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Subject: fail: ad.tex
Date: Mon, 23 Nov 1998 16:23:39 +0100 (MET)
From: Lyudmila Turowska <turowska@math.chalmers.se>
To: Yurii Samoilenko <yurii_sam@imath.kiev.ua>

\section{}
The method of dynamical systems developed in section 2.5 can be
carried over to other classes of operator relations. The idea
behind is .....

Here we consider a class of such operator relations the
representations of which will be needed below.

2.6.1. Let $A=A^*$, $X$, $X^*$ be bounded operators which satisfy
relations of the form:
\begin{eqnarray}
&AX=XF_1(A),\label{d1}\\
&X^*X=F_2(A,XX^*),\label{d2}
\end{eqnarray}
where $F_1(\cdot):{\Bbb R}\rightarrow{\Bbb R}$,
$F_2(\cdot,\cdot):{\Bbb R}^2\rightarrow{\Bbb R}$ are measurable
mappings.
It follows from (\ref{d1}) that the operators $A$,
$XX^*$ commute and hence $F_2(A,XX^*)$ is well defined.

As before, let us consider  the polar decomposition  of the
operator $X^*=UC$ where $C=C^*=(XX^*)^{1/2}$, $U$ is a partial
isometry such that $\ker U=\ker C$. Using relations
(\ref{d1})--(\ref{d2}) one can obtain
\begin{eqnarray}
&AU^*=U^*F_1(A),\label{d3}\\
&C^2U^*=U^*F_2(A,C^2)\label{d4}
\end{eqnarray}
and $U$ is a centered operator.
Conversely, any  triple of the operator $A=A^*$, $C\geq 0$ and
the centered partial isometry $U$ satisfying
(\ref{d3})--(\ref{d4}) and such that $\ker U=\ker C$, define a
representation $A$, $X=CU^*$ of relations (\ref{d1})--(\ref{d2}).

Let $F=(F_1,F_2):{\Bbb R}^2\rightarrow{\Bbb R}^2$. For $k\in N$ we
shall denote by $F^{\circ k}(\cdot)$ the $k$-th iteration of $F$
and for $\lambda\in{\Bbb R}$, $n=1,2$ by $F_n^{\circ k}(\lambda)$
the $n$-th coordinate of $F^{\circ k}(\lambda)$.

Analogously, to relations (\ref{d1})--(\ref{d2}) we shall
associate the two-dimensional dynamical system $F(\cdot):{\Bbb
R}^2\rightarrow{\Bbb R}^2$. The possibility to classify all
irreducible representations of the relations depends on the
properties of the dynamical system.
\begin{proposition}
Let ($A=A^*$, $X$) be a representation of (\ref{d1})--(\ref{d2})
in a space $H$. Then $H$ can be decomposed into orthogonal
subspaces $H_1$ and $H_2$, invariant with respect to $A$, $X$,
$X^*$ such that the phase $U$ of $X$ is unitary in $H_1$ and $\ker
U\cup\ker U^*\ne \{0\}$ in $H_2$.
\end{proposition}
Similarly to the case of relation (\ref{}),
irreducible representations of (\ref{d1})--(\ref{d2}) in $H_2$ can
be completely described.  There is a correspondence between
irreducible representations and orbits of the dynamical
system going through a point with zero second coordinate.
Moreover, since $C^2\geq 0$, the spectral measure of the pair
($A$, $C^2$) is concentrated on that part of the orbit where the
second coordinates are non-negative.  Namely, we have the
following description of irreducible representations.
\begin{proposition} Any irreducible representation ($A$, $X$) of
(\ref{d1})--(\ref{d2}) such that $\ker X\cup\ker X^*\ne \{0\}$ is
unitarily equivalent to one of the following:

(i) $H={\Bbb C}^n$, $n\in {\Bbb N}$
$$A=\left(
\begin{array}{cccc}
\lambda&&&0\\
&F_1(\lambda,0)&&\\
&&\ddots&\\
0&&&F^{\circ (n-1)}_1(\lambda,0)
\end{array}\right),\quad
X=\left(
\begin{array}{cccc}
0&&&0\\
F_2(\lambda,0)&\ddots&&\\
&\ddots&0&\\
0&&F^{\circ (n-1)}_2(\lambda,0)&0
\end{array}\right),$$
where $\lambda$ belongs to the set $\sigma_n=\{\lambda\in{\Bbb
R}\mid F_2^{\circ k}(\lambda,0)>0, k=1,\ldots,n-1, F_2^{\circ
n}(\lambda,0)=0\}$;

(ii) $H=l_2({\Bbb N})$
$$Ae_k=F_1^{\circ (k-1)}(\lambda,0)e_k,\quad Xe_k=F_2^{\circ
k}(\lambda,0)e_{k+1},$$
where $\lambda$ belongs to the set
$\sigma_{\infty}=\{\lambda\in{\Bbb R}\mid F_2^{\circ
k}(\lambda,0)>0, k\in {\Bbb N}\}$;

(iii) $H=l_2({\Bbb N})$
$$Ae_k=\lambda_ke_k,\quad
Xe_k=\mu_{k-1}e_{k-1},$$ where
$\lambda_k=F_1(\lambda_{k+1})$,
$\mu_k=F_2(\lambda_{k+1},\mu_{k+1})$, $\mu_1=0$ and $\mu_k>0$,
$k=2,\ldots$.
\end{proposition}
\begin{remark} Note that finite dimensional representations are
not necessary to be related to cycles of the dynamical system.
\end{remark}
 The
 possibility of description of irreducible representations in
 $H_1$ depends on the topological properties of the
 two-dimensional dynamical system. We assume here that $F(\cdot)$ is
one-to-one.
\begin{proposition} If the
 dynamical system $F(\cdot):{\Bbb R}^2\rightarrow{\Bbb R}^2$ has a
 measurable section then any irreducible representation is
unitarily equivalent to one of the following:

(i) $H={\Bbb C}^n$, $n\in {\Bbb N}$
$$A=\left(
\begin{array}{cccc}
\lambda&&&0\\
&F_1(\lambda,\mu)&&\\
&&\ddots&\\
0&&&F^{\circ (n-1)}_1(\lambda,\mu)
\end{array}\right),\
X=\left(
\begin{array}{cccc}
0&&&e^{i\varphi}\mu\\
F_2(\lambda,\mu)&\ddots&&\\
&\ddots&0&\\
0&&F^{\circ (n-1)}_2(\lambda,\mu)&0
\end{array}\right),$$
where $\lambda,\mu$ belongs to the set
$\sigma_n=\{\lambda\in{\Bbb R}\mid F_2^{\circ k}(\lambda,\mu)>0,
k=1,\ldots,n-1, F^{\circ n}(\lambda,\mu)=(\lambda,\mu)\}$,
$\varphi\in[0,2\pi)$;

(ii) $H=l_2({\Bbb Z})$
$$Ae_k=\lambda_ke_k,\quad
Xe_k=\mu_{k-1}e_{k-1},$$ where
$\lambda_k=F_1(\lambda_{k+1})$,
$\mu_k=F_2(\lambda_{k+1},\mu_{k+1})$,  and $\mu_k>0$,
$k\in {\Bbb Z}$.

\end{proposition}
\begin{remark} If there exists an ergodic quasi-invariant measure
which is not concentrated on a single orbit provided that all
second coordinates of the points of the orbit are positive, then
one can construct factor representations of the relation which are
not of type I.
\end{remark}

{\bf $q$-Deformation of $U_q(so(3,{\Bbb C}))$}.

$q$-Deformation of the orthogonal Lie algebra $so(3)$ was proposed by
Fairlie \cite{}. This nonstandard $q$-analog $U_q(so(3,{\Bbb C}))$ is
constructed  starting from $so(3,{\Bbb C})$
defined by generating elements $I_1$, $I_2$, $I_3$. Namely,
$U_q(so(3,{\Bbb C}))$ is the associative algebra generated by $I_1$,
$I_2$, $I_3$ satisfying the relations:
\begin{eqnarray}\label{soq3}
q^{1/2}I_1I_2-q^{-1/2}I_2I_1=I_3\nonumber\\
q^{1/2}I_2I_3-q^{-1/2}I_3I_2=I_1\\
q^{1/2}I_3I_1-q^{-1/2}I_1I_3=I_2\nonumber
\end{eqnarray}
Note that the Lie algebras $sl(2,{\Bbb C})$ and $so(3,{\Bbb C})$ are
isomorphic. However, the quantum algebra $U_q(sl(2,{\Bbb C})$ which is
a Hopf algebra, differs from $U_q(so(3,{\Bbb C}))$.

Let us describe $*$-structures  (involutions) over the algebra
$U_q(so(3,{\Bbb C}))$. It is clear that an involution
in an algebra with generators and relations is completely defined
by its values on the generators.
An involution $*$ may send linear combinations of generators to linear combinations of
generators. In this case $*$ is said to be an involution of the first order or a linear
involution. On the other hand, there might exist involutions
which map linear combinations of generators to the polynomials
in generators of the degree higher then one. We will
call such involutions nonlinear. If linear combinations of generators
are  mapped by
an involution to the polynomials of  the second degree  then we will call such
involutions quadratic.

Here we will consider, for the algebras $ U_q(so(3,{\Bbb C})) $,
all involutions of the first order and some quadratic involutions.

\begin{theorem} \label{th-inv1-eq}
\begin{itemize}
\item[$1)$] If $ q\in{\Bbb R} $, $|q|\ne 1$ then all involutions of the
first order  in the algebra $U_q(so(3,{\Bbb C})) $ are equivalent to
the following  involution:
\item[$a)$] $I_1^*=I_2$, $I_2^*$,
$I_3^*=\left\{
\begin{array}{ll}
I_3,&q>0\\
-I_3,&q<0
\end{array}\right.$.

\item[$2)$] If $ |q| = 1, q \neq \pm 1 $, then all involutions of the
first order  in the algebra $ U_q(so(3,{\Bbb C})) $ are equivalent to
the following
two inequivalent involutions:
\item[$a)$] $ I_1^* = I_1 $ , $ I_2^*=-I_2 $ , $ I_3^*=I_3 $ ,
\item[$b)$] $ I_1^* = -I_1 $ ,$ I_2^*=-I_2 $, $ I_3^*=-I_3 $ ,
\item[$3)$] If $q=-1$, then each involution of the first order
in the algebra $ U_q(so(3,{\Bbb C})) $ is either equivalent to 1a), or  2a),
or  2b)
\end{itemize}
\end{theorem}
\begin{proof}
By definition any involution of the first order in the algebra
$ U_q(so(3,{\Bbb C}))$
is defined, on the generators $I_1$, $I_2$ and $I_3$, by formulas of the form
\begin{eqnarray}
I_1^{*}=c_{11}I_1+c_{12}I_2+c_{13}I_3, \nonumber \\
I_2^{*}=c_{21}I_1+c_{22}I_2+c_{23}I_3, \label{invconst}\\
I_3^{*}=c_{31}I_1+c_{32}I_2+c_{33}I_3, \nonumber
\end{eqnarray}
where $C=[c_{jk}]$ is a complex $3\times3$ matrix.
The condition $(A^*)^*=A$ , $A\in U_q(so(3,{\Bbb C}))$, is satisfied if
and only if it
is satisfied for generators $I_1$, $I_2$ and $I_3$,
that is if and only if
\begin{equation}\label{invdoublmatr}
\overline{C}C=I,
\end{equation}
where $\overline{C}=[\overline{c_{jk}}]$ is the matrix obtained
from $C$ by the complex conjugation of all elements.
Applying  involution to the commutation relations (\ref{soq3})
and using the axioms for $*$ and
(\ref{invconst}) we get three noncommutative
polynomials in variables $I_1$, $I_2$ and $I_3$
which must be zero in $U_q(so(3,{\Bbb C}))$. Using the commutation
relations (\ref{soq3})
these three polynomials can be
rewritten in the degree graded lexicographically ordered form,
that is in the form where  $I_1$
does not appear after  $I_2$ and $I_3$
and $I_2$  does not appear after $I_3$, and the monomials of the higher
total degree appear first. These polynomials written  in the
lexicographically ordered form  will
be equal to zero if and only if
all their coefficients are zero, since it can be shown that the lexicographically
ordered monomials
form a basis in  $U_q(so(3,{\Bbb C}))$, that is $U_q(so(3,{\Bbb C}))$
is the PBW-type algebra.
So, we obtain an additional set of
equations for the constants $c_{jk}$. These equations and (\ref{invdoublmatr})
form a nonlinear system
of equations. Solving this system we  get all possible involutions of the first
order in $U_q(so(3,{\Bbb C}))$.
Finally, the obtained involutions can be classified up to isomorphism
of $*$-algebras.
The described calculations are lengthy and we leave them out.
\end{proof}

%Let $ |q|=1, q\neq 1 $ and $ q = e^{i\phi} $ for some
%$ \phi \in ( 0, 2\pi ) $.
It is possible to choose
hermitian  generators
in all the $*$-algebras.  Indeed,  the new generators
$a_1=(I_1+I_2)/2$,
$a_2=i(I_1-I_2)/2$,
$a_3=I_3$
are hermitian in $R_q^{1a}$. Moreover, it is easy to check that they
satisfy the relations:
\begin{eqnarray}
(q^{\frac{1}{2}}-q^{-\frac{1}{2}})(a_1^2+a_2^2)+i(q^{\frac{1}{2}}+
q^{-\frac{1}{2}})[a_1,a_2]=a_3,\nonumber\\
q^{\frac{1}{2}}a_1 a_3-q^{-\frac{1}{2}}a_3 a_1+i(q^{\frac{1}{2}}a_2
a_3-q^{-\frac{1}{2}}a_3 a_2)=a_1,\\
q^{\frac{1}{2}}a_3 a_1-q^{-\frac{1}{2}}a_1 a_3-i(q^{\frac{1}{2}}a_3
a_2-q^{-\frac{1}{2}}a_2 a_3)=a_2.\nonumber
\end{eqnarray}
In $R_q^{2a}$ the new generators
$ a_1=I_1$,
$ a_2=iI_2 $,
$ a_3=I_3 $
are hermitian and  satisfy
the following  relations:
\begin{eqnarray}
q^{\frac{1}{2}}a_1 a_2-q^{-\frac{1}{2}}a_2 a_1&=&ia_3, \nonumber \\
q^{\frac{1}{2}}a_2 a_3-q^{-\frac{1}{2}}a_3 a_2&=&ia_1, \label{rel:Mqa} \\
q^{\frac{1}{2}}a_3 a_1-q^{-\frac{1}{2}}a_1 a_3&=&-ia_2   \nonumber .
\end{eqnarray}
In $R_{q}^{2b}$ the new generators
$ a_1=iI_1$,
$ a_2= iI_2 $,
$ a_3=iI_3 $
are hermitian and satisfy the relations:
\begin{eqnarray}
q^{\frac{1}{2}}a_1 a_2-q^{-\frac{1}{2}}a_2 a_1&=&ia_3, \nonumber \\
q^{\frac{1}{2}}a_2 a_3-q^{-\frac{1}{2}}a_3 a_2&=&ia_1, \label{rel:Mqb} \\
q^{\frac{1}{2}}a_3 a_1-q^{-\frac{1}{2}}a_1 a_3&=&ia_2   \nonumber .
\end{eqnarray}

It turns out that the algebras
$U_q(so(3,{\Bbb C}))$ have  also some  quadratic involutions.

Let us substitute $I_3$ in the second and third relation in
(\ref{soq3})
by the left hand side of the first relation.
Then we will get the following two relations of the third order
for $I_1$ and $I_2$:
\begin{equation} \label{rel:Mq2}
\begin{tabular}{l}
$I_2^2 I_1 - (q+q^{-1}) I_2 I_1 I_2 + I_1 I_2^{2} = - I_1$ \\
$I_1^2 I_2 - (q+q^{-1}) I_1 I_2 I_1 + I_2 I_1^{2} = -I_2.$
\end{tabular}
\end{equation}

Let us find all involutions of the first degree in the algebra
with generators $I_1$ and $I_2$ and relations (\ref{rel:Mq2}).
The element $I_3$ is defined by relations (\ref{soq3}).
Applying the arguments given in the proof of Theorem~\ref{th-inv1-eq}
we get
the following statement
\begin{proposition}
Any  such involution is isomorphic either to involutions from
Theorem~\ref{th-inv1-eq} or to the following ones:
\begin{itemize} \item[$1)$] $q\in{\Bbb R}$, \hspace{0.6cm}
$I_1^*=I_1$, $I_2^*=I_2$, and $I_3^*=\left\{ \begin{array}{ll}
q^{\frac{1}{2}}I_2I_1-q^{-\frac{1}{2}}I_1I_2,&q>0,\\
-(q^{\frac{1}{2}}I_2I_1-q^{-\frac{1}{2}}I_1I_2),&q<0,
\end{array}\right.$

\item[$2)$] $q \in {\Bbb R}$, \hspace{0.6cm} $I_1^*=I_1$,
$I_2^*=-I_2$, and $I_3^* =\left\{
\begin{array}{ll}
-(q^{\frac{1}{2}}I_2I_1-q^{-\frac{1}{2}}I_1I_2),&q>0,\\
q^{\frac{1}{2}}I_2I_1-q^{-\frac{1}{2}}I_1I_2,&q<0,
\end{array}\right.$

\item[$3)$] $q \in {\Bbb R}$, \hspace{0.6cm} $I_1^*=-I_1$,
$I_2^*=-I_2$, and $I_3^* =\left\{
\begin{array}{ll}
(q^{\frac{1}{2}}I_2I_1-q^{-\frac{1}{2}}I_1I_2),&q>0,\\
-(q^{\frac{1}{2}}I_2I_1-q^{-\frac{1}{2}}I_1I_2),&q<0,
\end{array}\right.$

\item[$4)$] $|q|=1$, \hspace{0.6cm} $I_1^*=I_2$,
$I_2^*=I_1$, $I_3^* =
q^{-\frac{1}{2}}I_2I_1-q^{\frac{1}{2}}I_1I_2$,

\end{itemize}
\end{proposition}

\input{soq3}