%Paper: hep-th/9406116
%From: "Zapletal" <zapletal@omega.physik.fu-berlin.de>
%Date: Fri, 17 Jun 1994 15:26:07 +0100 (MET DST)

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\title{Highest Weight $U_q[sl(n)]$ Modules\\
 and Invariant Integrable $n$-State Models\\
 with Periodic Boundary Conditions}
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\author{A. Zapletal
\thanks{Supported by DFG, Sonderforschungsbereich 288 'Differentialgeometrie
        und Quantenphysik'}
\thanks{e-mail: zapletal@omega.physik.fu-berlin.de}
\\M. Karowski}
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\date{\small\it Institut f\"{u}r Theoretische Physik\\
      Freie Universit\"{a}t Berlin\\Germany}
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\maketitle
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\begin{abstract}
The weights are computed for the Bethe vectors of an RSOS type
model with periodic boundary conditions obeying $U_q[sl(n)]$
($q=\exp(i\pi/r)$) invariance.
They are shown to be highest weight vectors.
The q-dimensions of the corresponding irreducible representations
are obtained.
\end{abstract}
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In the last years considerable progress has been made on
the ''quantum symmetry" of integrable quantum chain models as the
XXZ-Heisenberg model and its generalizations. In \cite{p1} we constructed
an $sl_q(n)$ invariant RSOS type model with periodic boundary conditions.
In the present paper we prove for this model
the highest weight property of the Bethe states,
calculate the weights and the q-dimensions
of the representations and classify the irreducible ones.
For the case of open boundary conditions see e.g.~\cite{xxzopen},
\cite{devegahw} and \cite{georg}.

The model of \cite{p1} is defined by the transfer matrix
$\tau=\tau^{(n)}$ where
\beq
\label{transfer}
\tau^{(k)}(x,\x^{(k)})=\tr_q({\cal T}^{(k)}(x,\x^{(k)})=\sum_\alpha
q^{n+1-2\alpha}({\cal T}^{(k)})_\alpha^\alpha(x,\x^{(k)}),\quad
k=1,\dots,n.
\eeq
The ``doubled" monodromy matrix is given by
\beq
\label{monodromy}
{\cal T}^{(k)}_0(x,\x^{(k)})=\tilde T^{(k)}_0\cdot T^{(k)}_0(x,\x^{(k)})=
(R_{01}\dots R_{0N_k})\cdot(R_{N_k0}(x_{N_k}/x^{(k)})\dots
R_{10}(x_1/x^{(k)})).
\eeq
For the purpose of the nested algebraic Bethe ansatz in addition to
${\cal T}(x)={\cal T}^{(n)}(x)$ the monodromy matrices
for all $k\le n$ are needed.
The $sl_q(k)$ R-matrix is given by
\beq
\label{rmatrix}
R(x)=xR-x^{-1}PR^{-1}P,\quad
R=
\sum_{\alpha\neq \beta}E_{\alpha\alpha}\ot E_{\beta\beta}+q\sum_{\alpha}
E_{\alpha\alpha}\ot E_{\alpha\alpha}+(q-q^{-1})
\sum_{\alpha>\beta}E_{\alpha\beta} \ot E_{\beta\alpha},
\eeq
The Yang-Baxter equation reads
\beq
\label{ybe}
R_{12}(y/x){\cal T}_1(x)R_{21}{\cal T}_2(y)=
{\cal T}_2(y)R_{12}{\cal T}_1(x)R_{21}(y/x).
\eeq
The model is quantum group invariant, i.e. the transfer matrix
commutes with the generators of $U_q[sl(n)]$. These are obtained from
the monodromy matrices $T(x)$ in the limits
$x$ to $0$ or $\infty$ (up to normalizations)
\beq
\label{generator1}
T=\left( \ba{cccc}
	1&0&\cdots &0\\
	\alpha E_1 &1&\ddots&\vdots\\
	&\ddots &\ddots & 0 \\
	{*} & &\alpha E_{n-1}&1
\ea\right) q^{\bf W},\quad
%
T_\infty=q^{-\bf W}
\left(\ba{cccc} 1&-\alpha F_1 & &*\\
0&1&\ddots& \\&\ddots&\ddots&-\alpha F_{n-1}\\
0&&0&1\ea\right),
\eeq
where $\alpha=q-q^{-1}$ and the matrix
${\bf W}=\mbox{diag}\{W_1,\ldots,W_n\}$ contains the $U_q[gl(n)]$
Cartan elements.
Analogously to eq.~(\ref{generator1}), we introduce
as a limit of ${\cal T}^{(n)}(x,\x^{(n)})$ for $x$ to $0$
% defined by eq.~(\ref{monodromy})
\beq
\label{generator2}
{\cal T}=\tilde T\cdot T,\quad{\rm where}\quad \tilde T=T^{-1}_\infty,
\eeq
here and in the following operators without argument denote these limits
for $x$ to $0$.\\
We write the doubled monodromy matrices as
$k\times k$ block-matrices of operators
\beq
\label{block}
{\cal T}^{(k)}(x)=\left( \ba{ll} \amod^{(k)}(x) & \bemod^{(k)}(x) \\
\cmod^{(k)}(x) & \dmod^{(k)}(x) \ea \right).
\eeq
We also introduce the
reference states $\Phi^{(k)}$ with $\cmod^{(k)}(x)\Phi^{(k)}=0$.
The eigenstates of the transfer matrix $\tau(x)$ are the Bethe ansatz states
$\Psi=\Psi^{(n)}$ obtained by the nested procedure
\beq
\label{bethevector}
\Psi^{(k)}=\bemod^{(k)}_{\alpha_1}(x^{(k-1)}_1)\dots
\bemod^{(k)}_{\alpha_{N_{k-1}}}(x^{(k-1)}_{N_{k-1}})\Phi^{(k)}
\Psi^{(k-1)}_{\underline \alpha},\quad (k=2,\dots,n),\
\Psi^{(1)}=1.
\eeq
The sets of parameters $x_j^{(k)}=\exp\left(i\theta_j^{(k)}-(n-k)
\gamma/2\right)$ $(q=e^{i\gamma})$ fulfil the Bethe ansatz equations:
for $j=1,\dots,N_k$ and $k=1,\dots,n-1$
\beq
\label{bae}
 q^{2+w_{n-k}-w_{n-k+1}}\prod_{l=k \pm 1}\prod_{i=1}^{N_{l}}
\frac{\sinh\frac{1}{2}\left(\theta_j^{(k)}-\theta_i^{(l)}-i\gamma\right)}
{\sinh\frac{1}{2}\left(\theta_j^{(k)}-\theta_i^{(l)}+i\gamma\right)}
\prod_{i=1}^{N_{k}}
\frac{\sinh\frac{1}{2}\left(\theta_j^{(k)}-\theta_i^{(k)}+2i\gamma\right)}
{\sinh\frac{1}{2}\left(\theta_j^{(k)}-\theta_i^{(k)}-2i\gamma\right)}=-1
\eeq
where, below, the $w_i=N_{n-i+1}-N_{n-i}$ will turn out to be the
weights of the state $\Psi$, i.e.~the eigenvalues of the $W_i$'s defined
by eq.~(\ref{generator1}).

\bigskip
\noindent{\bf Theorem}: {\em
The Bethe ansatz states are highest weight states, i.e.}
\[ E_i\psi=0\quad (i=1,\dots,n-1). \]
The analogous statement for the case of open boundary conditions has been
proved in [3].

\medskip
\noindent\underbar{\it Proof}:
First we prove ${\cal T}^\alpha_\beta\Psi=0$
and then  $T^\alpha_\beta\Psi=0$ for $\alpha >\beta$.
The Yang-Baxter relation (\ref{ybe}) implies
\beq
\label{com1}
\tmod^\alpha_\beta \bemod_\gamma(x)=
\bemod_{\gamma'}(x)\tmod^{\alpha'}_{\beta'}
R^{\gamma' \alpha}_{\gamma'' \alpha'} R^{\beta' \gamma''}_{\beta\gamma} ,
\quad \mbox{(for $\alpha>1$, else see eq.~(\ref{cvert}))}.
\eeq
We apply the technique of the nested algebraic Bethe ansatz and commute
${\cal T}$ through all the ${\cal B}$'s of  $\Psi$ in eq.~(\ref{bethevector})
\beq
\label{com2}
\tmod^\alpha_\beta
\bemod_{\alpha_1}(x_1)\dots\bemod_{\alpha_{N_{n-1}}}(x_{N_{n-1}})\Phi^{(k)}
\Psi^{(n-1)}_{\underline \alpha}=
\bemod_{\alpha_1}(x_1)\dots\bemod_{\alpha_{N_{n-1}}}(x_{N_{n-1}})\Phi^{(k)}
\left(\tmod^{(n-1)}\right)^\alpha_\beta\Psi^{(n-1)}_{\underline \alpha}.
\eeq
Iterating this procedure $\beta-1$ times we arrive at
${\cal C}^{(k)}_{\alpha'}\Psi^{(k)}$ with $k=n-\beta+1$ and
$\alpha'=\alpha-\beta$.
At this stage we use, as usual (see e.g.~ref.~\cite{devegan}),
the commutation rule
%//cvert
\beq
\label{cvert}
\cmod^{(k)}_{\alpha'}\bemod^{(k)}_{\gamma}(x)=q^{-1}R^{\ga' \alpha'}
_{\ga \alpha''} \bemod^{(k)}_{\ga'}(x) \cmod^{(k)}_{\alpha''}
+(1-q^{-2})\left((\dmod^{(k)})^{\alpha'}_{\gamma}\amod^{(k)}(x)-
(\dmod^{(k)})^{\alpha'}_{\alpha''}(\dmod^{(k)})^{\alpha''}_{\gamma}(x)\right),
\eeq
to prove that the Bethe ansatz equations (\ref{bae}) imply
\beq
\label{zero1}
{\cal C}^{(k)}_{\alpha'}\Psi^{(k)}=0\quad (k=2,\dots,n)
\quad{\rm and\ therefore}\quad
{\cal T}^\alpha_\beta\Psi=0 \quad{\rm for}\quad \alpha >\beta.
\eeq
%
Finally we show $T^\alpha_\beta\Psi=0$ for $\alpha >\beta$.
We have with eqs.~(\ref{generator1}) and (\ref{zero1}) for all $\beta<n$
\beq
\label{zero2}
\tmod^n_\beta\Psi = \tilde{T}^n_nT^n_\beta\Psi =0
\eeq
and because
$\tilde{T}^n_n$ is an invertible operator it follows that
$T^n_\beta\Psi=0$. Now we consider the previous row, where $\beta<n-1$:
\beq
\label{zero3}
\tmod^{n-1}_\beta\Psi=\left(\tilde{T}^{n-1}_{n-1}T^{n-1}_{\beta}+
\tilde{T}^{n-1}_{n}(T_0)^{n}_{\beta} \right)\Psi=0,
\eeq
and therefrom along with the foregoing result we get $(T_0)^{n-1}_\beta
\Psi=0$.
By iteration we find $T^\alpha_\beta\Psi=0$ for $\alpha >\beta$ and
hence from eq.~(\ref{generator1}) $E_i\Psi=0$ for all
$i$, this proves the highest weight property of the Bethe vectors.\par

Next we compute the weights of the Bethe ansatz states $\Psi$.
We consider first
%
%//decomp
\beq \label{decomp}
\tmod^\alpha_\alpha\Psi=\tilde{T}^\alpha_\alpha T^\alpha_\alpha\Psi+
\sum_{\beta>\alpha}
\tilde{T}^\alpha_\beta T^\beta_\alpha \Psi.
\eeq
Again shifting $\tmod^\alpha_\alpha$ to the right as in eq.~(\ref{com2})
we get the operator $(\tmod^{(n-1)})^\alpha_\alpha$.
By iteration we arrive at $\amod^{(k)} \Psi^{(k)}$ for $k=n-\alpha+1$.
The Yang-Baxter relation (\ref{ybe}) and eqs.~(\ref{monodromy}) and
(\ref{rmatrix}) imply
\beq
\label{weight1}
{\cal A}^{(k)}{\cal B}^{(k)}_\beta(x)=
q^{-2}{\cal B}^{(k)}_\beta(x){\cal A}^{(k)},\quad
{\cal A}^{(k)}\Psi^{(k)}=q^{2N_{k}}\Psi^{(k)}
\eeq
and therefore
\beq
\label{weight2}
\amod^{(k)} \Psi^{(k)}=q^{2(N_{k}-N_{k-1})}\Psi^{(k)}.
\eeq
 From eqs.~(\ref {generator1}) and (\ref{generator2}) we have
\beq
\label{weight3}
\tilde{T}^i_i=T^i_i=q^{W_i}\quad{\rm and}\quad{\cal T}^i_i=q^{2W_i}.
\eeq
and finally
\beq
\label{weight4}
q^{2W_i}\Psi=q^{2w_i}\Psi\quad{\rm with}\quad
w_i= N_{n-i+1}-N_{n-i}.
\eeq
%
So any Bethe ansatz solution is characterized by a weight vector
%//weightvector
\beq \label{weightvector}
w=(w_1,\dots,w_n)=(N_{n}-N_{n-1},\dots,N_{2}-N_{1},N_{1})
\eeq
with the usual highest weight condition
\beq \label{hw}
w_1\ge\cdots\ge w_n\ge 0.
\eeq
Here $N=N^{(n)}$ is the number of lattice sites and $N^{(k)}$
($k=n-1,\dots,1$) is the
number of roots in the $k$-th Bethe ansatz level.
The highest weight condition (\ref{hw}) may be shown as usual.
The result (\ref{weightvector}) is consistent with the ``ice rule"
fulfilled by the $R$-matrix
(\ref{rmatrix}). This means that each operator ${\cal B}^{(k)}_\alpha(x)$
reduces $w_k$ and lifts $w_\alpha$ by one.

The q-dimension of a representation $\pi$ with representation space $V$
is obtained from the ``Markov trace" (see e.g.~ref.~\cite{resht})
\beq \label{qdim}
\dim_q \pi=\tr_V \left(q^{-\sum_{i>j}(W_j-W_i)} \right).
\eeq
As is well known for the case of $q$ being a root of unity
the generators $E_i$ and $F_i$ become nilpotent
%//nil
\beq\label{nil}
(E_i)^r=(F_i)^r=0,\quad q=\exp(i\pi/r),\quad r=n+2,n+3,\ldots.
\eeq
A highest weight module is equivalent to the corresponding one of $sl(n)$,
if this relation does not concern it.
%If this property does not concern the highest weight module, the situation is
%similar to $sl(n)$.
These representations remain still irreducible and will be called good ones.
%
The other representations are called bad and up to special irreducible
cases with vanishing q-dimension
they are reducible but not decomposable.

For the irreducible representations $\pi_w$ with
highest weight
vector $w$ eq.~(\ref{qdim}) gives
\beq
\label{qdim1}
\dim_q \pi_w=   %\tr_{V_w}\left(
		%q^{-\sum_{i>j} (W_j-W_i)} \right)=
\prod _{\alpha \in \Phi_+} \frac
{\left[(w+g,\alpha)\right]_q}{\left[(g,\alpha)\right]_q}=
\prod_{i>j}\frac{\left[w_j-w_i+i-j\right]_q}{\left[
i-j \right]_q},
\eeq
where $\quad [x]_q=(q^x-q^{-x})/(q-q^{-1})$ is a q-number,
$\Phi_+$ denotes the set of positive roots and $g$ is the
Weyl vector $g=\frac{1}{2} \sum_{\alpha \in \Phi_+} \alpha$.
	%\beq  \label{qdim2}
	%\dim_q \pi_w=\prod_{i>j}\frac{\left[w_j-w_i+i-j\right]_q}
	%{\left[i-j\right]_q}.
	%\eeq
The good representations are characterized by positive q-dimensions.
{}From eq.~(\ref{qdim1}) it follows that their weight patterns are restricted
in their length
\beq
w_1-w_n \leq r-n,
\eeq
The q-dimensions of bad representations vanish.

It is an interesting question, how these good representations are
characterized in the language of the Bethe ansatz.
In ref.~\cite{georg}
it is shown for $sl_q(2)$ that these are given by all Bethe ansatz solutions
with only positive parity strings (in the language of Takahashi
\cite{takahashi}).
In a forthcoming paper we will show
how this classification extends to q-symmetries of higher rank.
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   %	       			REFERENCES		        	   %
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\begin{thebibliography}{99}
%
\bibitem{p1} M. Karowski and A. Zapletal, Nucl. Phys.
{\bf B} (1994) in press.
%//p1
%
\bibitem{xxzopen}
L. Mezincescu and R.I. Nepomechie, Mod. Phys. Lett. {\bf A6} (1991)
2497;\\
C. Destri and H.J. de Vega, Nucl. Phys. {\bf B361}
(1992) 361;\\
A. F\"orster and M. Karowski, Nucl. Phys. {\bf B408} (1993) 512;\\
H.J. de Vega and A. Gonz\'{a}lez-Ruiz, Nucl. Phys. {\bf B417} (1994) 553.
%//xxzopen
%
\bibitem{devegahw}
H.J. de Vega and A. Gonz\'{a}lez-Ruiz, `Highest Weight Property for the
$SU_q(n)$ Invariant Spin Chains', LPTHE-PAR 94/13 (1994).
%
%
\bibitem{georg} G. J\"{u}ttner and M. Karowski, `The ``Good" Bethe
Ansatz Solutions of Quantum Group Invariant Heisenberg Models',
SFB 288 preprint (1994).
%//georg
%
\bibitem{devegan} H.J. de Vega, Int. J. Mod. Ph. A {\bf 4} (1989) 2063.
%//devegan
%
\bibitem{resht} N. Yu. Reshetikhin `Quantized Univ. Envel. Algebras, YBE
and Invariants of Links (Part I)', LOMI E-4-87 (1987);\\
%
%\bibitem{pasquier}
V. Pasquier and H. Saleur, Nucl. Phys. {\bf B330} (1990) 523;\\
%//pasquier
%
N. Reshetikhin and V.G. Turaev, Invent. math. 103 (1991) 547;\\
	%\bibitem{fuchs}
J. Fuchs, `Affine Lie Algebras and Quantum Groups',
Cambridge Univ. Press (1992).
%//resht
%
\bibitem{takahashi} M. Takahashi and M. Suzuki, Progr. Th. Ph., Vol. 48,
No. 6b (1972) 2187.
\end{thebibliography}
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