% Latex file
% On the principal subalgebra of quantum enveloping algebras
% ${\rm gl}_q(l+1)$
% J. Van der Jeugt
% J. Phys. A: Math. Gen. 25 (1992), L213-L218.
%
\documentstyle[12pt]{article}
%
\setlength{\topmargin}{0cm}
\setlength{\oddsidemargin}{0cm}
\setlength{\evensidemargin}{0cm}
\setlength{\textheight}{22.5cm}
\setlength{\textwidth}{16cm}
%
\def\beq{\begin{equation}}
\def\eeq{\end{equation}}
\def\nn{\nonumber}
%
\begin{document}
%\addtolength{\baselineskip}{3mm}
%
\title{On the principal subalgebra of quantum enveloping algebras
${\rm gl}_q(l+1)$}
\author{
J.\ Van der Jeugt\thanks{Research Associate of the NFWO
(National Fund for Scientific Research of Belgium)} \\
{\normalsize Laboratorium voor Numerieke Wiskunde en Informatica,}\\
{\normalsize Universiteit Gent, Krijgslaan 281-S9, B9000 Gent,
Belgium}}
%
\maketitle
%
\vskip 1.5cm
\noindent Classification numbers~: 02.20, 03.65
\vskip 2cm
\begin{abstract}
The existence of a principal subalgebra of type ${\rm sl}_q(2)$ for
quantum enveloping algebras ${\rm gl}_q(l+1)$ or ${\rm sl}_q(l+1)$ is
investigated.
Surprisingly, only when $l=2$ and when all relations are restricted
to symmetric representations such a principal subalgebra happens to exist.
This case, ${\rm sl}_q(3)\supset {\rm sl}_q(2)$, is the $q$-deformation of the
classical ${\rm su}(3)\supset {\rm so}(3)$ embedding for symmetric
${\rm su}(3)$ representations, and is analysed in more detail, giving
a connection with $q$-deformed spherical harmonics.
\end{abstract}

\newpage

\section{Introduction}

Principal three-dimensional subalgebras for simple Lie algebras were
introduced by Dynkin (1957) and Kostant (1959).
They have many important applications im mathematics, being related to
the exponents of simple Lie groups (Kostant 1959) and to various
combinatorial results obtained by Hughes (1977) and later generalised
by Stanley (1980).
In various physical models, the principal three-dimensional subalgebra plays
a crucial role, since it is usually the subalgebra describing the angular
momentum of the system (Hamermesh 1962).
As examples, we mention here~: Elliott's model ${\rm SU}(3)\supset
{\rm SO}(3)$ (Elliott 1958);
quadrupole vibrations of the nucleus (Bohr 1952, Chac\'on {\sl et al} 1976)
or octupole vibrations in which the chains ${\rm U}(5)\supset{\rm O}(5)
\supset{\rm O}(3)$ or ${\rm U}(7)\supset{\rm O}(7)\supset G_2
\supset{\rm O}(3)$ appear (these appear also in atomic spectroscopy
(Judd 1963));
the interacting boson model (Arima and Iachello 1976) has dynamical
symmetries in which ${\rm U}(3)\supset{\rm O}(3)$  and ${\rm U}(5)
\supset{\rm O}(5)\supset{\rm O}(3)$ appear.

Let $G_l$ be a simple Lie algebra of rank $l$, with Chevalley
generators $\{e_i,f_i,h_i | i=1,2,\ldots l\}$. A principal
three-dimensional subalgebra of $G_l$ is a subalgebra $A$
of type ${\rm sl}(2)$, with basis $\{E,F,H\}$ satisfying
\beq
[H,E]=2E,\qquad [H,F]=-2F,\qquad [E,F]=H,
\eeq
such that the number of irreducible components occurring in the
complete reduction of the adjoint representation of $G_l$ with
respect to $A$ is equal to $l$ (Kostand 1959).
The Lie algebra $G_l$ has an involutive antiautomorphism $\sigma$
defined by $\sigma(h_i)=h_i$, $\sigma(e_i)=f_i$ and $\sigma(f_i)=e_i$,
which is related to Hermitian conjugation; if the principal
subalgebra $A$ is required to be invariant under $\sigma$,
i.e.~$\sigma(H)=H$, $\sigma(E)=F$ and $\sigma(F)=E$, then the
elements of $A$ have a unique expression in terms of the
generators $\{e_i,f_i,h_i\}$. For $G_l=A_l={\rm sl}(l+1)$,
one obtains
\begin{eqnarray}
H & = & \sum_{i=1}^l i(l+1-i)h_i, \nn \\
E & = & \sum_{i=1}^l \sqrt{i(l+1-i)} e_i, \label{2} \\
F & = & \sum_{i=1}^l \sqrt{i(l+1-i)} f_i. \nn
\end{eqnarray}
For ${\rm gl}(l+1)$, the Cartan subalgebra contains $l+1$ basis
elements $N_0,N_1,\ldots,N_l$, which are related to the $l$ basis
elements $h_i$ of ${\rm sl}(l+1)$ by $h_i=N_{i-1}-N_i$ ($i=1,2,\ldots,l)$.
Thus the principal subalgebra of ${\rm gl}(l+1)$ has the same form
as~(\ref{2}), except that the diagonal element becomes
\beq
H = \sum_{i=0}^l (l-2i) N_i.
\label{3}
\eeq

Quantum enveloping algebras are certain $q$-deformations of
enveloping algebras of simple Lie algebras, being at the center of
much attention recently (e.g.~Doebner and Hennig 1990).
So far, however, very little work has been done in studying
non-trivial subalgebras of quantum enveloping algebras (see
e.g.~Dobrev 1990).
In this Letter the investigation of principal subalgebras of
quantum enveloping algebras of type ${\rm gl}_q(l+1)$ or
${\rm sl}_q(l+1)$ is initiated. The algebra ${\rm gl}_q(l+1)$ is
the associative algebra spanned by generators $e_i, f_i$ ($i=1,2,\ldots,l$)
and $N_i$ ($i=0,1,\ldots,l$) subject to the relations (Jimbo 1986)
\begin{eqnarray}
&&[N-i,N-j]=0, \nn\\
&&[N_i,e_j]=(\delta_{i,j-1}-\delta_{ij})e_j,\quad
  [N_i,f_j]=-(\delta_{i,j-1}-\delta_{ij})f_j, \nn\\
&&[e_i,f_j]=\delta_{ij} [N_{i-1}-N_i] ,  \label{4} \\
&& \sum_{k=0}^{1+\delta_{i-1,j}+\delta_{i,j-1}}
 (-1)^k \left[{1+\delta_{i-1,j}+\delta_{i,j-1} \atop k} \right]
 e_i^{1+\delta_{i-1,j}+\delta_{i,j-1}-k} e_j e_i^k = 0 \quad (i\not=j),\nn\\
&& \sum_{k=0}^{1+\delta_{i-1,j}+\delta_{i,j-1}}
 (-1)^k \left[{1+\delta_{i-1,j}+\delta_{i,j-1} \atop k} \right]
 f_i^{1+\delta_{i-1,j}+\delta_{i,j-1}-k} f_j f_i^k = 0 \quad (i\not=j),\nn
\end{eqnarray}
where $[x]={q^x-q^{-x} \over q-q^{-1} }$,
$\left[ {x\over y}\right]= {[x]! \over [y]![x-y]!}$, and
$[x]!=[x][x-1]\cdots[1]$.
In the limit $q\rightarrow 1$, this reduces to the universal enveloping
algebra of ${\rm gl}(l+1)$.

A principal subalgebra of ${\rm gl}_q(l+1)$ is defined as follows~:
it is a subalgebra of ${\rm gl}_q(l+1)$ of type ${\rm sl}_q(2)$,
i.e.~its generators $\{E,F,H\}$ satisfy
\beq
[H,E]=2E,\qquad [H,F]=-2F,\qquad [E,F]=[H],
\label{5}
\eeq
and in the limit $q\rightarrow q$, this ${\rm sl}_q(2)$ subalgebra
reduces to the principal subalgebra of ${\rm gl}(l+1)$.

In this Letter it is proved that principal subalgebras of ${\rm gl}_q(l+1)$
do not exist in general. It is shown, however, that when all relations
are restricted to the totally symmetric representations of
${\rm gl}_q(l+1)$, a principal subalgebra does exist in the case $l=2$
but not for other $l$-values.
Some further aspects of ${\rm gl}_q(3)\supset {\rm sl}_q(2)$,
which is the $q$-deformation of ${\rm su}(3)\supset {\rm so}(3)$,
are discussed. A remarkable relation between $q$-numbers, eq.~(\ref{20}),
is obtained as a byproduct.

\section{A general principal subalgebra?}

Since $H$ is an ordinary diagonal operator in~(\ref{5}), and because
of the limiting case~(\ref{2}) or~(\ref{3}), it follows that the most
general form of a principal subalgebra is~:
\beq
H=\sum_{i=0}^l(l-2i)N_i,\quad
E=\sum_{i=1}^l \mu_i(N_0,\ldots,N_l)e_i,\quad
F=\sum_{i=1}^l f_i\mu_i(N_0,\ldots,N_l),
\label{6}
\eeq
where $\mu_i$ are ($q$-dependent) functions of $N_0,\ldots,N_l$,
and invariance uncer $\sigma$ has been assumed.
The first two relations in~(\ref{5}) are satisfied by~(\ref{6});
the crucial relation to be satisfied is the third relation of~(\ref{5}).
Let us concentrate for a moment on the case $l=2$, and rewrite~(\ref{6})
in the form
\begin{eqnarray}
H&=&2N_0-2H_2 , \nn\\
E&=& \alpha(N_0 N_1 N_2)e_1 + e_2 \beta(N_0 N_1 N_2), \label{7} \\
F&=& f_1 \alpha(N_0 N_1 N_2) + \beta(N_0 N_1 N_2) f_2. \nn
\end{eqnarray}
When calculating $[E,F]$, the terms in $e_1f_2$ and $e_2f_1$ must vanish,
leading to the following condition on the functions $\alpha$ and $\beta$~:
\beq
\alpha(N_0 N_1 N_2)\beta(N_0-1,N_1+1,N_2) =
\alpha(N_0,N_1+1,N_2-1) \beta(N_0 N_1 N_2) .
\label{8}
\eeq
Then, there comes
\begin{eqnarray}
\lefteqn{[E,F] =
  \alpha^2(N_0 N_1 N_2)e_1f_1 - \alpha^2(N_0+1,N_1-1,N_2)f_1e_1 } \nn \\
& & +\beta^2(N_0,N_1-1,N_2+1)e_2f_2 -\beta^2(N_0 N_1 N_2)f_2e_2.
\label{9}
\end{eqnarray}
The right hand side of~(\ref{9}) should be expressible in terms of
$N_0$, $N_1$ and $N_2$ only. It follows from~(\ref{4}) that this is the case
only when the coefficients of $e_if_i$ and $f_ie_i$ are equal, i.e.
\beq
\begin{array}{l}
\alpha(N_0+1, N_1-1, N_2) = \alpha(N_0 N_1 N_2), \\
\beta(N_0, N_1-1, N_2+1) = \beta(N_0 N_1 N_2).
\end{array}
\label{10}
\eeq
Then~(\ref{9}) becomes
\beq
[E,F]=\alpha^2(N_0 N_1 N_2)[N_0-N_1] + \beta^2(N_0 N_1 N_2) [N_1-N_2],
\label{11}
\eeq
such that the condition $[E,F]=[H]$ reduces to
\beq
\alpha^2(n_0 n_1 n_2)[n_0-n_1]+\beta^2(n_0 n_1 n_2) [n_1-n_2]
= [2n_0 - 2n_2],
\label{12}
\eeq
where we have substituted ordinary variables $n_i$ for the operators
$N_i$, which is allowed since the $N_i$ are commuting operators.
Replacing in~(\ref{12}) $n_0$ by $n_0+1$ and $n_1$ by $n_1-1$, and
using~(\ref{10}), leads to
\[
\alpha^2(n_0 n_1 n_2)[n_0-n_1+2]+\beta^2(n_0+1, n_1-1, n_2) [n_1-n_2-1]
= [2n_0 - 2n_2+2],
\]
or by iteration
\beq
\alpha^2(n_0 n_1 n_2)[n_0-n_1+2k]+\beta^2(n_0+k, n_1-k, n_2) [n_1-n_2-k]
= [2n_0 - 2n_2+2k].
\label{13}
\eeq
Putting $k=n_1-n_2$ leads to
\beq
\alpha^2(n_0 n_1 n_2) = q^{n_0+n_1-2n_2}+q^{-n_0-n_1+2n_2},
\eeq
which satisfies indeed the first equation of~(\ref{10}). Using this
in~(\ref{12}) then implies that
\beq
\beta^2(n_0 n_1 n_2) = q^{n_1-n_2} + q^{-n_1+n_2}.
\eeq
In conjunction with the second equation of~(\ref{10}), this leads to
a contradiction unless $q=1$. Thus we have shown that a general
principal subalgebra does not exist for ${\rm gl}_q(3)$.
For $l>2$, the conclusion is the same, but we shall not present a
detailed treatment here.

\section{The symmetric case for ${\rm gl}_q(3)$}

Although a general principal subalgebra does not exist for ${\rm gl}_q(3)$,
we shall show in this section that it does exist when all
relations are restricted to the completely symmetric representations
of ${\rm gl}_q(3)$.
These representations are labelled by $[N,0,0]$ ($N\in\{0,1,2,\ldots\}$),
and the basis vectors are of the form $|n_0n_1n_2\rangle$ with
$n_0+n_1+n_2=N$. The action of the ${\rm gl}_q(3)$ generators is
given by (Jimbo 1986)~:
\begin{eqnarray}
N_i|n_0n_1n_2\rangle & = & n_i|n_0n_1n_2\rangle, \nn\\
e_1|n_0n_1n_2\rangle & = & \sqrt{[n_0+1][n_1]}|n_0+1,n_1-1,n_2\rangle, \nn\\
e_2|n_0n_1n_2\rangle & = & \sqrt{[n_1+1][n_2]}|n_0,n_1+1,n_2-1\rangle,
 \label{16} \\
f_1|n_0n_1n_2\rangle & = & \sqrt{[n_0][n_1+1]}|n_0-1,n_1+1,n_2\rangle,  \nn \\
f_2|n_0n_1n_2\rangle & = & \sqrt{[n_1][n_2+1]}|n_0,n_1-1,n_2+1\rangle. \nn
\end{eqnarray}

The starting point is again~(\ref{7}), and the purpose is to find
functions $\alpha$ and $\beta$ such that~(\ref{5}) is satisfied on
basis vectors of the form $|n_0n_1n_2\rangle$. The condition
$[E,F]=[H]$ leads to the following two equations in $\alpha$
and $\beta$~:
\beq
\alpha(n_0 n_1 n_2)\beta(n_0-1,n_1+1,n_2) =
\alpha(n_0, n_1+1,n_2-1)\beta(n_0 n_1 n_2),
\label{17}
\eeq
\begin{eqnarray}
\lefteqn{\alpha^2(n_0 n_1 n_2)[n_0][n_1+1] - \alpha^2(n_0+1,n_1-1,n_2)
 [n_0+1][n_1] } \nn \\
& & +\beta^2(n_0,n_1-1,n_2+1)[n_1][n_2+1] -
 \beta^2(n_0 n_1 n_2)[n_1+1][n_2]  = [2n_0-2n_2] .
 \label{18}
\end{eqnarray}
It is remarkable that this set of equations in two unknown
functions $\alpha$ and $\beta$ has a solution, namely
\beq
\begin{array}{l}
\alpha(n_0n_1n_2) = q^{n_2- {1\over 2}n_1} \sqrt{q^{n_0}+q^{-n_0}} , \\
\beta(n_0n_1n_2) = q^{n_0- {1\over 2}n_1} \sqrt{q^{n_2}+q^{-n_2}} .
\end{array}
\label{19}
\eeq
The verification of~(\ref{18}) depends upon a rather intriguing
identity for $q$-numbers~:
\begin{eqnarray}
\lefteqn{ q^{2z-y}[2x][y+1] - q^{2z-y+1}[2x+2][y] } \nn \\
& & + q^{2x-y+1}[y][2z+2] - q^{2x-y}[y+1][2z]  = [2x-2z] .
 \label{20}
\end{eqnarray}
Thus we have shown that ${\rm gl}_q(3)$ contains three elements
\beq
\begin{array}{l}
H=2N_0-2N_2,\\
E=q^{N_2-{1\over 2}N_1} \sqrt{q^{N_0}+q^{-N_0}} e_1 +
  e_2 q^{N_0-{1\over 2}N_1} \sqrt{q^{N_2}+q^{-N_2}} , \\
F=f_1 q^{N_2-{1\over 2}N_1} \sqrt{q^{N_0}+q^{-N_0}}  +
   q^{N_0-{1\over 2}N_1} \sqrt{q^{N_2}+q^{-N_2}} f_2 ,
\end{array}
\label{21}
\eeq
which satisfy the relations~(\ref{5}) of an ${\rm sl}_q(2)$ algebra
when acting upon symmetric representations of ${\rm gl}_q(3)$,
and which tend to the principal ${\rm sl}(2)$ subalgebra of ${\rm gl}(3)$
in the limit $q\rightarrow 1$.

\section{The symmetric case for ${\rm gl}_q(l+1)$}

It is easy to verify that~(\ref{19}) is a solution for the set
(\ref{17})--(\ref{18}), but the reader may wonder how the solution~(\ref{19})
was obtained and if it is unique. The way we solved (\ref{17})--(\ref{18})
is as follows. For a chosen $N$-value, (\ref{17}) and (\ref{18}) are
written down for all vectors $|n_0 n_1 n_2\rangle$ with
$n_0+n_1+n_2=N$. This gives rise to a large system of non-linear
equations in a number of ordinary variables.
For example, when $N=1$ there remain three equations in two unknowns
$\alpha(1 0 0)$ and $\beta(0 0 1)$.
For every $N$, one can try to solve the system.
It turns out that for some small values of $N$ the solution is not
always unique, but as $N$ increases a unique solution emerges,
leading to~(\ref{19}).

This technique can be applied to ${\rm gl}_q(l+1)$ with $l>2$.
Explicitly, we looked at symmetric representations $[N,0,0]$ with
basis states $|n_0n_1n_2n_3\rangle$ of ${\rm gl}_q(4)$, with
$E$ and $F$ given by~(\ref{6}) in terms of three unknown functions
$\mu_1$, $\mu_2$ and $\mu_3$ (rather than two unknown functions
$\alpha$ and $\beta$ in the case of ${\rm gl}_q(3)$).
The condition $[E,F]=[H]$ gives rise to equations similar to
(\ref{17}) and (\ref{18}). When trying to solve these, a solution
was obtained for $N=1$ and $N=2$. However, for $N=3$, leading to a
system of 16 non-linear equations in 16 variables, we were able to
show that this system is inconsistent unless $q=1$ (this involved
the help of MACSYMA).
So ${\rm gl}_q(4)$ does not contain a principal subalgebra, even when
all relations are restricted to symmetric representations only.
For ${\rm gl}_q(l+1)$ with $l>3$ we have not performed any explicit
calculations, but the ${\rm gl}_q(4)$ case seems to indicate that none
of these algebras contains a principal subalgebra for the symmetric
representations.

\section{The $q$-deformation of ${\rm u}_q(3)\supset {\rm so}_q(3)$}

We continue here with the realisation given in Section~3.
In order to emphasize that we are dealing with a $q$-generalisation
of angular momentum, the states $|n_0 n_1 n_2\rangle$ shall now be denoted
by $|n_{+1} n_0 n_{-1}\rangle$, the index referring to angular
momentum projection.

For such vectors belonging to the totally symmetric representations
$[N,0,0]$ of ${\rm gl}_q(3)$ or ${\rm u}_q(3)$, there exists a
realisation in terms of $q$-boson operators (Macfarlane 1989,
Biedenharn 1989). Thus we assume there are three number
operators $N_{+1}$, $N_0$, $N_{-1}$ and three independent
$q$-boson operators $b_i$ and $b_i^+$ ($i=+1,0,-1$) satisfying
\beq
[N_i,b^+_i]=b^+_i,\qquad
[N_i,b_i]=-b_i,\qquad
b_ib^+_i-q^{-1}b^+_ib_i=q^{N_i}.
\label{22}
\eeq
The basis states are then of the form
\beq
|n_{+1}n_0 n_{-1}\rangle =
{ (b_{+1}^+)^{n_{+1}} (b_0^+)^{n_0} (b_{-1}^+)^{n_{-1}}
\over \sqrt{[n_{+1}]![n_0]![n_{-1}]!}  } |0\rangle ,
\label{23}
\eeq
with $b_i|0\rangle=0$ and $N_i|n_{+1}n_0 n_{-1}\rangle =
n_i|n_{+1}n_0 n_{-1}\rangle$. The principal subalgebra, here denoted
by ${\rm so}_q(3)$, follows from~(\ref{21})~:
\beq
\begin{array}{l}
L_0 = N_{+1}-N_{-1}, \\
L_{+1}=q^{N_{-1}-{1\over 2}N_0} \sqrt{q^{N_{+1}}+q^{-N_{+1}}} b^+_{+1}b_0 +
  b^+_0b_{-1} q^{N_{+1}-{1\over 2}N_0} \sqrt{q^{N_{-1}}+q^{-N_{-1}}} , \\
L_{-1}=b^+_0b_{+1} q^{N_{-1}-{1\over 2}N_0} \sqrt{q^{N_{+1}}+q^{-N_{+1}}}  +
   q^{N_{+1}-{1\over 2}N_0} \sqrt{q^{N_{-1}}+q^{-N_{-1}}} b^+_{-1}b_0 ;
\end{array}
\label{24}
\eeq
compared with~(\ref{21}), we have chosen a different factor for the diagonal
element, in order to realise the $q$-relations which are more familiar
to physicists~:
\beq
[L_0,L_{\pm 1}]=\pm L_{\pm 1}, \qquad [L_{+1},L_{-1}]=[2L_0].
\label{25}
\eeq
It is obvious that (\ref{24}) is the $q$-generalisation of the ${\rm so}(3)$
subalgebra of ${\rm u}(3)$.

In the classical case of ${\rm u}(3)\supset {\rm so}(3)$, the symmetric
representations $[N,0,0]$ decompose into ${\rm so}(3)$ representations $(L)$
with $L=N,N-2,\ldots,1\hbox{ or }0$. In the $q$-generalised case, this
decomposition is exactly the same. In fact, we have calculated the matrix
elements relating the ${\rm so}_q(3)$ basis to the $q$-boson basis~(\ref{23}).
For this purpose, the following operator can be introduced~:
\beq
s=(b^+_0)^2 q^{N_{+1}+N_{-1}+1} -
\sqrt{ {[2N_{+1}]\over [N_{+1}]} {[2N_{-1}]\over [N_{-1}]} }
b^+_{+1} b^+_{-1} q^{-N_0-1/2} .
\label{26}
\eeq
It can be verified that $s$ is an ${\rm so}_q(3)$ scalar, i.e.\
$[L_i,s]=0$ ($i=+1,0,-1$). Then we have, in terms of the states~(\ref{23}),
that~:
\begin{eqnarray}
v(N,L,M)&=& q^{-{(L+M)(L+M-1)\over 4}}
 \left\{ {[N+L]!![2L+1]\over [N-L]!![N+L+1]!} [L+M]![L-M]! \right\}^{1/2}
 \nn \\
&\times& \sum_x q^{(2L-1){x\over 2}} s^{N-L\over 2}
{|x,L+M-2x,x-M\rangle \over
\sqrt{[2x]!![L+M-2x]![2x-2M]!!} },
\label{27}
\end{eqnarray}
where $x$ runs from $\max(0,M)$ to $\lfloor {L+M\over 2} \rfloor$
in steps of one, $L=N,N-2,\ldots,1\hbox{ or } 0$, and
$M=-L,-L+1,\ldots,+L$. As usual, the symbol $[2t]!!$ stands for
$[2t][2t-2]\cdots[2]$. The vectors~(\ref{27}) are genuine
orthonormal $q$-generalised angular momentum states~:
\beq
\begin{array}{l}
L_0 v(N,L,M) = M v(N,L,M) , \\
L_{\pm 1} v(N,L,M) = \sqrt{[L\mp M][l\pm M+1]} v(N,L,M\pm 1).
\end{array}
\eeq
The classical analogon of~(\ref{27}) was given by Sharp {\em et al} (1969)
and by Moshinsky {\em et al} (1975). There, the ${\rm so}(3)$ states
are in an obvious way related to spherical harmonics. In a following
paper, we intend to give more details on the derivation of~(\ref{27})
and to relate the states $v(N,L,M)$ to functions which can be seen
as $q$-generalised spherical harmonics.

\section*{References}
\begin{description}
\item[]
Arima A and Iachello F (1976) {\sl Ann.\ Phys.} {\bf 99} 253
\item[]
Biedenharn L C (1989) {\sl J.\ Phys.\ A~: Math.\ Gen.} {\bf 22} L873
Bohr A (1952) {\sl Kgl.\ Dan.\ Videnskab.\ Selsk.\ Mat.\ Fys.\ Medd.}
 {\bf 26} 14
\item[]
Chac\'on E, Moshinsky M and Sharp R T (1976) {J.\ Math.\ Phys.}
 {\bf 17} 668
\item[]
Dobrev V K (1990) in {\sl Lecture Notes in Physics} {\bf 370} 107
\item[]
Doebner H-D and Hennig J-D (1990) ``Quantum Groups'' {\sl Lecture
 Notes in Physics} {\bf 370} (Springer-Verlag~: Berlin)
\item[]
Dynkin E B (1957) {\sl Am.\ Math.\ Soc.\ Transl., Ser.~2} {\bf 6} 111
\item[]
Elliott J P (1958) {\sl Proc.\ Roy.\ Soc.\ Ser.~A} {\bf 245} 128, 562
\item[]
Hamermesh M (1962) {\sl Group Theory and its Applications to Physical
 Problems} (Addison-Wesley~: London)
\item[]
Hughes J W B (1977) in {\sl Number Theory and Algebra} ed H Zassenhaus
 (Academic~: New York) 135
\item[]
Jimbo M (1986) in ``Field Theory, Quantum Gravity and String'' {\sl
 Lecture Notes in Physics} {\bf 246} eds H J de Vega and N S\'anchez
 (Springer-Verlag~: Berlin) 335
\item[]
Judd B R (1963) {\sl Operator techniques in atomic spectroscopy}
 (McGraw-Hill~: New York)
\item[]
Kostant B (1959) {\sl Am.\ J.\ Math.} {\bf 81} 973
\item[]
Macfarlane A J (1989) {\sl J.\ Phys.\ A~: Math.\ Gen.} {\bf 22} 4581
\item[]
Moshinsky M, Patera J, Sharp R T and Winternitz P (1975) {\sl Ann.\
 Phys.} {\bf 95} 139
\item[]
Sharp R T, von Baeyer H C and Pieper S C (1969) {\sl Nucl.\ Phys.\ A}
 {\bf 127} 513
\item[]
Stanley R P (1980) in Combinatorics, representation theory and statistical
 methods in groups,'' {\sl Lecture Notes in Pure and Applied Mathematics}
 vol {\bf 57} eds T V Narayana, R M Mathsen and J G Williams (Marcel
 Dekker~: New York) 127
\end{description}
\end{document}
