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\markboth{\small \em \quad M.~Chaichian and P.~Kulish \hfill}{\hfill
\small\em  Spin Hamiltonians \quad}
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\begin{document}
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\rightline{\mbox{\sl{In Memory of Professor M.S.~Marinov}}}
\medskip

\title{SPIN HAMILTONIANS, QUANTUM GROUPS
AND REACTION-DIFFUSION PROCESSES}

\author{MASUD CHAICHIAN}
%\vspace{0.4cm}
\address{Helsinki Institute of Physics \\
University of Helsinki, FIN-00014, Helsinki, Finland\\
email: chaichia@pcu.helsinki.fi}
\author{PETR P.~KULISH}
\address{St.~Petersburg
Department of Steklov Mathematical Institute,\\
Fontanka 27, 191011, St.~Petersburg, Russia\\
email: kulish@pdmi.ras.ru}



\maketitle

\vspace{0.4cm}

\abstracts{ Solvable spin models on one-dimensional
chain have found a variety of applications in the description
of stationary and non-equilibrium processes. It is pointed out
that some of these models being solved by the quantum inverse
scattering method can be transformed one to another by an
appropriate twist of the underlying dynamical symmetry quantum
algebra. Connections among spectra and eigenfunctions of the
twist related hamiltonians are discussed.}

\vspace{0.4cm}

\tableofcontents

\newpage
%%%%%%%%%%%%%%%%%%%%
%
   \section{Introduction}
%
Integrable models\,\cite{BAX} solved by the quantum inverse scattering
method (QISM)\,\cite{LD,KS,BIK} have a wide range of applications in quantum
field theory and statistical mechanics. Recently some solvable spin
models have found interesting applications in the description of non-equilibrium
processes\,\cite{GL}$^-$\cite{Ker98} and renormalization group
method.\cite{SIERRA}  One of the characteristic features of these
spin models is that corresponding hamiltonians are often
non-hermitian.\cite{RITT,KSTO} We point out in this paper, that
in the theory of quantum groups \,\cite{DR,DRQG}
there exists an interesting transformation (twist)
which relates at first sight different spin models
in the framework of the QISM.\cite{KSTO,KL} The explicit knowledge
of this transformation element ($F$-matrix) leads to explicit relations
of different physically relevant properties of  spin models (spectrum,
eigenvectors).

Particular twist transformations  were applied for the Heisenberg
chain of spin $1/2$ to get so-called 7-vertex model.\cite{KSTO,AHZ}
We write down general formulas of hamiltonian transformations
under the twist procedure, and
apply them to the spin models used in  Ref.\,\protect\citebk{RITT} to describe
certain reaction-diffusion processes. Although the changes of
hamiltonians due to the twisting are relatively simple, the structure
of their eigenstates (Bethe states) and
construction of these Bethe states  can be more complicated. An
isotropic sl(2) Gaudin model deformed by a jordanian classical $r$-matrix
is considered as an example.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%      %
   \section{Reaction-diffusion processes and spin systems}
%      %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Starting, probably, from the paper on the Ising model with random
interaction,\cite{GL} the spin systems on one-dimensional chain
were used to describe also different reaction-diffusion processes.
The basis vectors $|\alpha >, \alpha = 0, 1, \dots , N-1$ of
an $N$ dimensional state space $V$ at a given site
of the chain are identifying with different
species of particles or molecules $A_\alpha$, while  spin
hamiltonian coefficients are given by the transition rates of reactions
corresponding to the operator structure: diffusion (to the right or
to the left), coagulation, annihilation, etc
(see details in  Ref.\,\protect\citebk{RITT}). The state
of the system is given by a probability distribution
$P(\{\alpha_j \})$, which is in
one to one correspondence with normalized vectors
of the spin state space $\cal H$
$$
{\cal H} = \otimes_k V_k \,, \quad
|P\rangle = \sum P(\{\alpha_j \}) \otimes_k |\alpha_k \rangle
$$
with non-negative coefficients in the expansion over the basis vectors.
The main problems related to this non-equilibrium
system are its time evolution, given by a master equation, and the
evaluation of different expectation values or correlation functions.

There is a very special requirement on the spin hamiltonian $H$
entering the evolution equation (master equation):
\be %
 \frac{d}{dt}\, |P\rangle = - H |P\rangle  \; ,
\label{ME}
\ee %
where $|P\rangle$ is the state and $H$ is the spin hamiltonian.
This requirement (conservation of probability) reads:
\be %
\langle all| H = 0 \; , \quad  \langle all| =
\sum_{\{\sigma_j\}} \langle \{\sigma_j\}| \;,
\label{ALL}
\ee %
where $\langle \{\sigma_j\}|$ are all basis vectors of tensorial structure
$\otimes_j \, \langle \sigma_j|$. It is interesting to note that considering
quantities like the mean concentration of a type $A$ particles, are
given by expressions different from usual quantum mechanical
expectation values, e.g.
$$
c_A(t) = \langle all|\Pi_A \exp (- Ht)|P \rangle \,,
$$
where $\Pi_A = \sum_k |\alpha_A\rangle_k \langle \alpha_A|_k$
is the projectors
on the states corresponding to the
particle $A$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%      %
   \section{Twisting of quantum groups and integrable models}
%      %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%

One of the main algebraic ingredients of the quantum inverse scattering
method (QISM) is the theory of quantum groups.\cite{DRQG} A quantum group
( in this context better to call it a quantum algebra ${\cal A}$)
is a particular case of the Hopf algebra i.e. it is
an associative algebra with certain additional maps:
the counit $\epsilon$ , the antipode $S$ and the coproduct
$\Delta$. The latter one is an algebra homomorphism from the quantum algebra
${\cal A}$ to the tensor square of this algebra
$\Delta: {\cal A}\rightarrow {\cal A}\otimes {\cal A}$
so that 
$$
\Delta (a)= \sum b_i\otimes c_i\,, \quad
a, b_i, c_i \in {\cal A}\;.
$$
The coproduct map $\Delta$ satisfies several
axioms and as a consequence it defines a representation of ${\cal A}$ in the
tensor product of its representations ${\cal V}_{\rho} \otimes {\cal V}_{\pi}$.
One can consider composition of any number of coproducts $\Delta^{(n)}:
{\cal A}\rightarrow {\cal A}^{\otimes n}$,
\be %
 \Delta^{(n)} = (id^{\otimes n-2} \otimes \Delta ) \circ
\cdots (id \otimes \Delta ) \circ \Delta   \; ,
\label{COP}
\ee %
and the image does not depend on
the order of $(n-2)$ maps $\Delta$'s (coassociativity property), e.g.
$(id \otimes \Delta ) \circ \Delta (a) = (\Delta \otimes id) \circ \Delta (a)$.

A very special and characteristic property of  quantum algebras used in
the QISM, is the existence of a universal R-matrix ${\cal R}\in {\cal
A}\otimes {\cal A}$
intertwining the coproduct $\Delta$ with the opposite coproduct
$\Delta^{\prime} = \tau\circ \Delta$,
$\Delta^{\prime}(a) = \tau (\sum b_i\otimes c_i)=
\sum c_i\otimes b_i$: ${\cal R}\Delta =\Delta ^{\prime} {\cal R}$. By fixing
the set of representations $\rho_n$, $n=1,2,\cdots,M$,
of the quantum algebra, an $L$-operator defining
integrable model on chain with $M$ sites, can
be obtained from the universal R-matrix:
\begin{eqnarray}
L_{0n}&=& (\rho_0\otimes\rho_n) {\cal R}~, \quad
{\cal R} = \sum {\cal R}_i^{(1)} \otimes {\cal R}_i^{(2)} \,,
\quad  {\cal R}_i^{(j)} \in {\cal A}\,, \\[1mm]
\nonumber\\
T&=& L_{0N}L_{0N-1}\cdots L_{01}= (\rho_0 \otimes \prod_1^N\rho_n)
(id \otimes \Delta^{(N)}) {\cal R}~,
\end{eqnarray}

\noindent
where for simplicity the spectral parameter  dependence of the
$L$-operator
and the monodromy matrix $T(\lambda)$
is not displayed. The hamiltonian of the spin model is
extracted from the matrix $T(\lambda)$ by  quantum trace formulas.\cite{LD,KS,BIK} In particular, the generating function of mutually
commuting integrals of the motion (the transfer matrix) is given
just be the trace of $T(\lambda)$: $ t(\lambda) = tr T(\lambda)$.

There is quite a useful transformation of coproduct in the theory of quantum
groups which is called twisting.\cite{DR}
It is defined by an invertible element
 ${\cal F} \in {\cal A} \otimes {\cal A} $,
${\cal F}= \sum f_i^{(1)}\otimes f_i^{(2)} \,, f_i^{(j)} \in {\cal A}$
and the similarity
transformation of the original coproduct (the antipode $S$ is
also transformed)
\begin{eqnarray}
\Delta_t (a)&=& {\cal F}\Delta (a){\cal F}^{-1}~.
\end{eqnarray}

\noindent
However, to stay in the realm of quantum groups the twisted coproduct
has to satisfy the Hopf algebra axioms. It is sufficient that
the twisting element satisfies the equations
\begin{eqnarray}
(\epsilon \otimes id){\cal F} &=& (id \otimes \epsilon){\cal F}=1~.\\[2mm]
\nonumber\\
{\cal F}_{12}(\Delta \otimes id){\cal F}&=& {\cal F}_{23}(id\otimes \Delta )
{\cal F}~,\label{TE}
\end{eqnarray}

\noindent
where the indices define a non-trivial  embedding of ${\cal F}$ into
${\cal A} \otimes {\cal A} \otimes{\cal A}$, e.g.
${\cal F}_{23}=1 \otimes {\cal F}$.
The twist equation (\ref{TE}) is difficult to solve in full generality.
Still many twisting elements are known explicitly.
Few of them will be used below to explain connections
between different integrable models. A nontrivial character
of these connections is due to the QISM formalism and the following
transformation property of the universal $R$-matrix of the quantum
group\,\cite{DR}
\begin{eqnarray}
{\cal R}^{(t)} &=&  {\cal F}_{21}{\cal R} {\cal F}~.
\label{RT}
\end{eqnarray}

\noindent
Often the choice of representation (e.g. fundamental)  results  in
a simple structure of the $R$-matrix
(as in the case of hamiltonians found in
 Ref.\,\protect\citebk{RITT})
\begin{eqnarray}
 R (u)  &=&  (\rho\otimes \rho) {\cal R} = uR -\frac{1}{u}
R_{21}^{-1}~,
\end{eqnarray}

\noindent
where ${\cal R}$ is the universal $R$-matrix  of the quantum affine
algebra ${\cal A}$, while  $R$ is constant $R$-matrix of finite
dimensional quantum algebra ${\cal U}_q (g) \subset {\cal A}$.
The corresponding hamiltonian is expressed in terms of
$R$, provided $R-R_{21}^{-1} = \omega {\cal P}$
is permutation operator in $V_{\rho}\otimes V_{\rho}$ ,
\begin{eqnarray}
H  &=&  \sum_n  h_{n-1n} = \sum_n ({\cal P}_{n-1n} R_{n-1n}) =
\sum_n {\check{R}}_{n-1n}~.  \end{eqnarray} \noindent The
twisting element ${\cal F}$ in representation space $V_{\rho} \otimes
V_{\rho}$, is also a matrix $F = (\rho\otimes\rho) {\cal F}$, and
according to (\ref{RT}) the hamiltonian density changes by a
similarity transformation \begin{eqnarray} h_{n-1n}^{(t)}  &=&  {\cal
P}_{n-1n} (F_{nn-1}R_{n-1n}F_{n-1 n}^{-1})= F_{n-1n}
{\check{R}}_{n-1n}F_{n-1 n}^{-1}~.  \end{eqnarray}

\noindent
Although this similarity transformation depends on the site number
$n$ there exists a global operator $K$ acting on $\otimes_{n=1}^M V_n$
giving similarity transformation between original and twisted hamiltonians
with free end boundary conditions
\begin{eqnarray}
H^{(t)}\!\!\!\!  &=&  \sum_{n=2}^M  h_{n-1n}^{(t)}=
K (\sum_{n=2}h_{n-1 n}) K^{-1} ~,\\[1mm]
\nonumber\\
K &=&(\rho\otimes\rho\otimes\cdots\otimes \rho) \left({\cal
F}_{12}(\Delta\otimes id)
{\cal F}\cdots (\Delta^{(M-2)}\otimes id)\cal{F}\right)~.
\end{eqnarray}


\noindent
Once we know the operator $K$ or even just its existence, the
conclusion about the
coincidence of  spectra of $H^{(t)}$ and $H$ can be drawn.

Let us point out those models and/or R-matrices for which twist elements
relating
these models with well-known ones, can be found.\\
1. Non-hermitian free fermion model ($XX_\xi$) which was analyzed in detail in
 Ref.\,\protect\citebk{RITT} (B.5), (B.8), is related by a scaling
to the standard $XY$ model twisting  by the element
${\cal F}= exp(\xi \chi^-\otimes \chi^-)$, where $\chi^-$ is a
nilpotent generator of $gl_q(1\vert1)$ having a primitive coproduct
after an additional twist of Reshetikhin type and transformation of
$Z_2$-graded $R$-matrix to the usual one.\cite{KS} Due to the embedding
of $gl_q(1\vert1) \subset gl_q(m\vert n)$ one can apply this twist to
$R$-matrices corresponding  to the higher rank quantum super-algebras
(cf  Ref.\,\protect\citebk{RITT} (C.16)). Hence,
some integrable models 
have a hidden super-algebra symmetries.\cite{CHKU} \\
2. Spin model with three-dimensional space of states at each
site. Its hamiltonian is given by the Cremmer--Gervais $R$-matrix\,\cite{CDK}
$$
H^{(t)}  =  \sum_{1}^{M-1}  h_{nn+1}^{(t)}~,
$$
where the hamiltonian density is given by the matrix
acting in $C^3 \otimes C^3$ ($\omega= q-1/q$):
$$
qI - h_{nn+1}^{(t)} = \left(\begin{array}{ccccccccc}
q&0&0&0&0&0&0&0&0\\
0&0&0&1/p&0&0&0&0&0\\
0&0&0&0&q\nu&0&q/p^2&0&0\\
0&p&0&\omega&0&0&0&0&0\\
0&0&0&0&q&0&0&0&0\\
0&0&0&0&0&0&0&1/p&0\\
0&0&p^2/q&0&-p^2\nu/q&0&\omega&0&0\\
0&0&0&0&0&p&0&\omega&0\\
0&0&0&0&0&0&0&0&q\\
\end{array}\right)$$

\noindent
Using the basis matrices $(E_{ab})_{ij}= \delta_{ai}\delta_{bj}$, the
density can be written as:
$$\begin{array}{l}
q I - h_{nn+1}^{(t)} = \sum_{i=1}^3
qE_{ii}\otimes E_{ii} + (p E_{21}\otimes E_{12} + \frac{1}{p} E_{12}\otimes
E_{21}) \\[1mm]
\qquad +q\nu (pE_{12}\otimes E_{32} -(p/q)^2 E_{32}\otimes E_{12}) \\[1mm]
\qquad + (p^2/q \, E_{31} \otimes E_{13} + q/p^2 \,E_{13}\otimes E_{31})+
\omega \sum_{a<b} E_{bb} \otimes E_{aa}~. \\
\end{array}$$

\noindent
The universal twist element ${\cal F}$ has been proposed in  
Ref.\,\protect\citebk{KM97}.
It is interesting that
the probability conservation condition (\ref{ALL}) requires $p=q$,
while the parameter $\nu$ is arbitrary.

The extra non-zero elements in twisted $R$-matrices (e.g. with
$\nu$-dependence in the Cremmer--Gervais $R$-matrix) vs to the
standard quantum group $R$-matrices,\cite{LD,KS,BIK} result in
a more complicated structure of the commutation relations of the
monodromy matrix $T$ entries, and create difficulties in
constructing the eigenvectors of the hamiltonian by
an algebraic Bethe Ansatz (cf \mbox{Ref.\,\protect\citebk{T}--\protect\citebk{KM}}).
These difficulties will be discussed in the next Section using
the deformation of the isotropic $sl(2)$ Gaudin model by a jordanian
classical $r$-matrix. The latter model can be obtained from the
deformed $XXX$ spin chain\,\cite{KSTO} by a quasi-classical limit.

%%%%%%%%%%%%%%%%%%%%
%
   \section{On Bethe Ansatz for a deformed Gaudin model}
%

Gaudin models can be constructed using any classical $r$-matrix.
Let us start with the classical $r$-matrix of the trigonometric
$XXZ$-Gaudin model
\be %
 r_{XXZ}(\lambda) = (\sinh \lambda)^{-1}
(e^\lambda r_{DJ} + e^{-\lambda}(r_{DJ})_{21})  \; ,
\label{XXZ}
\ee %
where $r_{DJ} = h \otimes h + X^+ \otimes X^- $ is
the classical $r$-matrix of the quantum algebra $sl_q(2)$.
The similarity transformation with $A(y) = \exp y X^+$ gives
$$
{\tilde r_{XXZ}}(\lambda) = Ad(\exp y X^+)^{\otimes 2}  r_{XXZ}(\lambda) =
r_{XXZ}(\lambda) - 2y r_j \,,  
$$
$$
r_j = h \otimes X^+ -  X^+ \otimes h \,.
$$
The additional term, the jordanian $r$-matrix $r_j$, leads to
changes in the commutation relations for trigonometric loop
algebra
$$
[L_1(\lambda) ,\,L_2(\nu)] = -[{\tilde r_{XXZ}}(\lambda - \nu),
\,L_1(\lambda) + L_2(\nu) ]\,,
$$
where as usual the $L$-operator $L(\nu)$ is a matrix in the fundamental
representation
\be  \la{L}
\left(\begin{array}{cc}
h(\nu)&X^- (\nu)\\
X^+ (\nu)&-h(\nu)\\
\end{array}\right)\,.
\ee %
\noindent
However the resulting transfer matrix ${\tilde t}(\lambda) =
\frac 12 {\rm tr} L(\lambda)^2$ is just the similarity transformation
of the initial $t(\lambda)$. Hence, their spectra coincide, and
their eigenfunctions are related by the same similarity transformation
in the quantum space $Ad\, (\otimes_k A_k(y))$.

To get an isotropic $XXX$-Gaudin model\,\cite{S} from the trigonometric
model one has to do a scaling limit:
$$
\lambda \to \epsilon \lambda \,, \quad
y \to \xi/\epsilon  \,, \quad \epsilon \to 0 \,.
$$
The $L$-operator has the usual expression in terms of the loop
algebra generators $h(\lambda), X^+(\lambda), \, X^- (\lambda)$.
However due to the jordanian term in the $r$-matrix their
commutation relations are more complicated
\baa \la{CR}
\left[ h (\lambda) \,,\, h(\nu) \right] &=& -
\xi (X^+ (\lambda) - X^+ (\nu)) \,, \dnon
\left[ h (\lambda) \,,\, X^- (\nu) \right] &=&
\frac{X^- (\lambda) - X^- (\nu)}{\lambda - \nu} - 2 \xi\, h(\nu) \,, \dnon
\left[ h (\lambda) \,,\, X^+ (\nu) \right] &=& -
\frac{X^+ (\lambda) - X^+ (\nu)}{\lambda - \nu} \,, \dnon
\left[X^+ (\lambda) \,,\, X^- (\nu) \right]  &=&
-2\; \frac{h (\lambda) - h(\nu)}{\lambda - \nu} - 2 \xi\, X^+(\lambda) \,, \dnon
\left[X^- (\lambda) \,,\, X^- (\nu) \right]  &=& 2 \xi\, ( X^- (\lambda)
- X^- (\nu) ) \,.
\eaa
This algebra is a quasi-classical limit
of the deformed Yangian ${\cal Y}_\xi (sl(2))$.\cite{KST}

The transfer matrix $t(\lambda)$ has finite scaling limit, as well
as its spectrum despite of the singular behavior of $y \to \xi/\epsilon$. 
Hence the spectrum of the deformed $XXX$-Gaudin
model coincides with the original one. However, the similarity
transformation of the original creation operators $X^-(\nu)$ 
in the quantum space do not 
have finite scaling limit, and the same is true for the non-trivial
eigenvectors. By this reason their construction requires a recurrence
algebraic Bethe Ansatz.\cite{T,KM} Let us pointed out that realization
of the commutation relations (\ref{CR}) can be given as usual in terms
the local generators $h_k,\, X^{\pm}_k$ and the inhomogeneity parameters
$z_k$ related to the sites $k = 1, 2, \dots , N$ of the chain. The 
realization is 
\be %
h(\lambda) = \sum_{k=1}^N \left( \frac {h_k}{\lambda - z_k} \right) 
- \xi X_{gl}^+ \,, \quad 
X^-(\lambda) = \sum_{k=1}^N \left( \frac {X^-_k}{\lambda - z_k} \right) 
+2 \xi h_{gl} \,,  
\ee %
with unchanged $X^+(\lambda)$ and $X^+_{gl} = \sum_{k=1}^N X^+_k$. 
Taking the highest spin vector $\Omega$ as the bare vacuum (it is
stable under the similarity transformation)
\be %
h(\lambda)\Omega = \rho (\lambda) \Omega \,, \quad
X^+(\lambda) \Omega = 0\,, \quad  \otimes_k A_k(y) \Omega = \Omega \,,
\label{VAC}
\ee %
one can obtained the eigenvectors of $t(\lambda)$  acting on
$\Omega$ by the lowering generators $X^-(\nu_j)$. However, due to their
non-commutativity among themselves (\ref{CR})
the ``multi-particle''
creation operators have elaborated structure (cf  Ref.\,\protect\citebk{T}
and \protect\citebk{KM})
\be %
t(\lambda)\Psi(\nu_1, \nu_2, \dots \nu_M) = \Lambda (\lambda; \{\nu_j\})
\Psi(\nu_1, \nu_2, \dots \nu_M)\,,
\label{EV}
\ee %
\be %
\Psi(\nu_1, \nu_2, \dots \nu_M) =  B_M(\nu_1, \nu_2, \dots \nu_M)\Omega,
\label{CO}
\ee %
\be %
B_1(\nu) = X^-(\nu) \,, \quad
B_2(\nu_1, \nu_2) = X^-(\nu_1)X^-(\nu_2) - 2\xi X^-(\nu_1),
\label{B}
\ee %
$$
%\be %
B_3(\nu_1, \nu_2, \nu_3)= X^-(\nu_1)X^-(\nu_2)X^-(\nu_3) -
$$
$$
-2 \xi ( 2 X^-(\nu_1)X^-(\nu_2) + X^-(\nu_1)X^-(\nu_3)) +
8 \xi^2 X^-(\nu_1).
%\label{B3}
%\ee %
$$
These operators are obtained by commuting $t(\lambda)$ with the
ordered product of $X^-(\nu_j)$ and correcting the result by a lower
order products to get eigenvector provided the Bethe equations
on the set of quasimomenta $\{\nu_j\}_{j=1}^M$ are satisfied 
\be \la{Beq}
\rho (\nu_j) - \sum_{k \neq j}^M \frac {1}{\nu_j - \nu_k}
= 0 \;.
\ee 
The eigenvalue of $t(\lambda)$ is the same as for the $sl(2)$-invariant 
Gaudin model\,\cite{S} 
\be %
\Lambda (\lambda; \{\nu_j\}) = 
(\rho (\lambda) - \sum_{k =1}^M \frac {1}{\lambda - \nu_k})^2 
- \frac {\partial}{\partial \lambda}
(\rho (\lambda) - \sum_{k =1}^M \frac {1}{\lambda - \nu_k})\,.
\ee %
It is instructive to note that these creation operators are
symmetric functions of quasimomenta $\{\nu_j\}$.
The commutation relation of $t(\lambda)$ with the product of
$X^-(\nu_j)$ can be obtained by using the quantum $M$-operator
\be %
[t(\lambda),\, L(\nu)] = [M(\lambda -\nu),\, L(\nu)]\,, \quad
\label{M}
\ee %
where
\baa \la{Ma}
M(\lambda - \nu ) = - \underset 1 {\tr} \left(
r_{12}(\lambda - \nu) \underset 1 {L}(\lambda) \right) - \frac 12
\underset 1 {\tr} \left(  r_{12}^2 (\lambda - \nu)  \right) 
\eaa
with the second term as quantum corrections  due to non-commutativity 
of the quantum $L$-operator entries (\ref{L}). 
In particular, we have
$$
t(\lambda)\, X^-(\nu) = X^-(\nu)\,t(\lambda)\,+ 2\,
\frac{X^- (\lambda)h(\nu) - X^- (\nu)h(\lambda)}{\lambda - \nu}
$$
$$
- 2 \xi\, (X^-(\nu) - \xi\, )\, X^+(\lambda) -
2 \xi\, (h(\lambda)h(\nu) +h(\nu)h(\lambda)) \,.
$$
It is interesting to note that the transfer matrix $t(\lambda)$
commutes with the global generator $X^+_{gl}$, and the constructed
eigenstates $\Psi(\nu_1,...,\nu_M)$ are the highest spin vectors
$$
 X^+_{gl} \Psi(\nu_1,...,\nu_M) = 0\,.
$$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% new stuff %%%%%%%%%%%%%%%%%%%%%%%
The complete set of the creation operators and correlation functions,
as well as connection with the Knizhnik-Zamolodchikov equation and
deformed conformal field theory will be studied in the near future.


\section*{Acknowledgement}
\addcontentsline{toc}{section}{\numberline{}Acknowledgment}

The authors are grateful to N.M.~Bogoliubov,
R.~Kerner and A.I.~Mudrov  for
useful discussions and to the Academy of Finland for financial support
under the Project no. 163394, this research of PPK was also
partially supported by RFBR grant 99-01-00101 and by INTAS grant
99-01459.


\section*{References}
\addcontentsline{toc}{section}{\numberline{}References}

\begin{thebibliography}{**}

\bibitem{BAX}
R.J.~Baxter, {\em Exactly Solved Models in Statistical Mechanics}
(Academic Press, New York, 1982).
\bibitem{LD}
L.D.~Faddeev, in
{\em Quantum Symmetries/Symetries Quantiques}, Les Houche Lec\-tures,
Session LXIV, eds.\ A.~Connes, K.~Gawedzki and J.~Zinn-Justin 
(North-Holland, 1998) p.~149.
\bibitem{KS}
P.P.~Kulish and E.K.~Sklyanin, in {\em Integrable Quantum Field Theories},
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\end{thebibliography}

\end{document}

