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\begin{document}
\title{Operator relations, dynamical systems, and representations of a
class of Wick algebras}
\author{Vasyl Ostrovsky\u\i{}\thanks{Supported by CRDF grant
no.~UM1-311}, Daniil Proskurin}
\maketitle

\section*{Introduction}

Families of operators subjected to different kinds of relations appear
in different problems of mathematics and its application, mostly in
representation theory, and related problems of theoretical physics.
In general, such problem may be very complicated (wild); however, for
some classes of relations (for example, representations of Lie algebras
etc.) it may happen that complete description of all solutions is
possible up to a unitary equivalence.

In this paper, we consider a general class of operator relations connecting
operators from some family, whose
solutions can be described in terms of orbits of some dynamical system
acting on the spectrum of commuting subfamily.

\section{Operator relations and multi-dimensional dynamical systems}
We consider representations of a family of operator relations between
operators $a_j$, $j=1$, \dots, $n$, of the following form
\begin{align}
a_j^*a_j & = F_j(a_1a_1^*, \dots, a_na_n^*) \notag
\\
a_j^*a_k & = \mu_{jk}\, a_k a_j^*, \notag
\\
a_ja_k& = \lambda_{jk} \, a_k a_j, \label{rels-multi}
\end{align}
where $F_1(\cdot)$, \dots, $F_n(\cdot) \colon
\mathbb{R}^n \to \mathbb{R}$ are measurable mappings,
$\lambda_{jk}$, $\mu_{jk}>0$, $1\le j,k \le n$. Notice that the last two
relations in \eqref{rels-multi} imply that the operators $a_1a_1^*$,
\dots, $a_na_n^*$ commute; this makes clear the sense of taking
functions of them in the first relation.

Now we assume that $a_1$, \dots, $a_n$ are, in general, unbounded,
closed densely defined operators which satisfy relations
\eqref{rels-multi}. We need to take some care when studying relations
between unbounded operators; our approach is to rewrite them in formally
equivalent form, which involves only bounded operators (this is similar
to the case of unbounded representations of a real Lie algebra, which can
be
described in terms of unitary representations of the corresponding Lie
group). Write polar decompositions, $a_j = C_jU_j$, $j=1$, \dots, $n$,
where $C_j$ are non-negative, $U_j$ are partial isometries, and each $C_j$
is zeroe on vectors orthogonal to the range of $U_j$.

\begin{lemma}
Let the operators $a_j$, $j=1$,\dots, $n$ be bounded. Then relations
\eqref{rels-multi} are equivalent to the following
\begin{align}
C_j^2 U_k& = q_{jk} U_kC_j^2,\quad j\ne k, \notag
\\
C_j^2 U_j& = U_j F_j(C_1^2,\dots,C_n^2),\quad
j=1,\dots n, \notag
\\
U_jU_k&=U_kU_j,\quad U_jU_k^*=U_k^*U_j,\quad
j<k,\label{rels-cu}
\end{align}
where
\[
q_{jk} = \begin{cases}
 \mu_{jk}\lambda_{jk},& j<k\\
\mu_{jk}\lambda_{jk}^{-1},& j>k
\end{cases}.
\]
Moreover, operators $(U_j^*)^kU_j^k$, $U_l^m(U_l^*)^m$, $j$, $l=1$, \dots,
$n$;
$k$, $m=1$, $2$,~\dots all $U_j$, $j=1$, \dots, $n$, form a commuting
family
(in particular, all $U_j$ are centered partial isometries).
\end{lemma}

\begin{proof}
The proof is rather straightforward calculation involving the fact that
$U_lU_l^*=\sign C_l$.
\end{proof}

According to \cite{ostur} relations \eqref{rels-cu} can be rewritten in
the following form, involving only bounded operators. Introduce mappings
of $\mathbb{R}^n$ into itself
\begin{gather} \label{rels-ds}
\bold F_l(x_1,\dots,x_n)
=(q_{1l}x_1,\dots,q_{l-1\,l}x_{l-1},
F_l(x_1,\dots,x_n),q_{l+1\,l}x_{l+1},
\dots,q_{nl}x_n),
\\
l=1, \dots, n. \notag
\end{gather}
Then the relations are equivalent to the following
\[
E(\Delta) U_l = U_l E(F_l^{-1}(\Delta)),
\]
where $E(\cdot)$ is a joint resolution of identity of the commuting
family $C_1^2$, \dots, $C_n^2$, $\Delta$ ranges over all measurable
subsets of $\mathbb{R}^n$, $l=1$, \dots, $n$. The latter relations
include only bounded operators, and will be used as a precise version of
the relations in the unbounded case.

According to \cite{ostur}, it makes sense to consider such relations,
for which $\mathbf{F}_j(\mathbf{F}_k(\cdot))=
\mathbf{F}_k(\mathbf{F}_j(\cdot))$, $j\ne k$, which is equivalent to the
following equalities
\[
F_j(\mathbf{F}_k(x_1,\dots,x_n))= q_{jk}
F_j(x_1,\dots,x_n).
\]
In what follows, we are mostly interested in the case of the second order
relations, i.e., linear functions $F_j(\cdot)$, $j=1$, \dots, $n$. If
\[
F_j(x_1, \dots, x_n) = \sum_{l=1}^n
\phi_{jl}x_l +\alpha_jI, \quad
j=1,\dots,n,
\]
the conditions are
\begin{gather}
\phi_{jl}(q_{lk}-q_{jk})+\phi_{jk}\phi_{kl}=0,\quad l\ne j,\ l\ne k \notag
\\
\phi_{jk}\phi_{kj}=0,\quad \phi_{jk}(\phi_{kk} - q_{jk})=0, \notag
\\
\alpha_j(1-q_{jk}) +\alpha_k\phi_{jk}=0,
\label{ds-commute}
\end{gather}
for all $j$, $k=1$\dots, $n$, $j\ne k$.

In what follows, we will assume that the $n$-dimensional dynamical
system generated by the mappings $\mathbf{F}_1(\cdot)$, \dots,
$\mathbf{F}_n(\cdot)$, possesses a measurable section, a measurable set
which meets each orbit at a single point. In this case, for any
irreducible
representation of the relations, the spectral
measure of the commuting family $C_1^2$, \dots, $C_n^2$ is concentrated
on (a subset of) a single orbit, and we can classify all irreducible
repersentations up to unitary equivalence. In the case of more
complicated dynamical systems without a measurable section, non-trivial
ergodic measures may occur, which gives rise to much more complicated
structure of representations, including factor representations of type
$II$ etc. Notice that the linear dynamical system of the form
\eqref{rels-ds} always possesses a measurable section.

We proceed with a  more detailed study of irreducible collections $a_j$,
$j=1$, \dots, $n$ satisfying \eqref{rels-multi}, which correspond to an
orbit $\Omega$. Denote by $\Delta$ the support of the spectral measure
of the commuting family $C_j^2$, $j=1$, \dots, $n$. It is a general fact
that in the basis of eigenvectors of the commuting family the operators
$a_j$ act as weighted shift operators \cite{ostur}, but we need to take
into account that $C_j\ge0$, and that $U_jU_j^*$ is a projection on
co-kernel $(\ker C_j^2)^\perp$, $j=1$, \dots, $n$.

\begin{lemma}
For any $\mathbf{x} = (x_1, \dots, x_n) \in \Delta$ we have

i) $x_j\ge0$, $j=1$, \dots, $n$;

ii) either $\mathbf{F}_j(\mathbf{x})\in \Delta$, or
$(\mathbf{F}_j(\mathbf{x}))_j=0$;

iii) similarly, either $\mathbf{F}_j^{-1}(\mathbf{x})
\in \Delta$, or $\mathbf{x}_j=0$.
\end{lemma}

\begin{proof}
i) Indeed, since $C_j^2\ge 0$, we have $x_j\ge0$, $j=1$, \dots, $n$.

ii) If $\mathbf{F}_j (\mathbf{x}) \notin \Delta$, then $U_j
e_{\mathbf{x}} =0$, where $e_{\mathbf{x}}$ is the basis eigenvector of
the commuting family corresponding to the joint eigenvalue $\mathbf{x}$.
Then we also have $U_j U_j^* e_{\mathbf{F}_j(\mathbf{x})}=0$, and
\[
C_j^2e_{\mathbf{F}_j(\mathbf{x})} = (\mathbf{F}_j(\mathbf{x}))_j\,
e_{\mathbf{F}_j(\mathbf{x})} =0,
\]
which implies $\mathbf{F}_j (\mathbf{x}))_j =0$.

iii) Similarly, if $\mathbf{F}_j^{-1} (\mathbf{x})\notin \Delta$,
then $U_j^*\, e_{\mathbf{x}} =0$. Then $U_jU_j^*\, e_{\mathbf{x}}
=0$, and $C_j^2\, e_{\mathbf{x}} = x_j e_{\mathbf{x}}=0$, which gives
$x_j=0$.
\end{proof}

\begin{corollary}
If for some $\mathbf{x}=(x_1,\dots,x_n) \in \Omega$ $x_j>0$,
and $(\mathbf{F}_j(x))_j <0$, then $\mathbf{x}\notin\Delta$. This
condition implies that irreducible representations correspond only to
orbits, for which $x_j >0$ implies $(\mathbf{F}_j(\mathbf{x}))_j\ge0$,
$(\bold{F}_j^{-1}(\mathbf{x}))_j\ge0$. Notice also that \eqref{rels-ds}
implies that from $x_j>0$ follows $(\mathbf{F}_k(\mathbf{x}))_j >0$ for
$k\ne j$.
\end{corollary}

Consider possible types of orbits and describe corresponding irreducible
representations of \eqref{rels-multi}.

\begin{theorem}
Any irreducible representation can be realized in the space
$l_2(\Delta)$. For any $l=1$, \dots, $n$ ther can be one of the
following:

a). Mapping $\mathbf{F}_l(\cdot)$ possesses a stationary point
$\mathbf{x}\in \Delta$ (in this case all other points are also
stationary). If $x_l=0$, then $a_l=0$; otherwise, the operator $a_l$ has
the form
\[
a_l e_{\mathbf{x}} = \beta_l \, x_l \, e_{\mathbf{x}},
\]
where $\beta_l$ is a parameter equal to one by absolute value;

b). Mapping $F_l(\cdot)$ does not have stationary points. In this case
the operator $a_l$ has the form
\begin{equation}
a_le_{\mathbf{x}} = F_l(\mathbf{x})\, e_{\mathbf{F}_l(\mathbf{x})}.
\end{equation}
The kernel of the operator $a_l$ is generated by vectors
$e_{\mathbf{x}}$ such that $F_l(\mathbf{x})=0$; the kernel of $b_l^*$ is
generated by vectors $e_{\mathbf{x}}$ for which $x_l=0$.
\end{theorem}

\begin{proof}
The proof is essentially based on the following statements from
\cite{ostur}.

\begin{theorem}
Let the dynamical system on $\mathbb{R}^n$ generated by mappings
$\mathbf{F}_l$, $l=1$, \dots, $n$, possess a measurable section. Then for
each irrducible coolection of operators $C_j$, $U_j$, $j=1$, \dots, $n$
satisfying \eqref{rels-cu} the following holds.

i. There exists a unique orbit $\Omega$ of the dynamical system of full
spectral measure of the commuting collection $C_j$, $j=1$, \dots, $n$,
$E(\Omega)=1$;

ii. If $\ker U_l = \{0\}$, then the spectral measure is quasi-invariant
with respect to the mapping $\mathbf{F}_l(\cdot)$; in the case of
unitary $U_l$, the measure is quasi-invariant with respect to
$\mathbf{F}_l^{-1}(\cdot)$, too;

iii. The joint spectrum of the commuting family $C_j$, $j=1$, \dots,
$n$, is simple.
\end{theorem}

\begin{theorem}
Ireducible collection $C_j$, $U_j$, $j=1$, \dots, $n$, satisfying
\eqref{rels-cu} acts in the space $l_2(\Delta)$,  $\Delta \subset
\Omega$ is a subset of some orbit $\Omega$ (for unitary $U_l$, $l=1$,
\dots, $n$, $\Delta=\Omega$) by the following formulas
\[
C_l e_{\mathbf{x}} = x_k e_{\mathbf{x}}, \quad U_l e_{\mathbf{x}} =
u_l(\mathbf{x}) \, e_{\mathbf{F}_l(\mathbf{x})},
\]
where $u_l(\mathbf{x})$ are constants which determine the action of
$U_l$. The subset $\Delta$ has the following ``border conditions''
\begin{align}
u_l(\mathbf{x} & =0 \quad \forall \, \mathbf{x} \in \Delta \colon
\mathbf{F}_l (\mathbf{x}) \notin \Delta \notag
\\
u_l(\mathbf{F}_l^{-1}(\mathbf{x})) & =0, \quad \forall \, \mathbf{x} \in
\Delta \colon \mathbf{F}_l^{-1}(\mathbf{x}) \notin \Delta,
\end{align}
and is ``connected'' in the following sense: $u_l(\mathbf{x}) \ne 0$
$\forall \, \mathbf{x} \in \Delta \colon \mathbf{F}_l(\mathbf{x}) \in
\Delta$, $l=1$, \dots, $n$.
\end{theorem}


Let $\mathbf{x}$ be a stationary point of the mapping
$\mathbf{F}_l(\cdot)$. If $x_l =0$, then for all points $\mathbf{y} \in
\Delta$, the commutaton of $\mathbf{F}_l(\cdot)$ and
$\mathbf{F}_k(\cdot)$ implies $y_l =0$ as well. Then $a_l=0$. If $x_l
\ne 0$, then also $y_l \ne 0$ for all $\mathbf{y} \in \Delta$. In this
case, the operator $U_l$ commutes with all operators $a_j$, $a_j^*$, and
therefore, is a multiple of the identity.

In the case when the mapping $\mathbf{F}_l(\cdot)$ does not have
stationary points, the operator $U_l$ is unitary equivalent to the shift
operator; taking into account that $a_la_l^* =C_l^2$, we get the needed
formula for $a_l$.
\end{proof}

\newcommand{\tenz}{T_{ij}^{kl}}
\newcommand{\otenz}{T_{ik}^{lj}}
\newcommand{\lm}{\lambda_{ij}}
\newcommand{\lf}{\alpha_{ij}}
\newcommand{\qu}{q_{ij}}
\newcommand{\hb}{\mathcal{H}}
\newcommand{\ub}{\mathcal{U}}
\newcommand{\tpr}{e_{i}\otimes e_{j}}
\newcommand{\otpr}{e_{j}\otimes e_{i}}
\newcommand{\alp}{\alpha^{\pi}(i,j)}
\newcommand{\alg}{\mathcal{U}(A,\Lambda)}
\newcommand{\tbig}{T_{1}T_{2}\ , \cdots\ ,\ T_{n}}

\section{Wick algebras}

 * representations of wide class of *-algebras appeared in the
mathematical physics can be described by technic of multidimensional
dynamical systems.

In this section we consider some Wick algebras connected with
dynamical systems.

It was noted above that relations (1) are needed in some cosistency
condition. It was presented in previous section as a commutation of
vector functions obtained from the basic relations.

In this section we look for this consistency from the point of view of
Wick algebras.

Recall that Wick algebras are defined only by relations between
$a_i^*\ ,\ a_j$, and possible relations between $a_i\ ,\ a_j$ are
described by so-called Wick ideals (see \cite{jsw} and \cite{pro} for more details).


Let us consider the following class of Wick algebras:
 \begin{eqnarray*}
a_{i}^{*} a_{i}& = &1+\sum_{j=1}^{d} \lf a_{j}a_{j}^{*}\\
a_{i}^{*}a_{j} &=& \lm\qu a_{j}a_{i}^{*}, \qquad i\ne j,
 \end{eqnarray*}
$0<\alpha_{ii}<1$, $q_{ij}=q_{ji}\in R_{+}$,
$\overline{\lm}=\lambda_{ji}$, $|\lm| = 1$,
 denoted by $\ub(A,\Lambda,Q)$, where $ A=(\lf)$,
$\Lambda = (\lm)$, $Q = (\qu) $.


The purpose of this section is to describe algebras from this class
which have the quadratic ideal of the maximal possible rank
and to classify $*$-representations of these algebras by bounded operators.

\subsection{Quadratic ideals}
 Let $\ub=\ub(A,\Lambda,Q) $, then the operator $T$ (see \cite{jsw})
has a form:
    \begin{eqnarray*}
T e_{i}\otimes e_{i} &=&\alpha_{ii}e_{i}\otimes e_{i} \\
T \tpr\ &=&\lf\tpr +\lambda_{ji}q_{ji}\otpr
    \end{eqnarray*}
Then
     \begin{eqnarray*}
& \hb\otimes\hb = \oplus_{i=1}^{d}\hb_{i}\oplus_{i,j=1}^{d}\hb_{ij} \\
& \hb_{i}=<e_{i}\otimes e_{i}>, \hb_{ij}=< \tpr , \otpr >
     \end{eqnarray*}
 ``Linear conditon'' of \cite{jsw} means that $P$ must be projection on the
subspace, generated by eigenvectors of $T$ with eigenvalue $-1$. Since
$\alpha_{ii}\not = -1$ then rank $P$ is a maximal possible if and
only if equalities
   \begin{equation}
(\lf +1)(\alpha_{ji}+1)= q_{ij}q_{ji} \label{eq:max}
   \end{equation}
hold for all $i\not = j $ and
     \begin{displaymath}
P \hb\otimes\hb = < \otpr - \frac{\lm \qu}{\lf+1}\tpr , i<j >
     \end{displaymath}
 Denote the algebra $\ub(A,\Lambda,Q)$ for which the
equations~\ref{eq:max} hold, by $\ub(A,\Lambda)$.
 ``Quadratic condition'' takes the form:
   \begin{eqnarray}
\lf\alpha_{ji}=0 & i \ne j \nonumber\\
\lf(\lf+1-\alpha_{jj})=0 & i \ne j    \label{eqnarray:main}  \\
\alpha_{ik}(\alpha_{kj}-\lf)+\lf\alpha_{jk}=0 & i\ne j, \
 i\ne k ,\ j \ne k \nonumber
   \end{eqnarray}
It is convenient to consider \{$\lf$\} as a function
\[
\alpha\colon\mathcal{I}\times\mathcal{I}\mapsto R
\]
and to denote $\lf=\alpha(i,j)$
     \begin{remark}


If $\alpha$ is a solution of the system (~\ref{eqnarray:main}) then
for all $\pi\in S_{d}$, $\alp = \alpha(\pi(i),\pi(j))$ is also a solution,
and if $\hat a_{i}=a_{\pi_{i}}$, then the ``structural constants'' for
$\hat a_{i}$ are $\hat{\lambda}_{ij}=\alp$,
 $\hat{\lambda}_{ij}=\lambda^{\pi}(i,j)$.


Consequently it is suffice to describe solutions of the
 (~\ref{eqnarray:main} ) up to the action of $S_{d}$
     \end{remark}
   \begin{definition}
Solution $\alpha$ is called canonical if $\alpha(i,j) = 0$ for all $i<j$
   \end{definition}
  \begin{proposition}
Let $\alpha$ be an arbitrary solution of (~\ref{eqnarray:main}) then
exists $\pi\in S_{d}$ such that $\alpha^{\pi}$ is a canonical solution.
  \end{proposition}

We may suppose now that $\lf=0$ $\forall i<j$. Then the
(~\ref{eqnarray:main})
is reduced to the following:
   \begin{gather*}
\lf(\alpha_{jk}-\alpha_{ik})=0  , \  1\leq k<j<i\leq d\\
\lf(1+\lf-\alpha_{j})=0  , \  1\leq j<i\leq d\\
\alpha_{j}=\alpha_{jj},
   \end{gather*}
where the second equation means only the fact that all non-zero
$\lf$ are equal to the same parameter $\alpha_{j}-1$ for fixed $j$
and $i>j$.
    \begin{definition}
Canonical solution is called decomposable if
\[
\mathcal{I}=\mathcal{I}_{1}\cup\mathcal{I}_{2},
\mathcal{I}_{1}\cap\mathcal{I}_{2}=\emptyset,
\]
and for all $i\in\mathcal{I}_{1}$, $j\in\mathcal{I}_{2}$,
$\lf=\alpha_{ji}=0$.
    \end{definition}
\begin{remark}  \label{remark:posm}
 If a canonical solution is decomposable, then $\exists\pi\in S_{d}$
such that $\alpha^{\pi}$ is decomposable, canonical and
\[
\mathcal{I}_{1}=\{1,\dots ,m\}, \ \mathcal{I}_{2}=\{m+1,\ldots ,d\}
\]
\end{remark}
 It is clear that if $\alpha_{21}=\cdots=\alpha_{d1}=\alpha_{1}-1$,
then $\alpha$ is indecomposable.
  \begin{proposition}  \label{proposition:cal}
Let $\alpha$ be a canonical solution, then it indecomposable if and
only if $\alpha_{21}=\cdots=\alpha_{d1}=\alpha_{1}-1$.
  \end{proposition}
Let $\alpha$ be a canonical solution, $A=(\lf)$. It follows from
the proposition~\ref{proposition:cal} and the remark~\ref{remark:posm},
that we may suppose that for any fixed $j$ all non-zero
$\lf$, $i>j$ are placed before all zero.
 Consider $\vec{k}=(k_{1},\ldots , k_{d-1})$, where
$i\leq k_{i}\leq d$ natural numbers, which is constructed after the
following rule: if for a fixed $j$ and all $i>j$ $\lf=0$ then
$k_{j}=0$, else $k_{j}$ is the greatest number $l$ for which
$\alpha_{lj}=\alpha_{j}-1$.
 The characteristic property of $\vec{k}$:
\begin{proposition}
If $i>j$ and $i\leq k_{j}$, then $k_{i}\leq k_{j}$
\end{proposition}
Conversely, let $\vec{k}$ be a vector with the characteristic
property, and $A=(\lf)$ is a matirix, such that
   \[
\alpha_{ii}=\alpha_{i}, \lf=0, \ i<j
   \]
if \[ k_{j}=j \ \Rightarrow\lf=0, \ \forall i>j \]
else \[ \alpha_{lj}=\alpha_{j}-1, \ j<l\leq k_{j}; \ \alpha_{lj}=0,\
l>k_{j} \].
Then it is easy to verify, that $A$ is a matrix of the canonical
solution. We will denote such matrix by $A(\vec{k})$. We have proved
the following
\begin{theorem}
Let $\alpha$ be a solution of the system ($2$), then
$\exists\pi\in S_{d}$, and $\vec{k}$ with the characteristic property,
such that $(\lf^{\pi})=A(\vec{k})$. Conversely for any $\vec{k}$
with the characteristic property $A=A(\vec{k})$ gives a solution.
\end{theorem}

\subsection{Representations}
Let $A=A(\vec{k})$, $\ub=\alg$. Then $\ub$ has the largest quadratic
ideal, generated by
\[
A_{ij}=a_{j}a_{i} - \lm\qu a_{i}a_{j}, \ i<j
\]
\begin{theorem}
Let $\pi$ be any bounded representation of the $\alg$, then
\[ \pi(A_{ij})=0 \]
\end{theorem}
\begin{remark}
The basic steps of the proof coincide with the proof of the
analogous fact for the twisted commutation relations (see \cite{jsw}),
which are determined by $\Lambda = 1$, $A=A(d,\ldots , d)$ and
$\alpha_{j}=\mu^{2}$.
\end{remark}
Consequently, to describe irreducible representations of the
$\ub$, we must to describe the families of operators
$\{A_{i},\ i=1,\ldots , d\}$, for which the relations hold:
\begin{align}
A_{i}^{*}A_{i}&=1+\alpha_{i}A_{i}A_{i}^{*}+\sum_{j<i} \lf A_{j}A_{j}^{*}
\nonumber\\
A_{i}^{*}A_{j}&=\lm\qu A_{j}A_{i}^{*}, \ i<j \label{align:sys}\\
A_{j}A_{i}&=\lm\qu A_{i}A_{j}, \ i<j \nonumber \\
\qu^{2}&=\alpha_{ji}+1, \ (\lf)=A(\vec{k}). \nonumber
\end{align}
Let $A_{i}^{*}=U_{i}C_{i}$ - polar decomposition.Then the
system~\ref{align:sys} can be rewrited in the equivalent form:
  \begin{gather*}
\vec{C}U_{i}^{*}=U_{i}^{*}\vec{\mathcal{F}}_{i}(\vec{C}), \
\vec{C}=(C_{1}^{2},\ldots ,C_{d}^{2}) \\
[C_{i},C_{j}]=0\ , \ U_{i}U_{j}=\overline{\lambda}_{ij}U_{j}U_{i}\ , \
U_{i}U_{j}^{*}=\lm U_{j}^{*}U_{i}\ , \ i<j \\
\vec{\mathcal{F}}_{i}(x_{1},\ldots ,x_{i},\ldots ,x_{d})=
(x_{1},\ldots ,x_{i-1},1+\alpha_{i}x_{i}+\sum_{j<i} \lf x_{j},
q_{ii+1}^{2}x_{i+1},\ldots ,q_{id}^{2}x_{d})
  \end{gather*}
 Using the dynamical systems technic we can reduce the problem of
the describing of irreducible representations of the $\alg$
to the analogous problem for the finite families of the unitary
operators $\{U_{i}\}$ which satisfy relatons
$U_{i}U_{j}=\lm U_{j}U_{i}$, $i<j$.First, we must introduce
some notations: $D(\mu)$ denotes the operator in $l_{2}(N)$:
\[ D(\mu)e_{n}=\mu^{n-1}e_{n}, \ n\in N \].
$D(j,k_{i})=1$, if $j>k_{i}$, $D(j,k_{i})=D(\alpha_{j})$, if $j\leq
k_{i}$.
$S$-the unilateral schift. $D(f_{j})e_{n}=f_{j}^{n-1}(0)$, where
$f_{j}(x)=1+\alpha_{j}x$, and $f^{n}$ denotes the $n$-th iteration
of $f$.
 Let $1\leq i_{1}\leq\cdots\leq i_{l}\leq d$ - natural numbers, such
that
\[ k_{i_{j}}+1\leq i_{j+1},\ j=1,\ldots ,l-1 \].
Fix one of the such families. Denote
$\Phi=\bigcup_{j=1}^{l}\{i_{j}+1,\ldots ,k_{i_{j}}\}$.
 Let us construct the following irreducible representation for a
fixed family $\{i_{1}\ldots ,i_{l}\}$:
\begin{gather*}
C_{j}=U_{j}=0 \  \forall j\in\phi \\
C_{j}^{2}=\otimes_{i=1,i\not\in\Phi}^{j-1} D(j,k_{i})
\otimes D(f_{j})\otimes 1\cdots\otimes 1 , \ j\neq i_{k} \\
U_{j}^{*}=\otimes_{i=1,i\not\in\Phi}^{j-1} D(\lm)\otimes S
\otimes 1\cdots\otimes 1 , \ j\neq i_{k} \\
U_{i_{k}}^{*}=\otimes_{i<i_{k},i\not\in\Phi} D(\lambda_{ii_{k}})
\otimes\otimes_{i>i_{k},i\not\in\Phi} D(\overline{\lambda}_{ii_{k}})
\otimes \hat{U}_{i_{k}}^{*}, \ k=1,\ldots ,l \\
C_{i_{k}}^{2}=\frac{1}{1-\alpha_{i_{k}}}
\otimes_{i<i_{k},i\not\in\Phi} D(i,k_{i_{k}})\otimes 1\cdots \otimes 1,
\ k=1,\ldots ,l
\end{gather*}
where $\{\hat{U}_{i_{k}}\}$ is irreducible family of the unitary
operators,
which satisfy relations:
\[ \hat{U}_{i}\hat{U}_{j}=\lm\hat{U}_{j}\hat{U}_{j} \].
\begin{theorem}
All irreducuble representations can be obtained by the following
way, moreover two representations are unitary equivalent if and only
if they correspond to the same family $\{i_{1},\dots ,i_{l}\}$, and
the corresponded unitary families are unitary equivalent
\end{theorem}
\begin{remark}
\begin{enumerate}
\item If at least one of the $\lm$ is not a root from $1$, then
exists not of type one representation.
\item If all $\lm$ are roots from $1$, then problem of the classification
of families $\{U_{i}\}$ can be reduced to the case $\lm^{q}=1$,where
$q=p^{m}$ for some prime $p$. In this case families $\{U_{i}\}$ can be
described by simple reduction algorithm.
\end{enumerate}
\end{remark}


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