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Dorogoj Yurii Stefanovich,
Eta nedelya byla u menya nemnogo bespokojnoj, no ya napisala kakoj-to
tekst. Vyglyadit on, navernoe, dovol'no ubogo. Fantaziya u menya bednaya.
Vse zamechaniya i kritiku prinimayu i gotova pererabotat' tekst.

Kak u Vas prodvizheniya? Pishite.
Vsego samogo dobrogo. Vasha Luda T.


P.S. Ssylki ya napishu pozzhe. 

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\begin{document}
In last few years together with Lie algebras and their ${\Bbb
  Z}_2$-graded analogs, quantum groups, different deformations of
  universal enveloping algebra of Lie algebras, quantum superalgebras
  and quantum homogeneous spaces have attracted more interest and play
  an important role in various
  branches of morden physics.

Let us recall that quantized universal enveloping algebras, also
called $q$-algebras, refer to some specific deformations of Lie
algebras with Hopf algebraic structure and were introduced
independently by Jimbo \cite{} and Drienfeld \cite{} in 1986. They
can be considered
as interpolation between Lie algebras and ${\Bbb Z}_2$ graded Lie
algebras to which they reduce when the deformation parameter $q$ is
set equal to $q=1$ and $q=-1$ respectively. They has been found
suitable for the solutions of different models of mathematical
physics, in the theory of special functions, knot theory, models of 
$q$-deformed  quantum physics and quantum field theory. 
The simplest example of $q$-algebra, first introduced in 1983 by
Sklyanin  \cite{}, Kulish, Reshetihin \cite{}, is
$su_q(2)=U_q(su(2))$. 


More recently, different generalizations have also been proposed and
turns out to be useful in applications, for example, such nonlinear
generalizations of $su(2)$ , intensively studied in the literature, as
Witten's first and second deformations \cite{}, the Higgs algebras
\cite{},
the Fairlie $q$-deformation of $so(3)$ \cite{}, nonlinear $sl(2)$
algebras and others.

Let us recall some ``quantum'' objects which were described in Section
!!!!.

{\bf $q$-Canonical commutation relations} We fix a real constant $q$
and set 
\begin{equation}
XX^*-qX^*X=I.
\end{equation}
For $q=1$ we have the canonical commutation relations for bosons, for
$q=-1$, we have the canonical anticommutation relations for
fermions. The relations have been suggested by Creenberg and Bozejko,
Speicher as an interpolation between Bose and Fermi statistics.
  
{\bf Hermitian $q$-plane} is the free  $*$-algebra generated by elements $X$,
$X^+$ and relations
\begin{equation}\label{hqp}
XX^*-qX^*X=0,\quad q\in{\Bbb R}.
\end{equation}
Here $X^*=X^+$. For hermitian generators $A=(X+X^+)/2$,
$B=(X-X^+)/2i$
relation \ref{hqp} is equivalent to following:
$$[A,B]=i\frac{1-q}{1+q}(A^2+B^2).$$

{\bf Real quantum plane ${\Bbb R}_q^2$} is the free $*$-algebra
generated by two hermitian elements $A$, $B$ satisfying the relation
\begin{equation}
AB=qBA, \quad |q|=1
\end{equation}
For $q=1$ both $q$-deformed planes are isomorphic to $R^2$.

{\bf Real quantum hyperboloid $X_{q,\gamma}$} is the free $*$-algebra
with unit element $I$ which is generated by two hermitian elements $A$, $B$
and the relation
\begin{equation}
AB-qBA=\gamma(1-q)I, \quad |q|=1, \ \gamma\in{\Bbb R}.
\end{equation}

{\bf $q$-Deformation of $U_q(so(3,{\Bbb C}))$}.
 Another $q$ deformed object we would like to
 mention here and representations of which will study in
Section???? is $q$-deformed algebra $U_q(so(3,{\Bbb C}))$. 
$q$-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 not in terms of Chevalley basis as it was done within
standard approach (\cite{}), but 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}).
Applying the arguments given in the proof of Theorem~\ref{th-inv1-eq}
we get 
the following statement
\begin{prop}
Any  such involution is isomorphic to the following ones:
\begin{itemize}
\item[$1)$] $q\in{\Bbb R}$, \hspace{0.6cm} $I_1^*=I_1$, $I_2^*=I_2$, 
$I_3^*=\left\{
\begin{array}{ll}
q^{\frac{1}{2}}I_2I_1-q^{-\frac{1}{2}}I_1I_2,&q>0,\\
-(q^{\frac{1}{2}}I_2I_1-q^{-\frac{1}{2}}I_1I_2),&q<0,
\end{array}\right.$

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

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

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

and to involutions from Theorem~\ref{th-inv1-eq}
\end{itemize}
\end{prop}


\end{document}