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\begin{document}
\begin{center}
{\Large{\bf TOROIDAL $Z$-ALGEBRAS}} \\ [3.5cm]
{\bf S. Eswara Rao} \\
{\bf School of Mathematics} \\
{\bf Tata Institute of Fundamental Research} \\
{\bf Homi Bhabha Road} \\
{\bf Mumbai - 400 005}
 {\bf India} \\ [5mm]
{\bf email: senapati@math.tifr.res.in}
\end{center}

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\vskip 1cm
\noindent
{\Large{\bf Introduction}}
\vskip 5mm

The purpose of this paper is to construct faithful representation of
twisted toroidal Lie algebra of type ADE. Twisted toroidal Lie
algebras are universal central extensions of $N+1$ twisted muiltiloop
algebras.  If $N=0$ they are precisely the twisted affine Kac-Moody
Lie algebras.  In the process we recover the main results of [EM],
[MEY] (homogeneous picture), [B1] and [T] (principal picture) for the
non-twisted case.  These works have applications in the solutions of
differential equations.  See [B2], [ISW1] and [ISW2].  We expect
similar applications for the twisted case also.  Though the idea is the 
same our proofs are much shorter than those in the above works.  The paper 
can be seen as a generalisation of certain results of [LW].

Let ${\cal G}$ be the simple finite dimensional Lie algebra over the complex
numbers.  Let $A$ be a Laurent polynomial ring in $N+1$ commuting
variables.  Consider the multiloop algebra ${\cal G} \otimes A$, its
universal central extension $\tilde{\tau}$ the toroidal  Lie algebra.
Let $\theta$ be an automorphism of ${\cal G}$ of order $m$.  Then
$\theta$ can be 
extended to an automorphism of $\tilde{\tau}$  (Section 1).  Then
the subalgebra of $\theta$ fixed points inside $\tilde{\tau}$ is called
twisted toroidal Lie algebra $\tilde{L} ({\cal G}, \theta)$.  It is the  
universal central extension of the underlining multiloop algebra (See [BK]). 

In Section 1, we define a category $\underline{C}_k $ of $\tilde{L}
({\cal G},\theta)$ modules which satisfy a factorisation property first
introduced in [BY].  The factorisation property is not satisfied for
a general class of integrable modules.  But there are enough of
integrable  modules which satisfy the factorisation property.  For
example the vertex representation defined in [EM] and  the representation
considered in [BY] satisfy factorisation property. 


Next by following [LW] closely we define toroidal  $Z$-algebras (1.10) and
define a category $\underline{D}_k$-of $Z$-algebra modules.  We then
prove the important Proposition (2.6) which says that the categories
$\underline{C}_k$ and $\underline{D}_k$ are equivalent.  Thus by
constructing a $Z$- algebra module we get a module for
$\tilde{L}({\cal G},\theta)$. 

In section 3 we specialise to the homogeneous picture for the nontwisted 
case of type ADE.  We construct a module for the $Z$-toroidal Lie
algebra closely following the results of [LP].  Thereby constructing
a module for $\tilde{L} ({\cal G}, I_d) \cong \tilde{\tau}$ which  is
faithful.  This recovers the main result of [EM].  Our calculations
are certainly much shorter. 

In Section 4 we specialise to the principal picture.  This includes
the twisted and nontwisted toroidal Lie algebras.  We again
construct a module for the $Z$-toroidal Lie algebra by making use of
the corresponding results for the affine Kac-Moody Lie algebra from
[LW].  We have to consider the additional Fock space for this purpose.
Thus we get a module for our $\tilde{L} ({\cal G}, \theta)$. This result
recovers the main result of [B1] and [T].  Again our proof are much
shorter.  The twisted case is completely new.

In the process we have given the following realization of twisted
toroidal Lie-algebra.  Let $\pi$ be a Dynkin diagram automorphism of
${\cal G}$.   Define an automorphism $\theta$ of ${\cal G}$ as in the
section 4. 
Then we prove that $\tilde{\tau} ({\cal G}, \theta) \cong \tilde{L}
({\cal G}, \pi)$. 
This is what is called the principal realization in the affine case.  The
isomorphism is given explicitly in twisted case and it is completely
new even in the affine case (Proposition 4.10). 

\section*{Section 1}

Let ${\cal G}$ be a finite dimensional  semisimple Lie-algebra over the complex
numbers $\C$.  Let $<,>$ be a non-degenerate symmetric ${\cal G}$-invariant bilinear 
form on ${\cal G}$.  We fix a nonnegative integer $N$.  Let
$A=\C[ t^{\pm}, t_1^{\pm}, \cdots, t_N^{\pm}]$ be the ring of
Laurent polynomials in $N+1$ commuting variables.  Let
$\underline{r}=(r_1, \cdots, r_N) \epsilon \Z^N$.  Let $t^{\underline{r}} =
t_1^{r_1} t_2^{r_2} \cdots t_N^{r_N}$. Fix a positive integer $m$.  
Let $\Omega_A$ be a free
$A$-module of rank $N+1$  with basis $\{k_0, \cdots, k_N\}$.  Let
$d_A$ be the subspace of $\Omega_A$ spanned by elements of the form
$\frac{1}{m} r_0 t^{r_0} t^{\underline{r}} k_0+ \cdots + r_N t^{r_0}
t^{\underline{r}} k_N$.  Let $x (r_0, \underline{r}) = x \otimes
t^{r_0} t^{\underline{r}} \epsilon {\cal G} \otimes A$.  Then the toroidal
Lie-algebra $\tau = {\cal G} \otimes A \oplus \Omega_A /d_A$ is defined by
the following bracket.
$$
\begin{array}{lll}
[x (r_0,\underline{r}), y (s_0, \underline{s})] &=& \\
\left[ x,y \right] (r_0+s_0, \underline{r}+ \underline{s})+ \frac{<
x,y>}{m} r_0 
t^{r_0+s_0} t^{\underline{r}+ \underline{s}}k_0 
&+& <x,y> \displaystyle{\sum_{i=1}^{N}} r_i t^{r_0+s_0}
t^{\underline{r}+ \underline{s}} k_i
\end{array}
$$
 for $x,y \epsilon {\cal G},
\underline{r}, \underline{s} \epsilon \Z^N, r_0, s_0 \in \Z$.\\
 $\Omega_A/d_A$ is central.  

It is known that $\tau$ is the universal central extension of ${\cal G}
\otimes A$. (See [K], [MEY]).  (First note that the toroidal Lie
algebra defined by $m=1$ is isomorphic to the above).   Let $\underline{h}$ be a
Cartan subalgebra of ${\cal G}$.  Let $\theta$ be an automorphism of
${\cal G}$ such that 
$\theta (\underline{h}) =\underline{h}$ and of order $m$.  Assume
that the form $<,>$  is 
$\theta$-invariant that is $< \theta x, \theta y> =<x,y>$. Let
$\Z_m=\Z/_{m  \Z}$ 
be the cyclic group of order $m$.  Let $w$ be a primitive $m$ th root of
unity. 

\paragraph*{(1.2)} Let ${\cal G}_i = \{x \epsilon {\cal G} \mid
\theta x =w^ix \}$ for $i \epsilon 
\Z$.  Then ${\cal G} = \displaystyle{\oplus_{i \epsilon \Z_m}} {\cal G}_i$.
Note that $< 
{\cal G}_i, {\cal G}_j > =0$ unless $i+j \equiv 0 (m)$.  For $x
\epsilon {\cal G}$ write 
$x=\displaystyle{\sum_{i \epsilon \Z_m}} x_i$ where $\theta x_i=w^i x_i$.
Define $x_i=x_{\overline{i}}$ for $ i \epsilon \Z$ and $\overline{i}
\epsilon \Z_m$.  

Extend the automorphism $\theta$ to $\tau$ by $\theta (x
(r_0,\underline{r}))= {w}^{-r_o} \theta (x) (r_0, \underline{r})$
and $w (t_0^{r_o} t^{\underline{r}} k_i) = {w}^{-r_0}
t^{r_0} t^{\underline{r}} k_i, 0 \leq i \leq N$.  Let $\tilde{\tau}
= \tau \oplus D$ where $D$ is spanned by derivations $\{d_0, \cdots
d_N\}$ with bracket $[d_i, x(r_0, \underline{r})] =r_i x (r_0,
\underline{r})$, for $0 \leq i \leq N$.  Extend the automorphism $\theta$
to $\tilde{\tau}$ by  $\theta (d_i) =d_i$.  Let $(\Omega_A/_{d_A})_0$ be
the linear span of $t^{r_0} t^{\underline{r}} k_i$ where $r_0
\equiv 0 (m)$.  Consider the $\theta$ fixed points of $\tilde{\tau }$ say
$\tilde{L} ({\cal G}, \theta)$. 

\paragraph*{(1.3)}.  Let $L ({\cal G}, \theta) =
\displaystyle{\oplus_{\stackrel{i \epsilon \Z}{\underline{r} \epsilon \Z^N}}}
{\cal G}_i (i, \underline{r})$ and $\overline{L} ({\cal G}, \theta) =
L ({\cal G}, \theta) 
\oplus (\Omega_A/_{d_A})_0$.  Then clearly $\tilde{L} ({\cal G}, \theta) =
\overline{L} ({\cal G}, \theta) \oplus D$.

Since $<,>$ is non-degenerate and ${\cal G}$-invariant, its restriction to
$\underline{h}$ is also non-degenerate.  We identify $\underline{h}$
and $\underline{h}^*$ via this form.  Let $\phi$
be the root system of ${\cal G}$.  For $\beta \epsilon \phi$, choose the
corresponding non-zero root vectors $x_{\beta}$ such that
$[x_{\beta}, x_{- \beta}] =< x_{\beta},
{x}_{- \beta} > \beta$.  Let $\epsilon (\beta, \gamma)$ be a
non-zero number such that $[x_{\beta}, x_{\gamma}]= \epsilon (\beta,
\gamma) x_{\beta+\gamma}$.  Clearly the set of roots $\phi $ is $\theta$-
stable.  Then define $\eta (p, \beta)$ a non-zero scalar such that

\paragraph*{(1.4)} $\theta^p x_{\beta} = \eta (p, \beta) x_{\theta^p \beta}$.
For any vector space $V$ and for indeterminates
$\zeta_1, \cdots, \zeta_\ell$, denote $V \{\zeta_1, \cdots,
\zeta_{\ell}\}$ the space of formal Laurent series.  Further $V
[\zeta_1^{\pm 1}, \cdots, \zeta_n^{\pm 1}]$ denote finite formal Laurent
series.  We recall the following Proposition from [LW].  Definition
$\delta (\zeta) = \displaystyle{\sum_{i \in \Z}} \zeta^i \in
\C\{\zeta\}$.

\paragraph*{Proposition ~(1.5) (a)} (Proposition (2.2) of [LW]).  Let $f
(\zeta) \in V [\zeta, \zeta^{-1}]$. Then
$$f (\zeta) \delta (\zeta^m) = \displaystyle{\sum_{p \in \Z_m}} f (w^p)
\delta (w^{-p} \zeta).$$
For $x \epsilon {\cal G}, \underline{r} \epsilon \Z^N$
let $$x (\underline{r}, \zeta) = \sum_i x_i \otimes t^i t^{\underline{r}}
\zeta^i, \ k_i (\underline{r}, \zeta^m) = \sum_p t^{mp} t^r k_i
\zeta^{mp}.$$ 
For any infinite series $f (\zeta) = \sum b_i \zeta^i$, let $Df
\zeta= \sum i b_i \zeta^i$. 

\paragraph*{Proposition (1.5)}  The following relations hold for $x
(\underline{r}, \zeta)$ and $k_0 (\underline{r}, \zeta^m)$.  In fact
they define a Lie-algebra $\tilde{L} ({\cal G}, \theta)$.  For
$\beta_1, \beta_2 
\epsilon \phi, \underline{r}, \underline{s} \epsilon \Z^N$.
$$[x_{\beta_1} (\underline{r}, \zeta_1), x_{\beta_2} (\underline{s},
\zeta_2)] \leqno{(1)}$$
$$= \frac{1}{m} \displaystyle{\sum_{\theta^p \beta_1+ \beta_2 \epsilon \phi}}
\eta (p, \beta_1) \epsilon (\theta^p \beta_1, \beta_2) X_{\theta^p
\beta_1+ \beta_2} 
(\underline{r}+\underline{s}, \zeta_2) \delta (w^{-p} \zeta_1/\zeta_2)$$ 
$$- \frac{1}{m} < x_{\beta_2}, {x}_{- \beta_2} >
\displaystyle{\sum_{\theta^p \beta_1+ \beta_2 =0}} \eta
(p, \beta_1) \beta_2 (\underline{r}+ \underline{s}, \zeta_2) \delta
(w^{-p} \zeta_1/\zeta_2)$$

$$+ \frac{1}{m} < x_{\beta_2}, x_{- \beta_2} >
\displaystyle{\sum_{\theta^p \beta_1+ \beta_2 =0}} \eta
(p, \beta_1) (\displaystyle{\sum_{i=1}^{N}} r_i k_i (\underline{r}+
\underline{s}, \zeta_2^m) \delta (w^{-p} \zeta_1/\zeta_2)$$
$$+\frac{k_0(\underline{r}+ \underline{s}, \zeta^m_2)}{m} D \delta
({w}^{-p} \zeta_1/\zeta_2))$$
$$[\beta_1 (\underline{r}, \zeta_1), \beta_2 (\underline{s},
\zeta_2)] \leqno{(2)}$$
$$= \frac{1}{m} \displaystyle{\sum_{p \epsilon \Z_m}} < \theta^p \beta_1,
\beta_2> ( \displaystyle{\sum_{i=1}^{N}} r_i k_i
(\underline{r}+\underline{s}, \zeta_2^m) \delta ({w}^{-p}
\zeta_1/ \zeta_2)$$
$$+ \frac{k_0 (\underline{r}+ \underline{s}, \zeta^m_2)}{m} D \delta
(w^{-p} \zeta_1/ \zeta_2)).$$
$$[\beta_1 (\underline{r}, \zeta_1), x_{\beta_2} (\underline{s},
\zeta_2)] \leqno{(3)}$$
$$\frac{1}{m} \displaystyle{\sum_{p \epsilon \Z_m}} < \theta^p \beta_1,
\beta_2> x_{\beta_2} (\underline{r}+ \underline{s}, \zeta_2) \delta
(w^{-p} \zeta_1/\zeta_2)$$
$$\frac{1}{m} D k_0 (\underline{r}, \zeta^m)+
\displaystyle{\sum_{i=1}^{N}} r_i k_i (\underline{r}, \zeta^m) =0
\leqno{(4)}$$ 
$$[d_i, k_j (\underline{r}, \zeta^m)] = r_i k_j (\underline{r},
\zeta^m) \ {\rm for} \ 1 \leq i \leq N \ {\rm and} \ \ 0 \leq j \leq
N \leqno{(5)}$$ 
$$[d_0, k_j (\underline{r}, \zeta^m)]=D k_j (\underline{r}, \zeta^m),
\ 0 \leq j \leq N
\leqno{(6)}$$ 
$$x_{\beta} (\underline{r}, w^p \zeta) = \eta(p, \beta) x_{\theta^p
\beta} (\underline{r}, \zeta) \leqno{(7)}$$
$$k_i (\underline{r}, \zeta^m) \ {\rm is \ central \ for } \ 0 \leq i \leq
N. \leqno{(8)}$$

\paragraph*{Proof}  (4)  follows from the definition of
$(\Omega_A/d_A)_0$. (5), (6), (7) and (8) are easy to see.  
First consider the following:
$$[x (\underline{r}, \zeta_1), y (\underline{s}, \zeta_2)]=F+G_1+G_2.$$
Where 
$$F= \displaystyle{\sum_{i,j}}  [x_i, y_j] t_0^{i+j}
t^{\underline{r}+ \underline{s}}  \zeta_1^i \zeta_2^j,$$
$G_1=\frac{1}{m} \ \displaystyle{\sum_{i,j}} \ i < x_i,
y_j> t_0^{i+j} t^{\underline{r}+ \underline{s}} k_0   \zeta_1^i
\zeta_2^j$,
$$G_2 = \displaystyle{\sum_{\ell=1}^{N}} \displaystyle{\sum_{i,j}} r_{\ell}
<x_i, y_j > t_0^{i+j} t^{\underline{r}+\underline{s}} k_\ell \zeta_1^i
\zeta_2^{j}.$$
From the proof of Theorem (2.3) of [LW] it following that
$$F= \frac{1}{m} \displaystyle{\sum_{p \in \Z_m}} [\theta^p x, y]
(\underline{r}+ \underline{s}, \zeta_2) \delta (w ^{-p} \zeta_1 /
\zeta_2).$$
Now consider $ [\theta^p x_{\beta_1}, x_{\beta_2}]= \eta (p, \beta_1)
[x_{\theta^p \beta_1}, x_{\beta_2}]$.
$$=\eta (p, \beta_1) \epsilon (\theta^p \beta_1, \beta_2) x_{\theta^
p (\beta_1+\beta_2)} \ {\rm if} \ \theta^p \beta_1+ \beta_2 \in \Phi$$
$=-\eta (p, \beta_1)<x_{\beta_2}, x_{- \beta_2} > \beta_2 \ {\rm if} \
\theta^p \beta_1+ \beta_2=0$ \\
$=0$ if $ \theta^{\beta} \beta_1+ \beta_2 \neq 0$ and not a root.  

Thus for $x=x_{\beta_1}$ and $y= x_{\beta_2}, F$ equals to the first and 
second term of
RHS in (1).  For $G_1$ and $G_2$, first note that
$<x_i, y_j> =0$ if $i+j \neq 0 (m)$ by (1.2).  Thus
$$mG_1= \displaystyle{\sum_{i,p}} i<x_i, y_{-i+mp}> t_0^{mp}
t^{\underline{r}+\underline{s}} k_0 (\zeta_1/ \zeta_2)^i \zeta_2^{mp}.$$
$$=\displaystyle{\sum_{i,p}} i < x_i, y_{-i} > t_0^{mp}
t^{\underline{r}+\underline{s}} k_0 (\zeta_1/ \zeta_2)^i \zeta_2^{mp}$$
$$=\displaystyle{\sum_{i,p}} i <x_i, y> t_0^{mp}
t^{\underline{r}+\underline{s}} k_0 (\zeta_1/ \zeta_2)^i \zeta_2^{mp}$$
$$=D (\displaystyle{\sum_{i}} <x_i, y> (\zeta_1/ \zeta_2)^i) k_0
(\underline{r}+\underline{s}, \zeta_2^m)$$
$$=D(\displaystyle{\sum_{i=0}^{m-1}} <x_i, y> (\zeta_1/\zeta_2)^i \delta
((\zeta_1 / \zeta_2)^m).k_0 (\underline{r}+\underline{s}, \zeta_2^m)$$
$$= \frac{1}{m} \displaystyle{\sum_{p}} < \theta^p x,y>D \delta(w^{-p}
\zeta_1/\zeta_2) k_0 (\underline{r}+\underline{s}, \zeta_2^m). \eqno{1.5(1)} $$
Similarly
$$G_2= \displaystyle{\sum_{i=1}^{N}} r_{\ell} < \theta^p x,y> \delta 
(w^{-p} \zeta_1/ \zeta_2) k_{\ell} (\underline{r}+ \underline{s},
\zeta_2^m ) ~ \eqno{  1.5(2)}$$

Again for $x=x_{\beta_1}$ and $y=x_{\beta_2}, < \theta^p x, y> = \eta (p,
\beta_1) < x_{\theta^p \beta_1}, x_{\beta_2}>=0 \ {\rm if} \ \theta^p
\beta_1+\beta_2 \neq 0$
$$=\eta(p, \beta_1) <x_{-\beta_2},x_{\beta_2}> \ {\rm if} \ \theta^p
\beta_1+\beta_2=0.$$
This completes the proof (1).  To see (2), take $x=\beta_1$ and
$y=\beta_2$.  Then $F=0$.  From (1.5 (1) and 1.5(2), (2) will follow.
To see (3) take $x=\beta_1$ and $y= x_{\beta_2}$ and note that $G_1= 0$
and
$G_2=0$ and $F$ is equal to RHS of 3.  This completes the proof of the 
Proposition (1.5).

Now we define a category $\underline{C}_k $ of $\tilde{L} ({\cal G},
\theta)$- modules.  

\paragraph*{Definition (1.6)} A $\tilde{L} ({\cal G}, \theta)$-module $V$ is in
$\underline{C}_k$ if
\begin{enumerate}
\item[(1)] $k_o$ acts as $k$.
\item[(2)] $V= \displaystyle{\oplus_{z \epsilon \C}} V_z, V_z = \{ v \epsilon V
\mid d_0 v=zv \}$.  Assume for any $z$ there exists $\ell_0 \ni V_{z+ \ell}
=0$ for $ \ell > \ell_0$ 
\item[(3)] $\frac{1}{k} X ( \underline{r}, \zeta) k_0 (\underline{s},
\zeta^m) =X (\underline{r}+ \underline{s}, \zeta) $ \\
$\frac{1}{k} k_i (\underline{r}, \zeta^m) k_o (\underline{s},
\zeta^m) = k_i (\underline{r}+ \underline{s}, \zeta^m) \ {\rm for} \
0 \leq i \leq N$.
\end{enumerate}

\paragraph*{Remark (1.7)}  Condition (3) is not satisfied for most
of the modules.  But there are enough of them which are sufficient for
a realization of $L ({\cal G}, \theta)$.  For examples vertex operator
representation of [EM] satisfy the condition (3) as well as the representations
considered in [BY].

\paragraph*{(1.8)} Consider the Lie subalgebra of
$\tilde{L} ({\cal G}. \theta), \tilde{\underline{h}} =
\displaystyle{\oplus_{i \epsilon \Z}} h_i \otimes t^i \oplus \C k_0 \oplus
\C d_0 (h \epsilon
\underline{h})$. The bracket is given by 
$$[h_i \otimes t^i, h_j \otimes t^j] = i k_0 <h_i, h_j >
\delta_{i+j,0}.$$ 
Clearly $\tilde{\underline{h}}$ is
$\Z$-graded.  Let $M (k)$ be a Verma module of level $k$ for
$\tilde{\underline{h}}$.  Then it is a standard fact that $M(k)$ is
irreducible whenever $k$ is non-zero.

\paragraph*{Proposition (1.9)}  Any module $V $ in $\underline{C}_k (k
\neq 0)$ has the following decomposition as
$\tilde{\underline{h}}$-modules. 
$V \cong M (k) \otimes \Omega_V$ where 
$$\Omega_V = \{v \epsilon V \mid h_i \otimes t^i v=0 \  {\rm for} \ i >0 \}$$
see Proposition 5.4 of [LW].

We now define toroidal $Z$ algebras. Notation as earlier.  For
$\alpha \epsilon \phi, \underline{r} \epsilon \Z^N$ let $Z (\alpha,
\underline{r}, \zeta)$ be a series in $\zeta $ with integral powers.
For $\underline{r} \epsilon \Z^N$ let $k_i (\underline{r}, \zeta^m)$ be a
series in $\zeta^m$.  The toroidal $Z_k$-algebra or simply
$Z_k$-algebra is spanned by the components of $Z (\alpha,
\underline{r}, \zeta), k_i (\underline{r}, \zeta^m)$ and
$\underline{h}_0$, by the following relation. 

\paragraph*{(1.10) ~~ Relations} $\alpha, \beta, \beta_1, \beta_2 \in
\Phi, \underline{r}, \underline{s} \in \Z^N$.

\begin{enumerate}
\item[(1)] $\frac{1}{k} Z (\alpha, \underline{r}, \zeta) k_0
(\underline{s}, \zeta^m) = Z (\alpha, \underline{r}+ \underline{s},
\zeta)$ 
\item[(2)] $\frac{1}{k} k_o( \underline{r}, \zeta^m) \ k_i
(\underline{s}, \zeta^m) = k_i ( \underline{s}+ \underline{r},
\zeta^m), \  0 \leq i \leq N$ .
\item[(3)] $\displaystyle{\sum_{i=1}^{N}} r_i k_i (\underline{r},
\zeta^m)+ \frac{1}{m} D k_0 (\underline{r}, \zeta^m) =0$.
\item[(4)] $[d_0, Z (\beta, \underline{r}, \zeta)]=D Z (\beta,
\underline{r}, \zeta)$ 
\item[(5)] $[d_0, k_i (\underline{r}, \zeta^m)] = D k_i
(\underline{r}, \zeta^m), \ 0 \leq i \leq N$ 
\item[(6)] $[d_i, k_j (\underline{r}, \zeta^m)]= r_i k_j
(\underline{r}, \zeta^m), \ 1 \leq i \leq N, 0 \leq j \leq N$
\item[(7)] $ \displaystyle{\prod_{p \epsilon \Z_m}} (1-w^{-p}
\zeta_1/\zeta_2)^{< \theta^p \beta_1, 
\beta_2>} Z (\beta_1, \underline{r}, \zeta_1) Z (\beta_2,
\underline{s}, \zeta_2)$ \\
$-\displaystyle{\prod_{p \epsilon \Z_m}} (1-w^{-p}
\zeta_2/\zeta_1)^{<\theta^p \beta_1, \beta_2>} Z 
(\beta_2, \underline{s}, \zeta_2) Z(\beta_1, \underline{r}, \zeta_1)$ \\
$=\frac{1}{m} \displaystyle{\sum_{\theta^p \beta_1+ \beta_2 \in
\phi}} \eta (p, \beta_1) \epsilon  (\theta^p \beta_1, \beta_2) Z
(\theta^p \beta_1+\beta_2, \underline{r}+\underline{s}, \zeta_2),
\delta (w^{-p} \zeta_1/ \zeta_2)$ \\
$- \frac{1}{mk} < x_{\beta_2}, {x}_{-\beta_2}>
\displaystyle{\sum_{\theta^p \beta_1+ \beta_2=0}} \ \eta (p,
\beta_1) (\beta_2)_0 k_0 (\underline{r}+ \underline{s},
\zeta_2^m) \delta (w^{-p} \zeta_1/\zeta_2)$ \\
$+ \frac{1}{m} <x_{\beta_2}, {x}_{-\beta_2} > \
\displaystyle{\sum_{\theta^p \beta_1+ \beta_2=0}} \ \eta (p, \beta_1)
(\displaystyle{\sum_{i=1}^{N}} \ r_i k_i (\underline{r}+\underline{s}, \zeta_2)
\delta (w^{-p} \zeta_1/ \zeta_2)+\frac{1}{mk} k_0
(\underline{r}+\underline{s}, \zeta_2) D
\delta (w^{-p} \zeta_1/ \zeta_2))$.
\item[(8)] $[\alpha, Z (\beta, \underline{r}, \zeta)] = <\alpha,
\beta > Z (\beta, \underline{r}, \zeta), \ \alpha \epsilon \underline{h}_0$ 
\item[(9)] $Z (\beta, \underline{r}, \zeta) = \eta(p, \beta) Z
(\theta^p \beta, \underline{r}, \zeta)$
\item[(10)] $k_i (\underline{r}, \zeta^m)$ is central for \ \ $0 \leq i \leq
N$. 
\end{enumerate}

As it is we do not know whether a $Z_k$ algebra is non-zero or not but
certainly it is well defined. 

\paragraph*{(1.11) Definition} $A ~ Z_k$ module $W$ is said to be in
the category $\underline{D}_k$ if \\
(1) $ k_0$ acts by scalar $k$. \\
(2) $V = \displaystyle{\oplus_{z \epsilon \C}} V_z, V_z = \{v
\epsilon V \mid d_0 
v=z v \}$. Assume for any given $z$ there exists $ \ell_0 \ni \ell > \ell_0$
implies $V_{z+ \ell} = 0$.

\section*{Section 2} 

In this section we establish equivalence between the categories
$\underline{C}_k$ and $\underline{D}_k$.  The proof are very similar to
[LW].  In fact most of the results go through.

Let $V \epsilon \underline{C}_k$.  Define for $\beta \epsilon \phi$.
$$E^{\pm} (\beta, \zeta) =  {\rm exp} (\pm m \sum_{J>0} \beta_i \otimes
t^i \zeta^{\pm j}/jk)$$
and $Z (\beta, \underline{r}, \zeta) = E^- (\beta, \zeta) x_{\beta}
(\underline{r}, \zeta) E^+ (\beta, \zeta)$.  We first prove that
these $Z$-operators satisfy relations in (1.10).

We first recall the following from section 3 of [LW].  We are taking
$\underline{a} = \underline{h}$ and $\underline{m} =0 $ (in decomposition 
(3.5)) in [LW].

\paragraph*{Proposition (2.1)} $\alpha, \beta, \gamma \in \Phi$

\begin{enumerate}
\item[(1)]~~~~~  $i>0$
\item[(a)] $ [\alpha_i \otimes t^i, E^+ (\beta, \zeta)] =0$
\\
\item[(b)] $[\alpha_{-i} \otimes t^{-i}, E^- (\beta, \zeta)] =0$ \\
\item[(c)] $[ \alpha_i \otimes t^i, E^- (\beta, \zeta)]= - < \alpha_i,
\beta > \zeta^{-i} E^- (\beta, \zeta)$. \\
\item[(d)] $[ \alpha_{-i} \otimes t^{-i}, E^+ (\beta, \zeta) ]
=  -< \alpha_i, \beta > \zeta^i E^+ (\beta, \zeta)$ \\
\item[(2) (a)] $ ~~~ E^{\pm} (\beta+ \gamma, \zeta) =  E^{\pm} (\beta,
\zeta), E^{\pm}(\gamma, \zeta) $\\
\item[(b)] $E^{\pm} (\theta^p \beta, \zeta)  =  E^{\pm} (\beta, w^p
\zeta)$ \\
\item[(c)] $D E^{\pm} (\beta, \zeta) = \frac{m}{k} \beta (\zeta)^{\pm}
E^{\pm} (\beta, \zeta)$ \\
${\rm where} \ \beta (\zeta)^{\pm}  = \sum \beta_{\pm} \otimes
t^{\pm 1} \zeta^{\pm i}$ \\
\item[(3)] (a) ~~$  x_{\beta} (\underline{r}, \zeta) =E^- (- \beta, \zeta)
Z (\beta_1, \underline{r}, \zeta) E^+ (- \beta, \zeta)$ \\
\item[(b)] $Z (\beta, \underline{r}, w^p \zeta) = \eta (p, \beta) Z
(\theta^p \beta,
\underline{r}, \zeta)$ \\
\item[(4)] ~~~~~ $E^+ (\beta, \zeta_1) E^- (\gamma, \zeta_2) = E^-
(\gamma,
\zeta_2) E^+ (\beta, \zeta_1)$. \\
\item[(5)] $\displaystyle{\prod_{p \epsilon \Z_m}} (1- w^{-p} \zeta_1 /
\zeta_2)^{ \frac{< \theta^p \beta, \gamma>}{k}}$ \\
\item[(6)] ~~~ $E^+ (\beta, \zeta_1) x_{\gamma} (\underline{r}, \zeta_2)
= x_{\gamma} (\underline{r}, \zeta_2) E^+ (\beta, \zeta_1)$
\item[(7)] $\displaystyle{\prod_{p \in \Z_m}} (1- w^{-p}
\zeta_1/\zeta_2)^{\frac{-< \theta^p \beta, \gamma >}{k}}$ \\
\item[(8)] $ x_{\beta} (\underline{r}, \zeta_1) E^- (\gamma, \zeta_2) =E^-
(\gamma, \zeta_2) x_{\beta} (\underline{r}, \zeta_1)$ \\
$$\displaystyle{\prod_{p \in \Z_m}} (1- w^{p} \zeta_1/ \zeta_2)^{\frac{-<
\theta^p \beta, \gamma >}{k}}$$

\end{enumerate}
\paragraph*{Proof} (1) a and b follows from the definition.  To see 1(c)
consider
$$[\alpha_i \otimes t^i, -\displaystyle{\sum_{j<0}} \beta_j \otimes t^j /
j_k \zeta^j]$$
$$=- \frac{1}{i} < \alpha_i, \beta_{-i}> \zeta^{-i}.$$
Now
$$[\alpha_i \otimes t^i, \frac{(-\sum \beta_j \otimes t^j/j_k \zeta^j)^{\ell}}{\ell !}]$$
$$=-\frac{< \alpha_i, \beta_{-i}>}{(\ell -1)!} (-\sum \frac{\beta_j 
\otimes t^j}{j_k} \zeta^j)^{\ell-1}).$$
First note that $< \alpha_i, \beta_{-i} > =<\alpha, \beta>$ by (1.2).  Now 1(c)
follows from the definition. \\
(1)(d) follows from similar  argument.\\
(2) and (3) follows from definition. See also Proposition 3.2 and 3.3 of [LW]. \\
(4) follows from Proposition 3.4 of [LW]. \\
(5) follows from similar argument of Proposition 3.5 of [LW]. \\
\paragraph*{Corollary (2.2)}  Let $\underline{\tilde{h}}^{1} =
\displaystyle{\oplus_{i \neq 0}} \underline{h}_i \otimes t^i$.  Then
as operators the following hold.
(1) ~~ $[ \tilde{\underline{h}}^1, Z (\alpha, \underline{r}, \zeta)] = 0$ \\
(2) ~~ $ [a_0, Z (\alpha, \underline{r}, \zeta)]=  < a_0,
\alpha > Z (\alpha_1, \underline{r}, \zeta),  a \epsilon \underline{h}$.

\paragraph*{Proof}  (1) Follows from above. (2) is easy to see.

\paragraph*{Proposition (2.3)} (Proposition (3.9) of [LW]).

Let $W$ be a vector space and let $f (\zeta_1, \zeta_2) = \sum w_{ij}
\zeta_1^i \zeta_2^j$ where  each $w_{ij} \epsilon W$ and suppose for some
$n \epsilon \Z$ either $w_{ij}=0$ where one of $i$ or $j> n$ or $w_{ij} =0$
whenever $i$ or $j <n$.

Set
$$D_i f (\zeta_1, \zeta_2)= \zeta_i \frac{df}{d \zeta_i} (\zeta_1,
\zeta_2)$$
Then for $a \neq 0$
$$
\begin{array}{lll}
(1) ~~~ \delta (a \zeta_1/ \zeta_2) f (\zeta_1, \zeta_2) &=& \delta
(a \zeta_1 / \zeta_2) f (\zeta_1, a \zeta_1) \\
&=& \delta ( a \zeta_1/\zeta_2) f (a^{-1} \zeta_2, \zeta_2) \\
(2) ~~ D \delta (a \zeta_1/\zeta_2) f (\zeta_1, \zeta_2) &=& (D
\delta) (a \zeta_1 / \zeta_2) f (\zeta_1, a \zeta_1)
+ \delta (a \zeta_1/\zeta_2) D_2 f (\zeta_1, a \zeta_1) \\
&=& D \delta (a \zeta_1 / \zeta_2) f (a^{-1}  \zeta_2, \zeta_2) -
\delta (a \zeta_1/\zeta_2) D_1 f (a^{-1} \zeta_2, \zeta_2) 
\end{array}
$$

\paragraph*{Proposition  (2.4) } 
$$
\begin{array}{lll}
&& \displaystyle{\prod_{p \epsilon \Z_m}} (1-w^{-p}
\zeta_1/\zeta_2)^{< \theta^p 
\beta_1, \beta_2 > / k} Z (\beta_1, \underline{r}, \zeta_1) Z
(\beta_2, \underline{s}, \zeta_2) \\
&&-\displaystyle{\prod_{p \epsilon \Z_m}} (1-w^{-p}
\zeta_2/\zeta_1)^{< \theta^p 
\beta_2, \beta_1 > / k} Z (\beta_2, \underline{s}, \zeta_2) Z
(\beta_1, \underline{r}, \zeta_1) \\
&=& E^- (\beta_1, \zeta_1) E^- (\beta_2, \zeta_2) [X (\beta_1,
\underline{r}, \zeta_1), X (\beta_2, \underline{s}, \zeta_2)] \\
&&E^+ (\beta_1, \zeta_1) E^+ (\beta_2, \zeta_2)
\end{array}
$$

\paragraph*{Proof}  Consider
$$Z(\beta_1, \underline{r}, \zeta_1) Z (\beta_2, \underline{s}, \zeta_2)$$
$$=E^- (\beta_1, \underline{r}, \zeta_1) x_{\beta_1} (\underline{r}, \zeta_1) 
E^+ (\beta_1, \underline{r}, \zeta_1).$$
$$E^- (\beta_2, \underline{s}, \zeta_2) x_{\beta_2} (\underline{s}, \zeta_2)
E^+ (\beta_2, \underline{s}, \zeta_2)$$
$$= \displaystyle{\prod_{p \in \Z_m}} (1- w^{-p} \zeta_1 / \zeta_2)^{\frac{< 
\theta \beta_1, \beta_2>}{k}}.$$
$$E^- (\beta_1, \underline{r}, \zeta_1) x_{\beta_1} (\underline{r}, \zeta_1)
E^- (\beta_2, \underline{s}, \zeta_2)$$
$$E^+ (\beta_1, \underline{r}, \zeta_1) x_{\beta_2} (\underline{s}, \zeta_2)
E^+ (\beta_2, \underline{s}, \zeta_2)$$
from 4 (a) of Proposition (2.1).
$$= \displaystyle{\prod_{p \in \Z_m}} (1-w^{-p} \zeta_1/ \zeta_2)^{\frac{-< \theta \beta_1, \beta_2>}{k}}.$$
$$E^- (\beta_1, \underline{r}, \zeta_1) E^- (\beta_2, \underline{s}, \zeta_2)
X_{\beta_1} (\underline{r}, \zeta_1)$$
$$x_{\beta_2} (\underline{s}, \zeta_2) E^+ (\beta_1, \underline{r}, \zeta_1)
E^+ (\beta_2, \underline{s}, \zeta_2)$$
(by 5 (a) and (b) of Proposition (2.1).

Multiplying both sides by the inverse of the first factor on the right, and 
subtracting the expression obtained by interchanging the roles of the
subscripts 1 and 2 we have Proposition  (2.4).

\paragraph*{Proposition (2.5)}  For these $Z$ operators the relation at
(1.10) hold.

\paragraph*{Proof}  (2) to (6), holds from definition of $d_i$.
Since $V \epsilon \underline{C}_k$ we have
$$\frac{1}{k} X_{\beta} (\underline{r}, \zeta) k_0 (\underline{s},
\zeta^m) = X_{\beta} (\underline{r}+ \underline{s}, \zeta).$$
Thus (1) holds from  definition of $Z$ operator. (9) holds from
Proposition 
1.5 (6), (10)  and (8) are easy to see.  We only need to prove (7).
The RHS of the Proposition (2.4) and by using Proposition 1.5 (1)
 is equal to
$$E_1+ E_2 + E_3+E_4$$
$$
\begin{array}{lll}
{\rm where} \ E_1&=&E^- (\beta_1, \zeta_1) E^- (\beta_2, \zeta_2). \\
&&\frac{1}{m} \displaystyle{\sum_{\theta^p \beta_1+ \beta_2 \epsilon
\phi}} \ \eta (p, \beta_1) \epsilon ( \theta^p \beta_1, \beta_2)
x_{\theta^p \beta_1+ 
\beta_2}  (\underline{r}+ \underline{s}, \zeta_2) \delta (w^{-p}
\zeta_1/\zeta_2) \\
&&E^+ (\beta_1, \zeta_1) E^+ (\beta_2, \zeta_2)\\
E_2 &=&E^- (\beta_1, \zeta_1) E^- (\beta_2, \zeta_2).\\
&& (\frac{-1}{m}) <x_{\beta_2}, x_{- \beta_2} >
\displaystyle{\sum_{\theta^p \beta_1+ \beta_2 =0}} \eta (p, \beta_1)
\beta_2 (\underline{r}+ s, \zeta_2) \delta (w^{- p} \zeta_1 /
\zeta_2) \\
&&E^+ (\beta_1, \zeta_1) E^+ (\beta_2, \zeta_2)\\
E_3&=& E^- (\beta_1, \zeta_1) E^- (\beta_2, \zeta_2). \\
&&\frac{1}{m} < x_{\beta_2}, x_{-\beta_2} >
\displaystyle{\sum_{\theta^p
\beta_1+p_2=0}} \eta (p, \beta_1) \displaystyle{\sum_{i=1}^{N}} r_i k_i
(\underline{r}+\underline{s} , \zeta_2^m) \delta (w^{-p}
\zeta_1/\zeta_2) \\
&&E^+ (\beta_1, \zeta_1) E^+ (\beta_2, \zeta_2) \\
E_4 &=& E^- (\beta_1, \zeta_1) E^- (\beta_2, \zeta_2) \\
&&\frac{1}{m^2} < x_{\beta_2}, x_{-\beta_2}>
\displaystyle{\sum_{\theta^p \beta_1+ 
\beta_2=0}} \eta (p, \beta_1) k_0 (\underline{r}+ \underline{s},
\zeta_2^m) D \delta (w^{-p} \zeta_1/\zeta_2) \\
&&E^+ (\beta_1, \zeta_1) E^+  (\beta_2, \zeta_2)
\end{array}
$$
Now

$$E_1= \frac{1}{m} \displaystyle{\sum_{\theta^p \beta_1+ \beta_2 \in \Phi}} 
\eta (p, \beta_1)E^- (\theta^p \beta_1+ \beta_2, \zeta_2).$$
$$x_{\theta^p \beta_1+\beta_2} (\underline{r}+ \underline{s}, \zeta_2) \delta
 (w^{-p} \zeta_1/\zeta_2).$$
$$E^+ (\theta^p \beta_1+\beta_2, \zeta_2).$$
$$E_2= - \frac{1}{m} <x_{\beta_2}, x_{- \beta_2} > \displaystyle{\sum_{
\theta^p \beta_1+ \beta_2=0}} \eta (p, \beta_1) \beta_2 (\underline{r}+
\underline{s}, \zeta_2) \delta (w^{-p} \zeta_1 /\zeta_2)$$
$$E_3 = \pm \frac{1}{m} <x_{\beta_2}, x_{- \beta_2} > \displaystyle{\sum_{
\theta^p \beta_1+ \beta_2=0}} \eta (p, \beta_1)
\displaystyle{\sum_{i=1}^{N}} r_i k_i (\underline{r}+ \underline{s}, \zeta_2^m) \delta (w^{-p} \zeta_1/\zeta_2).$$
For $E_4$ we use Proposition 2.3 (2) $(a=w^{-p})$ and Proposition 2.1 (2) (b).
Thus we get
$$E_4= \frac{1}{m^2} < x_{\beta_2}, x_{-\beta_2} > 
\displaystyle{\sum_{\theta^p \beta_1+ \beta_2=0}} \eta (p, \beta_1) 
k_0 (\underline{r}+ \underline{s}, \zeta_2^m) D \delta (w^{-p} 
\zeta_1/\zeta_2) .$$  
$$\frac{- 1}{m^2} < x_{\beta_2} ,x_{- \beta_2}> \displaystyle{\sum_{\theta^p
\beta_1+ \beta_2=0}} \ \eta (p, \beta_1)\frac{m}{k} \displaystyle{\sum_{i \neq 0}} (\beta_1)_i \otimes t^i (w^p \zeta_2)^i k_0 (\underline{r}+ \underline{s},
\zeta_2^m) \delta (w^{-p} \zeta_1/\zeta_2)$$
We will use the fact that $\displaystyle{\sum_{i \neq 0}} (\beta_1)_i 
\otimes t^i \zeta_2^i= \beta_2 (\zeta) - (\beta_2)_0$
and the fact that
$$\frac{1}{k} \beta_2 (\zeta_2) k_0 (\underline{r}+ \underline{s}, \zeta_2^m)=
\beta_2 (\underline{r}+ \underline{s}, \zeta_2) \leqno{(2.6)}$$
So we get
$$E_4 = \frac{1}{m^2} <x_{\beta_2}, x_{- \beta_2}> 
\displaystyle{\sum_{\theta^p \beta_1+ \beta_2=0}}
\eta (p, \beta_1) k_0 (\underline{r}+ \underline{s}, \zeta_2^m)  
D \delta (w^{-p} \zeta_1 / \zeta_2)   $$
$$- \frac{1}{mk} <x_{\beta_2}, x_{-\beta_2} > \displaystyle{\sum_{\theta^p 
\beta_1+\beta_2 =0}} \ \eta (p, \beta_1) \displaystyle{\sum_{i \neq 0}}
(\theta^p \beta_1)_i \otimes t^i \zeta_2^i k_0  (\underline{r}+ \underline{s}, \zeta_2^m) \delta (w^{-p} \zeta_1/\zeta_2)$$
$$ =\frac{1}{m^2} <x_{\beta_2}, x_{-\beta_2} > \displaystyle{\sum_{\theta^p 
\beta_1+ \beta_2=0}} 
\eta (p, \beta_1) k_0 (\underline{r}+ \underline{s}, \zeta_2^m)
D (\delta w^{-p} \zeta_1/\zeta_2) \
$$
$$ \frac{-1}{mk} <x_{ \beta_2}, x_{-\beta_2} > \displaystyle{\sum_{\theta^p 
\beta_1+ \beta_2=0}} \ \eta (p, \beta) \theta^p \beta_1 
 (\zeta_2) k_0(\underline{r}+\underline{s},
\zeta_2^m) \delta ( w^{-p} \zeta_1/\zeta_2)$$
$$+ \frac{1}{mk} <x_{ \beta_2}, x_{-\beta_2} > \displaystyle{\sum_{\theta^p 
\beta_1+ \beta_2=0}} \ \eta (p, \beta_1) (\beta_1)_0  
k_0(\underline{r}+\underline{s},
\zeta_2^m) \delta  (w^{-p} \zeta_1/\zeta_2).
$$
Note that the second term is $-E_2$ by (2.6). Since $\theta^p \beta_1+ \beta_2=0$ we have $(\beta_1)_0+(\beta_2)_0=0$.  Thus $E_2+E_4$
$$=\frac{1}{m^2} < x_{\beta_2}, x_{-\beta_2} > \displaystyle{\sum_{\theta^p 
\beta_1+\beta_2 =0}} \ \eta (p, \beta_1) k_0 (\underline{r}+\underline{s},
\zeta_2^m) D \delta (w^{-p}  \zeta_1/\zeta_2)$$
$$- \frac{1}{m_k} < x_{\beta_2}, x_{-\beta_2} > \displaystyle{\sum_{\theta^p 
\beta_1+\beta_2 =0}} \ \eta (p, \beta_1) (\beta_2)_0k_0 (\underline{r}+
\underline{s},
\zeta_2^m) \delta (w^{-p}  \zeta_1/\zeta_2).$$
Now  adding $E_1, E_2, E_3$ and $E_4$ we get the defined result.
Thus we proved that $\Omega (V) \in \underline{D}_k$.  Conversely
assume that $W \in \underline{D}_k$.  Let $V= M(k) \otimes W$.
Define $X_{\alpha}( \underline{r}, \zeta) = E^- (-\alpha, \zeta) E^+
(- \alpha,\zeta) \otimes Z (\alpha, \underline{r}, \zeta)$.
$$\beta (\underline{r}, \zeta)  = \frac{1}{k} \beta (z) k_0
(\underline{r}, \zeta^m).$$
The central elements to be same.  Since $ W  \epsilon \Omega_k$.  The
operators $X_{\alpha} (\underline{r}, \zeta)$ and $k_0
(\underline{r}, \zeta)$ satisfy
$$ \frac{1}{k} X_{\alpha} (\underline{r}, \zeta) k_0 (\underline{s},
\zeta^m) =X (\underline{r}+ \underline{s}, \zeta) \ {\rm for \
all} \ X_{\alpha} \epsilon  {\cal G}.$$
Conditions 4 to 7 of Proposition (1.6) are easily satisfied as the
corresponding conditions are satisfied for $Z$-operators.

Condition (3) can be proved exactly as in the proof of Proposition
5.3 of [LW]. Condition (1)  is satisfied as the same relation holds
for $Z$-operators and $Z$ operator commutes with $E^{\pm}$-operators.
Consider $[\beta_1 (\zeta_1), \beta_2 (\zeta_2)]= \frac{1}{m^2} k_0
\displaystyle{\sum_{p \epsilon \Z_m}} < \theta^p \beta_1, \beta_2 > D \delta
(w^{-p} \zeta_1/\zeta_2)$. 
See Theorem 2.4 of [LW].
$$
\begin{array}{lll}
[\beta_1 (\underline{r}, \zeta_1), \beta_2 (\underline{s},
\zeta_2)]&=& [\beta_1 (\zeta_1), \beta_2 (\zeta_2)] k_0
(\underline{r}, \zeta^m_1) k_0 (\underline{s}, \zeta_2^m) \\
&=&\frac{1}{km^2} \displaystyle{\sum_{p \epsilon \Z_m}} < \theta^p
\beta_1, \beta_2> k_0 (\underline{r}, \zeta_1^m) k_0 (\underline{s},
\zeta^m_2) D \delta
(w^{-p} \zeta_1/\zeta_2) 
\end{array}
$$
Now by Proposition 2.3 (2) we have
$$
= \frac{1}{m^2} \displaystyle{\sum_{p \epsilon \Z_m}} < \theta^p \beta_1,
\beta_2> k_0 (\underline{r}+\underline{s}, 
\zeta^m_1)  D \delta (w^{-p} \zeta_1/\zeta_2) $$
$$ \frac{-1}{mk} \sum < \theta^p \beta_1, \beta_2>
D_1 k_0(\underline{r}, \zeta_1^m) \mid_{\zeta_1 = w^p \zeta_2} k_0 
(\underline{s}, \zeta_2^m) \delta( w^{-p} \zeta_1 /\zeta_2) $$
$$=\frac{1}{m^2} \displaystyle{\sum_{p \epsilon \Z_m}} < \theta^p \beta_1,
\beta_2> k_0 (r+s, \zeta_2^m) D \delta (w^{-p} \zeta_1/\zeta_2)$$
$$+\frac{1}{m} \displaystyle{\sum_{p \epsilon \Z_m}} < \theta^p
\beta_1, \beta_2> \displaystyle{\sum_{i=1}^{N}} r_i k_i (\underline{r}+\underline{s}, \zeta_2^m) \delta (w^{-p} \zeta_1/\zeta_2) $$
$$\displaystyle{\sum_{i=1}^{N}} r_i k_i (\underline{r}+\underline{s},
\zeta^m_2) \delta ( w^{-p} \zeta_1/\zeta_2)$$
Thus we have proved the following:

\paragraph{Proposition 2.6}  The category $\underline{C}_k$ of $\tilde{L}
({\cal G}, \theta)$-modules are equivalent to the category $\underline{D}_k$ of
$\Z_k$-modules. 

\section*{Section 3 (Homogeneous picture)} 

In this section our aim is to construct a faithful representation
for the untwisted toroidal Lie algebra $\tilde{\tau}$ coming from
simple, simply connected Lie-algebra ${\cal G}$.  (First note that on any
representation $\tau$ where centre $\Omega_A / d_A$ acts faithfully,
then $\tilde{\tau}$ acts faithfully). That is we are giving a
realization.  This recovers the main result of [EM].  For this we
give a representation for the $Z-$algebra such that the centre acts
faithfully. Thus we have a faithful  representation
for the toroidal Lie algebra $\tau$.

We take the automorphism $\theta = id$.  We first give a presentation for
the Lie-algebra ${\cal G}$.  Let $\stackrel{\circ}{Q}$ be the root lattice
spanned by simple roots.  The nondegenerate form is chosen  so that
$(\alpha, \alpha) =2$ for a highest root $\alpha$.  Then it is known that
$$\phi=\{\alpha \epsilon \stackrel{\circ}{Q} \mid (\alpha, \alpha) =2.\}$$ 
The following cocycle  on $\stackrel{\circ}{Q} \times 
\stackrel{\circ}{Q}$ is known to exists.
$$\epsilon : \stackrel{\circ}{Q} \times \stackrel{\circ}{Q} \to \{\pm
1\}$$ 
\paragraph*{3.1}
\begin{enumerate}
\item[(1)] $\epsilon (\alpha, \alpha) = (-1)^{ \frac{(\alpha, \alpha)}{2}} $ 
\item[(2)] $ \epsilon (\alpha, \beta) \epsilon (\beta, \alpha) = (-1)^{(\alpha,
\beta)}$ 
\item[(3)] $ \epsilon (\alpha+ \beta, \gamma) = \epsilon ( \alpha,
\gamma) \epsilon 
(\beta, \gamma)$
\item[(4)] $\epsilon (\alpha, \beta+ \gamma) = \epsilon (\alpha, \beta) \epsilon
(\alpha, \gamma)$
\end{enumerate}

Note that $\epsilon (\alpha, \alpha)= \epsilon (\alpha, - \alpha) = -1$ for
$\alpha \epsilon \phi$.   Then there exist vectors $x_{\alpha}, h_{\alpha}$
in ${\cal G} \ \alpha \epsilon \phi$ satisfying the following:

\paragraph{(3.2)}
$$
\begin{array}{lll}
(1) ~~~ [x_{\alpha}, x_{\beta}] &=&\epsilon (\alpha, \beta)
x_{\alpha+\beta}, \alpha+ \beta \epsilon \phi \\
(2) ~~~ [x_{\alpha_1} x_{-\alpha}] &=&0 \ \ {\rm if} \ \alpha+ \beta
\notin \phi \cup \{0 \} \\ 
&=& \epsilon (\alpha_1, -\alpha_1) h_{\alpha} \ {\rm if} \ \alpha + \beta =0 \\
(3) ~~~ [ h_{\alpha}, h_{\beta}] &=& 0 \\
(4) ~~~ \left[ h, x_{\alpha} \right] &=& \alpha (h) x_{\alpha} ~~~
\forall h \epsilon \underline{h}. 
\end{array}
$$

Let $\Gamma$ be a $\Z$-lattice spanned by $\alpha_1, \cdots,
\alpha_{\ell}, \delta_1, \cdots, \delta_N, d_1, \cdots, d_N$.  Define
a non-degenerate bilinear form on $\Gamma$ extending the one on
$\stackrel{\circ}{Q}$ by 
$$
\begin{array}{lll}
(\stackrel{\circ}{Q}, \delta_i) &=& (\stackrel{\circ}{Q}, d_i )=0 \\
(d_i, d_j) &=& (\delta_i, \delta_j) =0 \\
(\delta_i, d_j) = \delta_{ij}
\end{array}
$$
Any vector which is integral linear combination of $\delta_i$ is
called a null root.  For $\underline{r} \epsilon \Z^N$  define
$\delta_{\underline{r}} = \sum r_i \delta_i$.  Note that
$(\delta_{\underline{r}},\delta_{\underline{s}})= 0$ .  Let $Q$ be
the sub lattice spanned by $\stackrel{\circ}{Q}$ and $\delta_1,
\cdots, \delta_n$.  Extend the co-cycle $\epsilon $ to $Q$ by $\epsilon
(\alpha, \delta_{\underline{r}})=1$ fir $\alpha \epsilon Q$.  Now extend
$\epsilon$ to $Q \times \Gamma$ to be bimultiplicative in any convenient way.
Consider the group algebra $\C[\Gamma]$ and make $\C[\Gamma]$ a $\C
[Q]$ module by the following multiplication.
$$e^{\alpha} \cdot e^{\gamma} = \epsilon (\alpha, \gamma) e^{\alpha+ \gamma}.$$
Let $\overline{h}=Q  \otimes_{\Z} \C$.  Let $\overline{h}_{\pm} =
\displaystyle{\oplus_{n \stackrel{>}{<} 0}} \overline{h} \otimes t^n$.  
Consider the Fock space
$$V(\Gamma) = S (\overline{h}-) \otimes \C [\Gamma].$$
Define operators $\alpha(0)$ on $ V (\Gamma)$ by
$$\alpha (0) \cdot u \otimes e^{\gamma} = (\alpha, \gamma) u \otimes
e^{\gamma}, \alpha \epsilon Q.$$ 
For $\delta$ nullroot
$$\delta (n) u \otimes e^{\gamma} = \delta (n) u \otimes e^{\gamma},
n \neq 0$$
$\delta (n) u$ is multiplication if $n<0$ and differentiation if $n >0$.
This is the standard Fock space representation of $\overline{h}
\otimes \C [t, t^{-1}]$ on $V[\Gamma]$.

For a null root $\delta$ define $E^{\pm} (\delta, \zeta) = {{\rm
exp}} \displaystyle{\sum_{n \stackrel{>}{<}0}} \frac{ \alpha {(\pm
n)}}{\pm n}  \zeta^{\pm n}$.  Define
operators 
$$\zeta^{\alpha (0)} u \otimes e^{\gamma} =\zeta^{(\alpha,
\gamma)} u \otimes e^{\gamma}, \alpha \epsilon Q.$$
Consider the vertex operator
$$X (\delta, \zeta) = E^- (\delta, \zeta) \zeta^{\delta (0)} E^+ (\delta, \zeta).$$
Let $k_i (\delta, \zeta)= \delta_i (\zeta)X (\delta, z) $ for $ 1 \leq i \leq
N$ and $k_0 (\delta, z) = X (\delta, \zeta )$. From [EM] it is known that 
each $k_i (\delta, z)$ acts non trivially
and $Dk_0 (\delta_{\underline{\gamma}}, \zeta)+
\displaystyle{\sum_{i=1}^{N}} r_i k_i (\delta_{\underline{r}}, \zeta) =0$.
 Further any relation among $k_i (\delta, \zeta)$ is the one given above.
 (see Lemma $C$ of [EM]).
 
 We will now define $Z$ operators.  Define $Z (\alpha, 0, \zeta) =
\zeta^{\frac{(\alpha, \alpha)}{2}} \zeta^{-\alpha} e^{\alpha}, \alpha
\epsilon \phi$.  Then define $Z (\alpha, \underline{r}, \zeta) = Z
(\alpha, 0, \zeta) k_0 (\underline{r}, \zeta)$. $d_0, d_1, \cdots, d_N$
are defined naturally as grading on $Z (\alpha, \underline{r},
\zeta)$. 

We will now check the relation at (1.10) for the above $Z$-operator.
(1) to (6) are clearly satisfied from definition.  We will rewrite
the relation (7) using the fact that $\theta =Id$ and $m=1$.
\paragraph*{(3.3)}
$$(1- \zeta_1/\zeta_2)^{< \beta_1, \beta_2>}Z (\beta_1, \underline{r}, \zeta_1)
Z(\beta_2, \underline{s}, \zeta_2) - (1- \zeta_2/\zeta_1)^{< \beta_1,
\beta_2>}Z (\beta_2, \underline{s}, \zeta_2) Z (\beta_1, \underline{r},
\zeta_1)$$
$$=\epsilon (\beta_1, \beta_2) Z (\beta_1+\beta_2, \underline{r}+\underline{s},
\zeta_2) \delta (\zeta_1/\zeta_2)\ {\rm if} \ \beta_1+\beta_2 \in \Phi$$
$$=<x_{\beta_2}, x_{-\beta_2}> \beta_2 \frac{1}{k} k_0 (\underline{r}+
\underline{s}, \zeta_2) \delta (\zeta_1/\zeta_2)$$
$$+<x_{\beta_2},  x_{-\beta_2}> (\displaystyle{\sum_{i=1}^{N}} r_i k_i (\underline{r}+\underline{s}, \zeta_2). \delta (\zeta_1/\zeta_2)+ k_0 (\underline{r}+
\underline{s}, \zeta_2) D \delta (\zeta_1/\zeta_2)) \ {\rm if} \ \beta_1+\beta_2
=0$$
$$=0 \ {\rm if} \ \beta_1+\beta_2 \notin \Phi \cup \{0\}.$$
Suppose $\underline{r}=0$ and $\underline{s}=0$ then (3.3) follows from
Theorem 5.3 of [LM].  For general $\underline{r}$ and $\underline{s}$,
consider L.H.S of (3.3) which is equal to
$$(1- \zeta_1/\zeta_2)^{< \beta_1, \beta_2>} Z (\beta_1, 0, \zeta_1) k_0
(\underline{r}, \zeta_1) Z (\beta_2,0, \ \zeta_2) k_0 (\underline{s}, \zeta_2)$$
$$-(1- \zeta_1/\zeta_2)^{< \beta_1, \beta_2 >} Z (\beta_2, 0, \zeta_2) k_0
(\underline{s}, \zeta_2).Z (\beta_1, 0, \zeta_1) k_0 (\underline{r}, \zeta_1)$$
$$= k_0 (\underline{r}, \zeta_1) k_0 (\underline{s}, \zeta_2) Z (\beta_1,
\beta_2) \ {\rm where} \ Z (\beta_1, \beta_2) $$
$$=\epsilon (\beta_1, \beta_2) Z (\beta_1+ \beta_2, 0, \zeta_1)  
\delta (\zeta_1 / \zeta_2) \ {\rm if} \ \beta_1+ \beta_2 \in \Phi$$
$$+ < x_{\beta_2}, x_{- \beta_2} > (- \beta_2 \delta (\zeta_1 / \zeta_2)+
kD \delta  (\zeta_1/\zeta_2) \ {\rm if} \ \beta_1+ \beta_2 =0$$
$$=0 \ {\rm if} \ \beta_1+ \beta_2 \notin  \Phi \cup \{0 \}.$$
This follows from case $\underline{r} =0=s$.  Now the case $\beta_1+
\beta_2 \in \Phi $, (3.3) follows from Proposition 2.3 (1).  The case
$\beta_1+ \beta_2 \notin \Phi \cup \{0\}$ is trivial.  For the case $\beta_1+
\beta_2 =0$,$< x_{\beta_2}, x_{-\beta_2}> k_0 (\underline{r}, \zeta_1) k_0
(\underline{r}, \zeta_2) \beta_2 \delta (\zeta_1 / \zeta_2)$ is
equal to the first term of 3.3 which follows from Proposition 2.3(1). \\
Now \\ 
$< x_{\beta_2}, x_{- \beta_2} > k_0 (\underline{r}, \zeta_1) k_0
(\underline{s}, \zeta_2). k D \delta (\zeta_1/\zeta_2)$\\
$ =< x_{\beta_2}, 
x_{- \beta_2} >( k_0 (\underline{r}+ \underline{s}, \zeta_2) D 
\delta (\zeta_1/ \zeta_2)+  \sum r_i k_i (\underline{r}+ \underline{s}, 
\zeta_2) \delta (\zeta_1 / \zeta_2)$. \\
This completes the proof of (3.3).

\section*{Section 4 ~~Principal realization}

Let ${\cal G}$ be a simple finite dimensional Lie-algebra.  Let $\pi$ be an
automorphism of order  $K(=1,2$ or $3$) induced by an automorphism of
the Dynkin diagram of ${\cal G}$ with respect to some Cartan 
subalgebras $h$ of ${\cal G}$.  Let $\epsilon$ be $K$-th primitive root.  We
will now extend the automorphism $\pi$ to ${\cal G} \otimes A \oplus
\Omega_A / d_A= \tau$ by
\paragraph*{(4.1)}  
$$\pi (x t^{r_0} t^{\underline{r}}) = \epsilon^{-r_0} \pi (x) t^{r_0}
t^{\underline{r}} $$
$$\pi ( t^{r_0} t^{\underline{r}}) = \epsilon^{-r_0} t^{r_0}
t^{\underline{r}}, 
\ \  0 \leq i \leq N. $$
The aim of this section is to prove that $\overline{L} ({\cal G}, \pi)
\cong \overline{L} ({\cal G}, \theta)$ where $\theta$ is a special automorphism
depending on $\pi$.  This is a generalisation of the standard
principal realization of affine Lie-algebras given in [KKLW].

To do this we first have to define the automorphism $\theta$. For $i \in
\Z$, let ${\cal G}_{[i]}$ be the $\epsilon^i$ eigenspace of ${\cal
G}$.  Then the  fixed point space ${\cal G}_{[0]}$ is a simple
Lie-subalgebra of ${\cal G}$ and 
${\cal G}_{[0]}$ 
module ${\cal G}_{[1]}$ and ${\cal G}_{[-1]}$ are irreducible and
contragradient. 

Fix a Cartan subalgebra $\underline{t}$ of of ${\cal G}_{[0]}$ inside 
$\underline{h}$.  Let $H_j, E_j, F_j \ (1 \leq j \leq \ell)$ be a
corresponding 
set of canonical generators of ${\cal G}_{[0]}$.  Let $E_0$ be the lowest
weight vector of ${\cal G}_{[0]}$ module ${\cal G}_{[1]}$, and let $F_0$ be the
highest weight vector, ${\cal G}_{[0]}$-module ${\cal G}_{[-1]}$, normalised so
that $[H_0, F_0]=2 F_0$ where $H_0 = [E_0, F_0]$.  Let $\psi_1,
\cdots, \psi_{\ell} \epsilon \underline{t}^*$ be simple roots of
${\cal G}_{[0]}$, and let $\psi_0 \in \underline{t}^*$ be the
lowest weight 
of the ${\cal G}_{[0]}$ module ${\cal G}_{[1]}$.  For $i,j =0,1, \cdots, \ell$
set $ A_{ij} = \psi_j (H_i)$.  Then it is known that $A=(A_{ij})$ is
an indecomposable  affine Cartan matrix (see [LW] and [KKLW]).  Let $a_0,
\cdots, a_{\ell}, a_0^1, \cdots, a_{\ell}^1$  be  positive
integers such that

\paragraph*{(4.2)}
$$
\begin{array}{lll}
A(a_0, \cdots a_{\ell})^T &=& 0 \\
(a_0^1, \cdots, a_{\ell}^1) A &=& 0 \ {\rm and} \\
g.c.d (a_0, \cdots, a_{\ell}) &=& 1 \\
g.c.d (a_0^1, \cdots , a_{\ell}^1 ) &=& 1.
\end{array}
$$

Then $a_0, a_1, \cdots, a_{\ell}$ are precisely the indices of the
Dynkin diagram of $A$.  (see Table $K$ of [KKLW]).

\subsection*{(4.3)}

Note that from above tables we see that $a_0=1$ always.  Recall from [LW] that
\paragraph*{(4.4)}
$$
\begin{array}{lll}
\displaystyle{\sum_{j=0}^{\ell}} a_j \psi_j &=&0 \ {\rm and} \\
\displaystyle{\sum_{j=0}^{\ell}} a_j^1 H_j &=& 0 
\end{array}
$$
\paragraph*{(4.5) Proposition (KKLW)}  The Lie subalgebra $G$ of ${\cal G}
\otimes \C[t, t^{-1}] \oplus \C$ generated by $E_i, F_i, H_i (1 \leq
i \leq \ell), E_0 \otimes t, F_0 \otimes t^{-1}, H_0$ and $C$ is
isomorphic to the affine Lie algebra corresponding to $A$.

Let $\underline{s}= (s_0, \cdots, s_{\ell})$ be a sequence of
non-negative integers, not all  $0$.
$${\rm Take} \ m= K \displaystyle{\sum_{j=0}^{\ell}} s_j a_j.$$

Define an automorphism $\theta$ of  ${\cal G}$ by the condition
$$\theta H_i = H_i, \theta E_j= w^{s_j} E_j.$$
Then $\theta$ defines an automorphism of ${\cal G}$ of order $m$.  See [LW].
Then from section 1 we can define $\overline{L} ({\cal G}, \theta)$.  Our aim
in this section is to prove $\overline{L} ({\cal G}, \pi) \cong
\overline{L} ({\cal G}, \theta)$ which is what we call principal realization of
toroidal Lie algebras.

\paragraph*{(4.6)}

As earlier $<,>$ be a non-degenerate ${\cal G}$-invariant bilinear form 
on ${\cal G}$ which is necessarily $\theta$ and $\pi$ invariant.  This form 
$<,>$ remains non-singular on the Cartan subalgebra $\underline{t}$ of 
${\cal G}_{[0]}$. See section 8 of [LW].

Using the restricted form we identify $\underline{t}$ and
$\underline{t}^*$.  We normalise the form $<,>$ such that

\paragraph*{(4.7)}
$$< \psi_0, \psi_0>= \frac{2 a_o^1}{K}.$$
Then we have $a_j^1=K < \psi_j, \psi_j> a_j/2$ for $j=0, \cdots,
\ell$.  (see [LW]).

Now we have the following $\underline{s}$-realization of the affine Lie
algebra $G$ from [KKLW].

\paragraph*{(4.8) ~~ Proposition } ([LW], [KKLW])

Let $e_j= E_j \otimes t^{s_j}, f_j = F_j \otimes t^{-s_j}, \ h_j = H_j
\otimes 1 + 2 s_j <\psi_j, \psi_j>^{-1} m^{-1} C$ inside
$\widehat{{\cal G}}$.   Then there is an isomorphism of affine
Lie-algebras $\varphi:G \to \widehat{{\cal G}}$ defined by
$$
\begin{array}{lll}
\phi (E_i \otimes 1) &=& e_i ~~ 1 \leq i \leq \ell \\
\varphi (F_i \otimes 1) &=&  f_i, 1 \leq i \leq \ell \\
\varphi (E_0 \otimes t) &=& e_0 \\
\varphi (F_0 \otimes t^{-1}) &=& f_0 \\
\varphi (H_i) &=& h_i ~~ 1 \leq i \leq \ell \\
\varphi (H_0+ \frac{< E_0, F_0>}{K} C) &=& h_0.
\end{array}
$$
Further $e_i, f_i, h_i \ (0 \leq i \leq \ell)$ forms a  set of canonical
generators for the affine Lie-algebra $\widehat{{\cal G}}$.  Here the Lie
bracket $G$ is defined by the bilinear form $\frac{1}{K} < >$.

As we are interested in the principal realization we take
$\underline{s}=(1,1, \cdots, 1)$.
\paragraph*{(4.9) ~~ Remark.}  The $\pi$-invariants of
$\underline{h}$ equal to $\underline{t}$.  In particular they are
spanned by $H_i, 1 \leq i \leq \ell$.

\paragraph*{(4.10) ~~ Proposition}  Let $w$ be the Chevalley involution
automorphism of ${\cal G}$.  Let $\phi: G \to \widehat{{\cal G}}$ be the
isomorphism of Lie-algebras given earlier.  Then the following hold.
\begin{enumerate}
\item[(1)] $\varphi (C) =C$
\item[(2)] $\varphi (x_{\alpha} \otimes t^{r_0}) = x_{\alpha} \otimes
t^{N(\alpha) + \frac{m}{k} r_o} $ where $x_{\alpha} \otimes t^{r_0}$
is a real root vector of $G$.  $N(\alpha)$ is an integer independent
of $r_0$ but depends on $\alpha$.
\item[(3)] Let $ h_{\alpha}= [x_{\alpha}, w
(x_{\alpha})]$ then
$$\varphi (h_{\alpha} \otimes t^{r_0}) = h_{\alpha} \otimes
t^{\frac{m}{k} r_0} +< x_{\alpha}, w (x_{\alpha})>
\frac{N_{\alpha}}{m} \delta_{r_0,0}$$
\item[(4)] $<x_{\alpha}, x_{\beta}> \neq 0$ implies $N(\alpha) +N
(\beta) =0$ 
\item[(5)] $<x_{\alpha}, x_{\beta}> = 0$ then
\end{enumerate}
$$\varphi ([x_{\alpha}, x_{\beta}] \otimes t^{r_0+s_0}) =
[x_{\alpha}, x_{\beta}] \otimes t^{(r_0+s_0) \frac{m}{k}+N
(\alpha)+N(\beta)}.$$ 
Here $[x_{\alpha}, x_{\beta}]$ could be part of real or imaginary
root. 

\paragraph*{Proof}  Let $x_{\alpha} \otimes t^{r_0}$ be a real root
vector of $G$.  Then $G$ is spanned by $x_{\alpha} \otimes^{r_0},
[x_{\alpha}, x_{\beta}] \otimes t^{r_0}$ and $C$.  We have from (4.4)
$$H_0 =- \displaystyle{\sum_{J=1}^{\ell}} \ \frac{a_j^1}{a_{o}^{1}}
H_j.$$
From Proposition (4.8) we have 
$$H_0 \otimes 1+2 <\psi_0, \psi_0>^{-1} m^{-1} C=h_0=[E_0t,
F_0t^{-1}]=[E_0,F_0]+< E_0, F_0> \frac{1}{m} C.$$
\paragraph*{(4.11)}

This implies $<E_0, F_0> = \frac{2}{< \psi_0, \psi_0>}$.  Consider
$$
\begin{array}{lll}
\phi (H_0) &=&- \displaystyle{\sum_{j=1}^{\ell}} \frac{a_j^1}{a_0^1}
\phi (H_j) \\
&=&- \displaystyle{\sum_{j=1}^{\ell}} \frac{a_j^1}{a_0^1} ( H_j+
\frac{2}{<\psi_j, \psi_j>} \frac{1}{m} C) \\
&=& H_0 - \displaystyle{\sum_{j=1}^{\ell}} \frac{a_j^1}{a_0^1}
\frac{1}{m} \frac{a_j}{a_j^1} K C \\
&=&H_0 - \displaystyle{\sum_{j=1}^{\ell}} \frac{a_j KC}{a_0^1 m} \\
&=& H_0 - \frac{(m-a_0k)}{a_o^1m} C \\
{\rm But} \ (H_0+ \frac{<E_0, F_0>}{K}  C) &=& H_0+ \frac{2}{<\psi_0,
\psi_0>} \frac{1}{m} C
\end{array}
$$
(by Proposition 4.8)
$$
\begin{array}{lll}
\varphi (C) \frac{2}{K < \psi_0, \psi_0 >} &=& H_0+ \frac{2}{<\psi_0,
\psi_0>} \frac{1}{m} C - \varphi (H_0) \\
&=& \frac{2}{(\psi_0, \psi_0)} \frac{1}{m} C+ \frac{1}{a_o^1} C -
\frac{a_0K}{a_0^1m}C \\
\frac{\varphi (C)}{a_0^1} &=& (\frac{K}{a_0^1} \frac{1}{m}+
\frac{1}{a_o^1} - \frac{a_0K}{a_0^1m} C)
\end{array}
$$
(We are using (4.10), (4.7).  This implies $\varphi (C) =C$ by (4.3).
\\
{\bf (2)} ~~ Clearly $\varphi (x_{\alpha} \otimes t^{r_0}) =
x_{\alpha} \otimes t^N$ for some integer.  Write $N= N_{\alpha}+ r_0
\frac{m}{K}$ for some integer $N_{\alpha}$ which may depend on $r_0$.
Clearly $w (x_{\alpha})t^{- r_0} = w (x_{\alpha}) \otimes t^{- r_0
\frac{m}{k} - N_{\alpha}}$.  Let $h_{\alpha} = [x_{\alpha}, w
(x_{\alpha})]$.  Consider the following in $G$.
$$[x_{\alpha} \otimes t^{r_0}, w(x_{\alpha}) \otimes t^{- r_0}]=
h_{\alpha}+ \frac{<x_{\alpha}, w (x_{\alpha})> r_0 C}{K}$$
Now apply $\varphi$ both sides and the bracket takes place in $\widehat{{\cal G}}$
$$\varphi (h_{\alpha}+\frac{r_0}{K} <x_{\alpha}, w (x_{\alpha})>)
 =h_{\alpha}+ \frac{<x_{\alpha}, w (x_{\alpha})>}{m}
(N_{\alpha}+ \frac{m}{K} r_0) C).$$
Since $\varphi (C)=C$ we have
$$\varphi (h_{\alpha}) = h_{\alpha} + <x_{\alpha}, w(x_{\alpha})>
\frac{N_{\alpha}}{m} C.$$
As $h_{\alpha}$ is independent of $r_0$ it follows that $N_{\alpha}$
does not depends on $r_0$. Now consider the following in $G$.
$$[x_{\alpha} \otimes t^{r_o}, w (x_{\alpha}) \otimes t^{s_0}]=
h_{\alpha} \otimes t^{r_0+s_0}+(x_{\alpha}, w(x_{\alpha}) r_0 \delta
_{r_0+s_0,0} C.$$
As earlier apply $\varphi$ both sides
$$\varphi (h_{\alpha} \otimes t^{r_0+s_0})=h_{\alpha} \otimes
t^{(r_0+s_0) \frac{m}{k}} + <x_\alpha, w (x_{\alpha})>
\frac{N_{\alpha}}{m} \delta_{r_0+s_0}C.$$
This proves (3).  For (4) suppose $<x_{\alpha}, x_{\beta}> \neq 0$.
Since $<,>$is $\underline{t}$-invariant, it follows that $\alpha+
\beta$ is zero root. (root with respect to $\underline{t})$.  Thus
$[x_{\alpha}, x_{\beta}]$ is a part of imaginary root and so
$[x_{\alpha}, x_{\beta}] \in \underline{h}$.  Since $\pi$ is an
automorphism we have 
$$< \pi (x_{\alpha}), \pi (x_{\beta})> = < x_{\alpha}, x_{\beta} >
\neq 0.$$
Let $\pi (x_{\alpha}) = e^i x_{\alpha}$ and $\pi (x_{\beta}) = e^j
x_{\beta}$.  Since $< x_{\alpha}, x_{\beta} > \neq 0$ it follows that
$i+j \equiv 0 (K)$.  Thus $[x_{\alpha}, x_{\beta}]$ is $
\pi$-invariant. Consider 
$$
\begin{array}{lll}
[x_{\alpha} t^{r_0}, x_{\beta} t^{s_0}]& =& [x_{\alpha},
x_{\beta}]t^{r_0+s_0} + <x_{\alpha}, x_{\beta} > r_0 \delta_{r_0+s_0}
C \\
\varphi([x_{\alpha}, x_{\beta}] t^{r_0+s_0}) &=& [x_{\alpha},
x_{\beta}] t^{(r_0+s_0) \frac{m}{k}+ N_{(\alpha)}+ N_{(\beta)}} \\
&+& <x_{\alpha}, x_{\beta}> N C \ {\rm for \ some } \ N.
\end{array}
$$
Now from (3) and (4.9) it follows that
$$\varphi [x_{\alpha}, x_{\beta}] t^{r_0+s_0} = [x_{\alpha},
x_{\beta}] t^{(r_0+s_0)} \frac{m}{k}.$$
This forces $N_{\alpha} +N_{\beta} =0$. \\
(5) is clear.

Now we define an isomorphism between $\overline{L} ({\cal G}, \pi)$ and
$\overline{L} ({\cal G}, \theta)$. 

\paragraph*{(4.12) ~~ Proposition}  The following map $\varphi$
define an isomorphism from $\overline{L} ({\cal G}, \pi)$ to $\overline{L}
({\cal G}, \theta)$.
$$\begin{array}{lll}
(1) ~~ \varphi (x_{\alpha} \otimes t^{r_0} t^{\underline{r}}) &=&
x_{\alpha} \otimes t^{r_0 \frac{m}{k}+ N_{\alpha}} t^{\underline{r}} \\
(2) ~~~ \varphi (h_{\alpha} t^{r_0} t^{\underline{r}}) &=& h_{\alpha}
t^{r_0 \frac{m}{k}+N_{\alpha}} t^{\underline{r}} \\
&+& <x_{\alpha}, w (x_{\alpha})> \frac{N_{\alpha}}{m}
t^{\frac{r_0m}{k}} t^{\underline{r}} k_0 \\ 
(3) ~~~ \varphi (t^{r_0} t^{\underline{r}} k_i) &=& t^{r_0
\frac{m}{k}} t^{\underline{r}} k_i, 0 \leq i \leq N \\
(4) ~~ {\rm if} \ <x_{\alpha}, x_{\beta}> &=& 0 \\
{\varphi} ([x_{\alpha}, x_{\beta}] t^{r_0} t^{\underline{r}}
&=& [x_{\alpha}, x_{\beta}] t^{r_0 \frac{m}{k}+ N (\alpha) +
N(\beta)} t^r.
\end{array}
$$
\paragraph*{Proof}  In view of earlier Proposition the RHS belongs to
$\overline{L} ({\cal G}, \theta)$ except possible for (3).  For (3) $t^{r_0}
t^{\underline{r}} k_i \in \overline{L} ({\cal G}, \pi)$ which means $r_0
\equiv (K)$.  Thus $r_0 \frac{m}{k} \equiv 0 (m)$ which means $t^{r_0
\frac{m}{k}} t^{\underline{r}} k_i$ belongs $ \overline{L} ({\cal G}, \theta)$.
 The fact that $\varphi$ defines an isomorphism follows by the
corresponding isomorphism of the earlier proposition.  We will verify
one bracket.  Consider
$$
\begin{array}{lll}
 {\bf 4.13} ~~ [x_{\alpha} t^{r_0}t^{\underline{r}}, x_{\beta} t^{s_0}
t^{\underline{s}}]  &=& [x_{\alpha}, x_{\beta}]t^{r_0+s_0}
t^{\underline{r}+\underline{s}} \\
&+&\frac{<x_{\alpha}, x_{\beta}>}{K} t^{r_0+s_0} t^{\underline{r}+
\underline{s}} k_0 
+<x_{\alpha},x_{\beta}> \sum r_i t^{r_0+s_0} t^{\underline{r+s}}{k_i}
\end{array}
$$
Suppose $<x_{\alpha}, x_{\beta}> =0$.  Then the $\varphi$ of both
sides are equal.  Suppose $<x_{\alpha}, x_{\beta}> \neq 0$, then by
previous proposition it follows that $[x_{\alpha}, x_{\beta}]$ is in
$\underline{h}$ and $\pi$-invariant.  Further $N (\alpha)+ N
(\beta)=0$. 
$$
\begin{array}{lll}
[\varphi (x_{\alpha} t^{r_0}t^{\underline{r}}),\varphi (x_{\beta} t^{s_0}
t^{\underline{s}})]&=& [x_{\alpha} t^{r_0 \frac{m}{k}+N_{\alpha}}
t^{\underline{r}}, x_{\beta} t^{s_0 \frac{m}{k}+N_{\beta}}
t^{\underline{s}}] \\
&=&[x_{\alpha}, x_{\beta}] t^{(r_0+s_0) \frac{m}{k}}
t^{\underline{r}+\underline{s}} \\
&&+ \frac{<x_{\alpha}, x_{\beta}>}{m} (N_{\alpha}+ \frac{r_0 m}{k})
t^{(r_0+s_0)} \frac{m}{k} t^{r+s} k_0 \\
&&+ <x_{\alpha}, x_{\beta} > \sum r_i t^{(r_0+s_o) \frac{r}{k}}
t^{r+s} k_i.
\end{array}
$$
 which is exactly equal to $\varphi$ of the RHS of (4.13).
 
 The remaining brackets are easily verified.   In this section while
defining Lie-bracket structure on $\overline{L} ({\cal G}, \pi)$ we have
taken the following lie bracket.
$$
\begin{array}{lll}
[xt^{r_0} t^{\underline{r}}, y t^{r_0} t^{\underline{s}}] &=&
[x,y]t^{r_0+s_0} t^{r+s} \\
&+& \frac{(x,y)}{K} t^{r_0+s_0} t^{r+s} k_0 \\
&+& (x,y) \sum r_i t^{r_0 + s_0} t^{r+s} k_i
\end{array}
$$
and $\Omega_A/d_A$ is span by $t^{r_0}t^{\underline{r}} k_i$ with
relation $\sum r_it^{r_0} t^{\underline{r}} k_i + \frac{1}{K}
r_0t^{r} t^{\underline{r}} k_0 =0$.  This algebra is clearly
isomorphic to $\overline{L} ({\cal G}, \pi)$ that has been defined in
section where we took $K=1$.  Thus $\overline{L} ({\cal G}, \pi)$ is
isomorphic to $\overline{L} ({\cal G}, \theta)$ without the
denominator $K$.  

\paragraph*{(4.14) ~~ Proposition} $\overline{L} ({\cal G}, \pi)$ is the
universal central extension of $L({\cal G}, \pi)$.  Follows from Remark (2.4)
of [BK].

In the next section we give a faithful realisation to $\overline{L}
({\cal G}, \theta)$ thereby giving a realization to $\overline{L}
({\cal G}, \pi)$ where the infinite dimension centre acts faithfully.
\vskip 5mm
\noindent
{\Large{\bf Section 5}} ~~~{\bf Principal picture. }

In this section we construct level one module for the toroidal
Lie-algebra of type $A_n^K, D_n^K$ and $E_n^K$.  What we do is to
construct representation for the $Z$ algebras where the centre acts
faithfully.  That in turn constructs module for toroidal algebras of
type ADE.  This also covers the twisted case which is new result. 

Notation as in section 4.  Consider the cyclic element $E=
\displaystyle{\sum_{i=1}^{\ell}} E_i \epsilon {\cal G}_{(1)}$.  We make the
assumption that the $\theta$-stable Cartan subalgebra $\underline{t}$ is
the centraliser of $E$.  (See [LW] for details).

Let $t_0=t \oplus \C c \oplus \C d$.  Let $L (\lambda)$ be a basic
module for $\tilde{L} ({\cal G}, \theta)$ where $\lambda \in t_0^*$.  We
renormalize the root vector $x_{\beta}, \beta \in \phi$ such that
$[x_{\beta}, {x}_{-\beta}] =- 2/ < \beta, \beta >$ and $\eta
(p, \beta) =1$ for all $p \in \Z_m$ and for all $\beta \in
\phi$.  As in theorem 8.7 of [LW] choose coset  representatives
$\beta_1, \cdots, 
\beta_{\ell}$ for the action $\theta$ on $\phi$ such that $(\beta_1)_0,
\cdots,  (\beta_{\ell})_0$ is a basis  for $\underline{t}$.  Let
$C_j = \lambda (x_{\beta_j})_0$. 

\paragraph*{Propostion (5.1)}  We have the following from section (8)
of [LW].

\begin{enumerate}
\item[(1)]  dim $\Omega_{L (\lambda)} =1$
\item[(2)] $Z  (\beta_J, \zeta) = C_j $ on $\Omega_{L(\lambda)}$ 
\item[(3)] $Z (\theta^p \beta_j, \zeta) =Z (\beta_j, w^p \zeta)$ 
\item[(4)] $ \displaystyle{\prod_{p \epsilon \Z_m}} (1-w^{-p}
\zeta_1/\zeta_2)^{< \theta^p \beta_1, \beta_2>} Z (\beta_1,\zeta_1) Z
(\beta_2, \zeta_2)$ 
\end{enumerate}
$$-\displaystyle{\prod_{p \epsilon \Z_m}} (1-w^{-p} \zeta_2/\zeta_1)^{<
\theta^p \beta_1, \beta_2 >} Z (\beta_2, \zeta_2) Z (\beta_1, \zeta_1)$$
$$= \frac{1}{m} \displaystyle{\sum_{\theta^p \beta_1+ \beta_2 \epsilon \phi}}
\epsilon (\theta^p \beta_1, \beta_2) Z (\theta^p \beta_1+ \beta_2,
\zeta_2) \delta (w^{-p} \zeta_1/\zeta_2)$$
$$-2m ^{-2} k < \beta_1, \beta_1>^{-1} \displaystyle{\sum_{\theta^p
\beta_1+ \beta_2=0}} D \delta (w^{-p} \zeta_1/ \zeta_2).$$ 

\paragraph{Proof} ~ (1), (2) follows from section 8 of [LW].  For
that just note that $\eta (p, \beta)=1$.  (4) follows from Theorem
8.7 of [LW] as $Z$ operator defined in (2) satisfy (8.21) of [LW].  We are
also using the fact that $\underline{a}_0=0$.
$\Gamma$ be $\Z$-lattice spanned by $\delta_1, \cdots \delta_N, d_1,
\cdots, d_n$. with bilinear form $(\delta_i, d_j) = \delta_{ij}$ and
$(\delta_i, \ \delta_j) = (d_{i}, d_j)=0$. Let $H = \oplus \C
\delta_i$ and $H_+ = \displaystyle{\bigoplus_{n \epsilon \Z_+,i}} \delta_i(n)$
Consider the symmetric algebra $S(H_+)$.

Consider the space $V(\Gamma) = S(H_+) \otimes e^\Gamma$ where
$e^\Gamma$ group algebra.

Let
$$
\begin{array}{lll}
E^+ (\delta, \zeta^m) &= & \rm{exp} \ \sum_{n > 0} \frac{\alpha(n)}{n}
\zeta^{m n}\\
E^-(\delta, \zeta^m) &=& \rm{exp} \ \sum_{n>0} \frac{\alpha(-n)}{-n}
\zeta^{-mn} 
\end{array}
$$

which act on $S(H_+)$ and on $V(\Gamma)$.

let $\delta(0)$ act on $V(\Gamma)$ by

$$
\begin{array}{lll}
\delta(0) u \otimes e^{r} & =& (\delta, r) u \otimes e^r\\[2mm]
\zeta^{\delta(0)} u \otimes e^r &=& \zeta^{(\delta, r)} u \otimes e^r\\[2mm]
C u \otimes e^r & =& 1 u \otimes e^r
\end{array}
$$
Consider $X( \delta, \zeta^m) = E^-(\delta, \zeta^m) z^{m \delta(0)}
E^+ (\delta, \zeta^m)$\\
Let $\underline{r} \epsilon \Z^N$ and let $ \delta_{\underline{r}} = \sum \
r_i \ \delta_i$\\
Let $K_0(\underline{r}, \zeta^m) = X (\delta_{ \underline{r}}, \ \zeta^m)$\\
Let $ \delta(\zeta^m) = \sum \ \zeta^{mn}$\\
Let $k_i(\underline{r}, \zeta^m) =\delta_i(\zeta^m) X (\delta
_{\underline{r}}, \zeta^m)$.
Then by standard argument one can prove that
$$
D k_0 (\underline{r}, \zeta^m) = -m \sum_{i=1}^N r_i k_i
(\underline{r}, \zeta^m). \leqno{(5.2)}
$$
Now define $Z(\alpha, \underline{r}, \zeta) = Z(\alpha, \zeta)
k_0(\underline{r}, \zeta)$. Now we will check all the relation define
at (1.10). Remember $k=1$.
(1)  is true by definition.  (2) is true by the fact that $X(\delta,
\zeta^m) X (\delta_2, \zeta^m) = X (\delta_1 + \delta_2, \zeta^m)$
(3) is just (5.2) (4), (5), (6) can be easily checked. (8) is clear,
(9) is true as $\eta(p, \beta) =1$. (10) is true by definition. Thus
it remains to prove (7). To see this multiply 4 of Prop 5.1
by $k_0 (\underline{r}, \zeta^m) k_0 (\underline{s}, \ \zeta^m_2)$.
Consider $k_0(\underline{s}, \zeta^m_1) k_0 (\underline{s}, \zeta^m_2) D \
\delta(w^{-p} \zeta_1, \zeta_2)$ which is equal to (from
Proposition 2.3(2)) 

$$
\begin{array}{lll}
& D \delta (w^{-p} \zeta_1 / \zeta_2) k_0(\underline{s}, \zeta^m_2) k_0
(\underline{s}, \zeta_2^m)\\
&-\delta(w^{-p} \zeta_1 \ \zeta_2) \frac{d}{d \zeta_1} (K_0
(\underline{r}, \zeta^m_1) k_0(\underline{s}, \zeta^m_2)
\mid_{ (w^{-p} \zeta_2, \zeta_1)} \\[2mm]
&= D \delta (w^{-p} \zeta_1 / \zeta_2) k_0 (\underline{r} +
\underline{s}, \zeta^m_2)\\[2mm]
&+ \delta (w^{-p} \zeta_1 / \zeta_2) m \sum r_i k_i(\underline{r},
\zeta^m) k_0(\underline{s}, \zeta^m)\\[2mm]
&= D \delta (w^{-p} \zeta_1 / \zeta_2) k_0 (\underline{r} +
\underline{s}, \zeta^m_2)\\[2mm]
&+ m \delta (w^{-p} \zeta_1 / \zeta_2) \sum r_i k_i ( \underline{r}
+ \underline{s}, \zeta^m)
\end{array}
$$
Thus (5.3) equals to the R.H.S. of (7) of (1.10)

\pagebreak
\begin{center}
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\end{center}
\vskip 6mm

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