% Latex file % Quantum algebra embeddings~: deforming functionals and % algebraic approach % J. Van der Jeugt % Canad. J. Phys. 72 (1994), 519-526. % \documentclass[12pt]{article} \usepackage{latexsym,amssymb} % \setlength{\topmargin}{0cm} \setlength{\oddsidemargin}{0cm} \setlength{\evensidemargin}{0cm} \setlength{\textheight}{22.5cm} \setlength{\textwidth}{16cm} \setlength{\headsep}{0cm} % \def\d{\dagger} % % % The Greek symbols defined by the first two letters of their name % \def\al{\alpha} \def\be{\beta} \def\ga{\gamma} \def\de{\delta} \def\ep{\epsilon} \def\vep{\varepsilon} \def\ze{\zeta} \def\et{\eta} \def\th{\theta} \def\io{\iota} \def\ka{\kappa} \def\la{\lambda} \def\rh{\rho} \def\si{\sigma} \def\ta{\tau} \def\ph{\phi} \def\ch{\raise 2pt\hbox{$\chi$}} % raise this a bit \def\ps{\psi} \def\om{\omega} \def\Ga{\Gamma} \def\De{\Delta} \def\Th{\Theta} \def\La{\Lambda} \def\Si{\Sigma} \def\Ph{\Phi} \def\Ps{\Psi} \def\Om{\Omega} % \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} % % the end-of-proof box \def\mybox{\hfill\llap{$\Box$}} % \def\Nat{\mathbb{N}} \def\Zah{\mathbb{Z}} \def\Real{\mathbb{R}} \def\Q{\mathbb{Q}} \def\C{\mathbb{C}} % \begin{document} \addtolength{\baselineskip}{1mm} % \begin{center} {\LARGE Quantum algebra embeddings~: deforming functionals and algebraic approach }\\[2cm] J. Van der Jeugt$^{a)}$ \\[.5cm] {\em Vakgroep Toegepaste Wiskunde en Informatica,\\ Universiteit Gent, Krijgslaan 281--S9, B-9000 Gent, Belgium}\\ \end{center} % % \vspace{3cm} \begin{abstract} The study of subalgebras of Lie algebras arising in physical models has been important for many applications. In the present paper we examine the $q$-deformation of such embeddings; the Lie algebras are then replaced by quantum algebras. Two methods are presented~: one based upon deforming functionals, and a direct algebraic approach. A number of examples are given, e.g.\ $su_q(2) \oplus su_q(2) \supset su_q(2)$ and $u_q(3)\supset so_q(3)$. For the last example, we give the $q$-boson construction, and the relevant overlap coefficients are related to a generalized basic hypergeometric function $_3\phi_2$. \end{abstract} \vspace{1cm} \noindent PACS numbers~: 02.20, 03.65F, 21.60F. % \vfill \noindent-----------------------------------\\ $^{a)}$ {\footnotesize Research Associate of the N.F.W.O. (National Fund for Scientific Research of Belgium)} \newpage % % % \section{Introduction} For classical Lie algebras, the study of subalgebras has been of great importance both in mathematics and in physics. Subalgebra chains arise in physical models which possess symmetries. For example, chains of Lie algebras $u(N)\supset so(N) \supset so(3) \supset so(2)$ make their appearance in many algebraic models in hadronic~\cite{iachello}, nuclear~\cite{elliott}, atomic~\cite{judd} and molecular physics~\cite{iachellol}. The study of subalgebra chains and their related mathematical quantities (branching rules, tensor products, generator matrix elements) are fundamental and bear a direct relation to the physical quantities of the model. It is in this field that Prof.\ Sharp -- to whom this paper is dedicated -- made numerous important contributions, a few of which will be mentioned and used here. The objects treated in this paper are, however, not Lie algebras but quantum algebras. Since the introduction of quantum groups and quantum enveloping algebras (quantum algebras, $q$-algebras, deformed algebras)~\cite{kulish,drinfeld,jimbo} these new mathematical structures are also finding applications in various branches in physics~\cite{zachos}. In a first approach, a quantum algebra can be seen as a deformation of the enveloping algebra of a (simple) Lie algebra; this structure can then be endowed with other quantities that turns it into a quasi-triangular Hopf algebra~\cite{majid}. The simplest quantum algebra is $su_q(2)$, an associative algebra with generators $\{L_-,L_+,q^{\pm L_0}\}$ and relations $q^{L_0}L_\pm=q^\pm L_\pm q^{L_0}$, $L_+L_--L_-L_+=(q^{2L_0}- q^{-2L_0})/(q-q^{-1})\equiv[2l_0]$, where $q$ is a deformation parameter. When $L_0$ is considered as an operator and $q^{L_0}$ as an exponential, these relations can actually be rewritten as $[L_0,L_\pm]=\pm L_\pm$ and $[L_+,L_-]=[2L_0]$, and in the limit $q\rightarrow 1$ this yields the relations satisfied in the enveloping algebra of $su(2)$. Quantum algebras have by now a well established role in 2-dimensional solvable models and inverse scattering theory~\cite{doebner}. In models of deformed molecules, atoms or nuclei, some research has been performed~\cite{bonatsosrrs,bonatsosar,bonatsosdk}, but only to a certain extent. In the study of such deformed models in physics, it is mainly the quantum algebra $su_q(2)$ (or $su_q(1,1)$) which has been used in the literature. The use of larger quantum algebras has so far been limited by the fact that many relevant properties of classical Lie algebras (e.g.\ subalgebra chains, decomposition rules) do not trivially extend to quantum algebras. In the present paper we mention some of the problems and solutions in the study of subalgebras of quantum algebras. We shall highlight two methods in the quantum algebra embedding problem~: the algebraic approach and the deforming functional approach. All methods will be illustrated by means of examples which not only have a pedagogical meaning, but are also potential candidates for applications in deformed physical models. In the following section we illustrate by means of the quantum algebra $u_q(3)$ the algebraic approach. Herein, the definition of quantum subalgebra is very natural. In Section~3 we use a well-known technique, of deforming functionals, to treat the deformation of embeddings. This technique has a number of shortcomings but nevertheless it can be applied in a straighforward way. A simple example, illustrating these shortcomings, is presented in Section~4. Section~5 returns to the quantum subalgebra chain given in Section~2, makes a correspondence with so-called $q$-boson creation and annihilation operators, and makes an explicit study of the transformation of $q$-boson states to $so_q(3)$ states labelled by the classical $(l,m)$-labels. The final section presents some conclusions and relations with other work. \section{Quantum (sub)algebras~: algebraic approach} Quantum algebras associated with simple Lie algebras are usually defined by means of relations between the Chevalley triples $\{h_i,e_i,f_i\}$. Considering the algebra generated by a subset of such triples (and with the corresponding relations), it is thus rather trivial to construct quantum subalgebras of the original quantum algebra. One could refer to this as the construction of subalgebras of quantum algebras in a Chevalley basis. In this case, also the Hopf structure (comultiplication, counit, and antipode) transfers trivially to the subalgebra. The construction of subalgebras of quantum algebras in a non-Chevalley basis is definitely a much harder problem, and so far only some partial solutions have been presented in the literature. What we mean by the algebraic approach for the construction of quantum subalgebras shall be illustrated in an example. Consider the quantum algebra $u_q(3)$~: this is the associative algebra with generators $e_1$, $e_2$, $f_1$, $f_2$, $k_i^{\pm}= q^{\pm N_i}$ ($i=0,1,2$), subject to the relations~\cite{jimbo246}~: \begin{eqnarray} &&\hbox{all }k_i^\pm \hbox{ and }k_j^\pm\hbox{ commute for }i\ne j,\nn\\ &&k_i^+k_i^-=1=k_i^-k_i^+ ,\nn\\ &&k_i^\pm e_j = q^{\pm(\de_{i,j-1}-\de_{ij})} e_j k_i^\pm,\nn\\ &&k_i^\pm f_j = q^{\mp(\de_{i,j-1}-\de_{ij})} f_j k_i^\pm, \label{rel} \\ &&e_if_j-f_je_i=\de_{ij} (k_{i-1}^+k_i^- - k_{i-1}^-k_i^+)/(q-q^{-1}),\nn\\ &&e_i^2e_j -(q+q^{-1})e_ie_je_i+e_je_i^2 =0 \quad (i\ne j),\nn\\ &&f_i^2f_j -(q+q^{-1})f_if_jf_i+f_jf_i^2 =0 \quad (i\ne j). \nn \end{eqnarray} Herein, the parameter $q$ is taken to be generic (not a root of unity). In the limit $q\rightarrow 1$, this reduces to the universal enveloping algebra of $u(3)$. The quantum enveloping algebra $u_q(3)$ as defined here can be made into a Hopf algebra, by defining a co-product, a co-unit and an antipode; however, for the purposes of this paper only the $q$-deformed algebra as determined by (\ref{rel}) is needed. In fact, the subalgebra structure introduced here will only be a $q$-deformed subalgebra structure (and not a Hopf subalgebra). To illustrate the remarks of the first paragraph of this section, it is easy to see that the triples $\{ e_i, f_i, h_i=N_{i-1}-N_i\}$ ($i=1,2$), together with the corresponding relations, generate an $su_q(2)$ subalgebra of $u_q(3)$. These are two examples of a subalgebra in the Chevalley basis. Another example is~\cite{dobrev,proc_sal}~: \beq \begin{array}{l} E=e_1e_2-qe_2e_1 ,\\ F=f_2f_1-q^{-1}f_1f_2 ,\\ H=N_0-N_2 . \end{array} \label{EFH} \eeq This example is less trivial but still simple since it corresponds to a regular $su(2)$ subalgebra of $u(3)$ in the nondeformed case. From (\ref{EFH}) it is also clear what we mean by the algebraic approach to the embedding problem~: a subalgebra of a quantum algebra ${\cal G}$ is a subalgebra of ${\cal G}$ as an associative algebra. When this subalgebra is generated by a number of generators with relations, such that it is itself a quantum algebra, we speak of a quantum algebra embedding. For instance, the three elements of (\ref{EFH}) belong to $u_q(3)$, and they satisfy \beq [H,E]=2E,\quad [H,F]=-2F,\quad [E,F]=[H]={q^H-q^{-H}\over q-q^{-1}} , \label{commEFH} \eeq the defining relations for $su_q(2)$. Hence, we have identified a quantum algebra embedding $u_q(3) \supset su_q(2)$. Another more important set of elements of $u_q(3)$ has been constructed in~\cite{proc_sal,princ}~: \beq \begin{array}{l} H=2N_0-2N_2 ,\\ E=\sqrt{2}q^{2N_2}e_1+e_2 q^{-N_1}(q^{N_2}+q^{-N_2})/\sqrt{2} ,\\ F=f_1q^{-N_1}(q^{N_0}+q^{-N_0})/\sqrt{2}+\sqrt{2} q^{2N_0} f_2 . \end{array} \label{soq3} \eeq These elements are of interest because in the limit $q\rightarrow 1$ they reduce to the generators of the principal $so(3)$ subalgebra of $u(3)$; it is therefore convenient to label the above elements as the generators of $so_q(3)$. However, the elements (\ref{soq3}) satisfy the last relation in (\ref{commEFH}) {\em only} when acting on the class of totally symmetric representations of $u_q(3)$. In this sence, we do not have a genuine quantum algebra embedding, and the embedding $u_q(3)\supset so_q(3)$ is valid {\em for symmetric representations only}~\cite{princ}. Nevertheless, the expressions (\ref{soq3}) are important and in Section~5 we shall show that they have a natural relation with so-called $q$-boson operators. Indeed, if a set of relations for generators is valid only for a certain class of representations, it means that there are in fact more relations than in the original set, or that the generators themselves can be expressed in terms of simpler objects (also satisfying some relations); here these simpler objects will be $q$-boson creation and annihilation operators. Note that the $(E,F,H)$-basis of (\ref{commEFH}) is the one used mainly by mathematicians; in the physics literature one identifies $(E,F,H)$ with $(L_+,L_-,2L_0)$ and then the relations become~\cite{sklyanin,jimbo} \beq [L_0,L_\pm]=\pm L_\pm,\quad [L_+,L_-]=[2L_0]={q^{2L_0}-q^{-2L_0}\over q-q^{-1}} , \label{commL} \eeq In the present section we have defined the concept of quantum subalgebras. A few examples have been given, from which it is clear that the problem of finding the ``$q$-analog'' of a Lie algebra embedding is a highly nontrivial task, in particular if the embedding is nonregular. In the following section we study the embedding problem by means of a technique which is commonly used in quantum algebras. \section{Deforming functionals} In a number of papers, deforming functionals have been introduced and studied for the quantum enveloping algebra $su_q(2)$ and for the quantum plane~\cite{curtrightz,curtrightgz,fairliez}. In the present section, we shall study how such deforming functionals can be used to treat the quantum subalgebra embedding problem. In the following section we shall give an example. In the framework of deforming functionals, explicit invertible functionals of the generators of $su(2)$ are given, which satisfy the defining relations of the $q$-deformed algebra $su_q(2)$. Let $l_0$, $l_+$ and $l_-$ be the basis of $su(2)$ satisfying \beq [l_0,l_\pm]=\pm l_\pm,\qquad [l_+,l_-]=2l_0. \label{su2} \eeq The Casimir operator is $I=l_+l_-+l_0^2-l_0\equiv l(l+1)$; then the positive operator $l$ is defined as $l=(-1+\sqrt{1+4I})/2$. Clearly, $l$ does not belong to the Lie algebra $su(2)$ nor to its enveloping algebra $U(su(2))$. However, $l$ acts as a scalar on all $su(2)$ modules $V$ generated by a vector $v$, i.e.~$V=U(su(2))v$, hence $l$ is well defined on such modules. Consider the category ${\cal C}$ of modules of the form $U(su(2))v$ with $v$ a weight vector with respect to $l_0$. Let $q$ be a generic real number, and denote again $(q^x-q^{-x})/(q-q^{-1})$ by $[x]$. The following deforming functionals~\cite{curtrightz} are well defined on modules belonging to ${\cal C}$~: \beq L_0=l_0,\qquad L_\pm=\sqrt{ [l\pm l_0][l\mp l_0+1]\over (l\pm l_0)(l\mp l_0+1)} l_\pm . \label{deform} \eeq Since the elements $L_0$, $L_+$, $L_-$ satisfy the standard relations (\ref{commL}) of the quantum enveloping algebra $su_q(2)$, the map (\ref{deform}) is said to be the deforming functional. For finite dimensional irreducible modules of $su(2)$ with basis $|l,m\rangle$ ($m=l,l-1,\ldots,-l$; $2l$ integer) and action $l_0|l,m\rangle= m|l,m\rangle$, $l_\pm|l,m\rangle=\sqrt{(l\mp m)(l\pm m+1)}|l,m\pm 1\rangle$, one can introduce the following basis~: \beq |n_1,n_2\rangle,\qquad\hbox{with}\qquad n_1=l+m,\;n_2=l-m, \eeq and denote the $su(2)$ generators by~: \beq l_0={1\over 2}(e_{11}-e_{22}),\quad l_+=e_{12},\quad l_-=e_{21} , \eeq with usual commutation relations \beq [e_{ij},e_{kl}]=\delta_{jk}e_{il}-\delta_{il}e_{kj}. \label{comm} \eeq The deforming functional (\ref{deform}) now takes the following form~: \beq E_{ii}=e_{ii}\;(i=1,2),\quad E_{12}=\sqrt{[e_{11}][e_{22}+1]\over (e_{11})(e_{22}+1)} e_{12},\quad E_{21}=\sqrt{[e_{22}][e_{11}+1]\over (e_{22})(e_{11}+1)} e_{21}. \eeq These expressions suggest a way to define deforming functionals for $u(N)$ or $su(N)$. The basis of $u(N)$ is $e_{ij}$ with $i,j=1,\ldots,N$, and the commutation relations are given by~(\ref{comm}). The actual generators to be deformed are $e_{ii}$ and $e_{ij}$ with $j=i\pm 1$. The deformed generators, $E_{ii}$ and $E_{ij}$ with $j=i\pm 1$, of $u_q(N)$ should satisfy~\cite{drinfeld,jimbo,sklyanin}~: \begin{eqnarray} &&[E_{ii},E_{jj}]=0, \label{rel1} \\ &&[E_{ii},E_{jk}]=(\delta_{ij}-\delta_{ik})E_{jk}, \label{rel2} \\ &&[E_{i,i+1},E_{j+1,j}]=\delta_{ij} [E_{ii}-E_{i+1,i+1}], \label{rel3} \end{eqnarray} and the Serre relations, which will not be repeated here. If one defines~: \beq E_{ii}=e_{ii},\quad E_{ij}=\sqrt{[e_{ii}][e_{jj}+1]\over (e_{ii})(e_{jj}+1)} e_{ij},\quad (j=i\pm 1), \eeq then (\ref{rel1}) and (\ref{rel2}) are satisfied, but it is easily verified that~(\ref{rel3}) is in general not valid. However, if all the relations are restricted to the class ${\cal S}$ of totally symmetric representations of $u_q(N)$, then~(\ref{rel3}) and also the Serre relations are automatically satisfied. An irreducible representation of ${\cal S}$ is labelled by an integer $n$, and has basis states $|n_1,n_2,\ldots,n_N\rangle$ with $\sum_{i=1}^N n_i=n$; the action of the $u_q(N)$ generators is~: \begin{eqnarray} E_{ii}|n_1,n_2,\ldots,n_N\rangle&=& n_i|n_1,n_2,\ldots,n_N\rangle,\nn\\ E_{i,i+1}|n_1,n_2,\ldots,n_N\rangle&=& \sqrt{[n_i+1][n_{i+1}]}|n_1,\ldots,n_i+1,n_{i+1}-1,\ldots,n_N\rangle, \label{action}\\ E_{i+1,i}|n_1,n_2,\ldots,n_N\rangle&=& \sqrt{[n_i][n_{i+1}+1]}|n_1,\ldots,n_i-1,n_{i+1}+1,\ldots,n_N\rangle.\nn \end{eqnarray} In the following, we shall assume that we are always working in the class ${\cal S}$. Then the deforming functionals for $u(N)$ or $su(N)$ are well defined and the corresponding generators of $u_q(N)$ or $su_q(N)$ satisfy the defining relations for these quantum enveloping algebras. As previously, the fact that we are working in the class of symmetric representations means that the deforming functionals are in fact those defined for $q$-boson operators of $u_q(N)$. Having defined the deforming functionals, we shall now consider the problem of constructing quantum subalgebras by means of these deforming maps. Let ${\cal G}=u(N)$ or $su(N)$, and let ${\cal G}'\subset{\cal G}$ be a subalgebra of ${\cal G}$, also of type $u(K)$ or $su(K)$. In particular, with a view on defining deforming functionals, we shall assume that in the reduction of an irreducible symmetric representation of ${\cal G}$ with respect to ${\cal G}'$, only symmetric representations of ${\cal G}'$ appear (this is certainly the case if ${\cal G}'=su(2)$). Let ${\cal S}$ be the class of symmetric representations of ${\cal G}$, and $V\in{\cal S}$. The standard (orthogonal) basis vectors $|n_1,n_2,\ldots,n_N\rangle$ of $V$ shall be denoted as $v_\la$, and the action of the generators $g_i$ of ${\cal G}$, which is well known, will be denoted by~: \beq g_i v_\la=\sum_\mu (g_i)_\la^\mu v_\mu. \label{gi} \eeq The generators $G_i$ of the quantum enveloping algebra ${\cal G}_q$ are then defined in terms of the $g_i$ by means of the deforming functionals. Hence, $V$ is a representation for ${\cal G}_q$ and the action of $G_i$ on the basis $v_\la$ follows from~(\ref{gi}) and the explicit form of the deforming functionals. Here, this action will be denoted by~: \beq G_i v_\la=\sum_\mu (G_i)_\la^\mu v_\mu. \label{Gi} \eeq Next, consider the branching ${\cal G}\rightarrow{\cal G}'$. In this reduction, the symmetric representations $W^{(l)}$ of ${\cal G}'$ will be labelled by the symbol $l$, and the internal labels for the (orthogonal) basis vectors will be denoted by $m$. Let the reduction of $V$ in ${\cal G}\rightarrow{\cal G}'$ be \beq V \rightarrow \oplus_l W^{(l)}, \label{decomp} \eeq where multiplicities can occur. We now assume that for the representation $V$ the decomposition coefficients are known explicitly, i.e.~: \beq v_\la=\sum_{l,m} T_\la^{lm} w_{lm}. \label{vw} \eeq Since we are dealing with orthogonal bases, it also follows that \beq w_{lm} = \sum_\la T_\la^{lm} v_\la. \label{wv} \eeq The action of the generators $g_k'$ of ${\cal G}'$ on the representations $W^{(l)}$ are known, and since the deforming functionals for ${\cal G}'$ are also known, the action of the generators $G_k'$ of ${\cal G}_q'$ follows~: \beq G_k' w_{lm}= \sum_{m'} (G_k')^{lm'}_{lm} w_{lm'} . \label{Gk} \eeq Next, we use (\ref{vw}), (\ref{Gk}) and (\ref{wv})~: \begin{eqnarray} G_k' v_\la &=& \sum_{l,m} T_\la^{lm} G_k' w_{lm} \nn\\ &=& \sum_{l,m} \sum_{m'} T_\la^{lm} (G_k')^{lm'}_{lm} w_{lm'} \nn\\ &=& \sum_\mu \left(\sum_{l,m} \sum_{m'} T_\la^{lm} (G_k')^{lm'}_{lm} T_\mu^{lm'}\right) v_\mu . \label{result} \end{eqnarray} This last equation gives the action of the elements $G_k'$ of ${\cal G}_q'$ on the standard basis $v_\la$ of the representations $V$ of ${\cal G}_q$. From such a relation it is, at least in principle, possible to express the $G_k'$ in terms of the $G_i$. However, it does not necessarily follow that ${\cal G}_q'$ is a subalgebra of ${\cal G}_q$ (seen as an associative algebra). This will be clear from a simple example treated in the following section. \section{Example~: $su_q(2)\oplus su_q(2)$} Note that the construction of the previous section is also valid for direct sums of $su(N)$. Consider now the case ${\cal G}=su(2)\oplus su(2)$ and ${\cal G}'=su(2)$ (the diagonal $su(2)$). The basis elements of $su(2)\oplus su(2)$ are $\{s_i,t_i (i=0,+1,-1)\}$, with commutation relations~: \begin{eqnarray*} &&[s_i,t_j]=0,\\ &&[s_0,s_\pm]=\pm s_\pm,\qquad [s_+,s_-]=2s_0,\\ &&[t_0,t_\pm]=\pm t_\pm,\qquad [t_+,t_-]=2t_0. \end{eqnarray*} The irreducible representations $V^{(s,t)}$ are labelled by two half-integers, and have basis states $|s,t,m_s,m_t\rangle$, with $m_s=s,s-1,\ldots,-s$ and $m_t=t,t-1,\ldots,-t$. The subalgebra has generators $l_0=s_0+t_0$ and $l_\pm=s_\pm+t_\pm$, with commutation relations~(\ref{su2}). Here, the decomposition~(\ref{decomp}) is given by~: \[ V^{(s,t)}\rightarrow W^{(s+t)}\oplus W^{(s+t-1)}\oplus \cdots\oplus W^{(|s-t|)}. \] The decomposition coefficients (\ref{vw}) are the well known $su(2)$ coupling coefficients~\cite{edmonds}~: \[ |s,t,m_s,m_t\rangle=\sum_{l,m}\langle l\,m|s\,t\,m_s\,m_t\rangle\; |l,m\rangle, \] or inversely~: \[ |l,m\rangle=\sum_{m_s,m_t}\langle l\,m|s\,t\,m_s\,m_t\rangle\; |s,t,m_s,m_t\rangle. \] Consider now the quantum algebras ${\cal G}_q$ and ${\cal G}_q'$ with generators $\{S_i, T_i (i=0,+1,-1)\}$ and $\{L_i (i=0,+1,-1)\}$ respectively, obtained from the nondeformed algebras $su(2)\oplus su(2)$ and $su(2)$ by means of the deforming functionals. For the generator $L_+$, for instance, it follows that the action~(\ref{Gk}) is given by \[ L_+|l,m\rangle = \sqrt{[l-m][l+m+1]} |l,m+1\rangle. \] Then (\ref{result}) implies that \begin{eqnarray} &&L_+|s,t,m_s,m_t\rangle = \nn\\ &&\sum_{m_s',m_t'} \left(\sum_{l,m}\langle l\,m|s\,t\,m_s\,m_t\rangle \sqrt{[l-m][l+m+1]}\langle l\,m+1|s\,t\,m_s'\,m_t'\rangle \right) |s,t,m_s',m_t'\rangle.\nn \\ && \label{Lplus} \end{eqnarray} This result is remarkable for the following reasons. When the nondeformed generator $l_+$ acts on $|s,t,m_s,m_t\rangle$, it yields only two contributions, namely in \beq |s,t,m_s+1,m_t\rangle\qquad\hbox{and}\qquad|s,t,m_s,m_t+1\rangle. \label{2states} \eeq When the deformed generator $L_+$ acts on $|s,t,m_s,m_t\rangle$, it follows from~(\ref{Lplus}) that in general it gives contributions in \beq |s,t,s,m_s+m_t-s+1\rangle,\;|s,t,s-1,m_s+m_t-s+2\rangle,\ldots, |s,t,m_s+m_t-t+1,t\rangle. \label{allstates} \eeq The coefficients accompanying these states are $q$-dependent. In the limit $q\rightarrow 1$ most of the coefficients of the states~(\ref{allstates}) tend to zero, such that only contributions in~(\ref{2states}) remain. One can now try to express the $L_+$ action in terms of the generators of ${\cal G}_q$. It follows that $L_+$ would take the following form~: \begin{eqnarray} L_+&=&a_1S_+ + a_2S_+^2T_- + a_3S_+^3T_-^2 + a_4S_+^4T_-^3 + \cdots\nn\\ &&+b_1T_+ + b_2S_+T_-^2 + b_3S_+^2T_-^3 + b_4S_+^3T_-^4 + \cdots, \label{series} \end{eqnarray} where $a_k$ and $b_k$ are coefficients still depending upon the diagonal generators $S_0$ and $T_0$. This implies that the ${\cal G}_q'$ algebra obtained this way is not a genuine subalgebra of ${\cal G}_q$ considered as an associative algebra. For the present example, it is rather surprising that the deforming functionals approach gives rise to such complicated objects. This is in strong contrast to the algebraic approach which for this case can also successfully be applied, and yields very simple expressions. The quantum algebra ${\cal G}_q$ of this example does indeed contain a quantum subalgebra of type $su_q(2)$, namely, \beq L_0=S_0+T_0,\qquad L_\pm=S_\pm q^{T_0}+T_\pm q^{-S_0}. \label{sub} \eeq This is a genuine subalgebra which tends in the limit $q\rightarrow 1$ to ${\cal G}'$, just as ${\cal G}_q'$ does. It may be surprising that the technique of quantum deformations does not give rise to the subalgebra~(\ref{sub}). The underlying reason is because we are dealing with a nonregular embedding ${\cal G} \supset{\cal G}'$. Recently, the deforming functionals approach has also been applied for the (symmetric representations) of $u_q(3)\supset so_q(3)$~\cite{feng,delsol}. In both papers, the treatment falls within the framework of our present Section~3. The conclusions are similar as in the example presented here, i.e.~one does not find a genuine subalgebra. \section{Boson realization of $u_q(3)\supset so_q(3)$} In this last section we comment further on the embedding discussed in Section~2. Since this embedding holds only for the class of symmetric representations, it is natural that there is a link with a boson realization. On the other hand, we are dealing with deformed structures and the bosons that appear are not ordinary bosons but so-called $q$-bosons~\cite{biedenharn,macfarlane}. The quantum enveloping algebra $u_q(3)$ can be realized by means of three independent $q$-boson operators $(N_i,b_i,b^\d_i)$, where $i=+,0,-$. These satisfy the relations~: \beq [N_i,b^\d_i]=b^\d_i,\qquad [N_i,b_i]=-b_i,\qquad, b_ib_i^\d-q^{\pm 1}b_i^\d b_i= q^{\mp N_i}, \eeq and the remaining commutators are zero. The orthonormal $n$-boson states are of the form~\cite{princ} \beq |n_+,n_0,n_-\rangle = { (b^\d_+)^{n_+} (b^\d_0)^{n_0} (b^\d_-)^{n_-} \over [n_+]![n_0]![n_-]!} |0\rangle, \label{nstate} \eeq where the vacuum $|0\rangle$ is defined by $b_i|0\rangle=0$ for $i=+,0,-$, and $[x]!=[x][x-1]\cdots[1]$. An orthonormal basis for the symmetric $u_q(3)$ representation labelled by $n$ consists of the states $|n_+,n_0,n_-\rangle$ with $n_++n_0+n_-=n$. It is common to introduce the operator $N= N_++N_0+N_-$ with eigenvalue $n$ on the states (\ref{nstate}). The Chevalley generators of $u_q(3)$ are given by $N_i$ (strictly speaking $q^{\pm N_i}$), $e_1=b^\d_+b_0$, $e_2=b^\d_0b_-$, $f_1=b^\d_0b_+$ and $f_2=b^\d_-b_0$, satisfying the usual relations (\ref{rel}). The $so_q(3)$ subalgebra can then also be written in this realization by means of (\ref{soq3}). Note that, if we accept square roots of operators which are acting diagonally on representations, the operators (\ref{soq3}) can be written in a more symmetric form and one obtains the following~: \begin{eqnarray} L_0 &=& N_{+}-N_{-}, \label{l0}\\ L_{+}&=&q^{N_{-}-{1\over 2}N_0} \sqrt{q^{N_{+}}+q^{-N_{+}}}\; b^\d_{+}b_0 + b^\d_0b_{-} q^{N_{+}-{1\over 2}N_0} \sqrt{q^{N_{-}}+q^{-N_{-}}} ,\label{l+} \\ L_{-}&=&b^\d_0b_{+} q^{N_{-}-{1\over 2}N_0} \sqrt{q^{N_{+}}+q^{-N_{+}}} + q^{N_{+}-{1\over 2}N_0} \sqrt{q^{N_{-}}+q^{-N_{-}}}\; b^\d_{-}b_0 . \label{l-} \end{eqnarray} These operators, defined in terms of the $u_q(3)$ realization, satisfy the usual $so_q(3)$ relations (\ref{commL}). It is possible to simplify the expressions for $L_i$ by making the following transformation~\cite{quesne,def-u3}~: \beq \tilde b^\d_i=\left(q^{N_i}+q^{-N_i}\over q+q^{-1}\right)^{1/2} b^\d_i,\qquad \tilde b_i=b_i\left(q^{N_i}+q^{-N_i}\over q+q^{-1}\right)^{1/2} \hbox{ for } i=+,-. \eeq This map transforms a $q$-boson into what could be called a $q^2$-boson, since \beq [N_i,\tilde b^\d_i]=\tilde b^\d_i,\qquad [N_i,\tilde b_i]=-\tilde b_i,\qquad, \tilde b_i\tilde b_i^\d-q^{\pm 2}\tilde b_i^\d\tilde b_i= q^{\mp 2N_i}. \eeq Note that we make this transformation only for $i=+$ and $i=-$. In terms of the two $q^2$-bosons and the single $q$-boson operators, the expressions for $L_i$ take the simple form~: \begin{eqnarray} L_0 &=& N_{+}-N_{-}, \\ L_{+}&=&\sqrt{[2]}\left(q^{N_{-}-{1\over 2}N_0} \tilde b^\d_{+}b_0 + b^\d_0\tilde b_{-} q^{N_{+}-{1\over 2}N_0}\right) ,\label{nl+}\\ L_{-}&=&\sqrt{[2]}\left(b^\d_0\tilde b_{+} q^{N_{-}-{1\over 2}N_0} + q^{N_{+}-{1\over 2}N_0} \tilde b^\d_{-}b_0 \right). \label{nl-} \end{eqnarray} So, in a sense one could say that the most natural embedding of $so_q(3)$ in $u_q(3)$ occurs for the case when $b_0^{(+)}$ is a $q$-boson and $b_+^{(+)}$ and $b_-^{(+)}$ are $q^2$-bosons. Finally, we shall explicitly study the decomposition of $u_q(3)$ boson representations in the $so_q(3)$ basis. In the classical case of $u(3)\supset so(3)$, the symmetric representations labelled by $n$ decompose into $so(3)$ representations $(l)$ with $l=n,n-2,\ldots,1$ or $0$. In the $q$-generalized case, this turns out to be exactly the same decomposition. In the basis (\ref{nstate}) the states of the $u_q(3)$ symmetric representation labelled by $n$ are given by $|n_-,n_0,n_+\rangle$ with $n_-+n_0+n_+=N$. In the $so_q(3)$ basis we shall use the classical labels $l$ and $m$, where $m$ is the eigenvalue of $L_0$ and $[l][l+1]$ is the eigenvalue of the $so_q(3)$ Casimir operator $L_+L_-+[L_0][L_0-1]$; the states are then labelled by $w(n,l,m)$ (plus additional labels, if necessary). Consider first the state $|l,0,0\rangle$. Since this state has $L_0$ eigenvalue $l$ and $N$ eigenvalue $n$, and $L_+|l,0,0\rangle=0$, it follows that $w(l,l,l)=|l,0,0\rangle$. Next, one uses the lowering operator $L_-$ to obtain states with quantum number $m