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\documentclass[a4paper,12pt]{article}

\usepackage{amsthm}
\usepackage{amsfonts}
\usepackage{amsmath}
\usepackage{eucal}
\newtheorem{definition}{Definition}
\newtheorem{proposition}{Proposition}
\newtheorem{lemma}{Lemma}
\newtheorem{theorem}{Theorem}
\newtheorem{corollary}{Corollary}

\theoremstyle{remark}
\newtheorem{remark}{\bf Remark}
\newtheorem{example}{\bf Example}

\title{Representations of Wick analogue of CCR}
\author{O. Ostrovska, D. Proskurin}

\begin{document}

\maketitle

\section{}
We consider Wick $*$-algebra WCCR generated by relations of the form
\begin{align}\label{wccr}
a_i^* a_i- a_i a_i^* & =1\\
a_1^* a_2  =\lambda & a_2 a_1^*,\ \mid \lambda\mid=1,\ \lambda\in\mathbb{C}.\nonumber
\end{align}
Recall that the largest quadratic Wick ideal $\mathcal{J}_2$, see \cite{jsw}, of WCCR  is generated by element
\[
A=a_2a_1-\lambda a_1a_2
\]

By the main result of \cite{proams} the largest cubic Wick ideal $\mathcal{J}_3\subset\mathcal{J}_2$ and is generated by
by elements $A a_1 -\lambda a_1 A$ and
$A a_2 -\overline{\lambda}a_2 A$.

Note that the unique irreducible (well-behaved) representation of WCCR annihilating the largest quadratic ideal is the Fock one acting on $l_2(\mathbb{Z}_+)\otimes l_2(\mathbb{Z}_+)$ by the following formulas
\begin{equation}\label{fockrep}
a_1 = a\otimes 1,\ a_2= d(\lambda)\otimes a,
\end{equation}
where $a\colon l_2(\mathbb{Z}_+)\rightarrow l_2(\mathbb{Z}_+)$ is the creation operator of Fock representation of CCR with one degree of freedom, i.e. $a e_n=\sqrt{n+1}e_{n+1}$, and $d(\lambda) e_n=\lambda^n e_n$, $n\in\mathbb{Z}_+$ for standard orthonormal basis of $l_2(\mathbb{Z}_+)$.

In this note we study the representations of WCCR annihilating the largest cubic ideal. That is we will suppose the following relations to be satisfied
\begin{align*}
 a_i^* a_i- a_i a_i^* & =1,\ a_1^* a_2  =\lambda a_2 a_1^*\\
a_1^* A & = \lambda A a_1^*,\ a_2^* A=\overline{\lambda} A a_2^*\\
 A a_1 & = \lambda a_1 A,\  A a_2=\overline{\lambda}a_2 A
\end{align*}
\begin{theorem}
Irreducible $*$-representations of WCCR, annihilating the largest qubic ideal have, up to the unitary equivalence, the following form:
\begin{align*}
a_1&=a\otimes\mathbf{1},\\
a_2&=\sqrt{1+c}d(\lambda)\otimes a+e^{i\phi}\sqrt{c}d(\lambda)a^*\otimes d(\overline{\lambda}), 
\end{align*}
where the space of representation $\mathcal{H}=l_2(\mathbb{Z}_{+})\otimes l_2(\mathbb{Z}_{+})$, $c\ge 0$ and $\phi\in [0,2\pi)$. Representations corresponding to different tuples $(c,\phi)$ with $c>0$ are non-equivalent. Representations corresponding to $(0,\phi)$ are equivalent to the Fock one. 
\end{theorem}

\begin{thebibliography}{99}
\bibitem{jsw}  {P.E.T.J{\o }rgensen, ,L.M. Schmitt , and R.F.Werner},
Positive representations of general commutation relations allowing
Wick ordering. \textit{J. Funct. Anal.} \textbf{134} (1995),
33-99.

\bibitem{proams} D. Proskurin, Homogeneous idelas in Wick *-algebras, \textit{  Proc. Amer. Math. Soc.}, {\bf 126} (1998), 3371-3376. 
\end{thebibliography}
\end{document}