{\baselineskip=26pt \begin{center} \Large #1 \\ \ \\ \end{center}}
\begin{center} Wen-Li Yang${}^{a,b}$ ~ Alexander Belavin${}^c$ and~Ryu Sasaki${}^d$\end{center}
The central elements of the algebra of monodromy matrices associated with the $\Zb_n$ R-matrix are studied. When the crossing parameter $w$ takes a special rational value $w=\frac{n}{N}$, where $N$ and $n$ are positive coprime integers, the center is substantially larger than that in the generic case for which the ``quantum determinant" provides the center. In the trigonometric limit, the situation corresponds to the quantum group at roots of unity. This is a higher rank generalization of the recent results by Belavin and Jimbo. \vspace{1truecm} \noindent {PACS:} 03.65.Fd; 05.30.-d \noindent {Keywords}: Integrable models; Elliptical quantum goup and quantum group; Central elements.
hep-th0404089.tex


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     \elevenmib Yukawa\, Institute\, Kyoto\\
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     YITP-04-21\\
     {\tt hep-th/0404089} \\ April 2004
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   \large \bf Wen-Li Yang${}^{a,b}$
    ~ Alexander Belavin${}^c$ and~Ryu Sasaki${}^d$\end{center}}
\newcommand{\Address}{\begin{center}


     ${}^a$ Institute of Modern Physics, Northwest University
     Xian 710069, P.\,R. China\\
     ~~\\
     ${}^b$ Department of Mathematics, University of Queensland,
     Brisbane, QLD 4072, Australia\\
     ~~\\
     ${}^c$ Landau Institute for Theoretical Physics, Chernogolovka,
     142432, Russia\\
     ~~\\
     ${}^d$ Yukawa Institute for Theoretical Physics,\\
     Kyoto University, Kyoto 606-8502, Japan
\end{center}}
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\Title{Central elements of the elliptic $\Zb_n$ monodromy matrix
algebra at roots of unity} \Author

\Address
\vspace{1cm}

\begin{abstract}
The central elements of the algebra of monodromy matrices
associated  with the $\Zb_n$ R-matrix are studied. When the
crossing parameter $w$ takes a special rational value
$w=\frac{n}{N}$, where $N$ and $n$ are positive coprime integers,
the center is substantially larger than that in the generic case
for which the ``quantum determinant" provides the center.  In the
trigonometric limit, the situation corresponds to the quantum
group at roots of unity. This is a higher rank generalization of
the recent results by Belavin and Jimbo.





\vspace{1truecm}
\noindent {\it PACS:} 03.65.Fd; 05.30.-d

\noindent {\it Keywords}: Integrable models; Elliptical quantum goup and
quantum group; Central elements.
\end{abstract}
\newpage
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%  1. Introduction                                            %
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\section{Introduction}
\label{intro} \setcounter{equation}{0} The algebra of the
monodromy matrix (or Yang-Baxter algebra) is generated by
noncommutative matrix elements of the L-operator $L(u)$ satisfying
a quadratic ``RLL" relation \bea
R_{12}(u-v)L_1(u)L_2(v)=L_2(v)L_1(u)R_{12}(u-v),\label{RLL}\eea
where $L_1(u)=L(u)\otimes 1$, $L_2(u)=1\otimes L(u)$ and  $R(u)$
is the R-matrix, a solution of quantum Yang-Baxter equation \bea
R_{12}(u-v)R_{13}(u)R_{23}(v)= R_{23}(v)R_{13}(u)R_{12}(u-v).
\label{YBE-V}\eea We adopt the standard notation: $R_{ij}(u)$ is
an embedding operator of R-matrix in the tensor space
$\Cb^n\otimes\Cb^n\cdots$, which acts as identity on the factor
spaces except for the $i$-th and $j$-th ones. If $R(u)$ is a
rational R-matrix, the algebra after a proper specialization is a
Yangian  algebra \cite{Dri85}. For a trigonometric R-matrix, the
algebra is closely connected with a quantum group \cite{Dri86,
Jim85}. If $R(u)$ is an elliptic R-matrix, the algebra gives rise
to a Sklyanin algebra \cite{Skl83,Hou89}.

In this paper, we will address the issue of  the center of an
elliptic algebra associated with the $\Zb_n$ R-matrix \cite{Bel81}
when the crossing parameter takes rational values. In the
trigonometric limit, it corresponds to a quantum group at roots of
unity. So here we call this case  the ``{\it roots of unity \/}"
case as its trigonometric limit. As is well known, for the
trigonometric algebra (as a degenerate algebra of the elliptic
algebra) when the crossing parameter (deformation parameter) is in
roots of unity case the center is much larger than that of the
generic case \cite{Con90, Tar93}. Recently for the simplest case
of the $\Zb_2$ R-matrix (or eight-vertex R-matrix) \cite{Bax86},
the structure of the center in the ``roots of unity" case was
clarified  \cite{Bel02}. Here, we will consider the question with
a generic case $n\geq 2$.

The paper is organized as follows. In section 2 we give the
definition of the elliptic algebra and introduce appropriate
notation. We formulate the fundamental Boltzmann weights of the
$A_{n-1}$ type interaction-around-a-face (IRF) model (or the SOS
model) and give the corresponding face-vertex correspondence
relation in section 3. In section 4, we construct the symmetric
fusion procedure for the R-matrices, Boltzmann weights and
intertwining vectors, which are useful to handle  the center of
the algebra. Finally, we give the center of the elliptic algebra
with the  $\Zb_n$ R-matrix when the crossing parameter is in the
``roots of unity" case in section 5. Section 6 is for conclusions.


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%  2. The Algebra of Monodromy Matrices                       %
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\section{The Algebra of Monodromy Matrices   }
 \label{AMM} \setcounter{equation}{0}

%\subsection{$\Zb_n$ Belavin R-matrix}

Let us fix an integer $n$ ($n\geqslant 2$) and a complex number
$\tau$ such that $Im(\tau)>0$. Introduce the following elliptic
functions \bea &&\t\lt[
\begin{array}{c}
a\\b
\end{array}\rt](u,\tau)=\sum_{m=-\infty}^{\infty}
\exp\lt\{\sqrt{-1}\pi\lt[(m+a)^2\tau+2(m+a)(u+b)\rt]\rt\},\\
&&\t^{(j)}(u)=
\t\lt[\begin{array}{c}\frac{1}{2}-\frac{j}{n}\\[2pt]\frac{1}{2}
\end{array}\rt](u,n\tau),~~~
\s(u)=\t\lt[\begin{array}{c}\frac{1}{2}\\[2pt]\frac{1}{2}
\end{array}\rt](u,\tau),\label{Function}\\
&&\bar{\t}^{(j)}(u)=\exp\lt\{\sqrt{-1}\pi
u\rt\}\,\t^{(j)}(u).\label{Function1}
 \eea Among them the $\s$-function\footnote{Our
$\s$-function is the $\vartheta$-function $\vartheta_1(u)$
\cite{Whi50}. It has the following relation with the {\it
Weierstrassian\/} $\s$-function if denoted it by $\s_w(u)$:
$\s_w(u)\propto e^{\eta_1u^2}\s(u)$,
$\eta_1=\pi^2(\frac{1}{6}-4\sum_{n=1}^{\infty}\frac{nq^{2n}}{1-q^{2n}})
$ and $q=e^{\sqrt{-1}\tau}$.} satisfies the following
identity:\bea
&&\s(u+x)\s(u-x)\s(v+y)\s(v-y)-\s(u+y)\s(u-y)\s(v+x)\s(v-x)\no\\
&&~~~~~~=\s(u+v)\s(u-v)\s(x+y)\s(x-y).\no\eea

Let $\lt\{e_{i}~|~i=1,2,\cdots,n\rt\}$  be the orthonormal basis
of the vector space $\Cb^n$ such that $\langle e_i,~e_j
\rangle=\d_{ij}$. Let $R(u)\in End(\Cb^n\otimes\Cb^n)$ be the
$\Zb_n$ R-matrix given by  \bea
R(u)=\sum_{i,j,k,l}R^{kl}_{ij}(u)E_{ik}\otimes
E_{lj},\label{Belavin-R}\eea in which $E_{ij}$ is the matrix with
elements $(E_{ij})^l_k=\d_{jk}\d_{il}$. The coefficient functions
are \cite{Ric86, Jim87} \bea R^{kl}_{ij}(u)=\lt\{
\begin{array}{ll}
\frac{h(u)\t^{(i-j)}(u+w)}{\t^{(i-k)}(w)\t^{(k-j)}(u)}&{\rm
if}~i+j=k+l~{\rm mod}~n,\\[6pt]
0&{\rm otherwise,}
\end{array}\rt.\label{Belavin-R1}
\eea where a complex parameter $w$ is called the {\it crossing
parameter\/}. We have set \bea
h(u)=\frac{\prod_{j=0}^{n-1}\t^{(j)}(u)}
{\prod_{j=1}^{n-1}\t^{(j)}(0)}.\no\eea The R-matrix satisfies the
quantum Yang-Baxter equation (\ref{YBE-V}),  unitarity and
crossing-unitarity relations \cite{Ric86}.  It should be remarked
that our R-matrix given in (\ref{Belavin-R}) and
(\ref{Belavin-R1}) is the same as that of Ref.\cite{Jim87}, and
that our R-matrix $R(u)$ is related to the R-matrix $S(u)$ of
Richey and Tracy \cite{Ric86} by $R_{12}(u)=\frac{e^{\sqrt{-1}\pi
u}}{n}S^{t_1t_2}_{12}(u)$, where $t_i$ denotes the transposition
in the $i$-th space. Consequently,  the R-matrix enjoys the
following  property:
\begin{Lemma}
The R-matrix given in (\ref{Belavin-R}) and (\ref{Belavin-R1})
satisfies \bea R(w)=M\times P_2^{(+)},\label{Fusion} \eea where
$M$ is a non-degenerate matrix and  $P_2^{(+)}$ is the two-body
symmetrization operator given by $P_2^{(+)}=\frac{1}{2}(1+P_{12})$
in terms of the permutation operator $P_{12}$: $P_{12}(e_i\otimes
e_j)=e_j\otimes e_i$.
\end{Lemma}
This lemma tells that \bea Ker\lt(R(w)\rt)={\rm
the~antisymmetric~subspace~of
}~\Cb^n\otimes\Cb^n.\label{Fusion-P}\eea The above property
enables us to construct symmetric fusion of the R-matrix in higher
tensor space \cite{Che85}  and the intertwining vector in section
4.


\begin{Definition}
The algebra of monodromy matrices $\cal{A}$ is an associative
algebra generated  by the matrix elements of the $L$-operator
$L_{ij}(u)$, $i,j=1,\cdots,n$ with the defining relations
(\ref{RLL}).
\end{Definition}

One can introduce a natural coproduct $\D$: $\cal{A}\rightarrow
\cal{A}\otimes\cal{A}$ and counit $\e$: $\cal{A}\rightarrow \Cb$
\bea \D(L_{ij}(u))=\sum_{k}L_{ik}(u)\otimes L_{kj}(u),~~
\e(L_{ij})=\d_{ij},\no \eea making $\cal{A}$ a bi-algebra. Certain
realization of the algebra $\A$ can be expressed in terms of the
$\Zb_n$ Sklyanin algebra \cite{Skl83,Hou89} which is an elliptical
generalization of quantum group $U_q(gl_n)$.

In this paper, we investigate the center of the algebra $\A$,
i.e., the set of the elements commuting with all the elements of
$\A$. For a generic value of the crossing parameter $w$, the
so-called quantum determinant of the L-operator ($det_qL(u)$)
generates the center of the algebra $\A$ \cite{Hou90}. However, it
is known \cite{Bel02} that $\A$ for the eight-vertex model case
($n=2$) the situation is quite different when $w=\frac{2}{N}$, $N$
is a positive odd integer. In this case, in addition to the
``quantum determinant" $det_qL(u)$, there exist much more central
elements in $\A$. For the generic case of $n\geqslant 2$, the same
phenomenon has been observed for $\A$ in the trigonometric limit
\cite{Con90, Tar93}. We shall study the {\it extra} central
elements of $\A$ with a generic $n\geqslant 2$ in the ``roots of
unity" case: $w=\frac{n}{N}$, where $N$ and $n$ are  positive
coprime integers. Hereafter, we restrict the crossing parameter to
the ``roots of unity" case $w=\frac{n}{N}$, unless otherwise
stated.


\section{$A^{(1)}_{n-1}$ Face Model and Face-Vertex Correspondence}
\label{FVC} \setcounter{equation}{0} In this section, we formulate
the fundamental Boltzmann weights of the $A_{n-1}$ type SOS face
model and give the face-vertex correspondence relation.

An ordered pair $(a,b)$ $(a,b\in \Cb^n)$ is called {\it
admissible\/} if \bea
b-a=e_i,~~i=1,\cdots,n,\label{Admissible}\eea and denoted by
$a\stackrel{1}{\rightarrow}b$. A path
$p=(a^{(0)},a^{(1)},\cdots,a^{(l)})\in (\Cb^n)^{l+1}$ is called an
$l$-path from $a$ to $b$ if $a^{(0)}=a$, $a^{(l)}=b$, and all
pairs $(a^{(i-1)},a^{(i)})$ for $i=1,\cdots,l$ are admissible.

\begin{Definition}
An ordered pair $(a,b)$ $(a,b\in \Cb^n)$ is called {\it
l-admissible\/} if there exists an l-path from $a$ to $b$, and
denoted by $a\stackrel{l}{\rightarrow}b$.
\end{Definition}

Let $W\lt(\lt.\begin{array}{ll}a&b\\c&d\end{array}\rt|\,u\rt)$
denote the Boltzmann weight corresponding to a configuration
$a,\,b,\,d$, $c~\in \Cb^n$ around a face, ordered clockwise from
the NW corner. We call that
$W\lt(\lt.\begin{array}{ll}a&b\\c&d\end{array} \rt|\,u\rt)$ is
{\it admissible\/} if $(a,b)$, $(b,d)$, $(a,c)$, $(c,d)$ are all
{\it admissible\/}. By definition non-admissible weights are set
to $0$. The non-vanishing weights are given as follows \bea
&&W\lt(\lt.\begin{array}{ll}a&a+e_{\mu}\\a+e_{\mu}&a+2e_{\mu}\end{array}
\rt| u\rt)=\frac{\s(u+w)}{\s(w)},\label{W-matrix1}\\
[5pt]
&&W\lt(\lt.\begin{array}{ll}a&a+e_{\mu}\\a+e_{\mu}&a+e_{\mu}+e_{\nu}
\end{array}\rt| u\rt)=\frac{\s(a_{\mu\nu}w -u)}{\s(a_{\mu\nu}w)},~~\mu\neq \nu,
\label{W-matrix2}\\[5pt]
&&W\lt(\lt.\begin{array}{ll}a&a+e_{\nu}\\a+e_{\mu}&a+e_{\mu}+e_{\nu}
\end{array}\rt| u\rt)=\frac{\s(u)\s(a_{\mu\nu}w+w)}{\s(w)\s(a_{\mu\nu}w)},~~\mu\neq
\nu,\label{W-matrix3}\\[5pt]
&&a_{\mu\nu}=\langle
a,e_{\mu}-e_{\nu}\rangle,~~\mu,\nu=1,\cdots,n.\eea The weights
given in (\ref{W-matrix1})-(\ref{W-matrix3}) satisfy the
star-triangle relation (or dynamical Yang-Baxter equation)
\cite{Jim87} \bea
&&\sum_{g}\,W\lt(\lt.\begin{array}{ll}a&b\\f&g\end{array}
\rt|\,u+v\rt)\, W\lt(\lt.\begin{array}{ll}f&g\\e&d\end{array}
\rt|\,u\rt)\,
W\lt(\lt.\begin{array}{ll}b&c\\g&d\end{array} \rt|\,v\rt)\no\\
&&~~~~=\sum_{g}\,W\lt(\lt.\begin{array}{ll}a&g\\f&e\end{array}
\rt|\,v\rt)\, W\lt(\lt.\begin{array}{ll}a&b\\g&c\end{array}
\rt|\,u\rt)\, W\lt(\lt.\begin{array}{ll}g&c\\e&d\end{array}
\rt|\,u+v\rt).\label{STR} \eea


For a generic $\l\in \Cb^n$, define \bea
&&\l_i=\langle\l,e_i\rangle,
~~\bar{\l}_i=\langle\l,\bar{e}_i\rangle,~~\bar{e}_i=e_i-\frac{1}{n}\sum_ke_k,
\label{Def1}\\
&&\l_{ij}=\l_i-\l_j=\bar{\l}_i-\bar{\l}_j,~~i,j=1,\cdots,n.
\label{Def2}\eea Let us introduce an intertwining vector---an
$n$-component column vector $\phi_{\l,\l+e_{j}}(u)$ whose $k$-th
element is \bea
\phi^{(k)}_{\l,\l+e_{j}}(u)=\bar{\t}^{(k)}(u-nw\bar{\l}_j).\label{Interwiner}\eea
One can easily verify the following properties of the intertwining
vector from the definition (\ref{Interwiner}) and equation
(\ref{Def1}).
\begin{Lemma} \label{Shift}
The intertwining vector defined by (\ref{Interwiner}) satisfies
the following relation \bea
&&\phi_{\l+e_j,\l+e_i+e_{j}}(u)=\phi_{\l,\l+e_{i}}(u+w-n\d_{ij}w),~~i,j=1,\cdots,n,\label{F-S-P}\\
&&\phi_{\l,\l+e_i}(u+n)=\phi_{\l,\l+e_i}(u).\label{G-P}\eea
\end{Lemma}
The intertwining vector (\ref{Interwiner}) is different  from that
of Jimbo et al \cite{Jim87} by the definition of the
$\bar{\t}$-function given  in (\ref{Function1}). The present
definition is preferred because of its  periodicity (\ref{G-P})
which will play an important role in section 5. From the
definition of the Boltzmann weights
(\ref{W-matrix1})-(\ref{W-matrix3}) and using the face-vertex
correspondence  obtained in \cite{Jim87}, we have the following
relation.
\begin{Theorem} \label{VF} The intertwining vector satisfies
\bea &&R_{12}(u-v) \phi_{\l,\l+e_{i}}(u)\otimes
\phi_{\l+e_{i},\l+e_{i}+e_{j}}(v)\no\\
&&~~~~=\sum_{k}\,W\lt.\lt(\begin{array}{ll}\l&\l+e_k\\\l+e_i&\l+e_i+e_j\end{array}
\rt|\,u-v\rt) \phi_{\l+e_{k},\l+e_{i}+e_{j}}(u)\otimes
\phi_{\l,\l+e_{k}}(v). \label{Face-vertex}\eea
\end{Theorem}

Then the Yang-Baxter equation of the $\Zb_n$ R-matrix $R(u)$
(\ref{YBE-V}) is equivalent to the star-triangle relation
(\ref{STR}).

\section{Fusion Procedure}
 \label{FP} \setcounter{equation}{0}
The fusion procedure of the R-matrix and the intertwining vector
is essential  in handling the central elements of the algebra
$\A$.
\subsection{The generic $w$ case}
In this subsection, we consider the case that the crossing
parameter $w$ takes a generic value. Let us introduce operators
${\cal{L}}_{1\cdots N}(u)$ and ${\cal{R}}_{1\cdots N}(u)$  by \bea
&&{\cal{R}}_{1\cdots N}(u)\equiv {\cal{R}}_{1\cdots
N;0}(u)=R_{10}(u+(N-1)w)\cdots
R_{N0}(u),\label{F-R}\\
&&{\cal{L}}_{1\cdots N}(u)=L_1(u+(N-1)w)\cdots L_N(u).\label{F-L}
\eea We consider ${\cal{L}}_{1\cdots N}(u)$ ( ${\cal{R}}_{1\cdots
N}(u)$) as an operator acting on the so-called ``{\it auxiliary
space\/}" $V_1\otimes\cdots\otimes V_N$, $V_i=\Cb^n$ with the
entries belonging to $\A$ (with the entries of the operator
${\cal{R}}_{1\cdots N}(u)$ acting on an additional $n$-dimensional
space denoted by $0$-th tensor space). Define an operator $S_N$
acting on $V_1\otimes\cdots\otimes V_N$ as follow \bea
S_N=\prod_{i=1}^{N-1}\prod_{j=i+1}^{N}\,R_{ij}\lt((j-i)w\rt),
\label{S-op}\eea where both indices grow from left to right. For
example, if $N=3$, then \bea
S_3=R_{12}(w)R_{13}(2w)R_{23}(w).\no\eea
\begin{Proposition} \label{Invar}
$Ker(S_N)$ is an invariant subspace in $V_1\otimes\cdots\otimes
V_N$ of the operators ${\cal{L}}_{1\cdots N}(u)$ and
${\cal{R}}_{1\cdots N}(u)$.
\end{Proposition}
{\it Proof\/}. Using the Yang-Baxter equation (\ref{YBE-V}) and
$RLL$ relation (\ref{RLL}), one can derive \bea
&&S_N{\cal{R}}_{1\cdots N}(u)={\cal{R}}_{N\cdots 1}(u)S_N,\no\\
&&S_N{\cal{L}}_{1\cdots N}(u)={\cal{L}}_{N\cdots 1}(u)S_N,\no\eea
where ${\cal{R}}_{N\cdots 1}(u)$ and ${\cal{L}}_{N\cdots 1}(u)$
differ from ${\cal{R}}_{1\cdots N}(u)$ and ${\cal{L}}_{1\cdots
N}(u)$ by permutation of all factors to the opposite order. Hence,
${\cal{R}}_{1\cdots N}(u)\,Ker(S_N)\subset Ker(S_N)$,
${\cal{L}}_{1\cdots N}(u)\,Ker(S_N)\subset
Ker(S_N)$.~~~~~~~$\square$

Now, let us consider the subspace $Ker(S_N)$. Define \bea
W_0=\Sigma^{N-1}_{i=1}~ Ker\lt(R_{i,i+1}(w)\rt)\,\subset
V_1\otimes\cdots\otimes V_N.\label{KC}\eea Using the Yang-Baxter equation
(\ref{YBE-V}), one can show that the factors of $S_N$ can be
permuted into the order such that $R_{i,i+1}(w)$ for a given
$i=1,\cdots,N-1$ is in the rightmost position. For example, if
$N=3$ and $i=1$, \bea
S_3=R_{12}(w)R_{13}(2w)R_{23}(w)=R_{23}(w)R_{13}(2w)R_{12}(w).\no\eea
So, $Ker(S_N)\supset W_0$. Comparing the dimensions of the
subspaces in the trigonometric limit, one can derive that
$Ker(S_N)=W_0$ for the generic crossing parameter $w$. However,
when the crossing parameter $w$ is in the ``roots of unity" case
$w=\frac{n}{N}$, the subspace $Ker(S_N)$ will be larger than
$W_0$.

It follows from Proposition \ref{Invar} that $W_0$ is invariant
under  ${\cal{L}}_{1\cdots N}(u)$ and ${\cal{R}}_{1\cdots N}(u)$.
Thus these operators can be defined in the quotient space
$W_0^{\perp}$ which is defined as follow \bea
W_0^{\perp}=V_1\otimes\cdots\otimes V_N/W_0.\label{KC1}\eea This
also enables one to
construct the symmetric fusion of the R-matrix and the L-operator
in higher tensor space \cite{Che85}. Moreover, the property
(\ref{Fusion-P}) and the definitions of $W_0$ (\ref{KC}) and
$W_0^{\perp}$ (\ref{KC1})
imply that \bea
&&W_0^{\perp}= {\rm
the~symmetric~subspace~of~}
V_1\otimes\cdots\otimes V_N,\label{SYM}\\
&&{\rm dim}(W_0^{\perp})=\frac{(N+n-1)!}{N!(n-1)!},~~ {\rm
dim}(W_0)=n^N-\frac{(N+n-1)!}{N!(n-1)!}. \eea Define a complete
symmetrizer $P_N^{(+)}$ which is given in terms of the element of
$N$-body permutation group denoted by $\P_N$ as follows: \bea
P_N^{(+)}=\frac{1}{N!}\lt\{\sum_{P\in\P_N}\,P
\rt\}.\label{SYM1}\eea Proposition \ref{Invar} and the equation
(\ref{SYM}) lead to the following fusion properties.
\begin{Lemma}
\bea &&P^{(+)}_N{\cal{R}}_{1\cdots N;0}(u)
=P^{(+)}_N{\cal{R}}_{1\cdots
N;0}(u)P^{(+)}_N,\label{F-PR}\\
&&P^{(+)}_N{\cal{L}}_{1\cdots N}(u)=P^{(+)}_N{\cal{L}}_{1\cdots
N}(u)P^{(+)}_N.\label{F-PL}\eea
\end{Lemma}

Hence, one can define the {\it symmetric\/} fused $R$-matrix
${\cal{R}}^{(s)}_{1\cdots N}$ and $L$-operator
${\cal{L}}^{(s)}_{1\cdots N}$ acting in $W_0^{\perp}$ as follows
\bea &&{\cal{R}}^{(s)}_{1\cdots N}(u)\equiv
{\cal{R}}^{(s)}_{1\cdots N;0}(u) =P^{(+)}_N{\cal{R}}_{1\cdots
N;0}(u)P^{(+)}_N,\label{F-R1}\\
&&{\cal{L}}^{(s)}_{1\cdots N}(u)=P^{(+)}_N{\cal{L}}_{1\cdots
N}(u)P^{(+)}_N.\label{F-L1}\eea
%These operators corresponds to the
%Young diagram
%$\stackrel{N}{\overbrace{\Box\!\Box\!\Box\cdots\Box}}$.

For an {\it N-admissible\/} pair $(a,b)$, we choose one of its
$N$-paths from $a$ to $b$: $p=
(a,a+e_{i_1},a+e_{i_1}+e_{i_2},\cdots,a+\sum_{l=1}^Ne_{i_l})$,
$b=a+\sum_{l=1}^Ne_{i_l}$. Let us introduce a fused intertwining
vector $\Phi_{p;a,b}(u)$ as follows \bea &&
\phi^{i_1,\cdots,i_N}_{N;a,b}(u)=\phi_{a,a+e_{i_1}}(u+(N-1)w)
\otimes\cdots\otimes\phi_{a+\sum_{l=1}^{N-1}e_{i_l},a+\sum_{l=1}^{N}e_{i_l}}(u),
\label{Ph-1}\\
&&\Phi_{p;a,b}(u)=P^{(+)}_{N}\lt\{
\phi^{i_1,\cdots,i_N}_{N;a,b}(u)\rt\}.\label{F-In} \eea Noting
Lemma \ref{Shift}, we obtain:
\begin{Proposition} \label{Prop-P}The fused intertwining vector given in
(\ref{F-In}) satisfies\bea \Phi_{p;a,b}(u)=\frac{1}{N!}
\lt\{P^{(+)}_N\lt\{\sum_{P\in \P_N
}\phi^{i_{p_1},\cdots,i_{p_N}}_{N;a,b}(u)\rt\}\rt\},
\label{Main-E}\eea where $P$ is an element of $\P_N$:\bea
P=\lt(\begin{array}{llll}1&2&\cdots&N\\p_1&p_2&\cdots&p_N\end{array}\rt).
\eea
\end{Proposition}
The proof of this Proposition is relegated  to Appendix.

\noindent Then, we have
\begin{Corollary} \label{ID}
The fused intertwining vector given in (\ref{F-In}) is independent
of the choice of the $N$-path from $a$ to $b$ for any {\it
$N$-admissible\/} pair ($a$, $b$).
\end{Corollary}
Because of the path independence, we hereafter  denote the fused
intertwining vector given in (\ref{F-In}) by $\Phi_{N;a,b}(u)$ for
any {\it $N$-admissible\/} pair ($a$, $b$) instead of
$\Phi_{p;a,b}(u)$. From the construction of the fused intertwining
vector, one easily has \bea {\rm Span}\lt\{\Phi_{N;a,b}(u)|~{\rm
for~all~} N-admissible~{\rm pairs} ~(a,b) \rt\}=W_0^{\perp}.\eea

Now we establish an equivalence between the symmetric fused
R-matrix (\ref{F-R1}) and Boltzmann weights considered in
\cite{Jim88} through the fused intertwining vector.
\begin{Proposition} \label{F-W-P}The fused intertwining vector satisfies the
following fused face-vertex correspondence relation \bea &&
{\cal{R}}^{(s)}_{1\cdots N;0}(u-v) \Phi_{N;a,b}(u)\otimes
\phi_{b,b+e_{j}}(v)\no\\
&&~~~~=\sum_{k}\,W_{N1}\lt(\lt.\begin{array}{ll}a&a+e_k\\b&b+e_j\end{array}
\rt|\,u-v\rt) \Phi_{N;a+e_{k},b+e_{j}}(u)\otimes
\phi_{a,a+e_{k}}(v). \label{Face-vertex1}\eea Here the fused
Boltzmann weights $
W_{N1}\lt(\lt.\begin{array}{ll}a&b\\d&c\end{array}\rt|u\rt)$ are
non-vanishing  only for a configuration $a,b,c,d\in \Cb^n$ such
that $a\stackrel{1}{\rightarrow}b$, $b\stackrel{N}{\rightarrow}c$,
$a\stackrel{N}{\rightarrow}d$, $d\stackrel{1}{\rightarrow}c$ and
the non-vanishing weights are given as follows\bea
W_{N1}\lt(\lt.\begin{array}{ll}a&b\\d&c\end{array}\rt|u\rt)=
\lt(\prod_{j=1}^{N-1}\frac{\s(u+jw)}{\s(w)} \rt)
\frac{\s(u+c_{\mu}w-b_{\nu}w)\prod_{\r\neq
\mu}\s(c_{\r}w-b_{\nu}w+w)}
{\prod_{\r}\s(a_{\r\nu}w+\d_{\r\nu}w)},\label{E-F} \eea when
$b-a=e_{\nu}$ and $c-d=e_{\mu}$.
\end{Proposition}
{\it Proof\/}. Suppose $(a,a^{(1)},\cdots,a^{(N-1)},b)$ is an
$N$-path from $a$ to $b$. By the fundamental face-vertex
correspondence relation of Theorem \ref{VF}, one can derive \bea
&&{\cal R}_{1\cdots N;0}
(u-v)\lt(\phi_{a,a^{(1)}}(u+(N-1)w)\otimes \cdots\otimes
\phi_{a^{(N-1)},b}(u)\rt)\otimes \phi_{b,b+e_{j}}(v)\no\\[5pt]
&&~~=\hspace{-5mm}\sum_{a'^{(1)},\cdots,a'^{(N-1)},b'}\hspace{-6mm}
W\hspace{-1.5mm}\lt(\lt.\begin{array}{ll}a&a'^{(1)}\\
a^{(1)}&a'^{(2)}\end{array}\rt|u-v+(N-1)w\hspace{-1mm}\rt)\hspace{-1.5mm}
W\hspace{-1.5mm}\lt(\lt.\begin{array}{ll}a^{(1)}&a'^{(2)}\\
a^{(2)}&a'^{(3)}\end{array}\rt|u-v+(N-2)w\hspace{-1mm}\rt)\no\\[5pt]
&&~~~~~~\times \cdots W\lt(\lt.\begin{array}{ll}a^{(N-2)}&a'^{(N-1)}\\
a^{(N-1)}&b'\end{array}\rt|u-v+w\rt)
W\lt(\lt.\begin{array}{ll}a^{(N-1)}&b'\\
b&b+e_{j}\end{array}\rt|u-v\rt)\no\\[5pt]
&&~~~~~~\times
\lt(\phi_{a'^{(1)},a'^{(2)}}(u+(N-1)w)\otimes\cdots\otimes
\phi_{b',b+e_{j}}(u)\rt) \otimes \phi_{a,a'^{(1)}}(v).\label{Eq}
\eea The definition of the fused intertwining vector (\ref{F-In})
leads to \bea \phi_{a,a^{(1)}}(u+(N-1)w)\otimes \cdots\otimes
\phi_{a^{(N-1)},b}(u)=\Phi_{N;a,b}(u)~~ {\rm mod}~
W_0.\label{Pro}\eea Multiplying equation (\ref{Eq}) by
$P^{(+)}_{N}$ from the left  and using Proposition \ref{Invar}, we
have \bea &&{\rm LHS}=P^{(+)}_{N}{\cal R}_{1\cdots N;0}(u-v)
(\Phi_{N;a,b}(u)+W_0)\otimes \phi_{b,\,b+e_{j}}(v)\no\\[3pt]
&&~~~={\cal R}^{(s)}_{1\cdots N;0}(u-v) \Phi_{N;a,b}(u)\otimes
\phi_{b,\,b+e_{j}}(v),\\[3pt]
&&{\rm RHS}=\sum_{a'^{(1)}}\hspace{-1mm}\lt\{\hspace{-1mm}
\sum_{a'^{(2)},\cdots,a'^{(N-1)},b'}\hspace{-7mm}
W\hspace{-1mm}\lt(\lt.\begin{array}{ll}\hspace{-2mm}a&\hspace{-2mm}a'^{(1)}\\
\hspace{-2mm}a^{(1)}&\hspace{-2mm}a'^{(2)}\end{array}\hspace{-1mm}\rt|\hspace{-1mm}
u\hspace{-1mm}-\hspace{-1mm}v\hspace{-1mm}+\hspace{-1mm}
(N\hspace{-1mm}-\hspace{-1mm}1)w\hspace{-1mm}\rt)\hspace{-1mm}
\cdots
\hspace{-1mm}W\hspace{-1mm}\lt(\lt.\begin{array}{ll}\hspace{-2mm}a^{(N-1)}&
\hspace{-3mm}b'\\\hspace{-2mm}b&
\hspace{-3mm}b+e_{j}\end{array}\hspace{-1mm}\rt|\hspace{-1mm}u\hspace{-1mm}
-\hspace{-1mm}v\hspace{-1mm}\rt)\rt.\no\\[3pt]
&&~~~~\times
\lt.P^{(+)}_{N}\lt\{\phi_{a'^{(1)},a'^{(2)}}(u+(N-1)w)\otimes\cdots\otimes
\phi_{b',b+e_{j}}(u)\rt\} \mbox{\vphantom{\Huge L}}\rt\}
\otimes \phi_{a,a'^{(1)}}(v)\no\\[3pt]
&&~~= \sum_{a'^{(1)}}\lt\{
\sum_{a'^{(2)},\cdots,a'^{(N-1)},b'}W\lt(\lt.\begin{array}{ll}a&a'^{(1)}\\
a^{(1)}&a'^{(2)}\end{array}\rt|u-v+(N-1)w\rt)\cdots\rt.\no\\[5pt]
&&~~~~\times\lt.W\lt(\lt.\begin{array}{ll}a^{(N-1)}&b'\\
b&b+e_{j}\end{array}\rt|u-v\rt)\rt\}
\Phi_{N;a'^{(1)},b+e_{j}}(u)\otimes \phi_{a,a'^{(1)}}(v)\no\\[5pt]
&&~~=\sum_{a'^{(1)}}
W_{N1}\lt(\lt.\begin{array}{ll}a&a'^{(1)}\\
b&b+e_{j}\end{array}\rt|u-v\rt)
\Phi_{N;a'^{(1)},b+e_{j}}(u)\otimes \phi_{a,a'^{(1)}}(v). \eea We
have used Corollary \ref{ID} in the second last equality and Lemma
2.2 of Ref.\cite{Jim88} in the last equality. The expression of
the fused Boltzmann weight
$W_{N1}\lt(\lt.\begin{array}{ll}a&b\\d&c\end{array}\rt|u\rt)$ was
given in \cite{Jim88}. So, we complete the proof. ~~~~~~~$\square$






\subsection{The ``roots of unity" case}
From now on, we come back to the case that the crossing parameter
takes value of ``roots of unity" $w=\frac{n}{N}$. It was shown in
\cite{Bel02} that for the eight-vertex (or $\Zb_2$) case  the
subspace $Ker(S_N)$ is larger than $W_0$ by an additional
$2$-dimensional subspace in ``roots of unity" case. We shall show
that $Ker(S_N)$ is also larger than $W_0$ by an additional
$n$-dimensional subspace for the generic $\Zb_n$ case.

Let us introduce $n$ vectors $\Psi_a^i(u)\in
V_1\otimes\cdots\otimes V_N$, $i=1,\cdots,n,$  \bea
\Psi_a^i(u)=\Phi_{N;a,a+Ne_i}(u),~~{\rm for~a~generic~
vector~}a\in \Cb^n.\label{Bas} \eea We denote the $n$-dimensional
subspace spanned by the vectors $\{\Psi_a^i(u)|i=1,\cdots,n\}$ by
$\overline{W}$ which is independent of the choice of $a$.

\begin{Proposition} \label{Ker}
When the crossing parameter $w=\frac{n}{N}$, $
Ker(S_N)=\overline{W}\oplus W_0$.
\end{Proposition}
{\it Proof\/}. Using the Yang-Baxter equation (\ref{YBE-V}), one
can permute the order of the factors of $S_N$ in such way \bea
S_N=\cdots R_{1N}((N-1)w)R_{1N-1}((N-2)w)\cdots R_{12}(w).\no\eea
Acting it on the vector $\phi_{N;a,a+Ne_i}^{i\cdots i}(u)$ defined
in (\ref{Ph-1}) (There exists a unique $N$-path for the $N$-{\it
admissible\/} pair $(a,a+Ne_i)$ such as
$(a,a+e_i,a+2e_i,\cdots,a+Ne_i)$) and using Theorem \ref{VF}, one
can derive \bea &&S_N\phi_{N;a,a+e_i}^{i\cdots i}(u)=\cdots
R_{1N}((N-1)w)R_{1N-1}((N-2)w)\cdots
R_{12}(w)\phi_{N;a,a+Ne_i}^{i\cdots i}(u)\no\\
&&~~~~=\cdots R_{1N}((N-1)w)\cdots
R_{12}(w)\phi_{a,a+e_i}(u+(N-1)w)\otimes\cdots\phi_{a+(N-1)e_i,a+Ne_i}(u)\no\\
&&~~~~=\cdots\times\frac{\s(Nw)\s((N-1)w)\cdots\s(w)}{(\s(w))^N}
\phi_{a+(N-1)e_{i},\,a+Ne_{i}}(u+(N-1)w)\no\\
&&~~~~~~~~\otimes\phi_{a,\,a+e_{i}}(u+(N-2)w)\otimes\cdots\otimes
\phi_{a+(N-2)e_{i},\,a+(N-1)e_{i}}(u).\eea The fact that
$\phi_{N;a,a+Ne_i}^{i\cdots i}(u)\in Ker(S_N)$ follows from the
identity $\s(Nw)=\s(n)=0$. Noting that \bea
\Psi_a^i(u)=\phi_{N;a,a+Ne_i}^{i\cdots i}(u)~~{\rm
mod}~W_0,\no\eea and the fact $W_0\subset Ker(S_N)$, one can show
$\overline{W}\subset Ker(S_N)$. Evidently $\overline{W}\cap
W_0=\{0\}$. It follows that $\overline{W}\oplus W_0\subset
Ker(S_N)$. Then it remains to prove that \bea {\rm dim}~
Ker(S_N)={\rm dim}~ W_0\,+\, {\rm dim}~
\overline{W}.\label{D-I}\eea However, the dimensions of these
subspaces do not depend on the modulus $\tau$ of the torus. Taking
the trigonometric limit $\tau\longrightarrow +\sqrt{-1}\infty$,
the equation (\ref{D-I}) has been already proved in \cite{Tar93}.
So we complete the proof. ~~~~~~$\square$

Proposition \ref{Invar} and \ref{Ker} show that $\overline{W}$ is
invariant under the fused  R-matrix ${\cal{R}}^{(s)}_{1\cdots
N}(u)$ defined in (\ref{F-R1}) and the L-operator
${\cal{L}}^{(s)}_{1\cdots N}(u)$ defined in (\ref{F-L1}). Let
$\hat{R}(u)\in {\rm End}(\overline{W}\otimes \Cb^n)$ and $\L(u)\in
{\rm End}(\overline{W})\otimes \A$ be the restrictions of the
fused operators ${\cal{R}}^{(s)}_{1\cdots N}(u)$ and
${\cal{L}}^{(s)}_{1\cdots N}(u)$ on $\overline{W}$, explicitly,
\bea &&{\cal{R}}^{(s)}_{1\cdots N}(u)\Psi_a^i(u)=
\hat{R}(u)\Psi_a^i(u)=\sum_{j=1}^n\hat{R}_j^i(u)\Psi_a^j(u),~~
\hat{R}^i_j(u)\in{\rm End}\,(\Cb^n), \label{F-R2}\\
&&{\cal{L}}^{(s)}_{1\cdots
N}(u)\Psi_a^i(u)=\L(u)\Psi_a^i(u)=\sum_{j=1}^n\L_j^i(u)\Psi_a^j(u),~~\L_i^j(u)\in
\A .\label{F-L2}\eea


\section{The Center }
\label{CE} \setcounter{equation}{0}

Let us compute some of the fused Boltzmann weights with a
specially chosen configuration $a,b,c,d\in \Cb^n$. By using
Proposition \ref{F-W-P} and the identity
$\s(Nw\d_{i\nu})=\s(n\d_{i\nu})=0$,  we can derive for the
crossing parameter $w=\frac{n}{N}$ \bea
&&W_{N1}\lt(\lt.\begin{array}{ll}a&a+e_{\nu}\\a+
Ne_i&a+Ne_i+e_{\mu}\end{array}\rt|u\rt)\no\\
&&~~~~= \lt(\prod_{j=1}^{N-1}\frac{\s(u+jw)}{\s(w)} \rt)
\frac{\s(u+Nw\d_{i\mu})\prod_{\r\neq
\mu}\s(a_{\r\mu}w+Nw\d_{i\r})}
{\prod_{\r}\s(a_{\r\mu}w+\d_{\r\mu}w)}\d_{\mu\nu},\no\eea in which
the quasi-periodicity of the $\s$-function \bea
\s(u+Nw)=\s(u+n)=(-1)^n\s(u),~~Nw=n,\eea is used. This leads to a
simple expression of the fused Boltzmann weight \bea
&&W_{N1}\lt(\lt.\begin{array}{ll}a&a+e_{\nu}\\a+Ne_i&a+Ne_i+e_{\mu}\end{array}\rt|u\rt)=
\lt(\prod_{j=1}^{N}\frac{\s(u+jw)}{\s(w)}
\rt)\d_{\mu\nu}.\label{Main-i}\eea In order to compute the
restricted R-matrix $\hat{R}(u)$ defined in (\ref{F-R2}), we
evaluate the action of the fused R-matrix
${\cal{R}}^{(s)}_{1\cdots N}$ defined in (\ref{F-R1}) on the
vector $\Psi_a^i$ \bea &&{\cal{R}}^{(s)}_{1\cdots
N;0}(u-v)\Psi_a^i(u)\otimes
\phi_{a+Ne_i,a+Ne_i+e_j}(v)\no\\
&&~~~~=\lt(\prod_{k=1}^{N}\frac{\s(u-v+kw)}{\s(w)}\rt)~
\Psi_{a+e_j}^i(u)\otimes \phi_{a,a+e_j}(v).\eea Using the shift
property of the fundamental intertwining vector
$\phi_{\l,\l+e_i}(u)$ (\ref{F-S-P}), one can write the fused
intertwining vector explicitly as \bea
&&\Psi_a^i(u)=P^{(+)}_N\lt\{\mbox{\vphantom{\Huge L}}
\phi_{a,a+e_i}(u+(N-1)w)\otimes
\phi_{a,a+e_i}(u+(N-1)w-nw)\rt.\no\\[5pt]
&&~~~~~~~~~~~~\otimes\cdots\otimes\lt.\phi_{a,a+e_i}(u+(N-1)w-(N-1)nw)
\mbox{\vphantom{\Huge L}}\rt\}.\label{F-P-2}\eea The periodicity
(\ref{G-P})  enables one to further derive the following relation
\bea &&\phi_{a,a+e_j}(u+Nw)=\phi_{a,a+e_j}(u+n)
=\phi_{a,a+e_j}(u).\label{Eq-1}\eea The above equation and
(\ref{F-P-2}) imply the following relations
 \bea
&&\Psi_{a}^i(u+nw)=\Psi_a^i(u), \label{F-P-3}
\\&&\Psi_{a+e_j}^i(u)=\Psi_a^i(u+w).
\label{F-P-4}\eea Then
\begin{Proposition} \label{Main-L}When the crossing parameter is in ``roots of
unity" case $w=\frac{n}{N}$, the vector $\Psi_{a}^i(u)$ satisfies
the following relation \bea
\Psi_{a+e_j}^i(u)=\Psi_a^i(u+w)=\Psi_a^i(u).\eea
\end{Proposition}
{\it Proof\/}. The first equality is just equation (\ref{F-P-4}).
We are to prove the second equality $\Psi_a^i(u+w)=\Psi_a^i(u)$.
The expression of $\Psi_a^i(u)$ given in (\ref{F-P-2}) and
equation (\ref{Eq-1}) imply that \bea
\Psi_a^i(u+Nw)=\Psi_a^i(u+n)=\Psi_a^i(u).\label{Eq-2}\eea Due to
the fact that $N$ and $n$ are positive coprime integers,  one can
choose an  integer $l$ such that \bea 1+ lN =0~{\rm mod}~ n.\eea
The above property and relations (\ref{F-P-3}) and (\ref{Eq-2})
allow one to derive that \bea \Psi_a^i(u+w)=\Psi_a^i(u+w+
lNw)=\Psi_a^i(u+(1+ lN)w)=\Psi_a^i(u).\eea Hence, we complete the
proof. ~~~~~~$\square$


\noindent Using Proposition \ref{Main-L}, the action of the fused
R-matrix ${\cal{R}}^{(s)}_{1\cdots N}$ on the vector $\Psi_a^i$
can be given explicitly as
\begin{Proposition}\label{Main-Prop}
\bea &&{\cal{R}}^{(s)}_{1\cdots N;0}(u-v)\Psi_a^i(u)\otimes
\phi_{a+Ne_i,a+Ne_i+e_j}(v)\no\\
&&~~~~=\lt(\prod_{k=1}^{N}\frac{\s(u-v+kw)}{\s(w)}\rt)~
\Psi_{a}^i(u)\otimes \phi_{a,a+e_j}(v).\eea
\end{Proposition}

Now, we calculate the exchange relation among the fused
L-operators $\{\L^{l}_{k}(u)|\,k,l=1,\cdots,n\}$ and the
generators of $\A$ $\{L^j_i(u)|\,i,j=1,\cdots,n\}$. Firstly, one
can derive the following relation  by the  ``RLL" relation
(\ref{RLL}) \bea R_{1\cdots N;0}(u-v)L_{1\cdots
N}(u)L_0(v)=L_0(v)L_{1\cdots N}(u)R_{1\cdots N;0}(u-v).\eea
Proposition \ref{Invar} allows one to restrict the above equation
on $W_0^{\perp}$, namely in terms of the fused R-matrix
(\ref{F-R1}) and L-operator (\ref{F-L1}) \bea
{\cal{R}}^{(s)}_{1\cdots N;0}(u-v){\cal{L}}^{(s)}_{1\cdots
N}(u)L_0(v)=L_0(v){\cal{L}}^{(s)}_{1\cdots
N}(u){\cal{R}}^{(s)}_{1\cdots N;0}(u-v).\label{Main-E-R}\eea
Proposition \ref{Ker} and the fact that $\overline{W}\subset
W_0^{\perp}$ allow one further to restrict the equation on
$\overline{W}$. Finally, we have our main result.
\begin{Theorem}\label{Main-T}When the crossing parameter is in ``roots of
unity" case $w=\frac{n}{N}$, the elements
$\{\L^i_j(u)|\,i,j=1,\cdots,n\}$ defined in (\ref{F-L2}) are the
central elements of $\A$, namely, \bea [\L^i_j(u),\,
L^k_l(v)]=0.\eea
\end{Theorem}
{\it Proof\/}.  For a generic $a\in \Cb^n$, we can introduce an
$n\times n$  matrix $A$ with matrix elements $A^i_j$: \bea
A^i_j=\phi_{a,a+e_j}^{(i)}(u), ~~i,j=1,\cdots,n.\no\eea One can
verify that $det(A)\neq 0$. It means that \bea {\rm
Span}~\{\phi_{a,a+e_j}(u)|\,j=1,\cdots,n\}= \Cb^n
.\label{Comp-1}\eea This fact allows one to define matrix elements
$\tilde{L}^j_i(u)\in \A$ from the L-operator $L(u)$ as follows
\bea
L(u)\phi_{a,a+e_i}(u)=\sum_{j=1}^n\tilde{L}^j_i(u)\phi_{a,a+e_j}(u).
\label{Def3}\eea The conditions $w=\frac{n}{N}$ and (\ref{G-P})
allow one to derive the following relation \bea
&&\phi_{a+Ne_i,a+Ne_i+e_j}(u)=\phi_{a,a+e_j}(u+Nw)
=\phi_{a,a+e_j}(u).\label{P-F-P}\eea Acting both sides of equation
(\ref{Main-E-R}) on the vector $\Psi_{a}^l(u)\otimes
\phi_{a,a+e_j}(v)$ and using the definitions (\ref{F-L2}) and
(\ref{Def3}), we have\bea &&{\rm LHS}=\sum_{k,i=1}^n
{\cal{R}}^{(s)}_{1\cdots N;0}(u-v)\Psi_{a}^k(u)\otimes
\phi_{a,a+e_i}(v)\{\L_k^l(u)\tilde{L}^i_j(v)\},\no\\
&&{\rm RHS}=L_0(v){\cal{L}}^{(s)}_{1\cdots N}(v)
{\cal{R}}^{(s)}_{1\cdots N;0}(u-v)\Psi_{a}^l(u)\otimes
\phi_{a,a+e_j}(v).\no \eea Proposition \ref{Main-Prop} and the
relation (\ref{P-F-P}) imply that we can further derive that \bea
&&{\rm
LHS}=\lt\{\prod_{k=1}^N\frac{\s(u-v+kw)}{\s(w)}\rt\}\sum_{k,i=1}^n
\Psi_{a}^k(u)\otimes
\phi_{a,a+e_i}(v)\{\L_k^l(u)\tilde{L}^i_j(v)\},\no\\
&&{\rm RHS}=
\lt\{\prod_{k=1}^N\frac{\s(u-v+kw)}{\s(w)}\rt\}\sum_{k,i=1}^n
\Psi_{a}^k(u)\otimes \phi_{a,a+e_i}(v)
\{\tilde{L}^i_j(v)\L_k^l(u)\} .\no \eea Hence
$[\L_k^l(u),\,\tilde{L}^i_j(v)]=0$. Equation (\ref{Def3}) and
non-degeneracy of the matrix $(A_j^i)$ enable us to derive that
$[\L_k^l(u),\,L^i_j(v)]=0$. Therefore, we complete the proof.
~~~~~~$\square$



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%                   Conclusions                               %
%                                                            %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Conclusions}
\label{Con} \setcounter{equation}{0} In this paper we have
investigated the {\it extra\/} central elements (differing from
the so-called quantum determinant $det_q(L(u))$) of the algebra
$\A$ of monodromy matrices associated with the $\Zb_n$ R-matrix
when the crossing parameter takes special value $w=\frac{n}{N}$.
In the trigonometric limit, this case corresponds to the quantum
group $U_q(gl_n)$ at roots of unity. Our results show that in even
elliptic case the center of $\A$ is also extended and is generated
by $n^2$ elements $\{\L^i_j(u)\}$ in addition to  $det_q(L(u))$.
For the special case of $n=2$, our result recovers that of
\cite{Bel02}.


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%                                                             %
%  Acknowledgments                                            %
%                                                             %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section*{Acknowledgements}
W.\,-L. Yang is supported from  the Japan Society for the
Promotion of Science and Australian Research Council. A. Belavin is
partially supported by grant
RFBR-04-02-16027 and NATO grant PST.CLG.979008. He also would like
to thank Yukawa Institute for Theoretical Physics, Kyoto
University and Professor R. Sasaki for kind hospitality during his
visit to Kyoto.



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%                                                            %
%     Appendix                                               %
%                                                            %
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\section*{Appendix: Proof of Proposition \ref{Prop-P}}
\setcounter{equation}{0}
\renewcommand{\theequation}{A.\arabic{equation}}
The property (\ref{F-S-P}) allows one to derive the following
relation \bea
&&\phi_{\l+e_j,\l+e_i+e_{j}}(u)=\phi_{\l,\l+e_{i}}(u+w),~~{\rm
if}~ i\neq j.\label{A-1}\eea For a vector $B\in
V_1\otimes\cdots\otimes V_m$, the corresponding symmetrized vector
$B^{(s)}$ is given as \bea
B^{(s)}=P^{(+)}_m(B)=\frac{1}{m!}\{\sum_{P\in\P_m}P(B)\}.\no\eea
The terms obtained by non-trivial permutations of $B$ are called
{\it descendants\/}. It is easy to show that the symmetrized
vector satisfies \bea B^{(s)} =P^{(+)}_m(PB),~~\forall P\in
\P_m.\label{A-2}\eea Hereafter we shall always keep equations
(\ref{A-1}) and (\ref{A-2}) in mind, because they will play an
important role to proof the proposition.

Let us denote the vector
$\phi_{m;a,a+\sum_{l=1}^me_{i_l}}^{i_1\cdots i_m}$ by $i_1\cdots
i_m$, and the set of
$\{\phi_{m;a,a+\sum_{l=1}^me_{i_l}}^{i_{p_1}\cdots
i_{p_m}}|\,P\in\P_m\}$ by $\overbrace{i_1\cdots i_m}$. We shall
prove the proposition by induction.

\begin{itemize}
\item  Using the relation (\ref{A-1}), we can prove that the
equation (\ref{Main-E}) holds for the case of $N=2$.
\item Suppose that (\ref{Main-E}) holds for the case
of $2<N$. We are to prove it is satisfied for $N+1$.

\begin{enumerate}
\item In the case of $i_{N+1}\not\in \{i_1,\cdots,i_N\}$, \bea
&&\Phi_{N+1;a,a+\sum_{l=1}^{N+1}e_{i_l}}^{i_1\cdots
i_{N+1}}\propto
P^{(+)}_{N+1}\lt\{\begin{array}{cc}\overbrace{i_1\cdots
i_N}&~~i_{N+1}\\\vdots&\end{array} \rt\},\label{A-3}\eea where
$\vdots$ stands for the {\it descendants\/}. Using the induction
hypothesis and (\ref{A-1}) we can derive that (\ref{A-3}) is equal
to \bea
&&P^{(+)}_{N+1}\lt\{\hspace{-1mm}\begin{array}{cc}i_1\cdots
i_{N}&i_{N+1}\\\vdots&\end{array}\hspace{-1mm}+\hspace{-1mm}
\begin{array}{cc}i_1\cdots
i_{N+1}&i_{N}\\\vdots&\end{array}\hspace{-1mm}+\hspace{-1mm}
\begin{array}{cc}i_1\cdots
i_{N+1}&i_{N-1}\\\vdots&\end{array}\hspace{-1mm}+\hspace{-1mm}
\cdots \rt\} \no\\[5pt]
&&~~~~~~=P^{(+)}_{N+1}\lt\{\overbrace{i_1\cdots
i_{N+1}}\rt\}.\no\eea So, (\ref{Main-E}) holds for  $N+1$.
\item If $i_{N+1}\in\{i_1,\cdots,i_N\}$, without loss of
generality, let us  suppose that $i_{N+1}=i_N$.  Let
$[i_N]^{\perp}$ denote the set $\{i_l\neq i_N|\,l=1,\cdots N-1\}$.
Let $M$ be the number of elements in the set. Let
$[i_N]^{\perp}_l$, $l=1,\cdots,M$, denote the element of the set.
Similarly to the first case, we have
 \bea
&&\Phi_{N+1;a,a+\sum_{l=1}^{N+1}e_{i_l}}^{i_1\cdots
i_{N}\,i_N}\propto
P^{(+)}_{N+1}\lt\{\begin{array}{cc}\overbrace{i_1\cdots
i_N}&i_{N}\\\vdots&\end{array} \rt\}\no\\[5pt]
&&~~=P^{(+)}_{N+1}\lt\{\begin{array}{cc}i_1\cdots
i_{N}&i_{N}\\\vdots&\end{array}+\begin{array}{cc}i_1\cdots
i_{N}&[i_{N}]^{\perp}_1\\\vdots&\end{array}+\cdots+
\begin{array}{cc}i_1\cdots
i_{N}&[i_{N}]^{\perp}_{M}\\\vdots&\end{array}\rt\} \no\\[5pt]
&&~~=P^{(+)}_{N+1}\lt\{\overbrace{i_1\cdots i_{N}~i_{N}}\rt\}.\no
\eea Then (\ref{Main-E}) holds for $N+1$.
\end{enumerate}
\end{itemize}
Therefore (\ref{Main-E}) holds for any positive integer $N\geq 2$.
Finally we complete the proof of Proposition \ref{Prop-P}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                                                             %
%  References                                                 %
%                                                             %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\end{thebibliography}

\end{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  End                     %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%

hep-th0404089.extracted_bib
\begin{thebibliography}{99}

\bibitem{Dri85} V.\,G. Drinfeld, {\it Sov. Math. Dokl.\/} {\bf 32}
(1985), 254.
\bibitem{Dri86} V.\,G. Drinfeld, {\it Proc. ICM\/}, Berkeley
(1986), 798.
\bibitem{Jim85} M. Jimbo, {\it Lett. Math. Phys.\/} {\bf 10}
(1985), 63.
\bibitem{Skl83} E.\,K. Sklyanin, {\it Func. Anal. Appl.\/} {\bf 17}
(1983), 273.
\bibitem{Hou89} B.\, Y. Hou and H. Wei, {\it J. Math. Phys.\/} {\bf
30} (1989), 2750.
\bibitem{Bel81} A.\,A. Belavin, {\it Nucl. Phys.\/} {\bf B 180}
(1981), 189.
\bibitem{Con90} C. De Concini, V.\,G. Kac, {\it Colloque
Dixmier\/} (1989), 471.
\bibitem{Tar93} V. Tarasov, {\it Commun. Math. Phys.\/} {\bf 158} (1993),
459.
\bibitem{Bax86} R.\,J. Baxter, {\it Exactly Solved Models in
Statistical Mechanics\/}, Academic Press, London, 1986.
\bibitem{Bel02} A. Belavin and M. Jimbo, {\it Inter. J. Mod.
Phys.\/} {\bf A 19} (2004), 50.

\bibitem{Whi50} E.\,T. Whittaker and G.\,N. Watson, {\it A course
of modern analysis}: {\bf 4th edn.}, Cambridge University Press,
Cambridge, 2002.
\bibitem{Ric86} M.\,P. Richey and C.\,A. Tracy, {\it J. Stat.
Phys.\/} {\bf 42} (1986), 311.
\bibitem{Jim87} M. Jimbo, T. Miwa and M. Okado, {\it Lett. Math.
Phys.\/} {\bf 14} (1987), 123; {\it Nucl. Phys.\/} {\bf B 300}
(1988), 74.
\bibitem{Che85} I.\,V. Cherednik, {\it Funct. Anal. Appl.\/} {\bf
19} (1985), 77.
\bibitem{Hou90} B.\, Y. Hou and Y.\,K. Zhou, {\it J. Phys.\/} {\bf A
23} (1990), 1147.
\bibitem{Jim88} M. Jimbo, A. Kuniba, T. Miwa and M. Okado, {\it
Commun. Math. Phys.\/} {\bf 119} (1988), 543.










\end{thebibliography}