One Hat Cyber Team

Edit File: WITHDP.TEX

\documentclass[12pt]{article}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{makeidx}
\makeindex

%\textheight 8.5 in
%\textwidth 5.5in
\oddsidemargin 0pt
\evensidemargin 0pt
\topmargin0pt

\theoremstyle{plain}
\newtheorem{theorem}{Theorem}
\newtheorem{proposition}{Proposition}
\newtheorem{lemma}{Lemma}
\newtheorem{corollary}{Corollary}
\newtheorem{definition}{Definition}
\theoremstyle{remark}
\newtheorem{example}{Example}
\newtheorem{remark}{Remark}

\newcommand{\bbone}{\mathbf{1}}
\DeclareMathOperator{\rep}{Rep}
\DeclareMathOperator{\irrep}{Irrep}
\DeclareMathOperator{\range}{R}
\DeclareMathOperator{\orb}{Orb}
\DeclareMathOperator{\sign}{sign}
\let\emptyset\varnothing

\begin{document}
\maketitle

\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.

Let us present here some properties of Wick algebras.
\begin{definition}
Let $I=\{1,\dots,d\}$, and $T_{ij}^{kl}\in \mathbb{C}$, $i$, $j$,
$k$, $l\in I$, be such that $T_{ij}^{kl} =
\bar{T}_{ji}^{lk}$. The Wick algebra with coefficients
$\{T_{ij}^{kl} \}$ (see \cite{jsw}) is a $*$-algebra generated by
the elements $a_i$, $a_i^*$ and the defining relations
\[
a_i^* a_j = \delta_{ij}1+\sum_{k,l=1}^d T_{ij}^{kl} a_la_k^*
\]
\end{definition}
Denote by $\mathcal{H} = \langle e_{1},\ldots, e_d \rangle$ the
finite-dimensional space over $\mathbb{C}$, and by
$\mathcal{H}^*$ its formally dual.
$\mathcal{T}(\mathcal{H}, \mathcal{H}^*)$ will denote the
tensor algebra over $\mathcal{H}$, $\mathcal{H}^{*}$. Then
$\mathcal{W}$ can be canonically realized as
\[
\mathcal{T}(\mathcal{H},\mathcal{H}^*)\Big/\Bigl< e_i^*\otimes e_j -
\delta_{ij}1 - \sum T_{ij}^{kl} e_l\otimes e_k^* \Bigr>.
\]
In this realization, the subalgebra, generated by
$\{a_i\}$ is identified with $\mathcal{T}(\mathcal{H})$.

It is obvious that any element of $\mathcal{W}$ can be uniquely
represented as a polynomial in the noncommuting variables
$a_i$, $a_i^*$, where in each monomial, variables
$a_i$ are placed to the left from $a_j^*$. Such
monomials are called Wick ordered ones, and they form a
basis in $\mathcal{W}$.

When studying properties of $\mathcal{W}$, one can find
useful the following operators :
\begin{align*}
T\colon & \mathcal{H} \otimes \mathcal{H}\mapsto \mathcal{H}
 \otimes\mathcal{H}, \quad  T e_{k}\otimes e_{l} =
\sum_{i,j} T_{ik}^{lj}e_{i}\otimes e_{j},
\\
T_{i}\colon & \mathcal{H}^{\otimes n}\mapsto\mathcal{H}^{\otimes n},
\quad 
T_{i}=\underbrace{1\otimes \cdots\otimes 1}_{i-1}\otimes T
\otimes\underbrace{1\otimes\cdots\otimes 1}_{n-i-1},
\\
R_{n}\colon & \mathcal{H}^{\otimes n}\mapsto \mathcal{H}^{\otimes n},
\quad  
R_{n}=1+T_1+T_1 T_2+\cdots + T_1 T_2\cdots T_{n-1}.
\end{align*}
Namely, these operators determine the Wick ordering on $\mathcal{W}$
\begin{proposition}
Let $X\in\mathcal{H}^{\otimes n};$ then
\begin{equation} \label{commutation_rule}
e_i^*\otimes X = \mu (e_i^*) R_n X + \mu (e_i^*)\sum_{k=1}^{d}
T_1 T_2 \cdots T_n (X\otimes e_k) e_k^*,
\end{equation}
where $\mu (e_i^*)\colon\mathcal{T}(\mathcal{H})
 \mapsto\mathcal{T}(\mathcal{H})$ is defined as follows
\[
\mu (e_i^*) 1=0,\quad \mu (e_i^*) e_{i_1}\otimes \cdots\otimes e_{i_n}=
\delta_{ii_1} e_{i_2}\otimes\cdots\otimes e_{i_n}.
\]
\end{proposition}

Let us note that basic relations of Wick algebra determine commutation
rule between generators $a_i^*, a_j$ only, and we have not any
relation between $a_i, a_j$. However these relations are very useful
in the study of representations of $\mathcal{W}$. It is obvious that
any relation detemine two-sided ideal. So, additional relations
between generators $a_i, a_j$ can be described by the terms of special
ideals in $\mathcal{W}$.
\begin{definition}
A Wick ideal is a two-sided ideal $I\subset
\mathcal{T}(\mathcal{H})$, such that
$\mathcal{T}(\mathcal{H}^{*})\subset
I\mathcal{T}(\mathcal{H}^{*})$. If $I$ is generated by a set
$I_{0}\subset\mathcal{H}^{\otimes{n}}$, then $I$ is called
a homogeneous Wick ideal of degree $n$.
\end{definition}

The following statement is a generalization of the fact
established in \cite{jsw} for $n=2$.

\begin{proposition}
Let $P\colon\mathcal{H}^{\otimes {n}}
\mapsto\mathcal{H}^{\otimes {n}}$ be a projection. Then
$I_{n}= \langle P \mathcal{H}^{\otimes n} \rangle$ is a
Wick ideal if and only if

$1.$ $R_{n}P=0,$

$2.$ $[1\otimes (1-P)]T_{1}T_{2}\cdots T_{n}[P\otimes 1]=0.$

Moreover, if $T$ satisties the braid condition $T_{1}T_{2}T_{1}=
T_{2}T_{1}T_{2}$ and $P$ is a projection on $\ker R_{n}$,
then the condition $2$ holds automatically.
\end{proposition}
Let us note that all examples presented below are Wick algebras with
the braided operator $T$ and additional relations, reducing these
algebras to the dynamical systems form generate quadratic Wick ideals.
\subsection{``Direct sums''of one-dimensional dynamical systems.}
By the ``direct sum'' of one-dimensional systems we mean the dynamical 
system on $\mathbb{R}^d$ defined by the family of functions 
$\{\vec{\mathcal{F}}_i,\ i=1,\ldots ,d\}$ with the property that $i$-th 
function acts on the $i$-th coordinate only, i.e.:
\[
\mathcal{F}_i (\vec{x})=(x_1,\ldots ,x_{i-1},f_i (x_i),x_{i+1},\ldots ,x_d)
,\quad \vec{x}\in\mathbb{R}^d
\]

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 [1])
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 [1] 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 [1]),
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}

\begin{thebibliography}{99}

\bibitem{1} {\sc P.\ E.\ T. Jorgensen, L.\ M.\ Schmitt, and R.\ F.
Werner},
Positive representations of general commutation relations allowing Wick
ordering, {\em J.\ Funct.\ Anal}.\ {\bf 134} 1995, 33-99.

\bibitem{} {\sc W. Pusz and S. L. Woronowicz}, Twisted seqond
quantization,
{\em Reports.\ Math.\ Phys}.\ {\bf 27} (1989), 231-257.

\bibitem{} {\sc Yu.\ S. Samoilenko and E.Ye. Vaisleb}, Representations
of operator relations by unbounded operators and multi-dimensional
dynamical systems, {\em Ukrain.\ Math.\ J}.\ {\bf 42} (1990), 1011-1019.

\bibitem{} {\sc D.\ P. Proskurin}, Homogeneous ideals in Wick
*-algebras, {\em Proc.\ of \ AMS}.\ (1998) (to appear).

\end{thebibliography}

\end{document}

\bibliography{ref}
\bibliographystyle{amsplain}

\end{document}