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% Finite dimensional representations of the quantum group $GL_{p,q}(2)$ 
% using the exponential map from $U_{p,q}(gl(2))$
% R. Jagannathan and J. Van der Jeugt
% J. Phys. A: Math. Gen. 28 (1995) 2819-2831.
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\begin{center}
{\Large \bf Finite dimensional representations of \\ 
the quantum group $GL_{p,q}(2)$ using \\ 
the exponential map from $U_{p,q}(gl(2))$}    

\vspace{2cm}

{\bf R. Jagannathan\footnote{Permanent address: The Institute of 
Mathematical Sciences, Madras - 600113, India; 
E-mail address: jagan@imsc.ernet.in}
and      
J. Van der Jeugt\footnote{Senior Research Associate of N.F.W.O 
(National Fund for Scientific Research of Belgium); 
E-mail address: Joris.VanderJeugt@rug.ac.be}}  

\vspace{0.5cm}

Department of Applied Mathematics and Computer Science \\ 
University of Ghent, Krijgslaan 281-S9, B-9000 Gent, Belgium 
\end{center}

\vspace{2cm}

\noindent
{\bf Abstract}:  Using the Fronsdal-Galindo formula for the exponential 
mapping from the quantum algebra $U_{p,q}(gl(2))$ 
to the quantum group $GL_{p,q}(2)$, we show how the $(2j+1)$-dimensional 
representations of $GL_{p,q}(2)$ can be obtained by `exponentiating' the 
well-known $(2j+1)$-dimensional representations of $U_{p,q}(gl(2))$ for 
$j$ $=$ $1,\frac{3}{2},\ldots $;  
$j$ $=$ $\frac{1}{2}$ corresponds to the  
defining 2-dimensional $T$-matrix.  The earlier results on the 
finite-dimensional representations of $GL_q(2)$ and $SL_q(2)$ 
(or $SU_q(2)$)  
are obtained when $p$ $=$ $q$.  Representations of $U_{\bar{q},q}(2)$ 
$(q$ $\in$ $\C \backslash \R$ and $U_q(2)$ 
$(q$ $\in$ $\R \backslash \{0\})$ are also considered.  The structure   
of the Clebsch-Gordan matrix for $U_{p,q}(gl(2))$ is studied. 
The same Clebsch-Gordan coefficients are applicable in the
reduction of the
direct product representations of the quantum group $GL_{p,q}(2)$.   

\vspace{1cm}

\noindent
Short title: Representations of $GL_{p,q}(2)$  

\vspace{1cm}

\noindent
Phys. Abs. Class. Nos.~: 0210, 0220, 0365   

\newpage

\renewcommand{\theequation}{\arabic{section}.{\arabic{equation}}}
\setcounter{equation}{0}
\section{Introduction}

\noindent
The analysis of the bialgebraic duality relationship between  
the pair of Hopf algebras ${\cal A}$ $=$ $A\left(GL_{p,q}(2)\right)$ (or 
$Fun_{p,q}(GL(2))$) and ${\cal U}$ 
$=$ $U_{p,q}(gl(2))$ by Fronsdal and Galindo~\cite{FG} provides the first 
example of generalization of the exponential relationship, obtained   
between the classical Lie groups and algebras, through the construction of 
the form of such a mapping from $U_{p,q}(gl(2))$ to the quantum group 
$GL_{p,q}(2)$.  Following~\cite{FG}, Bonechi {\em et al.}~\cite{Bo-et} have 
studied the forms of the exponential relationship for a few other examples 
of quantum groups, namely, the quantized Heisenberg, Euclidean and Galilei 
groups.  More recently, Morozov and Vinet~\cite{MV} have obtained a 
generalization of the result of~\cite{FG} for any simple quantum
group with a single deformation parameter.  
For $GL_q(2)$, Finkelstein~\cite{F} has obtained the representations 
using theory of invariants and studied the converse map $GL_q(2)$ 
$\longrightarrow$ $U_q(gl(2))$ leading to the identification of the 
generators of $U_q(gl(2))$ as the infinitesimal generators of $GL_q(2)$: 
hence, the representations of $U_q(gl(2))$ are also derived.  

%newpara
Here, we demonstrate explicitly how the Fronsdal-Galindo theory leads 
to a straightforward derivation of the finite-dimensional representaions of  
$GL_{p,q}(2)$.  The earlier results on the finite-dimensional irrreducible  
representaions of $GL_q(2)$~\cite{F} and $SL_q(2)$ 
(or $SU_q(2)$)~\cite{W,VS,K,GKK,Ma-et,N,KK}  
leading to the $q$-analogues of the classical Wigner $d$-functions or    
spherical functions, are seen to follow in the special case when 
$p$ $=$ $q$.  We consider briefly also the quantum groups 
$U_{\bar{q},q}(2)$ $(q$ $\in$ 
$\C \backslash \R)$  
and $U_q(2)$ $(q$ $\in$ $\R \backslash \{0\})$.   

%newpara
The question of $q$-analogues of the Clebsch-Gordan coefficients (CGCs) 
arises naturally when one considers the representation theory of quantum 
groups.  The CGCs for $U_q(su(2))$ and $SU_q(2)$ have been discussed  
extensively (see~\cite{K,GKK,N,KK}).  In the case of 
$GL_{p,q}(2)$~\cite{Ku,De-et,S1,T}, the structure of its dual Hopf algebra,  
$U_{p,q}(gl(2))$, is known from studies using different approaches~\cite
{SWZ,D1,D2}. It is clear that one can have only $U_{p,q}(u(2))$   
nontrivially and any $(p,q)$-deformation of $U(su(2))$ would depend 
effectively only on a single parameter ($\sqrt{pq}$).  
So, the earlier studies on 
a $(p,q)$-deformed $su(2)$ algebra led naturally to the conclusion   
that the corresponding deformed CGCs depend only on $\sqrt{pq}$ 
(see~\cite{Q,MM}).  Here, we shall arrive at the structure of the  
Clebsch-Gordan matrix for $U_{p,q}(gl(2))$ based on Reshetikhin's general  
theory of quantum algebras with multiple deformation parameters~\cite{R}.  
The same Clebsch-Gordan matrices can be used for the reduction
of direct product representations of the quantum group
$GL_{p,q}(2)$. 

%newpara
Before proceeding to consider the representation theory of $GL_{p,q}(2)$, 
in particular, let us first recall some elements of the general theory of 
representations of quantum groups.  The Hopf algebra $A(G_q)$, 
algebra of functions on a quantum group $G_q$, has a nonabelian coordinate 
ring generated by the (variable) elements of a $T$-matrix satisfying the 
$RTT$-relation 
%
%1.1{RTT}
\beq
RT_1T_2 = T_2T_1R \qquad 
{\rm with}\ \ \ \ T_1 = T \otimes \one  \ \ \  
T_2 = \one \otimes T 
\label{RTT}
\eeq
%
where the $R$-matrix is a solution of the Yang-Baxter equation.  The 
coproduct maps of the elements of $T$, $\left\{ T_{mk} \right\}$, 
specified by 
%
%1.2{copro}
\beq
\Delta\left( T_{mk} \right) = \sum_j\,T_{mj} \otimes T_{jk} \qquad 
{\rm or}\ \ \ \ 
\Delta (T) = T \dot{\otimes} T 
\label{copro}
\eeq
%
provide a unique homomorphism of the $\C$-algebra 
generated by $\{ T_{mk} \}$ subject to the relations~(\ref{RTT}). 
One may choose a basis for $A(G_q)$ with any convenient 
parametrization and take the required coproduct maps as induced from 
(\ref{copro}).  A $\C$-vector subspace $V$ spanned by
$\{ V_m$$|$$m = $$1,$ $2,$$\ldots ,$$n$$\}$ of $A(G_q)$ 
is said to carry an $n$-dimensional representation of $G_q$ if it forms 
a left subcomodule of $A$ such that 
%
%1.3{defrepV}
\beq
\Delta (V) = {\cal T} \dot{\otimes} V \qquad 
{\rm or}\ \ \ \ 
\Delta\left( V_m \right) = \sum_{k=1}^n\,{\cal T}_{mk} \otimes V_k \quad 
\forall\,m = 1,2,\ldots ,n  
\label{defrepV}
\eeq
%
where $\{ {\cal T}_{mk} \in A \}$ are called the elements of 
the `representation matrix' ${\cal T}$.  Then, the comodule structure of 
$\{ V_m \}$ implies, as in (\ref{copro}),    
%
%1.4{defrepT}
\beq
\Delta ({\cal T}_{mk}) = \sum_{j=1}^n\,{\cal T}_{mj} \otimes 
{\cal T}_{jk} \qquad 
{\rm i.e.,}\ \ \ \ \Delta ({\cal T}) 
= {\cal T} \dot{\otimes} {\cal T}\,.  
\label{defrepT}
\eeq
%
For example, the elements of the first (or, any) column of the $T$-matrix 
constitute such a $\C$-vector subspace of 
$A(G_q)$ carrying the defining representation with ${\cal T}$  
$=$ $T$.   Essentially, a representation matrix ${\cal T}$ of $G_q$ is a 
generalization of the $T$-matrix, with $\{ {\cal T}_{mk}$ $\in$ 
$A(G_q) \}$ and the relation (\ref{defrepT}) 
satisfied.  

%newpara
The paper is organized as follows.  In section 2, we recall briefly the 
main properties of the quantum group $GL_{p,q}(2)$ and its dual Hopf algebra 
$U_{p,q}(gl(2))$.  In section 3, we describe the Fronsdal-Galindo method for   
exponentiating the representations of $U_{p,q}(gl(2))$ to obtain the 
representations of $GL_{p,q}(2)$.  Section 4 gives the explicit 
representation matrices $\{{\cal T}\}$ for $GL_{p,q}(2)$ and shows how 
the earlier results on the representations of $GL_q(2)$, $SL_q(2)$ and 
$SU_q(2)$ are obtained in the appropriate limits.  In section 5 we briefly 
remark on the quantum group $U_{\bar{q},q}(2)$.  Section 6 presents the 
solution to the problem of CGCs for $U_{p,q}(gl(2))$ and
$GL_{p,q}(2)$. Section 7 contains several concluding remarks.  

\section{$GL_{p,q}(2)$ and $U_{p,q}(gl(2))$}  
\setcounter{equation}{0}  

\noindent
For the quantum group $GL_{p,q}(2)$ the defining $T$-matrix, or the defining 
representation matrix, is specified by 
%
%2.1{defT}
\beq
T = \left( 
\ba{cc}
a & b \\
c & d \\
\ea \right)
\label{defT}
\eeq 
%
with the commutation relations 
%
%2.2{comrel}  
\bea
ab & = & qba \qquad cd = qdc \qquad ac = pca \qquad bd = p db \nn \\
bc & = & (p/q)cb \qquad ad-da = (q-p^{-1})bc
\label{comrel}
\eea  
%
following from the $RTT$-relation (\ref{RTT}) corresponding to 
%
%2.3{R}
\beq
R = Q^{\frac{1}{2}}\left(
\ba{cccc}
Q^{-1} & 0 & 0 & 0 \\
0 & \lambda^{-1} & Q^{-1}-Q & 0 \\
0 & 0 & \lambda & 0 \\
0 & 0 & 0 & Q^{-1} 
\ea \right)
\label{R}
\eeq
%
where 
%
%2.4{Qlambda}
\beq
Q = \sqrt{pq} \quad \quad \lambda = \sqrt{p/q}\,. 
\label{Qlambda}
\eeq
%
We are concerned here only with the case of generic values of $Q$,
$\lambda$ $\in$ $\C \backslash \{0\}$.    
Also, it may be noted that we shall use both  
the sets of notations $(Q,\lambda )$ and $(p,q)$ for the deformation 
parameters interchangably, according to convenience, with the relationship 
(\ref{Qlambda}) always implied.  The coproduct maps (\ref{copro}) are 
explicitly given by 
%
%2.5{delT}
\beq
\Delta (T) = \left(
\ba{cc}
\Delta (a) & \Delta (b) \\
\Delta (c) & \Delta (d) 
\ea \right) 
= \left( 
\ba{cc}
a \otimes a + b \otimes c & a \otimes b + b \otimes d \\
c \otimes a + d \otimes c & c \otimes b + d \otimes d 
\ea \right)\,. 
\label{delT}
\eeq
%
The quantum determinant of $T$ defined by 
%
%2.6{qdet}
\beq
{\cal D} = ad - qbc = ad - pcb
\label{qdet}
\eeq
%
satisfies the commutation relations 
%
%2.7{qdcomrrel}
\beq
{\cal D}a = a{\cal D} \qquad {\cal D}b = \lambda^{-2}b{\cal D} \qquad 
{\cal D}c = \lambda^2 c{\cal D} \qquad {\cal D}d = d{\cal D} 
\label{qdcomrel}
\eeq
%
and is a group-like element such that 
%
%2.8{delqdet}
\beq
\Delta ({\cal D}) = {\cal D} \otimes {\cal D}\,.
\label{delqdet}
\eeq

%newpara   
The algebra ${\cal U}$ dual to ${\cal A}$ $=$ $A(GL_{p,q}(2))$, 
namely $U_{p,q}(gl(2))$, may be presented in a standard form as follows.  
The generators $\{ J_0, J_\pm\,,Z \}$ can be taken to satisfy 
the algebra 
%
%2.9{alg}
\bea
[ J_0,J_\pm ] & = & \pm J_\pm \quad \quad 
[ J_+,J_- ] = [ 2J_0 ]_Q \nn \\   
{} [ Z ,J_0 ] & = & 0 \quad \quad
[ Z ,J_\pm ] = 0
\label{alg}
\eea
%
with
%
%2.10{q-no}
\beq
[X]_{\sf q} = \frac{{\sf q}^X-{\sf q}^{-X}}{{\sf q}-{\sf q}^{-1}}
\quad \quad \forall\ \ {\sf q},X\,.
\label{q-no}
\eeq
%
Whereas the algebraic relations (\ref{alg}) do not depend on $\lambda$, 
the coproducts do and are given by, up to equivalence, 
%
%2.11{coalg}
\bea 
\Delta ( J_\pm ) & = 
& J_\pm \otimes Q^{-J_0}\lambda^{\pm Z} + 
Q^{J_0}\lambda^{\mp Z} \otimes J_\pm \nn \\
\Delta ( J_0 ) & = 
& J_0 \otimes \one + \one \otimes J_0 \qquad 
\Delta (Z) = Z \otimes \one + \one \otimes Z\,.
\label{coalg}
\eea 
%
The associated universal ${\cal R}$-matrix, relating the coproduct $\Delta$ 
and the opposite coproduct $\Delta '$ $=$ $\sigma\,\Delta$, with 
$\sigma\,(u \otimes v)$ $=$ $v \otimes u$,\,\,$\forall$\,$u,v$
$\in {\cal U}$, 
through the equation 
%
%2.12{univR}
\beq
\Delta '(u) = {\cal R}\Delta (u){\cal R}^{-1} \qquad 
\forall \  u \in\  {\cal U} 
\label{univR}
\eeq
%
is 
%
%2.13{calR}
\bea 
{\cal R} & = & Q^{-2(J_0 \otimes J_0)} 
\lambda^{2(Z \otimes J_0 - J_0 \otimes Z)} \nn \\ 
&    &  \ \ \times \sum_{n=0}^{\infty}\,
\frac{(1-Q^2)^n}{[n]!}\,Q^{-\frac{1}{2}
n(n-1)}(Q^{-J_0}\lambda^Z J_+ 
\otimes Q^{J_0}\lambda^Z J_-)^n   
\label{calR}
\eea 
%
where 
%
%2.14{n!}
\beq
[n] = [n]_Q \qquad 
[n]! = [n][n-1]\ldots[2][1] \qquad
[0]! = 1\,.
\label{n!}
\eeq
%
Hereafter, unless otherwise specified, $[n] = [n]_Q$.  
The ${\cal R}$-matrix~(\ref{calR}), say ${\cal R}_{Q,\lambda}$, is easily 
obtained~\cite{CJ1} following the observation that, if we denote by 
$\Delta_{Q,\lambda}$ the coproduct defined by~(\ref{coalg}), then, 
%
%2.15{FdelF}
\beq
\Delta_{Q,\lambda}(u) = F\Delta_{Q,\lambda =1}(u)F^{-1} \qquad
\forall\  u \in {\cal U} \qquad 
{\rm with}\ \ 
F = \lambda^{(J_0 \otimes Z - Z \otimes J_0)}
\label{FdelF}
\eeq
%
and by Reshetikhin's theory~\cite{R}
%
%2.16{FRF}
\beq
{\cal R}_{Q,\lambda} = F^{-1}{\cal R}_{Q,\lambda =1}F^{-1}\,.
\label{FRF}
\eeq

%\newpara
Here, we are not concerned with the other aspects of the Hopf algebraic   
structure, namely, the counit and the antipode.  

\section{The exponential map from $U_{p,q}(gl(2))$ to $GL_{p,q}(2)$} 
\setcounter{equation}{0}

\noindent
Let us now describe the exponential map from $U_{p,q}(gl(2))$ to 
$GL_{p,q}(2)$, {\em \`{a} la} Fronsdal and Galindo~\cite{FG}.  To this 
end, we have to redefine the generators of $U_{p,q}(gl(2))$ as 
%
%\3.1{newJ} 
\beq
\hat{J}_+ = J_+Q^{-(J_0+\frac{1}{2})}
\lambda^{Z-\frac{1}{2}} \quad  
\hat{J}_- = Q^{(J_0+\frac{1}{2})}
\lambda^{Z-\frac{1}{2}}J_- \quad  
\hat{J}_0 = J_0 \quad 
\hat{Z} = Z  
\label{newJ}
\eeq
%
with the algebra 
%
%3.2{newalg}
\bea
[ \hat{J}_0,\hat{J}_\pm ] & = & \pm \hat{J}_\pm \quad \quad 
[ \hat{J}_+,\hat{J}_-] = 
\lambda^{2\hat{Z}-1}[2\hat{J}_0] \nn \\
{} [ \hat{Z},\hat{J}_0 ] & = & 0 \quad \quad 
[ \hat{Z},\hat{J}_\pm ] = 0 
\label{newalg}
\eea
%
and the induced coproducts
%
%3.3{newcoalg}
\bea
\Delta (\hat{J}_+) & = & \hat{J}_+ \otimes Q^{-2\hat{J}_0} 
\lambda^{2\hat{Z}} + \one \otimes \hat{J}_+ \quad   
\Delta (\hat{J}_-) = \hat{J}_- \otimes \one + 
Q^{2\hat{J}_0}\lambda^{2\hat{Z}} \otimes \hat{J}_- \nn \\
\Delta (\hat{J}_0) & = & \hat{J}_0 \otimes \one + 
\one \otimes \hat{J}_0 \quad \quad  
\Delta (\hat{Z}) = \hat{Z} \otimes \one + 
\one \otimes \hat{Z}\,.
\label{newcoalg}
\eea 

%newpara
Let the element `$a$' of $T$ be taken to be invertible~\cite{FG} and 
expressed as  
%
%3.4{a}
\beq
a = {\rm e}^\alpha\,.  
\label{a}
\eeq 
%
Since we are dealing only with nonsingular $T$-matrices we shall 
represent  
%
%3.5{phi-mu}
\beq
{\cal D} = {\rm e}^{-2\phi} = \xi^{-2}\,.
\label{phi-mu}
\eeq
%
Further, let us take 
%
%3.6{begadel}
\beq
\beta = a^{-1}b \qquad \gamma = ca^{-1} \qquad 
\delta = \alpha + 2\phi\,.
\label{begadel}
\eeq
%
The variables $\{ a,b,c,d \}$ can be expressed as 
%
%3.7{a=}
\beq
a = {\rm e}^\alpha \qquad b = {\rm e}^\alpha \beta \qquad 
c = \gamma {\rm e}^\alpha \qquad 
d = \gamma {\rm e}^\alpha \beta + {\rm e}^{-\delta}\,.
\label{a=}
\eeq 
%
The set of new variables $\{ \alpha ,\beta ,\gamma ,\delta \}$ are seen 
to form a Lie algebra:  
%
%3.8{Lie}
\bea
[ \alpha , \beta ] & = & (\rho -\theta )\beta \quad \quad  
[ \alpha , \gamma ] = (\rho + \theta )\gamma \nn \\
{} [ \delta , \beta ] & = & (\rho + \theta )\beta \quad \quad 
[ \delta , \gamma ] = (\rho - \theta )\gamma \nn \\
{} [ \alpha , \delta ] & = & 0 \quad \quad 
[ \beta , \gamma ] = 0 \nn \\
  &   &  \qquad {\rm with}\ \ \ \ 
Q = {\rm e}^\rho \qquad \lambda = {\rm e}^\theta\,.   
\label{Lie}
\eea 
%
Then, following Fronsdal and Galindo~\cite{FG}, one can write down a 
`universal ${\cal T}$-matrix' for $GL_{p,q}(2)$ as 
%
%3.9{calT}
\beq
{\cal T} = {{\cal E}xp}_{Q^{-2}}\{ \gamma \hat{J}_- \} 
{\rm exp}\{ \alpha (\hat{J}_0+\hat{Z}) + 
\delta (\hat{J}_0-\hat{Z}) \} 
{{\cal E}xp}_{Q^2}\{ \beta \hat{J}_+ \}
\label{calT}
\eeq
%
where 
%
%3.10{calExp}
\beq
{{\cal E}xp}_{{\sf q}^2}\{ X \} = \sum_{n=0}^{\infty}\,
\left\{ \prod_{k=1}^{n}({\sf q}^{2k}-1) \right\}^{-1} 
({\sf q}^2-1)^n X^n = \sum_{n=0}^{\infty}\,
\frac{{\sf q}^{-\frac{1}{2}n(n-1)}}{[n]_{\sf q}!}\,X^n\,.
\label{calE}
\eeq
%
The universal ${\cal T}$-matrix (\ref{calT}) embodies all the 
finite-dimensional representations of $GL_{p,q}(2)$: substituting the 
finite-dimensional numerical matrices representing 
$\{ \hat{J}_0,$$\hat{J}_{\pm},\hat{Z} \}$ in the 
expression (\ref{calT}) for ${\cal T}$, expanding it, and reexpressing 
the resulting matrix elements in terms of $\{a,$$b,$$c,$$d\}$ and 
$\xi$ $=$ ${\cal D}^{-\frac{1}{2}}$ using the relations 
((\ref{qdet}),(\ref{a})-(\ref{Lie})), one obtains all the 
finite-dimensional representation matrices $\{{\cal T}\}$ of 
$GL_{p,q}(2)$ in the sense of the representation 
theory outlined in the introduction.  Though the definition of the matrix 
elements of ${\cal T}$ by (\ref{calT}) involves $a^{-1}$, it is found  
possible to express them completely in terms of $\{a,$$b,$$c,$$d\}$ and 
the group-like $\xi$ (with $\Delta (\xi )$ $=$ $\xi \otimes \xi )$, using 
the relations ((\ref{qdet}),(\ref{a})-(\ref{Lie})).    

%newpara
Except for the additional central element $Z$, the algebra (\ref{alg}) is 
same as the standard $U_Q(sl(2))$ for which all the finite-dimensional 
irreducible representations are known~(see \cite{Ro,RA}): for generic 
values of $Q$$\in$$\C \backslash \{0\}$, these are the 
straightforward 
$Q$-analogues of the $(2j+1)$-dimensional (spin-$j$) representations of  
the classical $sl(2)$, with 
$j$ $=$ $0,$$\frac{1}{2},$$1,$$\frac{3}{2},$$\ldots\,.$  
So, with the extra central element as $Z$ $=$ $z \one $, 
the $(2j+1)$-dimensional irreducible representations of 
$\{ \hat{J}_0,$$\hat{J}_\pm ,\hat{Z} \}$ obeying (\ref{newalg}), 
say $\left\{ \Gamma^{(\mu )} \mid \mu = (j,z) \right\}$, 
are readily obtained (see also~\cite{D3} for more details on the dual Hopf 
algebras $A(GL_{p,q}(2))$ and $U_{p,q}(gl(2))$ and representation theory of 
quantized universal enveloping algebras $\{ U_q(G) \}$ in general).  
Let us call the matrix obtained by substituting the representation 
$\Gamma^{(\mu )}$ in the   
expression (\ref{calT}) for ${\cal T}$ as ${\cal T}^{(\mu )}$.  
The matrix elements are labeled as 
$\{ {\cal T}^{(\mu )}_{mk}\mid m,k=j,j-1,\ldots ,-(j-1),-j \}$.  

%newpara
The basic theory underlying the Fronsdal-Galindo formalism is as follows.    
Let $\{ x^A \}$ be a basis for $A(G_q)$ such that  
%
%3.11{xx}
\beq
x^Bx^C = \sum_{A}\,h^{BC}_A x^A \quad \quad 
\Delta (x^C) = \sum_{A,B}\,f^C_{AB} x^A \otimes x^B\,.
\label{xx}
\eeq 
%
Let $\{ X_A \}$ be the basis of $U_q(G)$ chosen such that   
% 
%3.12{XX}
\beq
X_AX_B = \sum_{C}\,f_{AB}^C X_C \quad \quad
\Delta (X_A) = \sum_{B,C}\,h_A^{BC} X_B \otimes X_C\,.
\label{XX}
\eeq
%
Then, the universal ${\cal T}$-matrix defined by 
%
%3.13{TauxX} 
\beq
{\cal T} = \sum_{A}\,x^AX_A 
\label{TauxX}
\eeq
%
satisfies the equation 
%
%3.14{DelcalTau}
\beq 
\Delta ({\cal T}) = \sum_{A}\,\Delta (x^A) X_A 
= {\cal T} \dot{\otimes} {\cal T}
\label{DelcalTau}
\eeq
%
in view of the duality relations ((\ref{xx}),(\ref{XX})).   
Equation (\ref{TauxX}) should be interpreted as defining a universal 
${\cal T}$-matrix in the sense that ${\cal T}^{(\mu )}$ $=$ 
$\sum_{A}\,x^AX^{(\mu )}_A$ $\in$ $A(G_q)$, for 
$\{ X^{(\mu )}_A \}$, the numerical representations of 
$\{ X_A \}$, in any representation $\Gamma^{(\mu )}$.  Thus, the  
elements $\{ {\cal T}^{(\mu )}_{mk}\in A(G_q)\}$ 
form a representation matrix for the quantum group $G_q$ and the 
formula (\ref{TauxX}) expresses abstractly the exponential map $U_q(G)$ 
$\longrightarrow$ $G_q$.  In the present context of $GL_{p,q}(2)$, the 
basis elements of $A(GL_{p,q}(2))$ and $U_{p,q}(gl(2))$ 
are given (see~\cite{FG} for details of derivation), respectively, by  
%
%3.15{defxX}  
\bea
x^A & = & \gamma^{a_1}\alpha^{a_2}\delta^{a_3}\beta^{a_4} \nn \\ 
X_A & = & 
\frac{Q^{\frac{1}{2}a_1(a_1-1)}\hat{J}_-^{a_1}} 
{[a_1]!}
\frac{(\hat{J}_0 + \hat{Z})^{a_2}}{a_2!}
\frac{(\hat{J}_0 - \hat{Z})^{a_3}}{a_3!} 
\frac{Q^{-\frac{1}{2}a_4(a_4-1)}\hat{J}_+^{a_4}}
{[a_4]!} \nn \\   
    &   &   a_1,a_2,a_3,a_4 = 0,1,2,\ldots\,.
\label{defxX}
\eea

%newpara
As noted by Bonechi {\em et al.}~\cite{Bo-et} the relations   
((\ref{xx}),(\ref{XX})) also imply 
%
%3.16{calTT}
\beq
{\cal T}_1{\cal T}_2 = 
\sum_{A}\,x^A\Delta (X_A) \qquad 
{\cal T}_2{\cal T}_1 = 
\sum_{A}\,x^A \Delta '(X_A)    
\label{calTT}
\eeq
%
where ${\cal T}_1 = {\cal T} \otimes \one$ and 
${\cal T}_2 = \one \otimes {\cal T}$.  Hence, one would have, in the 
present case, 
%
%3.17{RcalTT}
\beq
R^{(\mu \otimes \mu )}{\cal T}_1^{(\mu )}{\cal T}_2^{(\mu )} =  
{\cal T}_2^{(\mu )}{\cal T}_1^{(\mu )}R^{(\mu \otimes \mu )} 
\label{RcalTT}
\eeq 
%
in accordance with the FRT-formalism of quantum groups~\cite{FRT} where 
$R^{(\mu \otimes \mu )}$ is the $R$-matrix corresponding to the direct 
product of two representations $(\Gamma^{(\mu )})$ 
and obtained by substituting the respective representations in the formula 
(\ref{calR}) for the universal ${\cal R}$.  

%newpara
Now, we note that the $R$-matrix (\ref{R}) defining the $T$-matrix 
((\ref{defT}),(\ref{comrel})) is 
$R^{\left(\frac{1}{2},\frac{1}{2}\right) \otimes \left(\frac{1}{2},
\frac{1}{2}\right)}$.  
For a generic value of $z$ $\left( \neq \frac{1}{2} \right)$, 
$R^{\left(\frac{1}{2},z\right) \otimes \left(\frac{1}{2},z\right)}$  
has the same form as (\ref{R}) with $\lambda$ replaced by $\lambda^{2z}$ 
and thus defines a $T$-matrix whose elements will obey the commutation 
relations of the same form as (\ref{comrel}) with $p$ and $q$ replaced,  
respectively, by $p'$ $=$ $p^{z+\frac{1}{2}}/q^{z-\frac{1}{2}}$ and 
$q'$ $=$ $q^{z+\frac{1}{2}}/p^{z-\frac{1}{2}}$.  In general, it is clear 
{}from (\ref{FdelF}) and (\ref{FRF}) that $R^{(j,z)\otimes (j,z)}$ would 
have the same form as $R^{(j,\frac{1}{2})\otimes (j,\frac{1}{2})}$ with 
$\lambda$ replaced by $\lambda^{2z}$. 

\section{Explicit representation matrices 
$\{ {\cal T}^{(j,z)} \}$}  
\setcounter{equation}{0}

\noindent  
First, let us consider explicitly the matrices 
${\cal T}^{\left(\frac{1}{2},\frac{1}{2}\right)}$ and 
${\cal T}^{\left(1,\frac{1}{2}\right)}$ and, then, generalize the result.  
To verify that ${\cal T}^{\left( \frac{1}{2},\frac{1}{2} \right)}$ is 
the defining $T$-matrix (\ref{defT}) we have to substitute the matrices   
%
%4.1{2drep}
\beq
\hat{J}_+ = \left(
\ba{cc}
0 & 1 \\
0 & 0 
\ea \right) \quad 
\hat{J}_- = \left(
\ba{cc}
0 & 0 \\
1 & 0 
\ea \right) \quad 
\hat{J}_0 = \frac{1}{2} \left( 
\ba{cr}
1 & 0 \\
0 & -1 
\ea \right) \quad 
\hat{Z} = \frac{1}{2}\left( 
\ba{cc}
1 & 0 \\
0 & 1 
\ea \right)
\label{2drep}
\eeq
%
in (\ref{calT}) and expand.  The result is 
%
%4.2{Tau1/2}
\bea
{\cal T}^{\left( \frac{1}{2},\frac{1}{2} \right)} & = & 
{{\cal E}xp}_{Q^{-2}}\left\{ \left( 
\ba{cc}
0 & 0 \\
\gamma & 0 
\ea \right) \right\} 
{\rm exp}\left\{ \left(
\ba{cr}
\alpha & 0 \\
0 & -\delta 
\ea \right) \right\}
{{\cal E}xp}_{Q^2}\left\{ \left(
\ba{cc}
0 & \beta \\
0 & 0 
\ea \right) \right\} \nn \\
   &   &  \nn \\
   & = & \left(
\ba{cc}
1 & 0 \\
\gamma & 1 
\ea \right) \left( 
\ba{cc}
{\rm e}^\alpha & 0 \\
0 & {\rm e}^{-\delta} 
\ea \right) \left(
\ba{cc}
1 & \beta \\
0 & 1 
\ea \right) =   
\left( 
\ba{cc}
{\rm e}^\alpha & {\rm e}^\alpha \beta \\
\gamma {\rm e}^\alpha & \gamma {\rm e}^\alpha \beta + {\rm e}^{-\delta} 
\ea \right) \nn \\
   &   &  \nn \\
   & = & \left( 
\ba{cc}
a & b \\
c & ca^{-1}b+a^{-1}{\cal D}
\ea \right) = \left( 
\ba{cc}
a & b \\
c & d 
\ea \right) 
\label{Tau1/2}
\eea 
%
as expected; this is, in fact, the starting point of the Fronsdal-Galindo  
formalism.  It may be noted that at the level of the two-dimensional 
defining representation ${\cal E}xp\{\ \}$ can be replaced by ordinary 
${\rm exp}\{\ \}$ (see Finkelstein~\cite{F} and Akulov 
{\em et al.}~\cite{AGG} for the use of such a realization of the 
$2 \times 2$ $T$-matrix for $GL_q(2)$ and $SL_q(2))$.  The deformed 
structure of the exponential map (\ref{calT}), brought out by the 
Fronsdal-Galindo approach, is revealed only in dimensions $>2$.    

%newpara
For  $\Gamma^{\left(1,\frac{1}{2}\right)}$, with $[2]$ $=$ 
$Q+Q^{-1}$,   
%
%4.3{3drep}
\bea
\hat{J}_+ & = & [2]^{\frac{1}{2}}\left(
\ba{ccc}
0 & Q^{-\frac{1}{2}} & 0 \\
0 & 0 & Q^{\frac{1}{2}} \\
0 & 0 & 0 
\ea \right) \quad \quad 
\hat{J}_- = [2]^{\frac{1}{2}}\left(
\ba{ccc}
0 & 0 & 0 \\
Q^{\frac{1}{2}} & 0 & 0 \\
0 & Q^{-\frac{1}{2}} & 0 
\ea \right) \nn \\
   &   &    \nn \\
\hat{J}_0 & = & \left(
\ba{ccr}
1 & 0 & 0 \\
0 & 0 & 0 \\
0 & 0 & -1 
\ea \right) \quad \quad 
\hat{Z} = \frac{1}{2}\left(
\ba{ccc}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1 
\ea \right)\,.
\label{3drep}
\eea 
%
Substituting this representation in (\ref{calT}) we obtain 
%
%4.4{Tau1alpha}
\beq
{\cal T}^{\left( 1,\frac{1}{2} \right)} 
= \left( 
\ba{ccc}
{\rm e}^{\phi +2\alpha} & 
[2]^{\frac{1}{2}}Q^{-\frac{1}{2}}{\rm e}^{\phi +2\alpha}\beta &  
Q^{-1}{\rm e}^{\phi + 2\alpha}\beta^2 \\   
   &    &   \\
{}[2]^{\frac{1}{2}}Q^{\frac{1}{2}}
\gamma {\rm e}^{\phi +2\alpha} & 
[2]\gamma {\rm e}^{\phi + 2\alpha}\beta + {\rm e}^{-\phi} & 
{\ba{c}
([2]^{\frac{1}{2}}Q^{-\frac{1}{2}}\gamma 
{\rm e}^{\phi + 2\alpha}\beta^2  \\ 
 +[2]^{\frac{1}{2}}Q^{\frac{1}{2}}{\rm e}^{-\phi}\beta ) 
\ea} \\ 
   &    &    \\ 
Q\gamma^2{\rm e}^{\phi + 2\alpha} & 
{\ba{c}
([2]^{\frac{1}{2}}Q^{\frac{1}{2}}\gamma^2
{\rm e}^{\phi + 2\alpha}\beta   \\ 
 +[2]^{\frac{1}{2}}Q^{-\frac{1}{2}}\gamma {\rm e}^{-\phi})  
\ea} & 
{\ba{c}
(\gamma^2{\rm e}^{\phi + 2\alpha}\beta^2  \\ 
+[2]\gamma {\rm e}^{-\phi}\beta \\
 + {\rm e}^{-(3\phi +2\alpha )}) 
\ea} 
\ea \right) 
\label{Tau1alpha}
\eeq 
% 
Now, using the relations ((\ref{qdet}),(\ref{a})-(\ref{Lie})), we get 
%
%4.5{Tau1abcd}
\beq
{\cal T}^{(1,\frac{1}{2})} = \xi \left(
\ba{ccc}
a^2 & 
[2]^{\frac{1}{2}}Q^{-\frac{1}{2}}ab & 
\lambda^{-1}b^2 \\
  &   &  \\ 
{} [2]^{\frac{1}{2}}Q^{-\frac{1}{2}}ac & 
ad+Q^{-1}\lambda^{-1}bc & 
[2]^{\frac{1}{2}}Q^{-\frac{1}{2}}\lambda^{-1}bd \\
  &   &  \\
\lambda c^2 & 
[2]^{\frac{1}{2}}Q^{-\frac{1}{2}}\lambda cd & 
d^2 
\ea \right)\,.  
\label{Tau1abcd}
\eeq
%
Note that in deriving (\ref{Tau1abcd}) from (\ref{Tau1alpha}) the 
relations (\ref{Lie}) are used in the Heisenberg-Weyl form 
%
%4.6{HW}
\bea 
{\rm e}^\alpha \beta & = & Q\lambda^{-1} \beta {\rm e}^\alpha \quad
{\rm e}^\alpha \gamma = Q\lambda \gamma {\rm e}^\alpha \quad 
{\rm e}^\phi \beta = \lambda \beta {\rm e}^\phi \quad 
{\rm e}^\phi \gamma = \lambda^{-1} \gamma {\rm e}^\phi \nn \\
{\rm e}^\alpha {\rm e}^\phi & = & {\rm e}^\phi {\rm e}^\alpha \qquad 
\beta \gamma = \gamma \beta\,.   
\label{HW}
\eea 

%newpara
In the limit $p = q$, or $Q = q$ and $\lambda = 1$, ${\cal D}$ becomes a 
central element of ${\cal A}$ and $GL_{p,q}(2)$ $\longrightarrow$ 
$GL_q(2)$; further, choosing ${\cal D}$ $=$ $1$ (or $\xi$ $=$ $1$) leads 
to the quantum group $SL_q(2)$. In these cases, i.e., for $GL_q(2)$ and 
$SL_q(2)$, with $\lambda$ $=$ $1$, $z$ drops out of the picture and hence 
we may simply denote the $(2j+1)$-dimensional representation matrix as 
${\cal T}^{(j)}$ which is obtained by taking $\lambda$ $=$ $1$ in  
${\cal T}^{(j,\frac{1}{2})}$.  Then, for $SL_q(2)$ the matrix elements  
$\{ {\cal T}^{(1)}_{mk} \mid m,k = 1,0,-1 \}$ of 
${\cal T}^{(1)}$ in (\ref{Tau1abcd}), with $\lambda$ $=$ $1$ and 
$\xi$ $=$ $1$, are seen to coincide with the spin-$1$ representation 
functions $\{ d^1_{mk} \mid m,k=1,0,-1 \}$ given by 
Nomura~\cite{N} (Note: our $q$ is Nomura's $q^{-\frac{1}{2}}$).  

%newpara
For $SU_q(2)$ one has to take into account further relations among 
$\{ a,b,c,d \}$ due to the requirement of the existence of an involutional 
antihomomorphic $*$-operation satisfying 
%
%4.7{su*}
\beq
T^* = \left(
\ba{cc}
a^* & c^* \\
b^* & d^* 
\ea \right) = T^{-1} = \left(
\ba{cc}
d & -q^{-1}b \\
-qc & a 
\ea \right)
\label{su*}
\eeq
%
so that 
%
%4.8{suT}
\beq 
T = \left(
\ba{cc}
a & b \\
c & d 
\ea \right) = \left(
\ba{cc}
a & -qc^* \\
c & a^* 
\ea \right)
\label{suT}
\eeq
%
with $q \in \R \backslash \{0\}$ necessarily.  It may be noted that 
the unitarity condition (\ref{su*}), or $T^*T$ $=$ $TT^*$ $=$ $\one$, 
requires the relations 
%
%4.9{aa*}
\beq   
aa^* + q^2 c^*c = a^*a + c^*c = 1 
\label{aa*}
\eeq
%
to be satisfied, besides the commutation relations (\ref{comrel}). 

%newpara
Let us now generalize the above result.  For $\Gamma^{(j,\frac{1}{2})}$, 
the matrix elements are 
%
%\4.10{jmk}
\bea
\hat{J}_{\pm ,mk} & = & 
Q^{\mp m+\frac{1}{2}}
\{ [j \pm m][j+1 \mp m] \}^{\frac{1}{2}}
\delta_{m,k \pm 1} \nn \\
\hat{J}_{0,mk} & = & m\delta_{mk} \qquad 
\hat{Z}_{mk} = \frac{1}{2}\delta_{mk} \nn \\
   &   &  \qquad  m,k = j,j-1,\ldots ,-(j-1),-j\,.
\label{jmk}
\eea 
%
Substituting this representation (\ref{jmk}) in (\ref{calT}) it is found, 
after considerable algebraic manipulations, that one can write  
%
%4.11{Taumk}
\bea
{\cal T}^{\left( j,\frac{1}{2} \right)}_{mk} & = & \xi^{2j-1}
Q^{-\frac{1}{2}(m-k)(2j-m+k)}\lambda^{-\frac{1}{2}(m-k)(2j-m-k-1)} \nn \\ 
   &    &\,\times\,{\{[j+m]![j-m]![j+k]![j-k]!\}}^{\frac{1}{2}} 
          \nn \\  
   &    &\,\times\,\sum_{s}\,Q^{-s(2j-m+k-s)}\lambda^{-s(m-k+s)} \nn \\ 
   &    & \quad \quad \frac{a^{j+k-s}}{[j+k-s]!} \frac{b^{m-k+s}}{[m-k+s]!} 
                \frac{c^s}{[s]!} \frac{d^{j-m-s}}{[j-m-s]!}\,.
\label{Taumk}
\eea
%
where $s$ runs from $\max (0,k-m)$ to $\min (j+k,j-m)$.
In the limit $\lambda$ $=$ $1$, $Q$ $=$ $q$ and $\xi$ 
$=$ $1$, corresponding to $SL_q(2)$ (or $SU_q(2)$), the above expression 
(\ref{Taumk}) for ${\cal T}^{(j)}_{mk}$ coincides with Nomura's 
expression~\cite{N} for the quantum $d$-function $d^j_{mk}$ (with our $q$ 
replaced by Nomura's $q^{-\frac{1}{2}}$ as already noted).  Nomura~\cite{N}  
has also noted the RTT-relation (\ref{RcalTT}) for the representation 
matrix ${\cal T}^{(j)}$ of $SU_q(2)$.  When $\xi$ is not taken specifically 
to be unity the above matrices 
$\{ {\cal T}^{(j)} \mid j=0,\frac{1}{2},1,\frac{3}{2},\ldots \}$ 
(\ref{Taumk}), with $Q$ $=$ $q$ and $\lambda$ $=$ $1$, provide the 
representations of $GL_q(2)$ (see~\cite{F} which gives these 
representations with an equivalent expression for the rhs of (\ref{Taumk})).   
The rhs of (\ref{Taumk}) can be rearranged in several ways and so there 
exist various equivalent expressions for ${\cal T}^{(j)}_{mk}$ in terms of 
different $q$-special functions for the cases of $SL_q(2)$ and $SU_q(2)$ 
(see~\cite{VS,K,Ma-et,KK,N}).  

When $z \neq \frac{1}{2}$, the 
$\Gamma^{(j,z)}$-representation of $U_{p,q}(gl(2))$ is given by 
\bea
\hat{J}_{\pm ,mk} & = & \lambda^{z-\frac{1}{2}}Q^{\mp m+\frac{1}{2}} 
\{ [j \pm m][j+1 \mp m]\}^{\frac{1}{2}} \delta_{m,k\pm 1} \nn \\
\hat{J}_{0,mk} & = & m\delta_{mk} \qquad 
\hat{Z}_{mk} = z\delta_{mk} \nn \\
  &   &  \quad m,k = j,j-1,\ldots ,-(j-1),-j\,.
\label{4.10z}
\eea
Substituting this representation~(\ref{4.10z}) in (\ref{calT}) one gets 
\beq
{\cal T}^{(j,z)}_{mk} = \lambda^{(m-k)(z-\frac{1}{2})} \xi^{1-2z} 
{\cal T}^{(j,\frac{1}{2})}_{mk} \qquad m,k = j,j-1,\ldots ,-(j-1),-j\,.
\label{4.11z}
\eeq
As is seen from~(\ref{4.11z}), for generic $z$, the one and two 
dimensional representations are respectively given by 
\beq
{\cal T}^{(0,z)} = \xi^{-2z} = {\cal D}^{z}\,,
\label{1drep-z}
\eeq
and
\beq
{\cal T}^{(\frac{1}{2},z)} = {\cal D}^{z-\frac{1}{2}}\left( 
\ba{cc}
a & \lambda^{z-\frac{1}{2}}b \\
\lambda^{-(z-\frac{1}{2})}c & d 
\ea \right)\,.
\label{2drep-z}
\eeq
As already noted at the end of Section 3, it is seen that 
${\cal T}^{(\frac{1}{2},z)}$ corresponds to the fundamental 
$T$-matrix of $GL_{p',q'}(2)$ with $p'$ $=$ 
$p^{z+\frac{1}{2}}/q^{z-\frac{1}{2}}$ and $q'$ $=$ 
$q^{z+\frac{1}{2}}/p^{z-\frac{1}{2}}$.  


\section{$U_{\bar{q},q}(2)$}
\setcounter{equation}{0}

\noindent
It is not possible to have $SU_{p,q}(2)$ with $p \neq q$ and for $SU_q(2)$ 
it is necessary that $q$ $\in$ $\R \backslash \{0\}$.  But, we can 
have $U_{p,q}(2)$ with $p$ $=$ $\bar{q}$ (complex conjugate of $q$); 
we can have $U_{\bar{q},q}(2)$ for any $q$ $\in$ 
$\C \backslash \R $.  For $q$ $\in$ 
$\R \backslash \{0\}$ one gets $U_q(2)$ of which $SU_q(2)$ is the 
special case corresponding to unit value for the quantum determinant.  To 
see the features of $U_{\bar{q},q}(2)$ and $U_q(2)$ one has to study the 
consequences of imposing the unitarity condition on $GL_{p,q}(2)$ 
(see~\cite{CJ2} for some useful details in this regard).  

%newpara
The fundamental $T$-matrix of $U_{\bar{q},q}(2)$, for $q$ $=$ 
$|q|{\rm e}^{{\rm i}\theta}$, is given by  
%
%5.1{Tqq}
\beq
T = \left(
\ba{cc}
a & b \\
c & d 
\ea \right) = \left(
\ba{cc}
a & -\bar{q}\,{\cal D}c^* \\
c & {\cal D}a^* 
\ea \right) = \left( 
\ba{cc}
a & -qc^*{\cal D} \\
c & a^*{\cal D} 
\ea \right)
\label{Tqq}
\eeq
%
with   
%
%5.2{Tqqrel}
\bea
ac & = & \bar{q}ca \quad a{\cal D} = {\cal D}a \quad 
ac^* = qc^*a \quad {\cal D}c^* = {\rm e}^{2{\rm i}\theta}c^*{\cal D} 
\quad cc^* = c^*c \nn \\
{\cal D}^*{\cal D} & = & {\cal D}{\cal D}^* = 1 \quad 
aa^* + |q|^2c^*c = 1 \quad a^*a + c^*c = 1 
\label{Tqqrel}
\eea 
%
and their $*$-conjugate relations, satisfied ($q^* = \bar{q}$).  
Note that 
%
%5.3{TT*}
\beq
TT^* = T^*T = \one   
\label{TT*}
\eeq
%
in view of the relations (\ref{Tqqrel}), as required and   
${\cal D}$ may be representated as ${\rm e}^{{\rm i}\varphi}$ with 
$\varphi^*$ $=$ $\varphi$.  

%newpara
If $q \in \R \backslash \{0\}$ the above equations 
((\ref{Tqq})-(\ref{TT*})) hold with $\theta$ $=$ $0$ and ${\cal D}$ is a 
central element with values on the unit circle in 
$\C$: one gets $U_q(2)$.  If the value of ${\cal D}$ 
is fixed to be unity $U_q(2)$ $\longrightarrow$ $SU_q(2)$.  It is obvious 
that the representation matrices of $U_{\bar{q},q}(2)$ and $U_q(2)$ are 
given by the formula (\ref{Taumk}) with the relations (\ref{Tqq}) between 
$\{ a,b,c,d \}$ taken into account.  

\section{Clebsch-Gordan coefficients for $U_{p,q}(gl(2))$ and 
$GL_{p,q}(2)$}
\setcounter{equation}{0}
 

\noindent
Let ${\cal C}$ and ${\cal C}'$ be the Clebsch-Gordan matrices (CGMs) such 
that ${\cal C}^{-1}\Delta {\cal C}$ and ${\cal C}'^{-1}\Delta '{\cal C}'$ 
are reduced representations corresponding to the coproduct $\Delta$ in 
(\ref{copro}) and the opposite coproduct.  From the relation 
(\ref{univR}) it is clear that ${\cal C}'$ $=$ $R{\cal C}$ where $R$ is 
the $R$-matrix obtained from the universal ${\cal R}$ by substituting the 
corresponding irreducible representations involved in the coproduct 
(see~\cite{CGZ} for a detailed discussion on the relation between 
${\cal C}$s and $R$).  Now, let us make the  following observation on the 
CGMs for $U_{p,q}(gl(2))$ (or $U_{Q,\lambda}(gl(2))$): from (\ref{FdelF}) 
it is easy to see that 
%
%6.1{CFC}
\beq
{\cal C}_{Q,\lambda} = F{\cal C}_{Q,\lambda =1} \quad \quad
{\cal C}'_{Q,\lambda} = F^{-1}{\cal C}'_{Q,\lambda =1}\,.  
\label{CFC}
\eeq
Explicitly writing, one has the expressions for the $(p,q)$ (or 
$(Q,\lambda)$) CGCs as 
\bea
\langle j_1 z_1 m_1 , j_2 z_2 m_2 | j z m \rangle &  =  & 
\lambda^{m_1z_2-z_1m_2} \delta_{z,z_1+z_2} 
\langle j_1 m_1 , j_2 m_2 | j m \rangle _Q \nn \\
\langle j_1 z_1 m_1 , j_2 z_2 m_2 | j z m \rangle ' &  =  & 
\lambda^{z_1m_2-m_1z_2} \delta_{z,z_1+z_2} 
\langle j_1 m_1 , j_2 m_2 | j m \rangle '_Q 
\label{explicitCGC}
\eea
%
where $\{ \langle j_1 m_1 , j_2 m_2 | j m \rangle _Q \}$ and     
$\{ \langle j_1 m_1 , j_2 m_2 | j m \rangle '_Q \}$ are the  
$Q$-CGCs of $U_Q(gl(2))$ corresponding respectively to the 
coproducts $\Delta_{Q,\lambda =1}$ and $\Delta '_{Q,\lambda =1}$ 
(see (\ref{coalg})).   

%newpara
It is particularly interesting to consider the coproducts for 
$U_{\bar{q},q}(u(2))$ with 
$q$ $\in$ $\C \backslash \R$.  In this case, 
$Q$ $=$ $|q|$ and $\lambda$ $=$ ${\rm e}^{-{\rm i}\theta}$. Hence, the 
coproduct (\ref{coalg}), with Hermitian $Z$ and $J_0$, preserves also the 
Hermiticity property of the pair $(J_+,J_-)$ i.e., 
$J_\pm^\dagger$ $=$ $J_\mp$.  In the physical context this implies an   
addition of $q$-angular momenta of two particles, (1) and (2), 
according to the rule   
%
%6.2{NEWPHASE}
\bea 
\Delta (J_\pm ) & = & 
J_\pm (1)|q|^{-J_0(2)}{\rm e}^{\mp {\rm i}\theta Z(2)} + 
|q|^{J_0(1)}{\rm e}^{\pm {\rm i}\theta Z(1)}J_\pm (2) \nn \\
\Delta (J_0) & = & J_0(1) + J_0(2) \qquad 
\Delta (Z) = Z(1) + Z(2) 
\label{NEWPHASE}
\eea 
%
with some `phases' which may somehow be irremovable.  In fact, one can 
even have $|q|$ $=$ $1$ i.e., a modified addition rule for ordinary angular 
momenta with a new additive `phase' quantum number.  This aspect of the 
quantum algebra $U_{\bar{q},q}(u(2))$ may be worth probing further, 
particularly, in view of the interest in physical applications of 
$U_{p,q}(u(2))$ (see~\cite{BMK}).  

%newpara
Let us now look at the direct product representations of the quantum 
group $GL_{p,q}(2)$.  From (\ref{calTT}) it follows that   
%
%6.3{TauTau}
\bea
{\cal T}_1{\cal T}_2 & = & 
{{\cal E}xp}_{Q^{-2}}\{ \gamma \Delta (\hat{J}_-)\}
{\rm exp}\{ \alpha (\Delta (\hat{J}_0) + 
\Delta (\hat{Z})) \} \nn \\
  &   &  \ \ \ \ \times\,{{\cal E}xp}_{Q^2}
\{ \beta \Delta (\hat{J}_+) \} 
\nn \\
{\cal T}_2{\cal T}_1 & = & 
{{\cal E}xp}_{Q^{-2}}\{ \gamma \Delta '(\hat{J}_-)\}
{\rm exp}\{ \alpha (\Delta '(\hat{J}_0) +
\Delta '(\hat{Z})) \} \nn \\ 
  &   &  \ \ \ \ \times\,{{\cal E}xp}_{Q^2}
\{ \beta \Delta '(\hat{J}_+)\}
\label{TauTau}
\eea 
%
where $\Delta$ is given in (\ref{newcoalg}) and the representations 
involved in the coproducts are the relevant representations 
$\{ \Gamma^{\left( j,\frac{1}{2} \right)} \}$.  It is particularly 
easy to verify these relations in the case when 
$\Delta$ corresponds to the direct product of two identical 
$\Gamma^{\left( \frac{1}{2},\frac{1}{2} \right)}$-representations.    
It is obvious that ${\cal C}^{-1}{\cal T}_1{\cal T}_2{\cal C}$ and 
${\cal C}'^{-1}{\cal T}_2{\cal T}_1{\cal C}'$ will be in reduced forms.  
Since, for $U_{p,q}(gl(2))$, $\Delta^{(j_1,z_1) \otimes (j_2,z_2)} 
\longrightarrow \sum_{j'=|j_1-j_2|}^{j_1+j_2}\,\oplus 
\Gamma^{(j',z_1+z_2)}$, the direct product representation
${\cal T}_1{\cal T}_2$ $=$ $({\cal T}^{(j_1,z_1)} \otimes 
I^{(j_2,z_2)})(I^{(j_1,z_1)} \otimes {\cal T}^{(j_2,z_2)})$ (or 
${\cal T}_2{\cal T}_1$ corresponding to the opposite coproduct), 
where $I^{(j,z)}$ is the $(2j+1)$-dimensional identity matrix 
corresponding to the unity in the $(j,z)$-representation, will be 
reduced to the direct sum of the representation matrices 
 $\{ {\cal T}^{(j,z_1+z_2)} | j = |j_1-j_2|, ... ,j_1+j_2 \}$ and 
the corresponding CGCs are given by (\ref{explicitCGC}). 

As an interesting example, let us consider the direct products of the  
representations labeled by $(0,z-\frac{1}{2})$ and $(j,\frac{1}{2})$  
for both $U_{p,q}(gl(2))$ and $GL_{p,q}(2)$; note that, in general, 
the $(j,z)$-representation is defined by~(\ref{4.10z}) (or (\ref{newJ})).  
Using the coproduct (\ref{newcoalg}) and its opposite the corresponding direct
product representations of $U_{p,q}(gl(2))$ are given by, with 
$(j_1,z_1)$ $=$ $(0,z-\frac{1}{2})$ and $(j_2,z_2)$ $=$ 
$(j,\frac{1}{2})$, 
\bea
\Delta (\hat{J}_+)_{mk} & = & Q^{-m+\frac{1}{2}}\{ [j+m][j+1-m] \}^
{\frac{1}{2}}\delta_{m,k+1} \nn \\
\Delta (\hat{J}_-)_{mk} & = & \lambda^{2z-1} Q^{m+\frac{1}{2}} 
\{ [j-m][j+1+m] \}^{\frac{1}{2}}\delta_{m,k-1} \nn \\
\Delta (\hat{J}_0)_{mk} &= & m \delta_{mk} \qquad 
\Delta(\hat{Z})_{mk} = z \delta_{mk} \nn \\
  &    &  \quad m,k = j,j-1, \ldots ,-(j-1),-j 
\label{Deltarep-alg}
\eea
and
\bea
\Delta '(\hat{J}_+)_{mk} & = & \lambda^{2z-1} Q^{-m+\frac{1}{2}}
\{ [j+m][j+1-m] \}^ {\frac{1}{2}}\delta_{m,k+1} \nn \\     
\Delta '(\hat{J}_-)_{mk} & = & Q^{m+\frac{1}{2}} 
\{ [j-m][j+1+m] \}^{\frac{1}{2}}\delta_{m,k-1} \nn \\
\Delta ' (\hat{J}_0)_{mk} &= & m \delta_{mk} \qquad 
\Delta '(\hat{Z})_{mk} = z \delta_{mk} \nn \\
  &    &  \quad m,k = j,j-1, \ldots ,-(j-1),-j\,.   
\label{Delta'rep-alg}
\eea
Using these representations in (\ref{TauTau}) above it is seen that the 
corresponding direct product representations of 
$GL_{p,q}(2)$ are given by 
\beq
{\cal T}_1{\cal T}_2 = {\cal D}^{z-\frac{1}{2}}
{\cal T}^{(j,\frac{1}{2})} \qquad 
{\cal T}_2{\cal T}_1 = {\cal T}^{(j,\frac{1}{2})}
{\cal D}^{z-\frac{1}{2}}\,.
\label{Deltarep-group}
\eeq
It is easy to verify that the direct product representations of 
$U_{p,q}(gl(2))$ given by~(\ref{Deltarep-alg}) 
and~(\ref{Delta'rep-alg}) can be `reduced' to (or made equivalent  
to, in this case) the defining representation~(\ref{4.10z}) using   
the CGCs obtained from~(\ref{explicitCGC}) by taking the $Q$-CGMs to    
be identity matrices.  The same CGMs are seen to `reduce' also the 
direct product representations of $GL_{p,q}(2)$ given 
by~(\ref{Deltarep-group}) to the defining     
representation~(\ref{4.11z}).  It may also be noted that the direct   
product representations of $GL_{p,q}(2)$ given 
by~(\ref{Deltarep-group}) satisfy the $RTT$-relation (\ref{RcalTT}) with a 
$(2j+1)$-dimensional `$R$-matrix' with elements $\{ R_{mk} = 
\lambda^{(2z-1)m}\delta_{mk} | m,k = j,(j-1), \ldots ,-(j-1),-j \}$
in accordance with (\ref{FRF}). 


\section{Conclusion}

\noindent
Before closing, we may mention a few related points.  

%newpara
The converse of the exponential map, namely, the passage $GL_{p,q}(2)$ 
$\longrightarrow$ $U_{p,q}(gl(2))$ using the 
representation of $GL_{p,q}(2)$ close to identity, for    
infinitesimal values of the group parameters $\{ \alpha ,\beta ,\gamma 
,\delta \}$, follows by writing ${\cal T}$ $\approx$   
$1 + \gamma \hat{J}_- + \alpha (\hat{J}_0 + \hat{Z}) + 
\delta (\hat{J}_0 - \hat{Z}) + \beta \hat{J}_+$.  Thus, if 
one can obtain the representations of $GL_{p,q}(2)$ by some method 
directly, then, the representations of its infinitesimal generators 
$\{ \hat{J}_0,\hat{J}_\pm ,\hat{Z} \}$ forming the dual 
algebra $U_{p,q}(gl(2))$ can be derived.  This is how Finkelstein~\cite{F} 
obtains the relationship $GL_q(2)$ $\longrightarrow$ $U_q(gl(2))$, 
independent of~\cite{FG}, using theory of invariants to derive an 
equivalent form of the representation matrix (\ref{Taumk}) for the case 
$p=q$.   

%newpara 
Finkelstein's analysis of the representations of $GL_q(2)$~\cite{F} 
is motivated by the construction of a $GL_q(2)$ Yang-Mills theory in which 
one would regard the nonabelian group parameters 
$\{\alpha (x),\beta (x),\gamma (x),\delta (x) \}$ (the coproduct rules of  
which specify the group multiplication law as pointed out in~\cite{MV}) as 
space-time fields.  Akulov {\em et al.}~\cite{AGG}
have considered the differential calculus of the group parameter space for 
$SL_q(2)$ and studied a related field theory model.  The problem of   
realization of the group parameters as dependent on continuous 
classical variables (like space-time) has been addressed recently by 
Volovich~\cite{V} at the level of the variables $\{ a,b,c,d \}$ (see 
also~\cite{CJ2}).  
Let us observe an example of such a realization based on the 
relations (\ref{a=}) and (\ref{Lie}):  Using the well known Bogoliubov 
transformation of a pair of commuting sets of boson operators, we 
can write
\begin{eqnarray*}
&&\alpha =(\rho - \theta )\psi_1^\dagger (x)\psi_1(x)
+(\rho + \theta )\psi_2^\dagger (x)\psi_2(x),\\
&&\beta=  \psi_1^\dagger (x),\qquad \gamma=\psi_2^\dagger (x), \\
&&\delta =(\rho + \theta )\psi_1^\dagger (x)\psi_1(x)+(\rho - 
\theta )\psi_2^\dagger (x)\psi_2(x), 
\end{eqnarray*}
with $\psi_1(x)$ $=$ 
$({\rm cosh}x)a_1$ $-$ $({\rm sinh}x)a_2^\dagger$, $\psi_2(x)$ $=$ 
$({\rm cosh}x)a_2$ $-$ $({\rm sinh}x)a_1^\dagger$, $x$ $\in$ $\R$, 
$[a_1,a_1^\dagger ]$ $=$ $1$, $[a_2,a_2^\dagger ]$ $=$ $1$, 
$[a_1,a_2]$ $=$ $0$, $[a_1,a_2^\dagger ]$ $=$ $0$.  
In the context of building gauge theories based on 
quantum groups related to $GL(2)$ it is interesting to observe that in the 
case of $GL_{p,q}(2)$ the two-dimensional vector spaces carrying the 
fundamental representation ($T$) have commuting components if $p$ $=$ $1$ 
just like the Hilbert space of a two-level ordinary quantum mechanical 
system.  Besides the interpretation of 
the matrix elements of the representations of $SU_q(2)$ as wavefunctions of 
quantum symmetric tops~\cite{N}, generalization of quantum dynamics  
based on the representations of $SU_q(2)$ have also been  
considered~\cite{A-et}.  Since having one more parameter would provide   
more flexibility in model building, we believe that the study of   
representations of $GL_{p,q}(2)$ should prove useful.  Recently, 
generalization of the exponential map for the quantum supergroup  
$GL_{p,q}(1|1)$ has also been obtained~\cite{CJ3}.   

%newpara
There are several approaches to quantum groups (or quantum matrix 
pseudogroups) and quantum algebras (or quantized universal enveloping 
algebras) which are in duality with the quantum groups in the Hopf 
algebraic sense (see, e.g.,~\cite{M,DHL} for reviews of the subject).    
In the case of the quantum group 
$GL_{p,q}(2)$~\cite{Ku,De-et,S1,T} the structure of its dual Hopf algebra 
$U_{p,q}(gl(2))$ is known from the studies by 
Schirrmacher et al.~\cite{SWZ}, using the noncommutative differential 
calculus approach, and by Dobrev~\cite{D1,D2,D3} using the approach of 
Sudbery~\cite{S2}.  The recent analysis~\cite{FG} of the duality 
relations between Lie bialgebras, with particular reference to 
$A(GL_{p,q}(2))$ and $U_{p,q}(gl(2))$, has led to a generalization of the 
well-known exponential relationship between a classical Lie group and its  
Lie algebra and the explicit form of such an exponential map   
has been obtained between the quantum algebra $U_{p,q}(gl(2))$ and the 
corresponding quantum group $GL_{p,q}(2)$.  This relationship is given 
abstractly in terms of a universal ${\cal T}$-matrix, involving both the 
group parameters of $GL_{p,q}(2)$ and the generators of $U_{p,q}(gl(2))$,   
and for particular representations of $U_{p,q}(gl(2))$  this universal 
${\cal T}$-matrix gives the representations of $GL_{p,q}(2)$.  Using this 
Fronsdal-Galindo formalism we have derived explicitly the 
finite-dimensional representations of $GL_{p,q}(2)$, by exponentiating 
directly the well-known 
$(2j+1)$-dimensional irreducible representations of $U_{p,q}(gl(2))$, and 
the earlier results on the representations of $GL_q(2)$, $SL_q(2)$ and 
$SU_q(2)$ are found to be special cases in the appropriate limits.  We 
have also derived the CGCs for the quantum algebra $U_{p,q}(gl(2))$
and noted that the same CGCs can be used for the Clebsch-Gordan
reduction of the direct product representations
of the quantum group $GL_{p,q}(2)$.

\section*{Acknowledgements}

\noindent
We have pleasure in thanking Prof. T.D. Palev for stimulating discussions. 
One of us (RJ) is thankful to Prof. Guido Vanden Berghe for the kind 
hospitality at the Department of Applied Mathematics and Computer Science, 
University of Ghent.  RJ wishes to thank Prof. K. Srinivasa Rao for  
helping him in many ways during his stay abroad.  He also acknowledges 
the benefit of discussions with Prof. K. Srinivasa Rao, Dr. V. Rajeswari 
and Dr. R. Chakrabarti, some time ago, on the problem of CGCs for 
$U_{p,q}(gl(2))$.  This research was partly supported by the E.E.C 
(Contract No. CI1*-CT92-0101). 

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