_{n-2}$. Now throughout
the paper we assume that $\bar{\mu}=\mu|_{\mathfrak{h}(\underline{0})}\neq 0$.
Therefore there exists a simple root $\alpha\in\bigtriangleup_{0}$ such that
$\mu(h_{\alpha})\neq 0$.
We fix some $i,\;1\leq i\leq n,\;i\neq k,2k$ and consider the extended loop
algebra $\mathfrak{g}(\underline{0})\otimes\mathbb{C}[t_{i}^{\pm
m_{i}}]\oplus\mathbb{C}d_{i}$. Let $\theta$ be the highest root of
$\mathfrak{g}(\underline{0})$, $\theta^{\vee}$ be its co-root and $W_{0}$ be
the Weyl group of $\mathfrak{g}(\underline{0})$. Also assume that $W_{i}$ be
the Weyl group of the loop algebra
$\mathfrak{g}(\underline{0})\otimes\mathbb{C}[t_{i}^{\pm
m_{i}}]\oplus\mathbb{C}d_{i}$. We can see that $h.v={\mu}(h)v$ for all $v\in
M$ and $h\in\mathfrak{h}(\underline{0})$. Define
$t_{i,h}:(h(\underline{0})\oplus\mathbb{C}d_{i})^{*}\rightarrow(h(\bar{0})\oplus\mathbb{C}d_{i})^{*}$
by $t_{i,h}(\lambda)=\lambda-\lambda(h)\delta_{i}$, where
$h\in\mathbb{Z}(W_{0}\theta^{\vee})$. Note that
$\mathbb{Z}(W_{0}\theta^{\vee})$ lies in $\mathbb{Z}$ linear combination of
coroots for $\bigtriangleup_{0}$ and $\lambda(\theta^{\vee})>0$ for all weight
$\lambda$ of $M$. Therefore
$r_{\mu}=min_{h\in\mathbb{Z}(W_{0}\theta^{\vee})}\\{\mu(h):\mu(h)>0\\}\in\mathbb{N}$.
Let $r_{\bar{\mu}}=\mu(h_{0})$ for some
$h_{0}\in\mathbb{Z}(W_{0}\theta^{\vee})$. It is well known that $W_{i}$ is
semidirect product of $W_{0}$ and the commutative group generated by
$t_{i,h}$, $h\in\mathbb{Z}(W_{0}\theta^{\vee})$.

######  Lemma 3.1.

For all $s_{i}\in\mathbb{Z}$, there exists $w\in W_{i}$ such that
$w({\mu}+s_{i}\delta_{i})={\mu}+\bar{s_{i}}\delta_{i}$, where
$0\leq\bar{s_{i}}<r_{{\mu}}$.

###### Proof.

Let $s_{i}=\bar{s_{i}}+p_{i}r_{\mu}$. Then
$t_{i,h_{0}}^{p_{i}}.({\mu}+s_{i}\delta_{i})={\mu}+\bar{s_{i}}\delta_{i}$. ∎

Let $A_{n-2}(m)$ be the Laurent polynomial ring with $(n-2)$ variables
$t_{i}^{m_{i}}$ for $1\leq i\leq n$, $i\neq k,2k$. Now we consider the Lie
algebra $\tau_{n-2}=\mathfrak{g}(\underline{0})\otimes A_{n-2}(m)\oplus
Z_{n-2}/K_{n-2}(m)\oplus H_{n-2}(m)$. Let $W$ be the Weyl group of
$\tau_{n-2}$, then $W_{i}\subseteq W$ for $1\leq i\leq n$, $i\neq k,2k$.

######  Lemma 3.2.

Let $\delta_{r}=\displaystyle{\sum_{1\leq i\leq n,\,i\neq
k,2k}}r_{i}\delta_{i}$, where
$r=(r_{1},\dots,r_{k-1},r_{k+1},\dots,r_{n-1})\in\mathbb{Z}^{n-2}$. Let
$r_{i}=c_{i}+s_{i}m_{i}$, where $0\leq c_{i}<m_{i}$. Then there exists $w\in
W$ such that $w({\mu}+\delta_{r})={\mu}+\delta_{c}+\underset{1\leq i\leq
n,\;i\neq k,2k}{\sum}\bar{s_{i}}m_{i}\delta_{i}$, where
$0\leq\bar{s_{i}}<r_{{\mu}}$.

###### Proof.

The proof follows from Lemma 3.1. ∎

Now from Proposition 3.3, we see that $M$ is $\mathbb{Z}^{(n-2)}$-graded. Let
$M=\oplus_{r\in\mathbb{Z}^{(n-2)}}M_{r}$, where $M_{r}=\\{v\in
M:d_{i}.v=(\mu(d_{i})+r_{i})v,\;1\leq i\leq n,\;i\neq k,2k\\}$. Let
$N^{\prime}=\\{M_{r}:r_{i}=c_{i}+s_{i}m_{i},\;0\leq c_{i}<m_{i},\;0\leq
s_{i}\leq r_{\mu},1\leq i\leq n,i\neq k,2k\\}$, then by Lemma 3.1, we see that
weight spaces of $M$ are uniformly bounded by maximal dimension of elements of
$N^{\prime}$. Take $N=\oplus_{M_{r}\in N^{\prime}}M_{r}$, then $N$ is finite
dimensional. Now define $M^{\prime}(r)=\oplus_{r_{i}\leq
k_{i}<m_{i}+r_{i}}M_{k}$, then $M=\oplus_{r\in\Gamma}M^{\prime}(r)$ is
$\Gamma$-garded $\tau^{0}$ module.

Now we give existence of an irreducible integrable module for $\tau_{n-2}$
such that atleast one element of $M$ is injectively sits inside the module. We
consider the module $V_{1}=U(\tau_{n-2})M$ for $\tau_{n-2}$. If every proper
submodule of $V_{1}$ intersects $M$ trivially then take the quotient of
$V_{1}$ by sum of all proper submodules, the quotient space is such an
example. Now assume that $V_{1}$ has a proper $\tau_{n-2}$ submodule, say
$W_{1}$ such that $W_{1}\cap M\neq 0$. Take $M_{1}=U(\tau_{n-2})(W_{1}\cap
M)$, if every proper submodule of $M_{1}$ intersects $M$ trivially then
construct the quotient space as above. In the quotient space $W_{1}\cap M$
goes injectively. Again if $M_{1}$ has a proper submodule, say $W_{2}$ such
that $W_{2}\cap M\neq 0$, then construct $M_{2}=U(\tau_{n-2})(W_{2}\cap M)$.

######  Lemma 3.3.

The decreasing chain of $\tau_{n-2}$-submodules $M_{1}\supseteq M_{2}\supseteq
M_{3}\supseteq\dots$ terminates after a finite step.

###### Proof.

Let $M_{1}\supsetneq M_{2}$ be two submodules of $\tau_{n-2}$. Then by Lemma
3.2, we can find $N_{i}\subseteq N$ such that $M_{i}=U(\tau_{n-2})N_{i}$. Now
this Proposition follows from Proposition 5.2.6 of [24]. ∎

Let $M_{i}$ be such a minimal module. Then every proper submodule of $M_{i}$
intersect $M$ trivially. Let $\widetilde{M_{i}}$ be the quotient module of
$M_{i}$ by the sum of all proper submodules. So $M_{i}\cap M$ sits injectively
inside $\widetilde{M_{i}}$. Now $\widetilde{M_{i}}$ is an irreducible
integrable module on which every $K_{i}$ acts trivially for $i\neq k,2k$. Now
using Proposition (7.4) of [1], we can find a vector $v\in\widetilde{M_{i}}$
such that $(Z/K(m))\cap\tau_{n-2}$ acts trivially on $v$. Now being an ideal
of $\tau_{n-2}$ along with the irreducibility of $\widetilde{M_{i}}$,
$(Z/K(m))\cap\tau_{n-2}$ acts trivially on $\widetilde{M_{i}}$.

######  Theorem 3.2.

$(Z/K(m))\cap\tau_{n-2}$ acts trivially on $M$.

###### Proof.

The proof follows from previous discussion and irreducibility of $M$ over
$\tau^{0}$. ∎

Let $h_{\alpha}$ be as defined above, i.e. $\mu(h_{\alpha})\neq 0$. Then we
have the following Proposition.

######  Proposition 3.5.

  * (1)

$h_{\alpha}\otimes t^{k}$ acts injectively on $M$ for every $k\in\Gamma$.

  * (2)

$h_{\alpha}\otimes t^{r}.h_{\alpha}\otimes
t^{s}=\lambda_{r,s}h_{\alpha}\otimes t^{r+s}$ on $M$, where
$\lambda_{r,s}=\lambda$ for all $r\neq 0,s\neq 0,r+s\neq 0$,
$\lambda_{r,-r}=\mu$ for all $r\neq 0$ and
$\lambda_{0,r}=\bar{\lambda}(h_{\alpha})$ for all $r\in\Gamma$. Further we
have $\mu\lambda_{0,r}=\lambda^{2}\neq 0$.

  * (3)

dim $(M^{\prime}(r))=$ dim $(M^{\prime}(s))$ for all $r,s\in\Gamma$.

###### Proof.

Follows from Theorem 9.1 of [1]. ∎

##  4\. classification of integrable simple modules

Now recall that our Lie algebra reduces to $\tau=LT\oplus H_{n}(m)$ with
$\tau^{0}=\displaystyle{\bigoplus_{\begin{subarray}{c}r\in\mathbb{Z}^{n}\\\
r_{k}=r_{2k}=0\end{subarray}}}\mathfrak{g}(\underline{r},0)\otimes t^{r}\oplus
H_{n-2}(m^{\prime})\oplus\mathbb{C}d_{k}\oplus\mathbb{C}d_{2k}$. Note that
$d_{k},d_{2k}$ are central in $\tau^{0}$, hence they act by scalars on $M$.
Take
$\mathfrak{g}^{\prime}=\\{x\in\mathfrak{g}\;|\;[h,x]=0,\;\sigma_{k}(x)=\sigma_{2k}(x)=x,\>\forall\>h\in\mathfrak{h}(\underline{0})\\}$.
Now since $\mathfrak{g}^{\prime}$ is invariant under $\sigma_{i}$’s (where
$i\neq k,2k$ ), therefore $\mathfrak{g}^{\prime}$ is $\Lambda$ graded. It is
easy to see that $L(\mathfrak{g}^{\prime},\sigma)=LT\cap\tau^{0}$ ($=LT^{0}$,
say). Let us take $H_{n-2}^{\prime}(m^{\prime})=$
span$\\{D(\bar{r},r)-D(\bar{r},0)|\;r\in\Gamma\\}$. We can easily check that
$H_{n-2}^{\prime}(m^{\prime})$ is a Lie subalgebra of $H_{n-2}(m^{\prime})$.
Furthermore let us set $L=H_{n-2}^{\prime}(m^{\prime})\ltimes
L(\mathfrak{g}^{\prime},\sigma)$ and $W=$ span$\\{h_{\alpha}\otimes
t^{r}.v-v|r\in\Gamma,v\in M\\}$. We can see that $W$ is an $L$-module.

######  Lemma 4.1.

  * (1)

$W$ is a proper $L$-submodule of $M$.

  * (2)

$\widetilde{V}=M/W$ is a finite dimensional $L$-module.

###### Proof.

Let $z_{i}=h_{\alpha}\otimes t^{m}_{i}$ for each $i=1,\dots,n$ and $i\neq
k,2k$. Without loss of generality we can assume that $\lambda_{r,s}=1$ for
$r\neq 0,s\neq 0,r+s\neq 0$. Therefore we can say that $W=$
span$\\{z_{i}.v-v|v\in M\\}$. Now same as Proposition 5.4(3) of [19], we can
find a proper $LT^{0}$-submodule of $M$, which contains $W$. ∎

Let $\beta_{i}=\mu(d_{i})$ for $1\leq i\leq n$ and $i\neq k,2k$. Then
$\beta=(\beta_{1}\dots,\beta_{n})\in\mathbb{C}^{n-2}$. Then for any $L$ module
$V^{\prime}$, we can give a $\tau^{0}$ module structure on
$L(V^{\prime})=V^{\prime}\otimes A_{n-2}$ by

$x\otimes t^{k}.(v_{1}\otimes t^{s})=((x\otimes t^{k}).v_{1})\otimes t^{k+s}$.

$D(\bar{r},r).(v_{1}\otimes t^{s})=((D(\bar{r},r)-D(\bar{r},0)).v_{1})\otimes
t^{r+s}+(\bar{r},s+\beta)(v_{1}\otimes t^{r+s})$

for all $v_{1}\in V^{\prime},x\in\mathfrak{g}(\underline{k},0)$ and
$D(\bar{r},r)\in H_{n-2}(m^{\prime})$.
For $v\in M$, let $\bar{v}$ be the image of $v$ in $\widetilde{V}$. Now define

$\phi:M\rightarrow L(\widetilde{V})$

by $v\mapsto\bar{v}\otimes t^{k}$ for $v\in M(k).$

This map is clearly a nonzero $\tau^{0}$ -module homomorphism. Hence by
irreducibility of $M$ it follows that $M\cong\phi(M)$ is a $\tau^{0}$
submodule of $L(\widetilde{V})$.
Clearly $L$ is naturally $\Lambda$ graded. Now since $M$ and $W$ are $Z^{n-2}$
graded, therefore they are naturally $\Lambda$ graded and hence so is
$\widetilde{V}$. Therefore
$\widetilde{V}=\oplus_{\bar{p}\in\Lambda}\widetilde{V}(\bar{p})$.
Now for $\bar{p}\in\Lambda$, we set

$L(\widetilde{V})(\bar{p})=\\{v\otimes
t^{k+r+p}|v\in\widetilde{V}(\bar{k}),r\in\Gamma,k\in\mathbb{Z}^{n-2}\\}$

It can be easily verified that $L(\widetilde{V})(\bar{p})$ is a $\tau^{0}$
submodule of $L(\widetilde{V})$.
Let $I(\bar{r},r)=D(\bar{r},r)-D(\bar{r},0)$ for all $r\in\Gamma$. Define
$H_{n-2}^{\prime}(m^{\prime})=span\\{I(\bar{r},r):r\in\Gamma\\}$, this becomes
a Lie subalgebra of $H_{n-2}(m^{\prime})$. The following result can be deduced
similarly as in [4].

######  Proposition 4.1.

  * (1)

$M\cong L(\widetilde{V})(\bar{0})$ as $\tau^{0}$ -module.

  * (2)

$\widetilde{V}$ is $\Lambda$ -graded-irreducible module over $L$.

  * (3)

$\widetilde{V}$ is completely reducible module over $L$ and all its
irreducible components are mutually isomorphic as
$H_{n-2}^{\prime}(m^{\prime})\ltimes\mathfrak{h}(\bar{0})\otimes
A_{n-2}(m^{\prime})$ -modules.

Now we will concentrate on irreducible representation of $L$. Let $(W,\pi)$ be
a finite dimensional representation of $L$. Let
$\pi(L(\mathfrak{g}^{\prime},\sigma))=\mathfrak{g}^{1}$, then
$\pi(L)=\mathfrak{g}^{1}\oplus\mathfrak{g}^{2}$, where $\mathfrak{g}^{2}$ is
the unique complement of $\mathfrak{g}^{1}$ in $\mathfrak{gl}(W)$ (Proposition
19.1(b) of [8] ). So $W$ will be an irreducible module for
$\mathfrak{g}^{1}\oplus\mathfrak{g}^{2}$. Therefore $W\cong W_{1}\otimes
W_{2}$, where $W_{1}$ and $W_{2}$ are irreducible modules for
$\mathfrak{g}^{1}$ and $\mathfrak{g}^{2}$ respectively (see [9] ). Let
$\mathfrak{g}^{\prime}=\mathfrak{g}^{\prime}_{ss}\oplus R$, where
$\mathfrak{g}^{\prime}$ and $R$ are Levi and radical part of
$\mathfrak{g}^{\prime}$. Then as
$\sigma_{i}(\mathfrak{g}^{\prime})=\mathfrak{g}^{\prime}$ and
$\sigma_{i}(R)=R$ for $1\leq i\leq n$, we have
$L(\mathfrak{g}^{\prime},\sigma)=L(\mathfrak{g}^{\prime}_{ss},\sigma)\oplus
L(R,\sigma)$. Now $W_{1}$ is irreducible module for
$L(\mathfrak{g}^{\prime},\sigma)$. As $R$ is a solvable ideal, it follows that
$\pi(L(R,\sigma))$ lies in the center of $\pi(L)$, which is atmost one
dimensional. Hence $L(R,\sigma)$ acts as a scalar on $W$. So $W_{1}$ will be
an irreducible module for $L(\mathfrak{g}^{\prime}_{ss},\sigma)$.

Fix a positive integer $l$. For each $i$, let $a_{i}=(a_{i,1},\dots,a_{i,l})$
such that

$\displaystyle a_{i,j}^{m_{i}}\neq a_{i,t}^{m_{i}},\,for\,j\neq t.$ (4.1)

Let $\mathfrak{g}$ be a finite dimensional semisimple Lie algebra. Let
$\sigma_{1},\dots\sigma_{n}$ be finite order automorphisms on $g$ of order
$m_{1},\dots m_{n}$ respectively. Let $L(\mathfrak{g},\sigma)$ be the
corresponding multiloop algebra. Let $I=\\{(i_{1},i_{2},\dots,i_{n})|1\leq
i_{j}\leq l\\}$. Now for $S=(i_{1},i_{2},\dots,i_{n})\in I$ and
$r=(r_{1},r_{2},\dots r_{n})\in\mathbb{Z}^{n}$,
$a_{S}^{r}=a_{1,i_{1}}^{r_{1}}a_{2,i_{2}}^{r_{2}}\cdots a_{n,i_{n}}^{r_{n}}.$
Now consider the evaluation map $\phi:\mathfrak{g}\otimes
A\rightarrow\bigoplus\mathfrak{g}$ ($l^{n}$ copies), $\phi(X\otimes
t^{r})=(a_{I_{1}}^{r}X,a_{I_{2}}^{r}X,\dots,a_{I_{l^{n}}}^{r}X)$, where
$I_{1},I_{2},\cdots I_{l^{n}}$ is some order on $I$. Now consider the
restriction of $\phi$ to $L(\mathfrak{g},\sigma)$.

######  Theorem 4.1.

$($[10]$)$ Let $W^{\prime}$ be a finite dimensional irreducible representation
of $L(\mathfrak{g},\sigma)$. Then the representation factors through
$\bigoplus\mathfrak{g}$ $($ $l^{n}$ copies$)$.

Here $W_{1}$ is irreducible module for $L(\mathfrak{g}^{\prime}_{ss},\sigma)$,
so the representation will factors through $l^{n-2}$ copies of
$\mathfrak{g}^{\prime}_{ss}$.

######  Proposition 4.2.

Let $W_{1}$ be irreducible module for $L(\mathfrak{g}^{\prime}_{ss},\sigma)$
as above. Then the representation of $L(\mathfrak{g}^{\prime}_{ss},\sigma)$
factors through only one copy of $\bigoplus\mathfrak{g}^{\prime}_{ss}$. So
$\mathfrak{g}^{1}_{ss}\cong\mathfrak{g}^{\prime}_{ss}$.

###### Proof.

We know by Theorem 4.1, that the representation factors through $l^{n-2}$
copies, for some positive integer $l$. we will prove here that $l=1$. Choose
the $i$-th piece of $\mathfrak{g}^{\prime}_{ss}$ and choose the proection of
the map $\pi$, say $\pi_{i}$ onto it. Doing the same calculation as in [3] we
will get $\pi_{i}(H_{n-2}^{\prime}(m^{\prime}))=0$ and $a_{I_{i}}^{r}=1$ for
all $r\in\Gamma$. Now suppose there is atleast two pieces, say $i$-th and
$j$-th piece is there. Therefore $I_{i}$ and $I_{j}$ are two different element
of $I$ with $a_{I_{i}}^{r}=1=a_{I_{j}}^{r}$ for all $r\in\Gamma$. Let
$I_{i}=(i_{1},i_{2},\dots,i_{n-2})$ and $I_{j}=(j_{1},j_{2},\dots j_{n-2})$.
Therefore there is $k$ with $1\leq k\leq n-2$ such that $i_{k}\neq j_{k}$. Now
if we take $r=(0,\dots,m_{k},\dots,0)$ then $a_{I_{i}}^{r}=1=a_{I_{j}}^{r}$
will give $a_{k,i_{k}}^{m_{k}}=a_{k,j_{k}}^{m_{k}}$, a contradiction to
equation (4.1). So there is atmost one piece. ∎

Now we know $\pi_{i}(H_{n-2}^{\prime}(m^{\prime}))=0$, therefore
$\mathfrak{g}^{2}\subseteq\pi(H_{n-2}^{\prime}(m^{\prime}))$. Now our aim is
to understand finite dimensional irreducible modules for
$H_{n-2}^{\prime}(m)$. We are going to establish a relation between finite
dimensional $H_{n}^{\prime}(m)$ modules and $H_{n}(m)\ltimes A_{n}(m)$-modules
with finite dimensional weight spaces. In order to do that we follow the
method used in [3].

Let $W$ be an irreducible finite dimensional $H_{n}^{\prime}(m)$ module. We
define $A_{n}(m)\rtimes H_{n}(m)$ module action on $L(W)=W\otimes A_{n}(m)$ in
the following way

$D(\bar{r},r).(w\otimes t^{k})=(I(\bar{r},r).w)\otimes
t^{r+k}+(\bar{r},\beta+k)w\otimes t^{r+k}$,
$D(u,0)(w\otimes t^{k})=(u,\alpha+k)w\otimes t^{k}$,
$t^{r}.w\otimes t^{k}=w\otimes t^{r+k}$, for all $u\in\mathbb{C}^{n},\;r(\neq
0),k\in\Gamma,w\in W$ and some $\alpha,\beta\in\mathbb{C}^{n}$.

We denote this module as $(\pi_{\alpha,\beta},L(W))$.
It is clear that $L(W)=\displaystyle{\bigoplus_{k\in\Gamma}}W\otimes t^{k}$ is
the weight space decomposition of $L(W)$. Consider the $H_{n}^{\prime}(m)$
submodule of $L(W)$, $W_{0}=\\{w\otimes t^{r}-w\;|\;r\in\Gamma,\;w\in W\\}$,
then $\overline{L(W)}=L(W)/W_{0}$ is an $H_{n}^{\prime}(m)$ submodule. Let
$(\theta,\overline{L(W)})$ be its corresponding representation. Now we define
a new representation $(\theta_{\xi},\overline{L(W)})$ of $H_{n}^{\prime}(m)$
by the following action

$\theta_{\xi}(I(\bar{r},r)).v=\theta(I(\bar{r},r))v+(\bar{r},\xi)v$, where
$\xi\in\mathbb{C}^{n}$.

It is easy to see $\theta_{0}=\theta$.

######  Lemma 4.2.

Let $W$ be a finite dimensional irreducible $H_{n}^{\prime}(m)$ module. Then
$L(W)$ is irreducible $A_{n}(m)\rtimes H_{n}(m)$ module.

###### Proof.

Let $U_{0}$ be a non-zero submodule of $L(W)$. Since $L(W)$ is a weight module
, so $u_{0}\otimes t^{r}\in U_{0}$ for some non-zero $u_{0}\in U_{0}$ and
$r\in\Gamma$. Now the action of $A_{n}(m)$ on $L(W)$ implies that
$u_{0}\otimes A_{n}(m)\in U_{0}.$ Hence we have $U_{0}=U_{1}\otimes A_{n}(m)$
for some non-zero subspace $U_{1}$ of $W$. Therefore to complete the proof, it
is sufficient to show that $U_{1}$ is a submodule of $W$.
Let $r\in\Gamma$, $u\in U_{1}$ and consider the action

$D(\bar{r},r)(u\otimes t^{-r})=I(\bar{r},r).u+(\bar{r},\beta-r)u.$

This imples that $I(\bar{r},r).u\in U_{1}$. Hence $U_{1}$ is a submodule of
$W$. ∎

######  Lemma 4.3.

Let $(\pi_{\alpha,\beta},L(W))$ be a finite dimensional irreducible
$A_{n}(m)\rtimes H_{n}(m)$ module for a finite dimensional module $(\eta,W)$
of $H_{n}^{\prime}(m)$. Then $(\theta_{\xi},\overline{L(W)}$ is irreducible
for $H_{n}^{\prime}(m)$.

###### Proof.

Note that $(\theta_{\alpha-\beta},\overline{L(W)})\cong(\eta,W)$. Moreover one
can see that if $W_{0}$ is a nonzero proper submodule of $W$, then
$W_{0}\otimes A_{n}(m)$ is a nonzero proper submodule of $L(W)$. ∎

######  Lemma 4.4.

Let $W$ be finite dimensional irreducible module for $\mathfrak{sp}_{n}.$ Then
$W$ can be made into $H_{n}^{\prime}(m)$-module by the action:
$I(\bar{r},r).w=(r^{t}{\bar{r}})w+(\bar{r},\zeta)w$, where
$\zeta\in\mathbb{C}^{n}$ and $r^{t}$ denote the transpose of the row vector
$r\in\Gamma.$

###### Proof.

Note that for $r\in\Gamma,$ $r^{t}\bar{r}\in\mathfrak{sp}_{n}$ and hence the
action is well define. It is easy to see that this action define a module
structure on $W$.

∎

######  Theorem 4.2.

Suppose $W$ is a finite dimensional irreducible $sp_{n}$-module. Let
$\alpha,\beta\in\mathbb{C}^{n}$. Take $L(W)=W\otimes A_{n}(m)$ and consider
the action $D(\bar{r},r)(w\otimes t^{k})=(\bar{r},k+\beta)w\otimes
t^{k+r}+(r^{t}{\bar{r}}).w\otimes t^{k+r}$, for $r(\neq 0)\in\Gamma$ and
$D(u,0)(w\otimes t^{k})=(u,\alpha+k)w\otimes t^{k}$ and $t^{r}(w\otimes
t^{k})=w\otimes t^{k+r}$. Then $L(W)$ is an irreducible module for
$H_{n}(m)\ltimes A_{n}(m)$. Moreover, all irreducible representations of
$H_{n}(m)\ltimes A_{n}(m)$ with finite dimensional weight spaces occur in this
way.

###### Proof.

The proof follows from Lemma 4.2, 4.3,4.4 and Theorem [11]. ∎

We know $\widetilde{V}$ is completely reducible $L$ module. Therefore
$\widetilde{V}=\oplus_{i=1}^{K}\widetilde{V}_{i}$ for some $K\in\mathbb{N}$.
Then by the previous discussion each $\widetilde{V}_{i}\cong W_{1}^{i}\otimes
W_{2}^{i}$ as $\mathfrak{g}^{\prime}_{ss}\oplus\mathfrak{sp}_{n-2}$ module,
where $W_{1}^{i}$, $W_{2}^{i}$ are irreducible modules for
$\mathfrak{g}^{\prime}_{ss}$ and $\mathfrak{sp}_{n-2}$ respectively. Since
each component $\widetilde{V}_{i}$ is isomorphic as
$H_{n-2}^{\prime}(m^{\prime})\ltimes(\mathfrak{h}(\underline{0})\otimes A(m))$
module, we can take $W_{2}^{i}\cong W_{2}^{1}$ ($=W_{2}$ ,say) as
$\mathfrak{sp}_{n}$-modules for each $i\in\\{1,\dots,K\\}$. Now consider
$W_{1}=\sum_{i=1}^{K}W_{1}^{i}$, which is a
$L(\mathfrak{g}^{\prime}_{ss},\sigma)$-module, in particular
$\mathfrak{g}^{\prime}_{ss}$ module. Since each $W_{1}^{i}$ is irreducible,
without loss of generality we can assume that the above sum is direct. It is
easy to see that $L$ is $\Lambda$-graded with zero-th component
$H_{n-2}^{\prime}(m^{\prime})\ltimes(\mathfrak{h}(\underline{0})\otimes A(m))$
and since $\widetilde{V}$ is $\Lambda$ graded irreducible module (Proposition
4.1), we can take $W_{1}$ as $\Lambda$-graded irreducible
$L(\mathfrak{g}^{\prime}_{ss},\sigma)$ module and $W_{2}$ a zero graded as
$H_{n-2}^{\prime}(m^{\prime})$-module which lies inside the zero-th graded
component of $L$.
We now define a $\tau^{0}$-module structure on $W_{1}\otimes W_{2}\otimes
A_{n}$ by

$X\otimes t^{k}(w_{1}\otimes w_{2}\otimes t^{l})=Xw_{1}\otimes w_{2}\otimes
t^{k+l}$, for $k,l\in\mathbb{Z}^{n-2}$ and
$X\in\mathfrak{g}^{\prime}_{ss}(\underline{k})$.

$D(\bar{r},r)(w_{1}\otimes w_{2}\otimes t^{k})=(u,k+\beta)w_{1}\otimes
w_{2}\otimes t^{k+r}+w_{1}\otimes(r{\bar{r}}^{t}).w_{2}\otimes t^{k+r}$ for
$r\neq 0$,

$D(u,0)(w_{1}\otimes w_{2}\otimes t^{k})=(u,k+\alpha)w_{1}\otimes w_{2}\otimes
t^{k}$.

Now take any one dimensional representation of $L(R,\sigma)$ say $\psi$. Then
for $y\in R(\underline{k})$ we take $y\otimes t^{k}(w_{1}\otimes w_{2}\otimes
t^{l})=\psi(y)(w_{1}\otimes w_{2}\otimes t^{k+l})$. Since $W_{1}$ is $\Lambda$
graded, which is compatible with $\Lambda$-gradation of
$\mathfrak{g}^{\prime}_{ss}$, so the submodule
$V^{\prime}=\bigoplus_{k\in\mathbb{Z}^{n}}W_{1,k}\otimes W_{2}\otimes t^{k}$
will be irreducible module for $\tau^{0}$. One can easily check that
$L(\widetilde{V})(\bar{0})\cong V^{\prime}$ as $\tau^{0}$-module.

######  Theorem 4.3.

Let $V$ be an irreducible integrable $\tau$ module with finite dimensional
weight spaces, with $K_{k}$ acting as $c_{0}$ and $K_{i}$ acts trivially for
$1\leq i\leq n$ and $i\neq k$. Then $V\cong
U(\tilde{\tau})M/M^{\textit{Rad}}$.

###### Proof.

Follows from the previous discussion. ∎

## References

  * [1] S. Eswara Rao, Hamiltonian Extended Affine Lie Algebra and Its Representation Theory, Journal of Algebra (628) 71-97.
  * [2] Bruce Allison, Stephen Berman, John Faulkner, Arturo Pianzola, Multiloop realization of extended affine Lie algebras and Lie tori, C Trans. Am. Math. Soc.361(9) (2009) 4807-4842
  * [3] S. Eswara Rao, Sachin S. Sharma, Punita Batra, Integrable modules for twisted toroidal extended affine Lie algebras, Journal of Algebra 556 (2020) 1057-1072
  * [4] Punita Batra, Senapathi Eswara Rao, On integrable modules for the twisted full toroidal Lie algebra, J. Lie Theory 28 (1)(2018) 79-105
  * [5] S. Eswara Rao, Classification of irreducible integrable modules for toroidal Lie algebras with finite dimensional weight spaces, J. of Algebra 277(2004)318-348
  * [6] Vyjayanthi Chari, Andrew Pressley, Weyl modules for classical and quantum affine algebras, Represent. Theory 5(2001) 191-223.
  * [7] Souvik Pal, S. Eswara Rao, Classification of level zero irreducible integrable modules for twisted full toroidal Lie algebra, Journal of Algebra, 578 (2021) 1-29
  * [8] James E. Humphreys, Introduction to Lie Algebras and Representation Theory, Graduate Texts in Mathematics, vol. 9,Springer- Verlag, New York-Berlin, 1978, second printing, revised.
  * [9] Haisheng Li, On certain categories of modules for affine Lie algebras, Math. Z. 248 (3) (2004) 635-664.
  * [10] Erhard Neher, Alistair Savage, Prasad Senesi, Irreducible finite-diemnsional representation of equivariant map algebras, Trans. Am. Math. soc. 364(5)(2012) 2619-2646.
  * [11] John Talboom, category $\jmath$ modules for Hamiltonian vector fields on torus, J. Lie theory 28 (2018) 903-914.
  * [12] Xiangqian Guo, Genquiang Liu, Jet modules for the centerless Virasoro-like algebra, J. Algebra Appl. 18(1) (2019) 1950002.
  * [13] Katsuyuki Naoi, Multiloop Lie algebras and the construction of extended affine Lie algebras. J. Algebra, 323(8):2103-2129, 2010
  * [14] V. Chari, Integrable representations of affine Lie algebras, Invent. Math. 85(1986), no. 2, 317-335.
  * [15] Chen, Fulin; Li, Zhiqiang; Tan, Shaobin; Classification of integrable representations for toroidal extended affine Lie algebras, J. Algebra 514(2021), 1-37.
  * [16] Erhard Neher, Extended affine Lie algebras, C. R. Math. Acad. Sci. Soc. R. Can. 26 (3) (2004) 90-96.
  * [17] Yoji Yoshii, Lie tori-a simple characterization of extended affine Lie algebras, Publ. Res. Inst. Math. Sci. 42 (3) (2006) 739-762.
  * [18] M. Lau, Representations of multiloop algebras, Pacific J. Math. 245 (2010), no. 1, 167-184.
  * [19] Souvik Pal, Integrable modules for graded Lie tori with finite-dimensional weight spaces, J. Pure and Applied Algebra.285(9)(2021)
  * [20] E.Neher, Extended affine Lie algebras-an introduction to their structure theory, in: Gemometric representation theory and Extended Affine Lie Algebras, in: Fields Inst. Commun.,vol.59, Amer. Math.Soc., Providence,RI,2011,107-167.
  * [21] V. Chari, J. Greenstein, Graded Level Zero Integrable Representations of Affine Lie Algebras, Transactions of American Mathematical Society, Vol. 360, No. 6, 2923-2940.
  * [22] Yuly Billig, John Talboom, Classification of category J modules for divergence zero vector fields on a torus, J. Algebra 500 (2018) 498–516.
  * [23] Punita Batra, Representations of twisted multi-loop Lie algebras, Journal of Algebra, 272(2004) 404-416.
  * [24] Souvik Pal, On level zero integrable modules over extensions of Lie tori, ”PhD thesis”, Harish-Chandra Research Institute, 2021.