%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%% LATEX FILE %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% Author: J. Van der Jeugt %% Title: Representations and Clebsch-Gordan coefficients for the %% Jordanian quantum algebra % J. Phys. A: Math. Gen. 31 (1998), 1495-1508. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \documentstyle[12pt]{article} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % definitions concerning automatic numbering of theorems, lemmas, etc. % \newtheorem{theo}{Theorem} \newtheorem{defi}[theo]{Definition} \newtheorem{prop}[theo]{Proposition} \newtheorem{lemm}[theo]{Lemma} % % \setlength{\topmargin}{0cm} \setlength{\oddsidemargin}{0cm} \setlength{\evensidemargin}{0cm} \setlength{\textheight}{21.5cm} \setlength{\textwidth}{15cm} % % special definitions for the paper % \def\al{\alpha} \def\be{\beta} \def\ga{\gamma} \def\de{\delta} \def\ka{\kappa} \def\la{\lambda} \def\rh{\rho} \def\si{\sigma} \def\ta{\tau} \def\De{\Delta} \def\beq{\begin{equation}} \def\eeq{\end{equation}} \def\bea{\begin{eqnarray}} \def\eea{\end{eqnarray}} \def\beas{\begin{eqnarray*}} \def\eeas{\end{eqnarray*}} \def\nn{\nonumber} \def\ds{\displaystyle} \def\U{{\cal U}_h(sl(2))} % % automatic numbering of equations: % \renewcommand{\theequation}{\arabic{section}.\arabic{equation}} % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{document} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \addtolength{\baselineskip}{2mm} \begin{center} {\Large \bf Representations and Clebsch-Gordan coefficients}\\[2mm] {\Large \bf for the Jordanian quantum algebra ${\cal U}_h(sl(2))$}\\[2cm] J.~Van der Jeugt\footnote{Senior Research Associate of the Fund for Scientific Research -- Flanders (Belgium)} \\[8mm] Vakgroep Toegepaste Wiskunde en Informatica, Universiteit Gent,\\ Krijgslaan 281--S9, B9000 Gent, Belgium\\[2cm] q-alg/9703011 \end{center} \vskip 3cm \begin{abstract} Representation theory for the Jordanian quantum algebra $\U$ is developed. Closed form expressions are given for the action of the generators of $\U$ on the basis vectors of finite dimensional irreducible representations. It is shown how representation theory of $\U$ has a close connection to combinatorial identities involving summation formulas. A general formula is obtained for the Clebsch-Gordan coefficients ${\cal C}^{j_1,j_2,j}_{n_1,n_2,m}(h)$ of $\U$. These Clebsch-Gordan coefficients are shown to coincide with those of $su(2)$ for $n_1+n_2 \leq m$, but for $n_1+n_2 > m$ they are in general a nonzero monomial in $h^{n_1+n_2-m}$. \end{abstract} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newpage \section{Introduction} \setcounter{equation}{0} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Viewing a quantum group as a quantum automorphism group acting on a noncommuting space~\cite{Manin,Schupp,Dubois}, one often requires the extra condition of the existence of a central determinant in the quantum matrix group. For two by two matrices, this condition restricts the quantum groups essentially to only two classes~\cite{Kuperschmidt}, namely the standard $SL_q(2)$ quantum group and the Jordanian $SL_h(2)$ quantum group. The quantum group $SL_h(2)$ was introduced in~\cite{Zakrzewski}, and the corresponding quantum algebra (or quantised universal enveloping algebra) $\U$ was given in~\cite{Ohn}. A unversal $R$-matrix for $\U$ was constructed in~\cite{Ballesteros}. The main object of this paper is to develop representation theory of the Jordanian quantum algebra $\U$, and in particular construct Clebsch-Gordan coefficients. The finite dimensional highest weight representations of $\U$ were given in~\cite{Dobrev}, first by a direct construction, and then by factorising the corresponding Verma module. In~\cite{Dobrev} the action of the $\U$ generators on a finite dimensional representation was not given explicitly. An important construction was developed by Abdesselam {\em et al}~\cite{Abdesselam}~: they gave a nonlinear relation between the generators of $\U$ and the classical generators of $sl(2)$. As a consequence of this relation, they obtained expressions for the action of the $\U$ generators $H$, $X$ and $Y$ (see the following section for their definition) on basis vectors of the finite dimensional irreducible representations. These expressions are in closed form, except for the action of the generator $Y$. Using this nonlinear map, Aizawa~\cite{Aizawa} constructed finite and infinite dimensional representations of $\U$, and considered the tensor product of two representations. Moreover, he gives some examples of Clebsch-Gordan coefficients. Our present work is motivated by the fact that $sl(2)$ or $su(2)$ representations appear in many physical theories, and often their Clebsch-Gordan coefficients are fundamental in these theories. Since representation theory of $\U$ is so closely related to that of $su(2)$, and could be used as the algebraic structure underlying deformations of such physical models, an important aspect to study are the Clebsch-Gordan coefficients of $\U$. In the present paper it is shown how an explicit formula for $\U$ Clebsch-Gordan coefficients can be obtained. In Section~2, we give the defining relations for $\U$, and the nonlinear relation between $sl(2)$ generators and $\U$ generators. In Section~3, closed forms for the action of the three generators $H$, $X$ and $Y$ of $\U$ acting on basis vectors of finite dimensional irreducible representations are determined. For $H$ and $X$, these expressions correspond to those of Ref.~\cite{Abdesselam}; the determination of the explicit action of $Y$ is new and is found using a number of combinatorial identities (Lemmas~\ref{lemmYcoef} and~\ref{lemmzeilb}). In Section~4 the tensor product of two representations is considered. In this tensor product we show the existence of an auxiliary basis which behaves like the uncoupled basis vectors in the tensor product of two $su(2)$ representations. Using this auxiliary basis, the $\U$ Clebsch-Gordan coefficients are easily determined in Section~5, and some expamles and properties are discussed in Section~6. A curious aspect of the results in this paper (and more generally of $\U$ representation theory) is that they are closely related to nontrivial combinatorial identities (see Lemmas~\ref{lemmYcoef}, \ref{lemmzeilb} and~\ref{lemmgosper}). The identities needed here have on one side a (definite or indefinite) sum of hypergeometric terms, and a closed form expression on the other side. To find closed form expressions for such summations is a problem that can be solved completely algorithmically~\cite{AB}~: for indefinite summations this can be done by means of Gosper's algorithm; for definite summations this is done by means of Zeilberger's algorithm. For both algorithms, programs are available in Maple or Mathematica. In the Appendix, we comment on the proofs of these combinatorial identities. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Definition and relation to $sl(2)$} \setcounter{equation}{0} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% The Jordanian quantum algebra $\U$ is an associative algebra with unity and generated by $X$, $Y$ and $H$, subject to the relations \bea &&[H,X]=2 {\sinh hX \over h}, \qquad [H,Y]= -Y(\cosh hX) - (\cosh hX)Y, \nn \\ &&[X,Y]=H; \eea herein, $h$ is the deformation parameter. We shall assume that $|h|<1$. In the limit $h\rightarrow 0$, $\U$ reduces to the universal enveloping algebra of $sl(2)$. The Hopf algebra structure of $\U$ is given in~\cite{Ohn}; here, we are only interested in the comultiplication, which reads \bea &&\De(X)=X\otimes 1 + 1\otimes X,\nn \\ &&\De(Y)=Y\otimes e^{hX}+ e^{-hX}\otimes Y, \label{Del}\\ &&\De(H)=H\otimes e^{hX}+ e^{-hX}\otimes H. \nn \eea The irreducible finite-dimensional highest weight representations of $\U$ can be obtained by using the invertible map from $sl(2)$ to $\U$, given in~\cite{Abdesselam}. With the following definition \bea && Z_+ = {2 \over h} \tanh {hX \over 2},\nn\\ && Z_- = ( \cosh {hX \over 2}) Y (\cosh {hX\over 2}), \label{sl2toU} \eea it follows that the elements $\{H, Z_+, Z_-\}$ satisfy the commutation relations of a classical $sl(2)$ basis~: \beq [H,Z_\pm]=\pm 2 Z_{\pm}, \qquad [Z_+,Z_-]=H. \label{defrelsl2} \eeq The relations (\ref{sl2toU}) can be inverted, see the following section, and thus with every finite-dimensional irreducible $sl(2)$ representation there corresponds a finite-dimensional irreducible representation of $\U$. These $sl(2)$ representations are labeled by a number $j$, with $2j$ a non-negative integer; the representation space is $V^{(j)}$ with basis $e^j_m$, where $m= -j, -j+1,\ldots, j$. The action of $sl(2)$ on this basis is given by \bea && H e^j_m = 2m\; e^j_m, \nn\\ && Z_\pm e^j_m = \sqrt{ (j\mp m)(j\pm m+1) }\; e^j_{m\pm 1}. \label{sl2eaction} \eea For most of the computations in this paper, it is easier to work with another basis of $V^{(j)}$ related to the above basis by \beq v^j_m = \al_{j,m} e^j_m,\;\;\hbox{with } \al_{j,m} = \sqrt{ (j+m)!/(j-m)!}\; . \label{ve} \eeq The matrix elements of the $sl(2)$ generators is then given by \bea && H v^j_m = 2m\; v^j_m, \nn\\ && Z_+ v^j_m = v^j_{m+1}, \label{sl2vaction} \\ && Z_- v^j_m = (j+ m)(j- m+1)\; v^j_{m-1}, \nn \eea where $v^j_{j+1}=0$. Clearly, if for an operator the matrix elements in the $v$-basis have been determined, the matrix elements in the $e$-basis follow immediately using~(\ref{ve}). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Representations of $\U$} \setcounter{equation}{0} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% In this section we wish to give explicit expressions for the matrix elements of $H$, $X$ and $Y$ in the $v$-basis. For $H$, this is trivial, see~(\ref{sl2vaction}). For $X$, one first determines the action of $e^{hX}$. From relation~(\ref{sl2toU}) one finds that \beq e^{hX} = (1+{h\over 2} Z_+) (1-{h\over 2} Z_+)^{-1}. \label{XitoZ} \eeq Then the action of $Z_+$ in the $v$-basis implies \beq e^{hX} v^j_m = v^j_m + 2 \sum_{k=1}^{j-m} ({h\over 2})^k v^j_{m+k}. \label{actionehX} \eeq Thus in this representation one can write $e^{hX}=1+N_1$, with $N_1$ a nilpotent matrix. Then $hX=\log(1+N_1)=N_1-N_1^2/2 +N_1^3/3 -\cdots$, and one obtains the following action of $X$ in the representation space $V^{(j)}$~: \beq X v^j_m = \sum_{k=0}^{\lfloor (j-m-1)/2 \rfloor} {(h/2)^{2k} \over 2k+1} \; v^j_{m+1+2k}. \label{Xaction} \eeq Up to a scaling of the basis vectors, (\ref{Xaction}) coincides with \cite[eq.~(23)]{Abdesselam}. The action of $Y$ is more difficult to determine explicitly; in \cite[eq.~(35)]{Abdesselam} an expression is given but the matrix elements still involve a single sum. Here we shall show that it can actually be given in closed form. Let us use the relation \beq Y = ( \cosh {hX \over 2})^{-1} Z_- (\cosh {hX\over 2})^{-1}, \label{YitoZ} \eeq and first determine the matrix form of $(\cosh {hX\over 2})^{-1}$ in the $v$-basis. In this basis of $V^{(j)}$, one can write \beq (\cosh {hX\over 2})^2 = {1\over 2} (1 + \cosh hX) = {1\over 2} (1 + {e^{hX}+e^{-hX} \over 2}) = 1+N_2, \eeq where, by (\ref{actionehX}), $N_2$ is again a nilpotent matrix whose matrix elements follow from those of $e^{hX}$~: \beq N_2 v^j_m = \sum_{k=1}^{\lfloor(j-m)/2 \rfloor} (h/2)^{2k} v^j_{m+2k}. \eeq Then, in this representation \bea (\cosh {hX\over 2})^{-1} &=& (1+N_2)^{-1/2} \nn\\ &=& \sum_{n=0}^\infty (-1)^n (1/2)_n {N_2^n \over n!}, \label{expN2} \eea where $(a)_n$ is the notation for the Pochhammer symbol~: \beq (a)_n = \left\{ \begin{array}{lll} a(a+1)\cdots (a+n-1) & \hbox{if} & n=1,2,\ldots; \\ 1 & \hbox{if} & n=0. \end{array} \right. \label{poch} \eeq Using the explicit powers of $N_2$ in the action of (\ref{expN2}) on $v^j_m$, the contributions to the coefficient of $v^j_{m+2k}$ ($k>0$) in $(1+N_2)^{-1/2} v^j_m$ are \beq \left( h\over 2 \right)^{2k} \sum_{n=1}^{k} (-1)^n {(1/2)_n \over n!} \left({k-1 \atop n-1}\right). \eeq Next, we use \begin{lemm} For $k>0$ integer, $$ \sum_{n=1}^{k} (-1)^n {(1/2)_n \over n!} \left({k-1 \atop n-1}\right) = - {1\over 2^{2k-1}} {(2k-2)! \over k!(k-1)!}. $$ \label{lemmYcoef} \end{lemm} As a consequence, one finds the explicit action of $(\cosh {hX\over 2})^{-1}$ in the $v$-basis~: \beq (\cosh {hX\over 2})^{-1} v^j_m = v^j_m -2 \sum_{k=1}^{\lfloor(j-m)/2 \rfloor} t_k \left(h\over 4\right)^{2k} v^j_{m+2k}, \label{cosh-1} \eeq with $t_k={(2k-2)!\over k!(k-1)!}$. Using (\ref{YitoZ}), (\ref{cosh-1}) and (\ref{sl2vaction}), one determines the action of $Y$, \bea Yv^j_m &=& (\cosh {hX\over 2})^{-1} Z_- \left(v^j_m -2 \sum_{k\geq 1} t_k (h/4)^{2k} v^j_{m+2k} \right) \nn\\ &=& (\cosh {hX\over 2})^{-1} \Bigl((j+m)(j-m+1) v^j_{m-1} - \nn\\ && 2 \sum_{k\geq 1} t_k (h/4)^{2k} (j+m+2k)(j-m-2k+1) v^j_{m+2k-1} \Bigr)\nn\\ &=& (j+m)(j-m+1) \left(v^j_{m-1} -2 \sum_{l\geq 1} t_l (h/4)^{2l} v^j_{m+2l-1} \right) - \nn\\ && 2 \sum_{k\geq 1} t_k (h/4)^{2k} (j+m+2k)(j-m-2k+1)\nn\\ &&\qquad\times \left( v^j_{m+2k-1} -2 \sum_{l\geq 1} t_l (h/4)^{2l} v^j_{m+2k+2l-1} \right) . \label{tempresY} \eea In this last expression, the coefficient of $v^j_{m-1}$ is $(j+m)(j-m+1)$. The coefficient of $v^j_{m+1}$ is \bea &&-2(j+m)(j-m+1)t_1 h^2/16 - 2(j+m+2)(j-m-1)t_1 h^2/16 \nn\\ &&= h^2/4 -(j-m)(j+m+1) h^2/4. \eea Finally, we determine the coefficient of $v^j_{m+2s-1}$ ($s\geq 2$) in (\ref{tempresY}); it reads \bea &&-2(j+m)(j-m+1)t_s (h/4)^{2s} -2(j+m+2s)(j-m-2s+1) t_s (h/4)^{2s} + \nn\\ && \qquad 4 (h/4)^{2s} \sum_{k=1}^{s-1} (j+m+2k)(j-m-2k+1) t_k t_{s-k}. \label{coeffv} \eea In order to give a closed form for this coefficient, we need to find an expression for $\sum_k t_k t_{s-k} k^n$ with $n=0$, 1 and 2. This is done with the help of the following lemma~: \begin{lemm} Let $s\geq 2$ be integer, then $$ \sum_{k=1}^{s-1} {(2k-2)!\over k!(k-1)!} {(2s-2k-2)!\over(s-k)!(s-k-1)!} k^n = \left\{ \begin{array}{ll} \ds{(2s-2)! \over s!(s-1)!} & \hbox{if } n=0,\\[4mm] \ds{(2s-2)! \over 2(s-1)!^2} & \hbox{if } n=1,\\[4mm] \ds{s(2s-2)! \over 2(s-1)!^2}-4^{s-2} & \hbox{if } n=2. \end{array} \right. $$ \label{lemmzeilb} \end{lemm} Using these three results, it follows from (\ref{coeffv}) that the coefficient of $v^j_{m+2s-1}$ ($s\geq 2$) is $4^s (h/4)^{2s}=(h/2)^{2s}$. Thus we finally obtain the explicit action of $Y$ on the $v$-basis~: \bea Y v^j_m &=& (j+m)(j-m+1) v^j_{m-1} -(j-m)(j+m+1)\left(h\over 2\right)^2 v^j_{m+1} + \nn\\ &&\qquad \sum_{s=1}^{\lfloor(j-m+1)/2\rfloor} \left(h\over 2\right)^{2s} v^j_{m-1+2s}. \label{Yaction} \eea Thus the first equation of (\ref{sl2vaction}) together with equations~(\ref{Xaction}) and~(\ref{Yaction}) determine the action of the generators $H$, $X$ and $Y$ of $\U$ on the finite dimensional irreducible representations in closed form. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Tensor product of two representations} \setcounter{equation}{0} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% The action of any of the generators $H$, $X$ or $Y$ on the tensor product of two representations is determined by the comultiplication. The comultiplication rule on $H$, $X$ and $Y$ induces a comultiplication on $H$, $Z_\pm$. The purpose of this section is to show that the tensor product $V^{(j_1)}\otimes V^{(j_2)}$ decomposes into a direct sum of representations isomorphic to $V^{(j)}$, where $j=j_1+j_2, j_1+j_2-1, \ldots, |j_1-j_2|$, and to determine the Clebsch-Gordan coefficients in \beq e^{(j_1j_2)j}_m = \sum_{n_1,n_2} {\cal C}^{j_1,j_2,j}_{n_1,n_2,m}(h) \; e^{j_1}_{n_1} \otimes e^{j_2}_{n_2}, \eeq such that $e^{(j_1j_2)j}_m$ is a standard $e$-basis of $V^{(j)}$, i.e. \bea && \De(H) e^{(j_1j_2)j}_m = 2m\; e^{(j_1j_2)j}_m, \nn\\ && \De(Z_\pm) e^{(j_1j_2)j}_m = \sqrt{ (j\mp m)(j\pm m+1) }\; e^{(j_1j_2)j}_{m\pm 1}. \eea The first step towards this goal is to find expressions of $\De(H)$ and $\De(Z_+)$ in terms of $H$ and $Z_+$. For $H$, the problem is easy, and follows from (\ref{Del}) and (\ref{XitoZ}), see also~\cite[eq.~(3.2)]{Aizawa}~: \bea &&\De(H) = H\otimes e^{hX} + e^{-hX} \otimes H \nn\\ &&= H\otimes 1 + 1 \otimes H + 2H \otimes \sum_{n=1}^\infty \left(hZ_+\over 2 \right)^n + \sum_{n=1}^\infty \left(- hZ_+\over 2 \right)^n \otimes 2H. \label{DelH} \eea To determine $\De(Z_+)$, denote $t=e^{hX}$ and $z={h\over 2} Z_+$. Then $z=(t-1)(t+1)^{-1}$ and $t=(1+z)(1-z)^{-1}$. Moreover, \beq \De(t)=t\otimes t = (1+2\sum_{k=1}^\infty z^k) \otimes (1+2\sum_{k=1}^\infty z^k). \eeq Thus $\De(z)=\sum_{k,l=0}^\infty \la_{k,l} \; z^k\otimes z^l$, and the coefficients can be obtained from $$ ( 1\otimes 1 - \De(z)) \De(t) = 1\otimes 1 + \De(z), $$ which must hold since $(1-z)t=(1+z)$. This leads to $$ \De(z) = (1\otimes z + z \otimes 1)(1\otimes 1 - z\otimes z + z^2\otimes z^2 - \cdots), $$ and we obtain~: \beq \De(Z_+) = (1\otimes Z_+ + Z_+ \otimes 1)\left(\sum_{n=0}^\infty (-h^2/4)^n \; Z_+^n \otimes Z_+^n \right). \label{DelZ} \eeq An explicit expression for $\De(Z_-)$ is much more complicated (see, e.g.\ eq.~(5.3) of~\cite{Aizawa}), but we do not need it here. In the tensor product space $V^{(j_1)}\otimes V^{(j_2)}$, we now define an auxiliary basis that is expressed in terms of the $v$-basis $v^{j_1}_{m_1} \otimes v^{j_2}_{m_2}$. Define the coefficients (see~(\ref{poch}) for the notation of Pochhammer symbols) \beq b^{m_1,m_2}_{k,l} = \left\{ \begin{array}{ll} \ds{(-2m_1-k)_l (-2m_2-l)_k \over k!l!} & \hbox{if }k\geq 0\hbox{ and } l\geq 0 ; \\[2mm] 0 & \hbox{otherwise}, \end{array} \right. \label{defb} \eeq and \beq a^{m_1,m_2}_{k,l} = (-1)^k (h/2)^{k+l} (b^{m_1,m_2}_{k,l} - b^{m_1,m_2}_{k-1,l-1} ). \label{defa} \eeq The auxiliary vectors are defined as follows~: \beq w^{j_1,j_2}_{m_1,m_2} = \sum_{k=0}^{j_1-m_1} \sum_{l=0}^{j_2-m_2} a^{m_1,m_2}_{k,l} \; v^{j_1}_{m_1+k} \otimes v^{j_2}_{m_2+l}. \label{defw} \eeq Clearly, they also form a basis for $V^{(j_1)}\otimes V^{(j_2)}$ since the relation between the vectors $w^{j_1,j_2}_{m_1,m_2}$ and the vectors $v^{j_1}_{m_1} \otimes v^{j_2}_{m_2}$ is given by an upper triangular matrix with 1's on the diagonal. This auxiliary basis has been introduced because the action of $\De(H)$ and $\De(Z_\pm)$ on it is simple. The idea of introducing such an auxiliary basis comes {}from~\cite{Aizawa}~: however, the coefficients used in~\cite[eq.~(3.9)]{Aizawa} are single sum expressions and no closed forms. \begin{prop} The action of $\De(H)$ on the auxiliary basis vectors is given by $$ \De(H) w^{j_1,j_2}_{m_1,m_2} = 2(m_1+m_2) \; w^{j_1,j_2}_{m_1,m_2}. $$ \label{propH} \end{prop} To prove this, use (\ref{DelH}), (\ref{defw}), and the explicit action of $H$ and $Z_+$ on the $v$-basis~: \bea &&\De(H) w^{j_1,j_2}_{m_1,m_2} = \De(H) \left(\sum_{k,l\geq 0} a^{m_1,m_2}_{k,l} \; v^{j_1}_{m_1+k} \otimes v^{j_2}_{m_2+l} \right) \nn\\ &&=\sum_{k,l\geq 0} a^{m_1,m_2}_{k,l} \Bigl( 2(m_1+k+m_2+l) v^{j_1}_{m_1+k} \otimes v^{j_2}_{m_2+l} \; + \nn\\ && 4(m_1+k) v^{j_1}_{m_1+k} \otimes \sum_{n\geq 1} (h/2)^n v^{j_2}_{m_2+l+n} + 4(m_2+l) \sum_{n\geq 1} (-h/2)^n v^{j_1}_{m_1+k+n} \otimes v^{j_2}_{m_2+l} \Bigr) \nn\\ &&= 2(m_1+m_2) w^{j_1,j_2}_{m_1,m_2} + 2 \sum_{k,l\geq 0} v^{j_1}_{m_1+k} \otimes v^{j_2}_{m_2+l} \Bigl( (k+l) a^{m_1,m_2}_{k,l} + \nn\\ &&\quad 2(m_1+k) \sum_{n=1}^l (h/2)^n a^{m_1,m_2}_{k,l-n} + 2(m_2+l) \sum_{n=1}^l (-h/2)^n a^{m_1,m_2}_{k-n,l} \Bigr). \label{DelHw} \eea Thus the proposition is proved provided we can show that the coefficient of $v^{j_1}_{m_1+k} \otimes v^{j_2}_{m_2+l}$ in the last summation is zero. Consider first \beq \sum_{n=1}^l (h/2)^n a^{m_1,m_2}_{k,l-n} = (-1)^k (h/2)^{k+l} \sum_{n=1}^l (b^{m_1,m_2}_{k,l-n} - b^{m_1,m_2}_{k-1,l-n-1}); \eeq we shall show that this last sum can be performed and yields \beq \sum_{n=1}^l (b^{m_1,m_2}_{k,l-n} - b^{m_1,m_2}_{k-1,l-n-1}) = { 2m_1+k-l+1 \over 2m_1+k } b^{m_1,m_2}_{k,l-1}. \label{resb} \eeq Using the symmetry $b^{m_1,m_2}_{k,l}=b^{m_2,m_1}_{l,k}$, one can use the same result (\ref{resb}) to find \beq \sum_{n=1}^l (-h/2)^n a^{m_1,m_2}_{k-n,l} = (-1)^k (h/2)^{k+l} \Bigl( { 2m_2+l-k+1 \over 2m_2+l }\Bigr) b^{m_1,m_2}_{k-1,l}. \eeq Then the coefficient of $v^{j_1}_{m_1+k} \otimes v^{j_2}_{m_2+l}$ in the summation-part of (\ref{DelHw}) is, up to a factor $(-1)^k (h/2)^{k+l}$, equal to \bea &&(k+l)(b^{m_1,m_2}_{k,l} - b^{m_1,m_2}_{k-1,l-1}) + 2(m_1+k) \Bigl({ 2m_1+k-l+1 \over 2m_1+k }\Bigr) b^{m_1,m_2}_{k,l-1} + \nn\\ &&\qquad 2(m_2+l) \Bigl({2m_2+l-k+1 \over 2m_2+l }\Bigr) b^{m_1,m_2}_{k-1,l}, \eea and using the definition of the coefficients $b^{m_1,m_2}_{k,l}$ this is trivially shown to be zero. What remains to be proved is (\ref{resb}). This follows from the following lemma~: \begin{lemm} Let $\al,\be,\ga$ be arbitrary parameters, $l\geq 0$ integer, and $$ g(l)= {(\al)_l(\be)_l \over (l+1)! (\ga)_l } \bigl( (\ga-\al-\be)l+\ga-1-\al\be \bigr). $$ Then for $n\geq 0$ integer, $$ \sum_{l=0}^n g(l) = \ga-1-{ (\al)_{n+1}(\be)_{n+1} \over (n+1)! (\ga)_n }. $$ \label{lemmgosper} \end{lemm} The sum (\ref{resb}) then follows from this lemma by putting $\al=1-k-2m_1$, $\be=1+2m_2$ and $\ga=2-k+2m_2$. This completes the proof of Proposition~\ref{propH}. \begin{prop} The action of $\De(Z_+)$ on the auxiliary basis vectors is given by \beq \De(Z_+) w^{j_1,j_2}_{m_1,m_2} = w^{j_1,j_2}_{m_1+1,m_2} + w^{j_1,j_2}_{m_1,m_2+1}, \label{DelZw} \eeq where $w^{j_1,j_2}_{m_1,m_2}$ is interpreted as 0 if one of the indices $m_i>j_i$. \label{propZp} \end{prop} This is proved by direct computation, and does not involve any combinatorial identities. The left hand side of (\ref{DelZw}) yields \bea &&(1\otimes Z_+ + Z_+\otimes 1)\left( \sum_{k,l\geq 0} \sum_{n\geq 0} a^{m_1,m_2}_{k,l} (-h^2/4)^{n} v^{j_1}_{m_1+k+n}\otimes v^{j_2}_{m_2+l+n} \right) \nn\\ &&=(1\otimes Z_+ + Z_+\otimes 1)\left\{ \sum_{k,l\geq 0} \left(\sum_{n\geq 0} a^{m_1,m_2}_{k-n,l-n} (-h^2/4)^{n}\right) v^{j_1}_{m_1+k}\otimes v^{j_2}_{m_2+l} \right\}. \label{tempZw} \eea {}From the definition (\ref{defa}) of the coefficients $a^{m_1,m_2}_{k,l}$ it follows that \bea && \sum_{n\geq 0} a^{m_1,m_2}_{k-n,l-n} (-h^2/4)^{n} = \nn\\ &&(-1)^k (h/2)^{k+l} \sum_{n\geq 0} (b^{m_1,m_2}_{k-n,l-n} - b^{m_1,m_2}_{k-n-1,l-n-1} ) = (-1)^k (h/2)^{k+l} b^{m_1,m_2}_{k,l}. \eea Putting this back in (\ref{tempZw}) gives \bea &&\sum_{k,l\geq 0} (-1)^k(h/2)^{k+l} b^{m_1,m_2}_{k,l}\; v^{j_1}_{m_1+k} \otimes v^{j_2}_{m_2+l+1} + \nn\\ &&\qquad \sum_{k,l\geq 0} (-1)^k(h/2)^{k+l} b^{m_1,m_2}_{k,l} \; v^{j_1}_{m_1+k+1} \otimes v^{j_2}_{m_2+l} \nn\\ &&=\sum_{k,l\geq 0} (-1)^k(h/2)^{k+l-1} (b^{m_1,m_2}_{k,l-1} - b^{m_1,m_2}_{k-1,l}) \; v^{j_1}_{m_1+k} \otimes v^{j_2}_{m_2+l}. \eea On the other hand, the rhs of (\ref{DelZw}) leads to \bea &&\sum_{k,l\geq 0} (a^{m_1,m_2+1}_{k,l-1} - a^{m_1+1,m_2}_{k-1,l})\; v^{j_1}_{m_1+k} \otimes v^{j_2}_{m_2+l} = \sum_{k,l\geq 0} (-1)^k (h/2)^{k+l-1} \bigl(b^{m_1,m_2+1}_{k,l-1} - \nn\\ && b^{m_1,m_2+1}_{k-1,l-2}- b^{m_1+1,m_2}_{k-1,l}+ b^{m_1+1,m_2}_{k-2,l-1} \bigr) v^{j_1}_{m_1+k} \otimes v^{j_2}_{m_2+l}. \eea So, it remains to show that \beq b^{m_1,m_2}_{k,l-1} - b^{m_1,m_2}_{k-1,l} = b^{m_1,m_2+1}_{k,l-1} - b^{m_1,m_2+1}_{k-1,l-2}- b^{m_1+1,m_2}_{k-1,l}+ b^{m_1+1,m_2}_{k-2,l-1}, \eeq and this follows trivially from the definition (\ref{defb}) of the $b$-coefficients. \begin{prop} The action of $\De(Z_-)$ on the auxiliary basis vectors is given by \bea \De(Z_-) w^{j_1,j_2}_{m_1,m_2} &=& (j_1+m_1)(j_1-m_1+1)\; w^{j_1,j_2}_{m_1-1,m_2} + \nn\\ && \qquad (j_2+m_2)(j_2-m_2+1)\; w^{j_1,j_2}_{m_1,m_2-1}. \label{DelZ-w} \eea \label{propZm} \end{prop} The proof of this property is rather long and technical. Let $h$, $z_\pm$ be the standard basis of $sl(2)$, with \beq [h,z_\pm]=\pm 2 z_\pm,\qquad [z_+,z_-]=h; \eeq $sl(2)$ has an action on the basis elements $v^j_m$, given by the same expressions as in (\ref{sl2vaction}). The standard comultiplication for $sl(2)$, $\de(x)=x \otimes 1 + 1 \otimes x$ for every $x\in sl(2)$, induces an action on elements of $V^{(j_1)}\otimes V^{(j_2)}$~: \bea &&\de(h) v^{j_1}_{m_1}\otimes v^{j_2}_{m_2} = 2(m_1+m_2)\; v^{j_1}_{m_1}\otimes v^{j_2}_{m_2} ,\nn\\ &&\de(z_+) v^{j_1}_{m_1}\otimes v^{j_2}_{m_2} = v^{j_1}_{m_1+1}\otimes v^{j_2}_{m_2} + v^{j_1}_{m_1}\otimes v^{j_2}_{m_2+1}, \\ &&\de(z_-) v^{j_1}_{m_1}\otimes v^{j_2}_{m_2} = (j_1+m_1)(j_1-m_1+1)\; v^{j_1}_{m_1+1}\otimes v^{j_2}_{m_2} + \nn\\ &&\qquad\qquad\qquad (j_2+m_2)(j_2-m_2+1)\; v^{j_1}_{m_1}\otimes v^{j_2}_{m_2+1}. \nn \eea Thus the action of $\de(h)$ and $\de(z_+)$ on the basis $v^{j_1}_{m_1}\otimes v^{j_2}_{m_2}$ is the same as the action of $\De(H)$ and $\De(Z_+)$ on the auxiliary basis $w^{j_1,j_2}_{m_1,m_2}$, by Propositions~\ref{propH} and~\ref{propZp}. We shall show that this holds for $Z_-$ too. First of all, for $sl(2)$ we know how the tensor product decomposes, so let \beq v^{(j_1j_2)j}_m = \sum_{m_1+m_2=m} c^{j_1,j_2,j}_{m_1,m_2,m}\; v^{j_1}_{m_1}\otimes v^{j_2}_{m_2} , \label{vvv} \eeq where $j\in J=\{j_1+j_2,j_1+j_2-1,\ldots, |j_1-j_2|\}$, and $c^{j_1,j_2,j}_{m_1,m_2,m}$ are the Clebsch-Gordan coefficients in the $v$-basis, related to the usual $su(2)$ Clebsch-Gordan coefficients by \beq c^{j_1,j_2,j}_{m_1,m_2,m}= {\al_{j,m}\over \al_{j_1,m_1}\al_{j_2,m_2}} C^{j_1,j_2,j}_{m_1,m_2,m}, \eeq see (\ref{ve}). Then there holds \bea &&\de(h) v^{(j_1j_2)j}_m = 2m \;v^{(j_1j_2)j}_m ,\nn\\ &&\de(z_+) v^{(j_1j_2)j}_m = v^{(j_1j_2)j}_{m +1}, \\ &&\de(z_-) v^{(j_1j_2)j}_m = (j+m)(j-m+1) \; v^{(j_1j_2)j}_{m-1}. \nn \eea We also define \beq w^{(j_1j_2)j}_m = \sum_{m_1+m_2=m} c^{j_1,j_2,j}_{m_1,m_2,m} \; w^{j_1,j_2}_{m_1,m_2}, \label{ww} \eeq and by the remark that the action of $\de(h)$ and $\de(z_+)$ on the basis $v^{j_1}_{m_1}\otimes v^{j_2}_{m_2}$ is the same as the action of $\De(H)$ and $\De(Z_+)$ on the auxiliary basis $w^{j_1,j_2}_{m_1,m_2}$, it follows that \bea &&\de(H) w^{(j_1j_2)j}_m = 2m \; w^{(j_1j_2)j}_m ,\nn\\ &&\de(Z_+) w^{(j_1j_2)j}_m = w^{(j_1j_2)j}_{m +1}. \eea It remains to find the action of $\De(Z_-)$ on $w^{(j_1j_2)j}_m$; let us denote it by \beq \De(Z_-) w^{(j_1j_2)j}_m = \sum_{j'} \sum_{m'} \mu^{j,j'}_{m,m'} \; w^{(j_1j_2)j'}_{m'}, \eeq with $\mu^{j,j'}_{m,m'}$ the coefficients to be determined. By acting with the relation \beq \De(H)\De(Z_-)-\De(Z_-)\De(H)= -2\De(Z_-) \eeq on $w^{(j_1j_2)j}_m$, it follows that the coefficients $\mu^{j,j'}_{m,m'}$ are zero unless $m'=m-1$. Write $\nu^{j,j'}_m$ for $\mu^{j,j'}_{m,m-1}$; then we have so far \beq \De(Z_-) w^{(j_1j_2)j}_m = \sum_{j'} \nu^{j,j'}_{m} \; w^{(j_1j_2)j'}_{m-1}, \eeq where $j'\in J$ such that $j'\geq |m-1|$. Next, we shall use the relation \beq \De(Z_+)\De(Z_-) - \De(Z_-)\De(Z_+) = \De(H). \label{rel} \eeq Acting with (\ref{rel}) on $w^{(j_1j_2)j}_m$ yields~: \bea \hbox{for } j\ne j'&:& \nu^{j,j'}_m = \nu^{j,j'}_{m+1}, \label{nueq1}\\ \hbox{for }j'=j & : & \nu^{j,j}_m - \nu^{j,j}_{m+1} = 2m. \label{nueq2} \eea In particular, by acting with (\ref{rel}) on $w^{(j_1j_2)j}_j$, one finds that $\nu^{j,j}_j=2j$, and that $\nu^{j,j'}_j=0$ for $j'>j$. Now (\ref{nueq2}) is a difference equation in $m$ with boundary condition $\nu^{j,j}_j=2j$, so it has a unique solution given by \beq \nu^{j,j}_m = (j+m)(j-m+1). \eeq {}From (\ref{nueq1}) and $\nu^{j,j'}_j=0$ for $j'>j$ it follows that $\nu^{j,j'}_m=0$ for all $j'>j$. Using this, and acting with (\ref{rel}) on $w^{(j_1j_2)j}_{j-1}$ implies that $\nu^{j,j-1}_{j-1}=0$, thus by (\ref{nueq1}) that $\nu^{j,j-1}_{m}=0$ for all $m$. Similarly, acting with (\ref{rel}) on $w^{(j_1j_2)j}_{j-2}$ implies that $\nu^{j,j-2}_{j-2}=0$, thus by (\ref{nueq1}) that $\nu^{j,j-2}_{m}=0$ for all $m$. One can continue and thus show by induction that $\nu^{j,j'}_m=0$ also for all $j'0$, then one can see that at least one of the indices of $A$ in (\ref{cgc0}) is negative, thus in this case ${\cal C}^{j_1,j_2,j}_{n_1,n_2,n_1+n_2+p}(h)=0$. Finally, suppose that $p<0$. Now there can be a number of contributions in (\ref{cgc0}), and by (\ref{defA}) and (\ref{defa}) they all have the same power of $h$, namely $h^{-p}$. Thus we have the following property~: \begin{prop} The Clebsch-Gordan coefficients for $\U$ satisfy \begin{itemize} \item if $m=n_1+n_2$ then ${\cal C}^{j_1,j_2,j}_{n_1,n_2,m}(h) = C^{j_1,j_2,j}_{n_1,n_2,m}$ ; \item if $m>n_1+n_2$ then ${\cal C}^{j_1,j_2,j}_{n_1,n_2,m}(h) =0$; \item if $m