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% SUBALGEBRAS OF QUANTUM ENVELOPING ALGEBRAS AND APPLICATIONS
% Joris Van der Jeugt
% in Anales de Fisica 1,
% eds. M.A. del Olmo, M. Santander and J.M. Guilarte
% Ciemat/Rsef, Madrid, 1993; pp. 131-134.
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\begin{center}
SUBALGEBRAS OF QUANTUM ENVELOPING ALGEBRAS AND APPLICATIONS\\[3mm]
{\em Joris Van der Jeugt}\\
Research Associate of the NFWO \\
Applied Mathematics and Computer Science\\
University of Ghent\\
Krijgslaan 281-S9, 9000 Gent, BELGIUM
\end{center}
\vskip 4mm
For classical Lie algebras, the study of subalgebras has been of great
importance both in mathematics and in physics. Subalgebra chains often
arise in physical symmetries, and much effort has gone into constructing
techniques for ``branching rules'', i.e.\ rules for determining the
decomposition of an irreducible representation of a Lie algebra $G$ into
irreducible representations of a Lie subalgebra $G'\subset G$. One
often makes the distinction between regular subalgebras (where the
roots of $G'$ form a subset of the roots of $G$), and nonregular
subalgebras. A typical example of a regular subalgebra of $su(3)$ is
$su(2)$ (for which the adjoint of $su(3)$ decomposes as a vector,
two spinors, and a singlet); a typical example of a nonregular
subalgebra of $su(3)$ is $so(3)$ (for which the adjoint of $su(3)$
decomposes as a vector plus a quadrupole tensor).
The purpose of the present contribution is to make some comments on
the problem of subalgebra chains $G\supset G'$, where both $G$ and
$G'$ are quantum enveloping algebras. For the presentation we shall
restrict ourselves to the case where $G=u_q(3)=gl_q(3)$, which shall
be defined later. The subalgebra $G'$ will be of type $sl_q(2)$
($su_q(2)$ or $so_q(3)$).

Let us first return to the well known nondeformed case of $u(3)=gl(3)$, with
nine basis elements $E_{ij}$ ($i,j=0,1,2$) and commutation relations
\[
[E_{ij},E_{kl}]=\de_{jk}E_{il}-\de_{il}E_{kj} \;.
\]
Often, one denotes $\bar e_1=E_{01}$, $\bar e_2=E_{12}$, $\bar e_3=E_{02}$,
$\bar f_1=E_{10}$,
$\bar f_2=E_{21}$, $\bar f_3=E_{20}$, and $\bar N_i=E_{ii}$.
An algebra of type $sl(2)$ has three generators $\bar E$, $\bar F$, and
$\bar H$, with relations
\[
[\bar H,\bar E]=2\bar E,\quad [\bar H,\bar F]=-2\bar F,
 \quad [\bar E,\bar F]=\bar H \;.
\]
Several subalgebras of type $sl(2)$ can then be identified in $u(3)$~:
\[
\begin{array}{llll}
{\rm (a)} & \bar E=\bar e_1 & \bar F=\bar f_1 & \bar H=\bar N_0-\bar N_1 ,\\
{\rm (b)} & \bar E=\bar e_2 & \bar F=\bar f_2 & \bar H=\bar N_1-\bar N_2 ,\\
{\rm (c)} & \bar E=\bar e_3 & \bar F=\bar f_3 & \bar H=\bar N_0-\bar N_2 ,\\
{\rm (d)}& \bar E=\sqrt{2}(\bar e_1+\bar e_2)&\bar F=\sqrt{2}(\bar f_1+
\bar f_2)&\bar H=2\bar N_0-2\bar N_2 .
\end{array}
\]
The first three cases are all examples of $u(3)\supset su(2)$; in fact
the three subalgebras given in (a), (b) and (c) are conjugate. The
last case, (d), corresponds to $u(3)\supset so(3)$.

Let us now pay our attention to the $q$-deformed case. The quantum
enveloping algebra $gl_q(3)$ is defined as an 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 [1]~:
\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 $gl(3)$. The quantum enveloping algebra $gl_q(3)$ 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, we shall see later that the subalgebra structure here introduced
will be only a $q$-deformed subalgebra structure.

The subalgebras we shall try to identify in this case will be of
type $sl_q(2)$. Such an algebra is generated by the elements
$E$, $F$, $q^{\pm H}$, subject to the relations
\[
[H,E]=2E,\quad [H,F]=-2F,\quad, [E,F]=[H]={q^H-q^{-H}\over q-q^{-1}} .
\]
In principal, all these relations should be written in terms of $q^{\pm H}$,
but instead of $q^HE=q^2 E q^H$ one usually writes $[H,E]=2E$. Note
that we have adopted the common notation $[x]=(q^x-q^{-x})/(q-q^{-1})$.

We now come to the problem of constructing the $q$ analog of the
subalgebras given in (a)--(d). From the relations (\ref{rel}) it is
immediately obvious that there are two trivial subalgebras of
type $sl_q(2)$, namely (a) $E=e_1$, $F=f_1$, $H=N_0-N_1$, and
(b) $E=e_2$, $F=f_2$, $H=N_1-N_2$. Another $sl_q(2)$ subalgebra
can be identified as follows [2]~:
\[
\begin{array}{l}
E=e_1e_2-qe_2e_1 ,\\
F=f_2f_1-q^{-1}f_1f_2 ,\\
H=N_0-N_2 .
\end{array}
\]
This is clearly the $q$ analog of (c), since for the classical
case $\bar e_3=[\bar e_1,\bar e_2]$. Note that the expressions for
$E$ and $F$ are now quadratic in the generators of $gl_q(3)$.

The rest of this paper we shall be concerned with constructing the
$q$ analog of (d). This turns out to be a fairly difficult problem.
In order to find a solution, we shall first restrict ourselves to
the case when all generators are acting on the {\em totally symmetric
representations} of $gl_q(3)$, and later try to extend to the general
case. These symmetric representations of $gl_q(3)$ are labelled by
an integer $N$, and have basic vectors $|n_0,n_1,n_2\rangle$ with
$n_0+n_1+n_2=N$. The action of the $gl_q(3)$ generators is given
by [1]~:
\begin{eqnarray}
N_i|n_0n_1n_2\rangle & = & n_i|n_0n_1n_2\rangle, \qquad (i=0,1,2),\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}
If we next assume that the $sl_q(2)$ subalgebra elements of type (d)
have the correct weight structure, then they should take the following
form when acting on these symmetric representations~:
\begin{eqnarray}
H&=&2N_0-2N_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}
Herein, $\al$, $\al'$, $\be$ and $\be'$ are functions of $q^{N_i}$;
since the $q^{N_i}$ are diagonal in the representations considered here,
all analytic functions can be allowed. On the other hand, if one wants
$E$ and $F$ to be genuine elements of $gl_q(3)$, these functions
should be linear combinations of words in $q^{\pm N_i}$.
For (\ref{7}) one can verify that $[H,E]=2E$ and $[H,F]=-2F$ are automatically
satisfied, so one still has to require $[E,F]=[H]$. This leads to the
following conditions for the functions introduced in (\ref{7}) (here,
we have replaced the operators $N_i$ by ordinary variables $n_i$, which
is allowed since the $N_i$ are commuting)~:
\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
\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{17b}
\eeq
\begin{eqnarray}
\lefteqn{A(n_0 n_1 n_2)[n_0][n_1+1] - A(n_0+1,n_1-1,n_2)
 [n_0+1][n_1] } \nn \\
& & +B(n_0,n_1-1,n_2+1)[n_1][n_2+1] -
 B(n_0 n_1 n_2)[n_1+1][n_2]  = [2n_0-2n_2] .
 \label{18}
\end{eqnarray}
Herein, $A=\al\al'$ and $B=\be\be'$. The relation (\ref{18}) admits
the following solution [3]~:
\beq
A(n_0,n_1,n_2)=q^{2n_2-n_1}(q^{n_0}+q^{-n_0}),\qquad
B(n_0,n_1,n_2)=A(n_2,n_1,n_0).
\label{AB}
\eeq
Using this, one can write the following solution for the
$sl_q(2)$ generators~:
\[
\begin{array}{l}
H=2N_0-2N_2 ,\\
E=\sqrt{2}q^{2N_2}e_1+e_2 q^{-N_1}(q^{N_2}+q^{-N_2})/\sqrt{2} ,\\
F=f_1q^{-N_1}(q^{N_0}+q^{-N_0})/\sqrt{2}+\sqrt{2} q^{2N_0} f_2 .
\end{array}
\]
Note that these expressions, which can be seen as the $q$ analog of
(d), are now cubic expressions in terms of the $gl_q(3)$ generators.
However, one should remember that these elements satisfy the $sl_q(2)$
relations only when acting on symmetric representations of $gl_q(3)$.
In fact, one can show that expressions of type~(\ref{7}) cannot
form an $sl_q(2)$ subalgebra in the abstract sense. Therefore, with
the ansatz~(\ref{7}), one cannot obtain a general $sl_q(2)$ subalgebra;
introducing quadratic or higher order expressions in the $e_i$, $f_i$
could lead to a solution, but this possibility has not yet been
explored.

Here we are restricting the relations to symmetric representations
only, thus we can allow other functions of the $q^{N_i}$. For reasons
of symmetry, one can choose~:
\begin{eqnarray*}
\al(N_0,N_1,N_2)&=&\al'(N_0,N_1,N_2)=\sqrt{A(N_0,N_1,N_2)},\\
\be(N_0,N_1,N_2)&=&\be'(N_0,N_1,N_2)=\sqrt{B(N_0,N_1,N_2)}.
\end{eqnarray*}
The restriction to symmetric representations is equivalent to expressing
the $gl_q(3)$ in terms of simpler objects, namely $q$-boson operators.
Let us assume there are three number
operators $N_0$, $N_1$, $N_2$ and three independent
$q$-boson operators $b_i$ and $b_i^+$ ($i=0,1,2$) satisfying [4]
\beq
[N_i,b^+_i]=b^+_i,\qquad
[N_i,b_i]=-b_i,\qquad
b_ib^+_i-q^{\mp 1}b^+_ib_i=q^{\pm N_i}.
\label{22}
\eeq
The $q$-boson basis states are then of the form
\beq
|n_0 n_1 n_2\rangle =
{ (b_0^+)^{n_0} (b_1^+)^{n_1} (b_2^+)^{n_2}
\over \sqrt{[n_0]![n_1]![n_2]!}  } |0\rangle ,
\label{23}
\eeq
with $b_i|0\rangle=0$ and $N_i|n_0n_1 n_2\rangle =
n_i|n_0n_1 n_2\rangle$. The remaining $gl_q(3)$ generators are then realised
by $e_1=b_0^+ b_1$, $e_2=b_1^+ b_2$, $f_1=b_1^+ b_0$ and $f_2=b_2^+ b_1$.
The subalgebra of type $sl_q(2)$, here for obvious reasons denoted
by $so_q(3)$, now reads~:
\beq
\begin{array}{l}
H=2N_0-2N_2,\\[2mm]
E=q^{N_2-{1\over 2}N_1} \sqrt{q^{N_0}+q^{-N_0}}\; b_0^+b_1 +
  b_1^+b_2\; q^{N_0-{1\over 2}N_1} \sqrt{q^{N_2}+q^{-N_2}} , \\[2mm]
F=b_1^+b_0 \;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}}\; b_2^+b_1 .
\end{array}
\label{21}
\eeq
Physicists often use a different notation where $L_0=H/2$,
$L_{+1}=E$ and $L_{-1}=F$, with relations~:
\beq
[L_0,L_{\pm 1}]=\pm L_{\pm 1}, \qquad [L_{+1},L_{-1}]=[2L_0].
\label{25}
\eeq

A few questions now remain. First, one can wonder whether the solution
given by (\ref{AB}) is unique. The answer is negative; in fact, replacing
$q\rightarrow q^{-1}$ yields another solution. The breakdown of this
$q\rightarrow q^{-1}$ symmetry in (\ref{21}) is the main reason why the
Hopf algebra structure of $gl_q(3)$ cannot be taken over for $sl_q(2)$.
This is the reason why we have only a $q$ deformed subalgebra structure.
Apart from this, we have been able to
construct the following complicated $(q\rightarrow q^{-1})$-invariant
solution~:
\[
A(n_0,n_1,n_2)=B(n_2,n_1,n_0)={
\{n_0\} \{2n_1+2n_2\} \{n_1-2\} \over \{2n_1\} \{2n_1-2\} },
\]
where $\{x\}$ stands for $q^x+q^{-x}$, but here the appearance of
denominators imply that $E$ and $F$ can no longer be written as
genuine $gl_q(3)$ elements. For the case (\ref{21}), one
can determine the branching rules for $gl_q(3)\supset so_q(3)$ (for
symmetric representations) and also the corresponding overlap coefficients [3].
Finally, one can also
relate the $q$-boson operators $b^+_i$ and $b_i$ to rank 1 tensors
of the present $sl_q(2)$ for some adjoint action [5].
\vskip 2mm
\noindent {REFERENCES}
\begin{enumerate}
\item
M.\ Jimbo, in ``Field Theory, Quantum Gravity and Strings'' {\em
 Lecture Notes in Physics} {\bf 246}, eds.\ H.J.\ de Vega and
 N.\ S\'anchez (Springer-Verlag~: Berlin, 1986), 335.
\item
V.K.\ Dobrev, in {\em Lecture Notes in Physics} {\bf 370} (1990), 107.
\item
J.\ Van der Jeugt, {\em J.\ Phys.\ A~: Math.\ Gen.} {\bf 25} (1992), L213.
\item
L.C.\ Biedenharn, {\em J.\ Phys.\ A~: Math.\ Gen.} {\bf 22} (1989), L873;
A.J.\ Macfarlane, {\em J.\ Phys.\ A~: Math.\ Gen.} {\bf 22} (1989) 4581.
\item
J.\ Van der Jeugt, {\em in preparation} (1992).
\end{enumerate}
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