\documentclass[11pt]{article}

\usepackage{amssymb,amsfonts}
\usepackage{amsthm}
%\usepackage{graphicx}
\hyphenation{Hei-sen-berg}

\textheight=22 cm
\topmargin=0 cm
\textwidth= 16cm
\oddsidemargin=0 cm
\evensidemargin=0 cm


\def\ni{\noindent}
\def\nn{\nonumber}

\def \bc {\begin{center}}
\def \ec {\end{center}}
\def \bi {\begin{itemize}}
\def \ei {\end{itemize}}

\def \ba {\begin{array}}
\def \ea {\end{array}}

\def \bea {\begin{eqnarray}}
\def \eea {\end{eqnarray}}

\def \be {\begin{equation}}
\def \ee {\end{equation}}

\def \cL {{\cal L}}
\def \cG {{\cal G}}
\def\cO {{\cal O}}
\def \cas {c}
\def\kbar{{\mathchar'26\mkern-9mu\lambda}}
\def\um {\frac{1}{2}}
\def\W {\cal W}


\newtheorem{thm}{Theorem}[section]
\newtheorem{cor}[thm]{Corolary}
\newtheorem{lem}[thm]{Lemma}
\newtheorem{prop}[thm]{Proposition}
%\theoremstyle{definition}
\newtheorem{defn}[thm]{Definition}
\theoremstyle{remark}
\newtheorem{rem}[thm]{Remark}

%\renewcommand{\baselinestretch}{1.5}
\begin{document}
\begin{center}
{\LARGE {\bf Oscillator Realization of Higher-$U(N_+,N_-)$-Spin 
Algebras of ${\cal W}_{\infty}$-type and Quantized Simplectic Diffeomorphisms}}
%{\LARGE {\bf Group, Tensor Operator, Poisson, Simplectic Diffeomorphism, $W_{\infty}$-like and Higher-$U(N_+,N_-)$-Spin 
%Algebras}}
\end{center}
\bigskip
\bigskip

\centerline{{\sc Manuel Calixto}}


\bigskip

\bc
{\it Departamento de Matem\'atica Aplicada y Estad\'\i stica,
Universidad Polit\'ecnica de Cartagena, Paseo Alfonso XIII 56, 30203 Cartagena, Spain}

\bigskip

E-mail: Manuel.Calixto@upct.es
\ec

\bigskip

\bigskip
\begin{center}
{\bf Abstract}
\end{center}
\small


\begin{list}{}{\setlength{\leftmargin}{3pc}\setlength{\rightmargin}{3pc}}
\item This article is a further contribution to our research \cite{infdimal} into a class 
of infinite-dimensional Lie algebras $\cL_\infty(N_+,N_-)$ generalizing the standard $\W_\infty$ 
algebra, viewed as a tensor operator algebra of $SU(1,1)$ in a group-theoretic  
framework. Here we interpret $\cL_\infty(N_+,N_-)$ either as infinite continuations of 
pseudo-unitary symmetries or as ``higher-$U(N_+,N_-)$-spin extensions"  
of the diffeomorphism algebra diff$(N_+,N_-)$. We also  
provide a deeper mathematical interconnection between Poisson (and 
symplectic diffeomorphism) algebras of functions on coadjoint 
orbits of pseudo-unitary groups $U(N_+,N_-)$ and the classical limit of 
the corresponding tensor operator (and group) algebras. As potential applications we 
comment on the formulation of integrable higher-dimensional dynamical 
(field) systems and gauge theories of higher-extended objects. Some 
remarks on non-commutative geometry are also made.
\end{list}
\normalsize
%\setlength{\baselineskip}{14pt}

\noindent PACS: 
%02.20.-a group theory 
02.20.Tw, % Inf. dim. Lie gr., 
02.20.Sv, % Lie alg.
03.65.Fd, % Algebraic meth. (quant.mech.), 
02.40.Gh % non comm. geom.

\noindent MSC: 81R10, %inf. dim. alg., 
%17B65, % inf. dim. Lie alg. 
16S30, % univ env. alg., 
20C35, % group rep. phys.
%81R15 op. alg., 81R30 coh. stat., 
81S10, % geom \& quant.
46L65, %quant. def. 
53D55, %def. quant. 
81T05, %axiomatic qft, op. alg.
81R60, % nonc. geom., 


\noindent {\bf Keywords:} universal enveloping (factor) algebra, tensor operators, classical limit, 
simplectic diffeomorphisms, Poisson-Lie bracket, coadjoint orbit, 
quantum (Moyal) deformations, generalized Virasoro and 
${\cal W}_\infty$ symmetries, extended objects, 
non-commutative geommetry.  


\section{Introduction}
The long sought-for unification of all interactions and exact solvability of
(quantum) field theory and statistics parallels the quest for new symmetry
principles. Symmetry is an essential resource when facing
those two fundamental problems, either as a gauge guide principle or
as a valuable classification tool. The representation theory of infinite-dimensional
groups and algebras has not progressed very far, except for some important achievements
in one and two dimensions (mainly Virasoro, $\W_\infty$ and Kac-Moody symmetries), and
necessary breakthroughs in the subject remain to be carried out.

In the last decade, a large body of literature has been devoted to the study
of the so-called $\W$-algebras. These algebras were first introduced as
higher-conformal-spin $s> 2$ extensions \cite{Zamolodchikov} of the Virasoro algebra $(s=2)$ 
through the operator product expansion of the stress-energy
tensor and primary fields in two-dimensional conformal field theory.
$\W$-algebras have been widely used in two-dimensional physics, mainly in
condensed matter, integrable models
(Korteweg-de Vries, Toda), phase transitions in two dimensions,
stringy black holes and, at a more fundamental level, as the underlying
gauge symmetry  of two-dimensional gravity models
generalizing the Virasoro gauged symmetry in the light-cone discovered
by Polyakov \cite{Poly} by adding spin $s>2$ currents (see e.g.
\cite{Bergshoeff} and \cite{Shen,Hull} for a review). Only when all ($s\to\infty$) 
conformal spins $s\geq 2$ are considered,
the algebra (denoted by $\W_{\infty}$) is proven to be of Lie type;
moreover, currents of spin $s=1$ can also be included \cite{Pope2}, thus
leading to the Lie algebra ${\cal W}_{1+\infty}$, which plays a
determining role in the classification of all universality classes
of incompressible quantum fluids and the identification of the
quantum numbers of the excitations in the quantum Hall effect
\cite{Capelli}.

The process of elucidating the mathematical structure underlying $\W$ algebras has 
led to various directions. Geometric approaches identify the classical ($\hbar\to 0$) 
limit of $\W_\infty$ algebras with area-preserving 
(symplectic) diffeomorphism algebras of two dimensional surfaces \cite{Bakas,Witten}. These algebras 
possess a Poisson structure, and it is a current topic of great activity to 
recover the ``quantum commutator" through (Moyal-like) deformations of the Poisson bracket. 
There is a group-theoretic structure underlying these quantum deformations \cite{Pope}, 
according to which $\W$ algebras are just 
particular members of a one-parameter family $\cL_\cas(su(1,1))$
---in the notation of the present paper---
of non-isomorphic \cite{Hoppe2,Bergshoeff2} infinite-dimensional Lie-algebras of
$SU(1,1)$ tensor operators (when ``extended beyond the wedge" \cite{Pope} or 
``analytically continued" \cite{Fradkin2}). 
The (field-theoretic) connection with the theory of
higher-spin gauge fields in (1+1)- and (2+1)-dimensional anti-de Sitter space
AdS \cite{Fradkin2,Fradkin,Vasiliev} ---homogeneous spaces of $SO(1,2)\sim SU(1,1)$ and $SO(2,2)\sim
SU(1,1)\times SU(1,1)$, respectively--- is then apparent in this
group-theoretical context. Also, the relationship between area-preserving diffeomorphisms and 
$\W$ algebras emerges naturally in this group-theoretic picture; indeed, it is well known that 
coadjoint orbits of any semi-simple Lie group like $SU(1,1)\simeq SL(2,\mathbb{R})$ 
(cone and hyperboloid of one and two sheets) naturally define a symplectic manifold, and 
the symplectic structure inherited from the group can be used to yield a Poisson bracket, which 
leads to a geometrical approach to quantization. From an algebraic point of view, the Poisson bracket 
is the classical limit of the quantum commutator of ``covariant symbols" (see next section). However, 
the essence of the full quantum algebra is captured in a classical construction by extending the Poisson bracket 
to Moyal-like brackets. In particular, one can reformulate the (cumbersome) problem of calculating commutators of tensor 
operators of $\cL_\cas(su(1,1))$ in terms of (easier to perform) Moyal (deformed) brackets of 
polynomial functions on coadjoint orbits 
$\cO_\cas$ of $SU(1,1)$.  A further simplification then consists of taking advantage of the standard oscillator 
realization of semi-simple Lie algebra generators and replacing non-canonical by Heisenberg 
brackets. 

The classification and labelling of tensor operators of Lie groups
other than $SU(1,1)$ and $SU(2)$ is not an easy task in
general. In the letter \cite{infdimal}, the author provided an infinite 
set of tensor operators of $U(N_+,N_-)$ and
calculated the structure constants of this quantum associative operator algebra 
by taking advantage of the oscillator realization of 
the $U(N_+,N_-)$ Lie-algebra, in terms of $N=N_++N_-$ boson operators, and by using Moyal 
brackets. Tensor labelling coincides here with the standard Gel'fand -Weyl pattern for 
vectors in the carrier space of unirreps of $U(N)$.  Latter on, the particular case of ${\cL}_\infty(2,2)$ 
was identified in \cite{vp} as a 3+1-dimensional analogue of 
the Virasoro algebra, i.e. an infinite extension (promotion or ``analytic continuation" \cite{Fradkin2}) 
of the finite-dimensional conformal symmetry  $SU(2,2)\sim SO(4,2)$ in 
3+1D. Also, ${\cL}_\infty(2,2)$ was interpreted as a higher-conformal-spin extension of the diffeomorphism 
algebra diff$(4)$ of vector fields on a 4-dimensional manifold (just as ${\W}_\infty$ is a higher-spin 
extension of the Virasoro diff$(1)$ algebra), thus constituting a potential gauge guide principle towards the formulation 
of of induced conformal gravities (Wess-Zumino-Witten-like models) in realistic dimensions \cite{qg}. 
${\W}_\infty$-algebras also appear as central extensions of the algebra of (pseudo-)differential 
operators on the circle \cite{Bakas2}, and higher-dimensional analogues have been constructed 
in that context \cite{Ramosh}; although we do not find a clear connection with our construction.


In this article we focus on the mathematical interconnection between  
algebraic and geometric approaches to higher-$U(N_+,N_-)$-spin algebras, 
their classical limit and their relation with quantized symplectic 
diffeomorphisms on coadjoint orbits $\cO$ of $U(N_+,N_-)$.  The ideas of
Non-Commutative Geometry (NCG) apply perfectly in this picture,
providing ``granular" descriptions of the underlying ``quantum" space: the non-commutative analogue of $\cO$
(see \cite{Connes,Madore} for similar concepts). The
classical (commutative) case is recovered in the limit $\hbar\to 0$ (large scales) and
$\cas\to\infty$ (high density of points), so that ``volume elements" $C\sim \hbar\cas$ remain 
finite. The classical geometry is recovered as the classical limit of $G$-tensor operator algebras (see later on 
Sec. \ref{nctsec} and \ref{ncam}):
\[
\lim_{\hbar\to 0,\cas\to\infty} {\cal L}_\cas({\cal G})\simeq
C^\infty(\cO)\subset
{\rm sdiff}(\cO). 
\]

The organization of the paper is as follows. First we set the general 
context of our problem and remind some notions on the representation 
theory of Lie groups (in particular, we focus on pseudo-unitary groups). 
In Sec. \ref{su2su11} we discuss the two well known examples of tensor 
operator algebras of $SU(2)$ and $SU(1,1)$ and their relation to large-$N$ 
matrix models (and relativistic membranes) and ${\cal W}_{(1+)\infty}$ 
symmetries, respectively; some remarks on the non-commutative torus are 
made in Sec. \ref{nctsec}, in connection with $\W$-algebras. In Sec. 
\ref{ghsa} we extend these constructions to general pseudo-unitary groups 
and show how to build ``generalized $w_\infty$ algebras" and their quantum 
(Moyal) deformations. Then, in Sec. \ref{generalization}, we comment on the connection of these algebras with 
symplectic (volume-preserving) diffeomorphisms on higher-dimensional coadjoint orbits  
${\cO}$ and its potential role as residual gauge symmetries of 
extended objects (``$p$-branes ${\cO}$") in the light-cone gauge. Some comments on the potential 
relevance of the factor algebras  
$\cL_{\cas}({\cal G})$ on tractable 
non-commutative versions of algebraic (flag) manifolds $\cO$ of $\cal G^*$, like the 
maximal orbits $\cO_N=U(N)/U(1)^N$, are made in Sec. \ref{ncam}. Last 
Section is devoted to conclusions and outlook.

\section{Generalities}

Let us start by fixing notation and reminding some definitions and results on
group, tensor operator, Poisson and simplectic diffeomorphism algebras of a
Lie group $G$; in particular, we shall focus on pseudo-unitary groups:
\be G=U(N_+,N_-)=\{U\in M_{N\times N}(\mathbb{C})\,\,/\,\,U\Lambda
U^\dag=\Lambda\},\,\,\,N=N_++N_-,\ee
that is, groups of complex $N\times N$ matrices $U$ that leave invariant
the indefinite metric
$\Lambda={\rm diag}(1,\stackrel{N_+}{\dots},1,-1,\stackrel{N_-}{\dots},-1)$.
The Lie-algebra $\cG$ is generated by the step operators
$\hat{X}_{\alpha}^{\beta}$,
\be \cG=u(N_+,N_-)=\{\hat{X}_\alpha^\beta,\;\;{\rm with}\;\;
(\hat{X}_{\alpha}^{\beta})_{\mu}^{\nu}\equiv\hbar
\delta_{\alpha}^\nu\delta^{\beta}_\mu\},\label{pun}\ee
with commutation relations:
\be \left[{\hat{X}}_{\alpha_1}^{\beta_1},{\hat{X}}_{\alpha_2}^{\beta_2}\right]=\hbar
(\delta_{\alpha_2}^{\beta_1}{\hat{X}}_{\alpha_1}^{\beta_2}-
\delta_{\alpha_1}^{\beta_2}{\hat{X}}_{\alpha_2}^{\beta_1}).\label{sunmcom}\ee

There is a standard \emph{oscillator realization} of these step operators in
terms of $N$ boson operator variables ($\hat{a}^\dag_\alpha,\hat{a}^\beta$), given by:
 \be \hat{X}_{\alpha}^{\beta}=\hat{a}^\dag_\alpha \hat{a}^\beta,\,\,\,[\hat{a}^\beta,\hat{a}^\dag_\alpha]=
\hbar \delta_{\alpha}^{\beta}\mathbb{I},\,\,\,\,\alpha,\beta=1,\dots N,\label{bosoprea}\ee
which reproduces (\ref{sunmcom}) --we use the metric $\Lambda$ to raise and lower indices. 
Thus, for unitary irreducible representations of $U(N_+,N_-)$ we have the 
conjugation relation:
\be (\hat{X}_\alpha^\beta)^\dag=\Lambda^{\beta\mu}\hat{X}_\mu^\nu\Lambda_{\nu\alpha}.\label{conjrel}\ee
(sum over doubly occurring indices is understood unless otherwise stated). 
Sometimes it will be more convenient to use the generators 
$\hat{X}_{\alpha\beta}=\Lambda_{\alpha\mu}\hat{X}^\mu_\beta$ instead of $\hat{X}_\alpha^\beta$, 
for which the conjugation relation (\ref{conjrel}) is simply written as 
$\hat{X}_{\alpha\beta}^\dag=\hat{X}_{\beta\alpha}$, and the commutation relations (\ref{sunmcom}) adopt the form:
\be \left[{\hat{X}}_{\alpha_1\beta_1},{\hat{X}}_{\alpha_2\beta_2}\right]=\hbar
(\Lambda_{\alpha_2\beta_1}{\hat{X}}_{\alpha_1\beta_2}-\Lambda_{\alpha_1\beta_2}{\hat{X}}_{\alpha_2\beta_1}).\label{bosreal}\ee
The oscillator realization (\ref{bosoprea}) of $u(N_+,N_-)$ generators will be suitable for our 
purposes later on.

There are $N$ Casimir operators for $U(N_+,N_-)$, which are written as polynomials
of degree $1,2,\dots,N$ of step operators as follows:
\be \hat{C}_1=\hat{X}_{\alpha}^\alpha,\;\;\hat{C}_2=\hat{X}_{\alpha}^\beta \hat{X}_{\beta}^\alpha,\;\;
\hat{C}_3=\hat{X}_{\alpha}^\beta \hat{X}_{\beta}^\gamma \hat{X}_{\gamma}^\alpha,
\dots\label{Casimir}\ee

\begin{defn} Let $\cG^\otimes$ be the tensor algebra over $\cG$, and 
${\cal I}$ the ideal of $\cG^\otimes$ generated by $[\hat{X},\hat{Y}]-(\hat{X}\otimes 
\hat{Y}-\hat{Y}\otimes \hat{X})$ where $\hat{X},\hat{Y}\in\cG$. The universal enveloping algebra 
${\cal U}(\cG)$ is the quotient $\cG^\otimes/{\cal I}$.
\end{defn}
\begin{thm} (Poincar-Birkhoff-Witt) The elements 
$\hat{X}_{\alpha_1\beta_1}^{k_1}\dots\hat{X}_{\alpha_n\beta_n}^{k_n}$, 
with $k_i\geq 0$, form a basis of ${\cal U}(\cG)$.
\end{thm}
The universal enveloping algebra ${\cal U}(\cG)$ decomposes into 
\emph{factor or quotient Lie 
algebras} $\cL_{\cas}(\cG)$:
\begin{thm} Let 
\[{\cal I}_{\cas}=\prod_{\alpha=1}^N(\hat{C}_\alpha-\hbar^\alpha\cas_\alpha)
{\cal U}(\cG)\] 
be the ideal generated by the Casimir operators $\hat{C}_\alpha$. 
The quotient $\cL_{\cas}(\cG)={\cal U}(\cG)/{\cal I}_{\cas}$ is a Lie algebra (roughly speaking,
this quotient means that we replace $\hat{C}_\alpha$ by the complex c-number
$C_\alpha\equiv\hbar^\alpha\cas_\alpha$ whenever it appears in the commutators of elements of
${\cal U}(\cG)$). We shall refer to $\cL_{\cas}(\cG)$ as a $c$-tensor operator algebra.
\end{thm}

According to Burnside's theorem \cite{Kirillov},
for some critical values $\cas_\alpha=\cas^{(0)}_\alpha$ the infinite-dimensional Lie
algebra $\cL_{\cas}(\cG)$ ``collapses" to a finite-dimensional
one. In a more formal language:

\begin{thm}\label{Burnside}(Burnside) When $\cas_\alpha, \alpha=1,\dots,N$ coincide with the
eigenvalues of $\hat{C}_\alpha$ in a $d_{\cas}$-dimensional irrep $D_{\cas}$ of
$G$, there exists an ideal $\chi\subset\cL_{\cas}(\cG)$ such that
$\cL_{\cas}(\cG)/\chi=sl(d_{\cas},\mathbb{C})$,
or $su(d_{\cas})$, by taking a compact real form of the complex Lie
algebra.
\end{thm}

Another interesting structure related to the previous one is the \emph{group
$C^*$-algebra} $C^*(G)$ [in order to avoid some technical
difficulties, let us restrict ourselves to the compact $G$ case in the next 
discussion]:
\begin{defn} Let $C^\infty(G)$ be the set of analytic complex functions $\Psi$ on $G$ ,
\be C^\infty(G)=\left\{\Psi:G\to\mathbb{C},\,\,\, g\mapsto \Psi(g)\right\}.\ee
The group algebra $C^*(G)$ is a $C^*$-algebra with an invariant associative *-product
(convolution product):
\be (\Psi *\Psi')(g')\equiv\int_G d^Lg\,\Psi(g)
\Psi'(g^{-1}g'),\label{convoprod}\ee
($d^Lg$ denotes the left Haar measure) and an involution
$\Psi^*(g)\equiv\bar{\Psi}(g^{-1 })$.
\end{defn}

The following theorem (see \cite{Kirillov}) reveals a connection between group and enveloping 
algebras:
\begin{thm}\label{Schwartz}(L. Schwartz) The group algebra $C^*(G)$ is 
isomorphic to the enveloping algebra ${\cal U}(\cG)$.
\end{thm}
\noindent\emph{Proof:} we shall only sketch the main lines. Let us realize the Lie algebra $\cG$ by left invariant vector fields $\hat{X}^L$ on $G$ and 
consider the mapping $\Psi:\cG\to C^*(G), \hat{X}\mapsto\Psi_{\hat{X}}$, defined by the formula
\be
\langle\Psi_{\hat{X}}|\Psi\rangle\equiv(\hat{X}^L\Psi)(e),\;\;\forall\Psi\in C^*(G),
\ee
where $\langle\Psi|\Psi'\rangle\equiv\int_G d^Lg\,\bar{\Psi}(g)
\Psi'(g)$ denotes a scalar product and $(\hat{X}^L\Psi)(e)$ means the action of $\hat{X}^L$ on $\Psi$ 
restricted to the identity element $e\in G$. One can also verify that:
\be \langle\Psi_{\hat{X}_1}*\dots*\Psi_{\hat{X}_n}|\Psi\rangle=(\hat{X}^L_1\dots \hat{X}^L_n\Psi)(e), \,\,\forall \Psi\in
C^*(G). \blacksquare\ee


The classical limit of the convolution commutator $[\Psi,\Psi']=\Psi*\Psi'-\Psi'*\Psi$
corresponds to the Poisson-Lie bracket
\be  \{\psi,\psi'\}_{PL}(g)=
\lim_{\hbar\to 0}\frac{i}{\hbar^2}[\Psi,\Psi'](g)=
(\Lambda_{\alpha_2\beta_1}{x}_{\alpha_1\beta_2}-\Lambda_{\alpha_1\beta_2}
{x}_{\alpha_2\beta_1})
\frac{\partial \psi}{\partial x_{\alpha_1\beta_1}}
\frac{\partial \psi'}{\partial x_{\alpha_2\beta_2}} \label{poissonlie}\ee
between smooth functions $\psi\in C^\infty(\cG^*)$ on the coalgebra $\cG^*$, where
$x_{\alpha\beta}, \alpha,\beta=1,\dots,N$  denote a coordinate system in the coalgebra
${\cal G}^*=u(N_+,N_-)^*\simeq \mathbb{R}^{N^2}$, seen as a $N^2$-dimensional vector space.
The ``quantization map" relating $\Psi$ and $\psi$ is symbolically given by the
expression:
\be \Psi(g)=\int_{\cG^*}\frac{d^{N^2}\Theta}{(2\pi\hbar)^{N^2}}
e^{\frac{i}{\hbar}\Theta(\hat{X})}\psi(\Theta),\ee
where $g=\exp(\hat{X})=\exp(x^{\alpha\beta}\hat{X}_{\alpha\beta})$ is an element of
$G$ and $\Theta=\theta_{\alpha\beta}\Theta^{\alpha\beta}$ is an element of
$\cG^*$.

The constraints $\hat{C}_\alpha(x)=C_\alpha=\hbar^\alpha\cas_\alpha$ defined by the
Casimir operators (\ref{Casimir}) (written in terms of the coordinates
$x_{\alpha\beta}$ instead of $\hat{X}_{\alpha\beta}$) induce a foliation
\be\cG^*\simeq \bigcup_{C} \cO_{C}\ee
of the coalgebra $\cG^*$ into leaves $\cO_C$ (coadjoint orbits, algebraic (flag) 
manifolds). This foliation is the (classical) analogue of the (quantum) 
standard Peter-Weyl decomposition (see \cite{Landsman}) 
of the group algebra $C^*(G)$: 
\begin{thm}(Peter-Weyl) Let $G$ be a compact Lie group. The group algebra $C^*(G)$ 
decomposes,  
\be C^*(G)\simeq
\bigoplus_{\cas\in\hat{G}}\cL_{\cas}({\cal G}),\label{P-W}\ee 
into factor algebras $\cL_{\cas}({\cal
G})$, where $\hat{G}$ denotes the space of all (equivalence classes of)
irreducible representations  of $G$ of dimension $d_{\cas}$.
\end{thm}

The  leaves $\cO_{C}$ admit a symplectic
structure  $(\cO_{C},\Omega_{C})$, where $\Omega_{C}$
denotes a closed 2-form (a
K\"ahler form), which can be obtained from a K\"ahler potential
$K_{C}$ as:
\be \Omega_{C}(z,\bar{z})=\frac{\partial^2 K_{C}(z,\bar{z})}{\partial z_{\alpha\beta}
\partial \bar{z}_{\sigma\nu}}dz_{\alpha\beta}\wedge d\bar{z}_{\sigma\nu}=
\Omega^{\alpha\beta;\sigma\nu}_{C}(z,\bar{z})dz_{\alpha\beta}\wedge d\bar{z}_{\sigma\nu},
\ee
where $z_{\alpha\beta},\,\alpha>\beta$ denotes a system of complex
coordinates in $\cO_{C}$ (see \cite{harmanal} for more details).

After the foliation of $C^\infty(\cG^*)$ into Poisson algebras $C^\infty(\cO_{C})$,
the Poisson bracket induced on the leaves $\cO_{C}$ becomes:
\be
\left\{\psi_l^{\cas},\psi_m^{\cas}\right\}_P(z,\bar{z})=\sum_{\alpha_j>\beta_j}
\Omega_{\alpha_1\beta_1;\alpha_2\beta_2}^{\cas}(z,\bar{z})
\frac{\partial\psi_l^{\cas}(z,\bar{z})}{\partial z_{\alpha_1\beta_1}}
\frac{\partial\psi_m^{\cas}(z,\bar{z})}{\partial \bar{z}_{\alpha_2\beta_2}}=
\sum_{n}f_{lm}^n(\cas)\psi_n^{\cas}(z,\bar{z}),\label{poiscoad}
\ee
where $\psi_{l,m,n}^{\cas}$ belong to the carrier space ${\cal H}_{\cas}(G)$ 
of a given irrep of $G$.
The structure constants  for (\ref{poiscoad}) can be obtained through the scalar product $f_{lm}^{n}(c)=
\langle {\psi_n^{\cas}}|\{\psi_l^{\cas},\psi_m^{\cas}\}_P\rangle$, with 
integration measure (\ref{intmeasleaf}), 
when the set $\{{\psi}_n^{\cas}\}$ is chosen to be
orthonormal.

To each function $\psi\in
C^\infty(\cO_{C})$, one can assign its Hamiltonian vector field
${H}_\psi\equiv\{\psi,\cdot\}_P$, which is divergence-free and
preserves de natural volume form 
\be d\mu_{C}(z,\bar{z})=(-1)^{\left(\ba{c} n\\ 2\ea\right)}\frac{1}{n!}
\Omega_{C}^n(z,\bar{z}),\;\;2n={\rm dim}(\cO_C).\label{intmeasleaf}\ee

In general, any vector field $H$ obeying $L_H\Omega=0$ (with $L_H\equiv i_H\circ d+d\circ i_H$ the 
Lie derivative) is called locally Hamiltonian. The space ${\rm LHam}(\cO)$ of locally Hamiltonian vector fields is 
a subalgebra of the algebra ${\rm sdiff}(\cO_C)$ of symplectic (volume-preserving)
diffeomorphisms of $\cO_{C}$, and the space ${\rm Ham}(\cO_C)$ of Hamiltonian vector fields is an ideal 
of ${\rm LHam}(\cO)$. The two-dimensional case ${\rm dim}(\cO)=2$ is special because 
${\rm sdiff}(\cO_C)={\rm LHam}(\cO)$, and the quotient ${\rm LHam}(\cO)/{\rm LHam}(\cO)$ can 
be identified with the first de-Rham cohomology class $H^1(\cO,\mathbb{R})$ of $\cO$ via $H\mapsto i_H\Omega$.

Also, Poisson and symplectic diffeomorphism algebras of $\cO_{C_+}=S^2$ and
$\cO_{C_-}=S^{1,1}$ (the sphere and the
hyperboloid)  appear as the classical limit [small $\hbar$ and large (conformal-)spin
$\cas_\pm=s(s\pm 1)$, so that the curvature radius $C_\pm=\hbar^2c_\pm$ remains finite]:

\begin{eqnarray}\lim_{\stackrel{\cas_+\to\infty}{\hbar\to 0}}{\cal L}_{\cas_+}(su(2))&\simeq
&C^\infty (S^2)
\simeq {\rm sdiff}(S^2)\simeq su(\infty),\label{largeN}\\
\lim_{\stackrel{\cas_-\to\infty}{\hbar\to 0}}{\cal L}_{\cas_-}(su(1,1))&\simeq&
C^\infty (S^{1,1})
\simeq {\rm sdiff}(S^{1,1})\simeq
 su(\infty,\infty)\nn\end{eqnarray}
of factor algebras of $SU(2)$  and $SU(1,1)$, respectively 
(see \cite{Fradkin2,Bergshoeff2}).\footnote{The approximation ${\rm sdiff}(S^2)\simeq su(\infty)$ 
is still not well understood and additional work should be done towards 
its satisfactory formulation. In \cite{glinfty} the approach to approximate ${\rm sdiff}(S^2)$ and 
${\rm sdiff}(T^2)$ by $\lim_{N\to\infty}su(N)$ was studied and a weak uniqueness theorem was proved; however, 
whether choices of sets of basis functions on spaces with different topologies do in fact correspond to distinct 
algebras deserves more careful study.}


Let us clarify the classical limits (\ref{largeN}) by making use of the {\it operator (covariant) symbols} 
\cite{Berezin},
\be L^{\cas}(z,\bar{z})\equiv\langle{\cas} z|\hat{L}|{\cas} z\rangle,
\,\,\, \hat{L}\in\cL_{\cas}(\cG), \label{covsymb}
\ee
constructed as the mean value of a tensor operator $\hat{L}\in\cL_{\cas}(\cG)$ in the coherent states (CS) 
\be \psi^{\cas}_{\bar{z}}(z')\equiv\langle \cas z'|\cas z\rangle=\sum_{\{n\}}\psi^{\cas}_n(z')
\bar{\psi}^{\cas}_n(z),\ee
where we have made use of the completeness relation 
\be
\sum_{\{n\}}|\cas n\rangle\langle\cas n|=\mathbb{I}_{\cas},
\ee
with $|\cas n\rangle, \langle\cas n|$ the Dirac's notation for the vectors 
$\psi^{\cas}_n(z), \bar{\psi}^{\cas}_n(z)$, respectively,  which lower index $n$ denotes an 
integral $N\times N$ upper triangular matrix like (\ref{uppert}), and $\mathbb{I}_{\cas}$ 
denotes the identity operator in the irreducible representation of label 
$\cas$ (see \cite{harmanal} for more details about explicit irreducible 
representations of $U(N_+,N_-)$, coherent states and applications). Using 
the completeness relation 
\be \int_{\cO_C} |\cas u\rangle\langle \cas u| d\mu_{C}(u,\bar{u})=\mathbb{I}_{\cas}\ee
for coherent states, one can define the so called \emph{star multiplication of symbols} 
$L^\cas_1\star L^\cas_2$ as the symbol of the product $\hat{L}_1\hat{L}_2$ of two 
tensor operators $\hat{L}_1$ and $\hat{L}_2$:
\be
(L^\cas_1\star L^\cas_2)(z,\bar{z})\equiv\langle{\cas} z|\hat{L}_1\hat{L}_2|{\cas} z\rangle=
\int_{\cO_C} L^\cas_1(z,\bar{u}) L^\cas_2(u,\bar{z})
e^{-s^2(z,u)}d\mu_{\cas}(u,\bar{u}),\label{symbolprod}\ee
where we introduce the non-diagonal symbols
\be
L^\cas(z,\bar{u})=
\frac{\langle{\cas} z|\hat{L}|\cas u\rangle}{\langle\cas z|\cas 
u\rangle}\ee
and $s^2_{\cas}(z,u)\equiv -\ln|\langle \cas z|\cas u\rangle|^2$ can be 
interpreted as the square of the distance between the points $z,u$ on the 
coadjoint orbit $\cO_{C}$. Using general properties of coherent states, 
it can be easily seen that $s^2_{\cas}(z,u)\geq 0$ tends to infinity with 
$\cas\to\infty$, if $z\not=u$, and equals zero if $z=u$. Thus, one can conclude that, in that limit, the domain 
$u\approx z$ gives only a contribution to the integral (\ref{symbolprod}). 
Decomposing the integrand near the point $u=z$ and going to the 
integration over $w=u-z$, it is straightforward to see that (see \cite{harmanal}):
\be
L^\cas_1\star L^\cas_2-L^\cas_2\star L^\cas_1=
i\left\{L^\cas_1,L^\cas_2\right\}_P+{\rm 
O}(1/c_\alpha),\ee
where the quantities $1/c_\alpha\sim\hbar^\alpha$ (inverse Casimir eigenvalues) 
play the role of the Planck constant $\hbar$, and we have used that 
$ds^2_c=\Omega^{\alpha\beta;\sigma\nu}_{C}dz_{\alpha\beta} 
d\bar{z}_{\sigma\nu}$ (Hermitian Riemannian metric of $\cO_C$).



Before going to the general case, let us discuss the two well known
examples of $SU(2)$ and $SU(1,1)$.

\section{Tensor operator algebras of $SU(2)$ and $SU(1,1)$\label{su2su11}}
\subsection{Tensor operator algebras of $SU(2)$ and large-$N$ matrix
models\label{su2largeN}}

Let  $\hat{J}^{(N)}_i,\,\,i=1,2,3$ be three
$N\times N$ hermitian  matrices
with commutation relations:
\be \left[\hat{J}^{(N)}_{i},\hat{J}^{(N)}_{j}\right]=i \hbar\epsilon_{ijk}\hat{J}^{(N)}_{k},\ee
that is, a $N$-dimensional irreducible representation of the angular momentum
operator algebra $su(2)$. The Casimir operator $\hat{C}_2=
(\hat{{J}}^{(N)})^2=\hbar^2\frac{N^2-1}{4}$ behaves as a constant. The
factor algebra $\cL_{N}(su(2))$ is generated by the
$SU(2)$-tensor operators:

\be \hat{T}^I_m(N)\equiv \sum_{\stackrel{i_k=1,2,3}{k=1,\dots,I}}
\kappa^{(m)}_{i_1,\cdots,i_I} \, \hat{J}^{(N)}_{i_1}\cdots \hat{J}^{(N)}_{i_I},
\label{su2to}
\ee
where the upper index $I=1,\dots,N-1$ is the spin label,
$m=-I,\dots,I$ is the third component and the complex coefficients
$\kappa^{(m)}_{i_1,\cdots,i_I}$ are  the components of a symmetric and traceless tensor.
According to Burnside's theorem, the factor algebra  $\cL_{N}(su(2))$ is
isomorphic to $su(N)$. Thus, the commutation relations:
\be
\left[\hat{T}^I_m(N),\hat{T}^J_n(N)\right]=f^{IJl}_{mnK}(N)\hat{T}^K_l(N) \label{su2tocr}\ee
are those of the $su(N)$ Lie algebra, where $f^{IJl}_{mnK}(N)$ symbolize the
structure constants which, for the Racah-Wigner basis of tensor operators
\cite{Biedenharn}, can be written in terms of
Clebsch-Gordan and (generalized) $6j$-symbols \cite{Hoppe,Pope,Fradkin2}.

The formal limit $N\to\infty$ of the commutation relations (\ref{su2tocr}) coincides
with the Poisson bracket
\be \left\{Y^I_m,Y^J_n\right\}_P=
\frac{i}{\sin\vartheta}
\left(\frac{\partial Y^I_m}{\partial\vartheta}
\frac{\partial Y^J_n}{\partial\varphi}-\frac{\partial Y^I_m}{\partial\varphi}
\frac{\partial Y^J_n}{\partial\vartheta}
\right)=f^{IJl}_{mnK}(\infty)Y^K_l\label{spherepb}\ee
between spherical harmonics 
\be Y^I_m(\vartheta,\varphi)\equiv\sum_{\stackrel{i_k=1,2,3}{k=1,\dots,I}}
\kappa^{(m)}_{i_1,\cdots,i_I} \, x_{i_1}\cdots x_{i_I},\ee
which are defined in a similar way to tensor operators (\ref{su2to}), but replacing
the angular momentum operators $\hat{{J}}^{(N)}$ by the coordinates
${x}=(\cos\varphi\sin\vartheta,\sin\varphi\sin\vartheta,\cos\vartheta)$, i.e. its covariant symbols 
(\ref{covsymb}). Indeed, the
large-$N$ structure constants can be calculated through the scalar product (see \cite{Floratos}):
\begin{eqnarray*}\lim_{N\to\infty} f^{IJl}_{mnK}(N)&=&f^{IJl}_{mnK}(\infty)=
\langle Y^K_l|\{Y^I_m,Y^J_n\}_P\rangle\\ &=&\int_{S^2}\sin\vartheta
d\vartheta d\varphi\,
\bar{Y}^K_l(\vartheta,\varphi)\{Y^I_m,Y^J_n\}_P(\vartheta,\varphi).\end{eqnarray*}

The set of Hamiltonian vector fields $H^I_m\equiv \left\{Y^I_m,\cdot\right\}_P$ close the
algebra ${\rm sdiff}(S^2)$ of area-preserving diffeomorphisms of the
sphere, which can be identified with $su(\infty)$ in the (``weak convergence") sense of
\cite{glinfty} --see Eq. (\ref{largeN}).
This fact was used in \cite{Hoppe} to approximate the residual gauge symmetry
${\rm sdiff}(S^2)$ of the relativistic spherical membrane by $su(N)|_{N\to\infty}$.
There is an intriguing connection between this
theory and the quantum mechanics of space constant (``vacuum configurations") $SU(N)$
Yang-Mills potentials
\be A_\mu(x)^i_j=\sum_{a=1}^{N^2-1}A^a_\mu(x) (\hat{T}_a)^i_j, \,\,\,\,\hat{T}_a=\hat{T}^I_m(N),\label{sunympot}\ee
in the limit of ``large number of colours'' (large-$N$). Indeed, the low-energy limit of
the $SU(\infty)$ Yang-Mills action
\begin{eqnarray} {\cal S}&=&\int {\rm d}^4x \langle F_{\mu\nu}(x)|
F^{\mu\nu}(x)\rangle,\nn\\
F_{\mu\nu}&=&\partial_\mu A_\nu -\partial_\nu A_\mu+\{A_\mu,A_\nu\}_P,\label{suinftyaction}\\
A_\mu(x;\vartheta,\varphi)&=&\sum_{I,m}A^{Im}_\mu(x)
Y^I_m(\vartheta,\varphi),\nn\end{eqnarray}
described by space-constant $SU(\infty)$ vector potentials
$X_\mu(\tau;\vartheta,\varphi)\equiv A_\mu(\tau,\vec{0};\vartheta,\varphi)$,
turns out to reproduce the dynamics of the relativistic spherical
membrane (see \cite{Floratos}). Moreover, space-time constant $SU(\infty)$ vector potentials
$X_\mu(\vartheta,\varphi)\equiv A_\mu(0;\vartheta,\varphi)$ lead to the
Schild action density for (null) strings \cite{Schild}; the
argument that the internal symmetry space of the $U(\infty)$ pure
Yang-Mills theory must be a functional space, actually the space of
configurations of a string, was pointed out in Ref. \cite{Gervais}. Replacing the Sdiff$(S^2)$-gauge 
invariant theory (\ref{suinftyaction}) by a $SU(N)$-gauge invariant theory with vector potentials (\ref{sunympot}) 
then provides a form of regularization. 

We shall see later in Sec. \ref{generalization} how actions for
relativistic symplectic $p$-branes (higher-dimensional coadjoint orbits) can be defined for
general (pseudo-)unitary groups $SU(N_+,N_-)$ in a similar way.

\subsection{Tensor operator algebras of $SU(1,1)$ and ${\cal W}_{(1+)\infty}$ symmetry}

As already stated in the introduction, $\W$ algebras were first introduced as
higher-conformal-spin $(s> 2)$ extensions \cite{Zamolodchikov} of the Virasoro algebra $(s=2)$ 
through the operator product expansion of the stress-energy
tensor and primary fields in two-dimensional conformal field theory. 
Only when all ($s\to\infty$) conformal spins $s\geq 2$ are considered,
the algebra (denoted by $w_{\infty}$) is proven to be of Lie type;
moreover, currents of spin $s=1$ can also be included, thus
leading to the Lie algebra $w_{1+\infty}$, which plays a
determining role in the classification of all universality classes
of incompressible quantum fluids and the identification of the
quantum numbers of the excitations in the quantum Hall effect
\cite{Capelli}.

The algebras $w$ prove to have a space-time origin as (symplectic)
diffeomorphism algebras and Poisson algebras of functions 
on symplectic manifolds. For example, $w_{1+\infty}$ is related to
the algebra of area-preserving diffeomorphisms of the cylinder. In
fact, let us choose the next set of classical functions of the
bosonic (harmonic oscillator) variables
$a(\bar{a})=\frac{1}{\sqrt{2}}(q\pm ip)=\rho e^{\pm i\vartheta}$ (we
are using mass and frequency $m=1=\omega$, for simplicity):
\be\ba{l} L^I_{+|m|}\equiv\um
(a\bar{a})^{I-|m|}a^{2|m|}=\um\rho^{2I}e^{2i|m|\vartheta},\\
L^J_{-|n|}\equiv\um
(a\bar{a})^{J-|n|}\bar{a}^{2|n|}=\um\rho^{2J}e^{-2i|n|\vartheta},\ea\label{auaral}
\ee where $n,m\in \mathbb Z/2; I,J\in \mathbb Z^+/2$. 
A straightforward calculation from the basic Poisson bracket
$\{a,\bar{a}\}=i$ provides the following formal Poisson algebra:
\be
\{L^I_m,L^J_n\}=i\left( \frac{\partial L^{I}_{m} }{\partial a}
\frac{\partial L^{J}_{n}}{\partial \bar{a}}- \frac{\partial
L^{I}_{m} }{\partial \bar{a}} \frac{\partial L^{J}_{n}}{\partial
a}\right)=i(In-Jm)L^{I+J-1}_{m+n}, \label{auaralcom} \ee of
functions $L$ on a two-dimensional phase space (see
\cite{simplin}). As a distinguished subalgebra of
(\ref{auaralcom}) we have the set: \be su(1,1)=\{L_0\equiv
L^1_0=\um a\bar{a},\, L_+\equiv L^1_1=\um a^2,\, L_-\equiv
L^1_{-1}=\um \bar{a}^2\}, \ee which provides an oscillator realization of 
the $su(1,1)$ Lie algebra generators $L_\pm,L_0$, in terms of a single 
bosonic variable, with commutation relations (\ref{su11cr}). With this notation, the
functions $L^I_{m}$ in (\ref{auaral}) can be also written as:
\be
L^I_{\pm |m|}=2^{I-1}(L_0)^{I-|m|}(L_\pm)^{|m|}.\label{auarop} \ee
This expression will be generalized for arbitrary $U(N_+,N_-)$
groups in (\ref{auarop2}). Following on the analysis of distinguished
subalgebras of (\ref{auaralcom}), we have the subalgebra 
\be w_\wedge\equiv\{ L^I_m, \,\, 
I-|m|\in\mathbb N\}\label{wedge}\ee of 
polynomial functions on the bosonic variables $(a,\bar{a})$, which can be formally extended ``beyond the wedge" 
$I-|m|\geq 0$ by considering functions on the punctured complex plane, with $I\geq 
0$. To the last set belong the (conformal-spin-2)
generators $L_n\equiv L^1_n, \,n\in\mathbb Z$, which close the Virasoro algebra without
central extension,
\be
\{L_m,L_n\}=i(n-m)L_{m+n},
\ee
and the (conformal-spin-1) generators
$\phi_m\equiv L^{0}_m$, which close the non-extended Abelian Kac-Moody
algebra,
\be
\{\phi_m, \phi_n\}=0.
\ee
In general, the higher-$su(1,1)$-spin fields $L^{I}_n$ have
``conformal-spin" $s=I+1$ and ``conformal-dimension" $n$ (the eigenvalue of
$L^1_0$).

$w$-algebras have been used as the underlying gauge symmetry  of
two-dimensional gravity models, and induced actions for these
``$w$-gravities'' have been written (see for example
\cite{Bergshoeff}), which turn out to be constrained
Wess-Zumino-Witten models \cite{Nissimov}, as happens with
standard induced gravity. The quantization procedure {\it deforms}
the classical algebra $w$ to the quantum algebra $\W$ due to the
presence of anomalies ---deformations of Moyal type of Poisson and
symplectic-diffeomorphism algebras caused essentially by normal
order ambiguities (see bellow). Also, generalizing the
$SL(2,\mathbb{R})$ Kac-Moody hidden symmetry of Polyakov's induced
gravity, there are $SL(\infty,\mathbb{R})$ and
$GL(\infty,\mathbb{R})$ Kac-Moody hidden symmetries for
$\W_{\infty}$ and  ${\cal W}_{1+\infty}$ gravities, respectively
\cite{Popehidden}. Moreover, as already mentioned, the symmetry
${\cal W}_{1+\infty}$ appears to be useful in the classification
of universality classes in the fractional quantum Hall effect.

The group-theoretic structure underlying these $\W$ algebras was
elucidated in \cite{Pope}, where  $\W_{\infty}$ and
${\cal W}_{1+\infty}$ appeared to be distinct members ($\cas=0$ and
$\cas=-1/4$ cases, respectively) of the (``extension beyond the wedge" of the) 
one-parameter family
$\cL_\cas(su(1,1))$ of non-isomorphic \cite{Hoppe2,Bergshoeff2}
infinite-dimensional factor Lie-algebras of $SU(1,1)$ tensor
operators
\bea
\hat{L}^I_{\pm |m|}&\sim& \underbrace{\left[ \hat{L}_{\mp},\left[
\hat{L}_{\mp},\dots \left[
\hat{L}_{\mp},\right.\right.\right.}_{I-|m|\,\,{\rm times}}
\left.\left.\left. (\hat{L}_\pm)^I\right] \dots\right] \right] =
({\rm
ad}_{\hat{L}_{\mp}})^{I-|m|}(\hat{L}_\pm)^I \label{sl2rtensorop}\\
&\sim&\hat{L}_0^{I-|m|}\hat{L}_\pm^{|m|}+{\rm 
O}(\hbar),\nn
\eea where the $su(1,1)$ Lie-algebra generators 
$\hat{L}_+=\hat{X}_{12},\hat{L}_-=\hat{X}_{21},\hat{L}_0=(\hat{X}_{22}-\hat{X}_{11})/2$, 
fulfil the standard commutation relations
\be
\left[\hat{L}_\pm,\hat{L}_0\right]=\pm
\hbar\hat{L}_\pm\,,\;\;\;\;\;
\left[\hat{L}_+,\hat{L}_-\right]=2\hbar\hat{L}_0,\label{su11cr}
\ee
and $\hat{C}=(\hat{L}_0)^2-\um(\hat{L}_+\hat{L}_-+\hat{L}_-
\hat{L}_+)$ is the  Casimir operator of $su(1,1)$. The structure
constants for $\cL_\cas(su(1,1))$ can be written in terms of
$sl(2,\mathbb{R})$ Clebsch-Gordan coefficients and generalized (Wigner) $6j$-symbols
\cite{Pope,Fradkin2}, and they have the general  form: \be
\left[\hat{L}^I_m,\hat{L}^J_n\right]_\cas = \sum_{r=0}^\infty
\hbar^{2r+1}f^{IJ}_{mn}(2r;\cas)
\hat{L}^{I+J-(2r+1)}_{n+m}+\hbar^{2I}Q_I(n;\cas)
\delta^{I,J}\delta_{n+m,0}\mathbb{I},\label{infq}\ee where
$\mathbb{I}\sim\hat{L}^0_0$ denotes a central generator and the 
central charges $Q_I(n;\cas)$ provide for the existence of central extensions. For example,
$Q_1(n;\cas)=\frac{c}{12}(n^3-n)$ reproduces the typical central
extension in the Virasoro sector and $Q_I(n;\cas)$ supplies 
central charges to all  conformal-spins $s=I+1$. Quantum deformations  
of (\ref{wedge}) do not introduce true 
central extensions. The inclusion of central terms in (\ref{infq}) 
requires the formal extension of (\ref{wedge}) beyond the wedge 
$I-|m|\geq 0$ (see \cite{Pope}), that is, 
the consideration of non-polynomial functions (\ref{auaral}) on $a$ and $\bar{a}$.


Central charges
provide the essential ingredient required to construct invariant
geometric action functionals on coadjoint orbits of the
corresponding groups. When applied to Virasoro and ${\cal W}$
algebras, they lead to Wess-Zumino-Witten models for {\it induced
conformal gravities in $1+1$ dimensions} (see e.g. Ref.
\cite{Nissimov}). Also, local and non-local versions of the Toda systems emerge, as integrable dynamical systems, 
from the one-parameter family (\ref{Moyaleq}) of (``quantum tori Lie") subalgebras 
of $gl(\infty)$ (see \cite{HoppeDS}). Infinite-dimensional analogues of rigid tops are discussed in 
\cite{HoppeDS} too; some of these systems give rise to ``quantized" (magneto) hydrodynamic equations of an 
ideal fluid on a torus.


The leading order (${\rm O}(\hbar), r=0$) structure constants
$f^{IJ}_{mn}(0;\cas)=Jm-In$ in (\ref{infq}) reproduce the
classical structure constants in (\ref{auaralcom}). It is also precisely 
for the specific values of $\cas=0$ and $\cas=-\frac{1}{4}$ ($\W_{\infty}$ and
${\cal W}_{1+\infty}$, respectively) that  
the sequence of higher-order terms on the right-hand side of (\ref{infq}) turns out to be zero whenever 
$I+J-2r\leq 2$ and $I+J-2r\leq 1$, respectively. Therefore,  $W_{\infty}$ (resp. $W_{1+\infty}$) 
can be consistently truncated to a 
closed algebra containing only those generators with conformal-spins $I+1 \geq 2$ 
(resp. $I+1 \geq 1$).

The higher-order terms (${\rm O}(\hbar^3), r\geq 1$) can be captured
in a classical construction by extending the Poisson bracket
(\ref{auaralcom}) to the Moyal bracket
\be
\left\{L^{I}_{m},L^{J}_{n}\right\}_{{\rm M}}=
{L}^{I}_{m}\star {L}^{J}_{n}-{L}^{J}_{n}\star {L}^{I}_{m}=\sum_{r=0}^{\infty}
2\frac{(\hbar/2)^{2r+1}}{(2r+1)!}P^{2r+1}(L^{I}_{m},L^{J}_{n})\,,\label{Moyal}
\ee
where $L\star L'\equiv\exp(\frac{\hbar}{2} P)(L,L')$ is an
{\it invariant associative $\star$-product} and
\be
P^r(L,L')\equiv\Upsilon_{\imath_1\jmath_1}\dots\Upsilon_{\imath_r\jmath_r}
\frac{\partial^r L}{\partial x_{\imath_1}\dots\partial
x_{\imath_r}}\frac{\partial^rL'}{\partial x_{\jmath_1}\dots\partial
x_{\jmath_r}}\,,\label{star}
\ee
with $x\equiv(a,\bar{a})$ and  $\Upsilon\equiv
\left(\begin{array}{cc} 0 & 1 \\ -1 &0\ea\right)$. We set $P^0(L,L')\equiv L\cdot L'$,
the ordinary (commutative) product of functions. Indeed, Moyal brackets where
identified as the primary quantum deformation ${\cal W}_\infty$ of the
classical algebra $w_\infty$ of area-preserving diffeomorphisms of the
cylinder (see Ref. \cite{Fairlie}).  Also, the oscillator realization in (\ref{auaral}) of the $su(1,1)$ Lie-algebra
generators $L_\pm,L_0$ in terms of a single boson $(a,\bar{a})$ is related to the  the ``symplecton'' algebra
of Biedenharn and Louck $\cL_{(-\frac{3}{16})}(su(1,1))$ \cite{Biedenharn} and the higher-spin algebra 
hs(2) of Vasiliev \cite{Vasiliev}.

\subsubsection{Non-commutative torus\label{nctsec}}
Before finishing this section, let us illustrate briefly the ``collapse" phenomenon mentioned in theorem
\ref{Burnside}, with the simple example of (a subalgebra of) $\cL_\cas(su(1,1))$, and make some comments
on the ``noncommutative torus" \cite{nctorusref}.

Let us consider the following new set of classical functions of the oscillator variables
$a(\bar{a})=\frac{2\pi}{\ell}(x_1\pm ix_2)$:
\be {L}_{\vec{n}}\equiv e^{\frac{2\pi i}{\ell} \vec{n}\cdot\vec{x}} =
\sum_{I=0}^\infty\sum_{l=0}^I 2(-1)^{I}
\frac{(n_1+in_2)^{I+l}}{2^{I+l}(I+l)!}\frac{(n_1-in_2)^{I-l}}{2^{I-l}(I-l)!}
\,\,L^I_l, \label{torusfunc}
\ee
obtained from (\ref{auaral}), where $\vec{x}=(x_1,x_2)$
is a pair of real coordinates (modulo $\ell$)
and $\vec{n}=(n_1,n_2)$ is a pair of integer numbers.
We identify (\ref{torusfunc}) with the set $C^\infty(T^2)$ of
smooth functions on a two-dimensional torus,
which is embedded in the set of functions (\ref{auaral})
on a cylinder [note that the lower-index $l$ of $L^I_l$ in (\ref{torusfunc}) is
restricted by $0\leq l\leq I$, whereas it can take any positive (half-)integer
value in (\ref{auaral})]. We shall restrict ourselves to this
subset of the whole $w$-algebra  (\ref{auaralcom})
where the above-mentioned ``collapse" phenomenon will be more apparent.

The ordinary product of functions $L_{\vec{m}}\cdot L_{\vec{n}}=L_{\vec{m}+\vec{n}}$ defines
$C^\infty(T^2)$ as a commutative algebra. We can assign a (Hamiltonian) vector field
$H_{\vec{m}}\equiv \{ L_{\vec{m}},\cdot\}_{{\rm P}}$ to any function $L_{\vec{m}}$, where:
\be
\left\{  L_{\vec{m}}, L_{\vec{n}}\right\}_{{\rm P}}= P^1(
L_{\vec{m}}, L_{\vec{n}})= \Upsilon_{jk}\frac{\partial
L_{\vec{m}}}{\partial x_j} \frac{\partial  L_{\vec{n}}}{\partial
x_k}= \frac{4\pi^2}{\ell^2}\vec{n}\times\vec{m}\,
L_{\vec{m}+\vec{n}}\,, \label{Poissontorus} \ee denotes the Poisson
bracket.  The vector fields $H_{\vec{m}}$ constitute a basis of symplectic
diffeomorphisms sdiff$(T^2)$, that is, they preserve the area
element $dx_1\wedge dx_2$ of the torus.

The quantum analogue of $C^\infty(T^2)$ can be captured from a classical construction by extending the
Poisson bracket (\ref{Poissontorus}) to its deformed version, the Moyal bracket
(\ref{Moyal}):
\bea
\left\{ L_{\vec{m}}, L_{\vec{n}}\right\}_{{\rm M}}&=&
\sum_{r=0}^{\infty}
\frac{2}{(2r+1)!}\left(\frac{\kbar^2}{2\pi i}\right)^{2r+1}
P^{2r+1}( L_{\vec{m}}, L_{\vec{n}})\nn\\ &=&
2i\sin\left(2\pi \frac{\kbar^2}{\ell^2} \vec{m}\times\vec{n}\right)
\hat{L}_{\vec{m}+\vec{n}}, \label{Moyaleq}
\eea
where $\kbar$ has length dimension (``Planck length") and $P^r$ now adopts the following form:
\be
P^r(L,L')=\sum_{l=0}^r(-1)^l\left(\ba{c} r\\ l\ea\right)
[\partial^{r-l}_{x_1}\partial^{l}_{x_2} L]
[\partial^{r-l}_{x_2}\partial^{l}_{x_1} L']. \label{star2} \ee The
$\star$-product  (\ref{Moyal}) also admits an integral
representation:
\be
(L\star L')(\vec{x})=\frac{i\ell^2}{4\pi\kbar^6}\int_0^\ell
d\vec{x}'d\vec{x}''e^{-\frac{2\pi i}{\kbar^2}|\vec{x}\vec{x}'\vec{x}''|}
L(\vec{x}')L'(\vec{x}'') \label{convotorus}
\ee
with $|\vec{x}\vec{x}'\vec{x}''|\equiv \vec{x}\times\vec{x}'+
\vec{x}'\times\vec{x}''+\vec{x}''\times\vec{x}$,
which is interesting when one wants to extend the
$\star$-product to other symplectic manifolds like coadjoint orbits ${\cal
O}$ through the convolution product (\ref{convoprod}).

Thus, this $\star$-product equips the set (\ref{torusfunc}) with a
noncommutative $C^\star$-algebra structure,
which we shall denote by $C^\star_\cas(T^2)$.
Just as the standard geometry of $T^2$ can be described by using the
algebra $C^\infty(T^2)$ of smooth complex functions (\ref{torusfunc}) on
$T^2$ with the ordinary (commutative) product, a noncommutative
geometry for $T^2$ can be described
by using its ``quantum" analogue $C^\star_\cas(T^2)$.
Noncommutative geometry offers a broader spectrum of
possibilities. In fact,
let us see how the definition of ``quantum torus" is
richer than that of the (standard) ``classical torus",
which eventually comes up as a particular
limiting case of the former one.

Note that, when the surface of the torus $\ell^2$ contains an integer
number $q$ of times the {\it minimal cell} $\kbar^2$ (that is,
$\ell^2=q \kbar^2$), the infinite-dimensional algebra
(\ref{Moyaleq}) collapses to a finite-dimensional matrix algebra: the Lie
algebra of the unitary group $U(q/2)$ for $q$ even or
$SU(q)\times U(1)$ for $q$ odd (see \cite{Fairlie2}).
In fact, taking the quotient in (\ref{Moyaleq}) by the equivalence
relation  ${L}_{\vec{m}+q\vec{a}}\sim
{L}_{\vec{m}},\,\forall \vec{a}\in {\mathbb Z}\times \mathbb Z$, it can be seen that
the following identification 
\be {L}_{\vec{m}}=\sum_{k}
e^{\frac{2\pi i}{q}m_1k}\hat{X}_{k,k+m_2}\label{trigbasis}\ee
implies a change of basis in the Lie-algebra (\ref{pun})
of $U(q)$ step-operators.

Thinking of $\cas=\frac{\ell^2}{\kbar^2}$ as a ``density of quantum points",
we can conclude that: for the critical values $\cas^{(0)}=q\in \mathbb Z$, the Lie algebra
(\ref{Moyaleq}) is {\it finite};
that is, the quantum analogue of the torus has a ``finite number $q$ of
quantum points". It is in this sense that we talk about a
``cellular structure of space". Actually, given the formal basic commutator
$\left[ \hat{x}_1,\hat{x}_2\right]=-i\kbar^2/\pi$ between position operators on the torus,
this cellular structure is a
consequence of the absence of localization expressed by the Heisenberg
uncertainty relation $\Delta \hat{x}_1 \Delta \hat{x}_2 \geq \kbar^2/(2\pi)$.
In the (classical) limit of large number of points $\cas\to\infty$ and
$\kbar\to 0$ (such that the size $\ell^2=\cas\kbar^2$ of the torus remains finite) we
recover the original (commutative) geometry on the torus.
For example, it is easy to see that
\be
\lim_{\cas\to \infty,\,\kbar\to 0}
\frac{i\pi}{\kbar^2}\left\{{L}_{\vec{m}},{L}_{\vec{n}}\right\}_M
=\frac{4\pi^2}{\ell^2}\vec{n}\times\vec{m}\,{L}_{\vec{m}+\vec{n}}=
P^1({L}_{\vec{m}},{L}_{\vec{n}})
\ee
coincides with the (classical) Poisson bracket (\ref{Poissontorus}) of functions in
$C^\infty(T^2)$. In particular, the last equality states that the Poisson
algebra (\ref{Poissontorus}) formally coincides with the Lie algebra of the
group of infinite unitary matrices $U(\infty)$ \cite{Fairlie2}. This is just
a facet of the general problem of approximating
infinite-dimensional Lie algebras of symplectic diffeomorphisms on
homogeneous manifolds by large-$N$ matrix algebras \cite{glinfty}.

This simple example gives us a taste of what should happen in
higher dimensions and less trivial quantum manifolds, capturing
features of the difference between an ordinary manifold and a
`noncommutative space'. Recent interest in such geometries has
occurred in the physics literature in the context of their
relation to M-Theory \cite{ConnesMT}. Although the general subject
of Noncommutative Geometry is rather developed (see e.g.
\cite{Connes,Madore}), there is still a technical difficulty
partially due to the lack of tractable (yet non-trivial)
noncommutative versions of curved spaces. We shall come to this
issue Sec. \ref{ncam}.

\section{Extending the previous constructions to $U(N_+,N_-)$\label{ghsa}}

\subsection{Generalized $w_\infty$ algebras}

The generalization of previous constructions to arbitrary unitary
groups proves to be quite unwieldy, and a canonical classification
of $U(N)$-tensor operators has, so far, been proven to exist only
for $U(2)$ and $U(3)$ (see \cite{Biedenharn} and references
therein). Tensor labeling is provided in these cases by the
Gel'fand-Weyl pattern for vectors in the carrier space of the
irreps of $U(N)$.

In the letter \cite{infdimal}, a quite appropriate basis of
$U(N_+,N_-)$-tensor operators was provided and the
Lie-algebra structure constants, for the particular case of the
oscillator realization (\ref{bosoprea}), were calculated through Moyal
bracket (see later on Sec. \ref{qnPoissonsec}). 
The chosen set of operators in ${\cal U}(u(N_+,N_-))$ is
the natural generalization of the $su(1,1)$-tensor operators
(\ref{auarop}), as follows:
\bea \hat{L}^{I}_{+|m|}&\equiv&
\prod_{\alpha}(\hat{X}_{\alpha\alpha})^{I_\alpha-(\sum_{\beta>\alpha}
|m_{\alpha\beta}|+\sum_{\beta<\alpha}|m_{\beta\alpha}|)/2}
\prod_{\alpha<\beta} (\hat{X}_{\alpha\beta})^{|m_{\alpha\beta}|}, \nn\\
 \hat{L}^{I}_{-|m|}&\equiv&
\prod_{\alpha}(\hat{X}_{\alpha\alpha})^{I_\alpha-(\sum_{\beta>\alpha}
|m_{\alpha\beta}|+\sum_{\beta<\alpha}|m_{\beta\alpha}|)/{2}}
\prod_{\alpha<\beta}
(\hat{X}_{\beta\alpha})^{|m_{\alpha\beta}|},\label{auarop2} \eea
where the the upper (generalized spin) index
$I\equiv(I_1,\dots,I_N)$ of $\hat{L}$ in (\ref{auarop2}) represents a
$N$-dimensional vector which, for the present, is taken to lie on
an half-integral lattice $I_\alpha\in \mathbb N/2$; the lower
index (``3th component") $m$ symbolizes now an integral
upper-triangular $N\times N$ matrix,
\be
m=\left(\ba{ccccc} 0 & m_{12}& m_{13} & \dots & m_{1N}\\
0 & 0 & m_{23} & \dots & m_{2N} \\ 0 & 0 & 0 & \dots & m_{3N} \\
 \vdots & \vdots & \vdots & \ddots  & \vdots \\
\vdots & \vdots & \vdots & \ddots  & 0 \ea \right)_{N\times N},
m_{\alpha\beta}\in \mathbb Z\label{uppert}\ee and $|m|$ means
absolute value of all its entries. Thus, the operators $\hat{L}^I_m$
are labelled by $N+N(N-1)/2=N(N+1)/2$ indices, in the same way as
wave functions $\psi^I_m$ in the carrier space of irreps of $U(N)$
(see \cite{harmanal}). We shall not restrict ourselves to polynomial subalgebras 
\be {\cL}_\wedge(N_+,N_-)\equiv\{\hat{L}^I_m,\,\,\,I_\alpha-(\sum_{\beta>\alpha}
|m_{\alpha\beta}|+\sum_{\beta<\alpha}|m_{\beta\alpha}|)/2\in\mathbb N\}\label{wedge2}\ee
and we shall consider ``extensions beyond the wedge" (\ref{wedge2}), by taking $I_\alpha\geq 0$, as 
done for standard $\W$ algebras (\ref{wedge}). This way, we are giving the possibility of  
true central extensions to the Lie algebra (\ref{commu}).\footnote{This 
claim deserves more careful study. So far, it is just a naive 
extrapolation of what happens to $\W_\infty$, Virasoro and Kac-Moody 
algebras, where Laurent (and not Taylor) expansions provide couples of 
conjugated variables (positive and negative modes).} 

The manifest expression of the structure constants $f$ for the
commutators
\be
\left[\hat{L}^{I}_{m},\hat{L}^{J}_{n}\right]=
\hat{L}^{I}_{m}\hat{L}^{J}_{n}-\hat{L}^{J}_{n}\hat{L}^{I}_{m}=
f^{IJl}_{mnK}\hat{L}^{K}_{l}\label{commu}
\ee
of a pair of operators (\ref{auarop2}) 
entails a cumbersome and awkward computation, because of inherent
ordering problems.  However,
the essence of the full ``quantum" algebra (\ref{commu}) 
can be still captured in a classical construction by extending
the Poisson-Lie bracket (\ref{poissonlie})
of a pair of functions $L^{I}_{m},L^{J}_{n}$ on the
commuting coordinates $x_{\alpha\beta}$
to its deformed version, in the sense of Ref. \cite{Bayen}.
To perform calculations with (\ref{poissonlie}) is still rather complicated because of non-canonical
brackets for the generating elements $x_{\alpha\beta}$. A way out to this technical problem is
to make use of the classical analogue of the standard oscillator 
realization, 
$x_{\alpha\beta}= \bar{a}_\alpha {a}_\beta$, of the generators of
$u(N_+,N_-)$, which simplifies things greatly. Indeed, it is not difficult  to
compute the standard Poisson bracket
 \begin{eqnarray}
\left\{L^{I}_{m},L^{J}_{n}\right\}_{{\rm P}}&=&i\Lambda_{\alpha\beta}\left(
\frac{\partial L^{I}_{m} }{\partial a_{\alpha}}
\frac{\partial L^{J}_{n}}{\partial \bar{a}_{\beta}}-
\frac{\partial L^{I}_{m} }{\partial \bar{a}_{\beta}}
\frac{\partial L^{J}_{n}}{\partial a_{\alpha}}\right)\nn\\
&=&i\Lambda^{\alpha\beta}(I_\alpha n_\beta-J_\alpha m_\beta)
L^{I+J-\delta_{\alpha}}_{m+n},\label{nPoisson}
\end{eqnarray}
for the Heisenberg-Weyl algebra, where
\be 
m_\alpha\equiv(\sum_{\beta>\alpha}m_{\alpha\beta}-
\sum_{\beta<\alpha}m_{\beta\alpha})\label{vecupper}
\ee
are the components of a
$N$-dimensional integral vector linked to the integral
upper-triangular matrix $m$ in (\ref{uppert}), and
\be
\delta_{\alpha}\equiv(\delta_{\alpha}^1,\dots,\delta_{\alpha}^N)
\ee
is a $N$-dimensional vector with the $\alpha^{\rm th}$ entry equal to one
and zero elsewhere. There is a clear resemblance between the $w_\infty$ 
algebra (\ref{auaralcom}) and (\ref{nPoisson}), although the last one is far
richer, as we shall show bellow. We shall refer to (\ref{nPoisson}) as 
$\ell_{\infty}(N_+,N_-)$ ---or ``generalized $w_\infty$"--- algebra. 

Note that the Poisson bracket (\ref{nPoisson}) does not distinguish between
polynomials like $x_{\alpha_1\beta_1} x_{\alpha_2\beta_2}$ and
$x_{\alpha_1\beta_2}x_{\alpha_2\beta_1}$, which admit the same
form when written in terms of oscillator variables
$a_\alpha,\bar{a}_\beta$. The difference between the 
Poisson-Lie bracket (\ref{poissonlie}) and the standard Poisson
bracket (\ref{nPoisson}) entails a minor ordering problem. In fact, it
is not difficult to prove that:
\begin{lem} Combinations like:
\be
\hat{X}_{\alpha_1\beta_1\alpha_2\beta_2}\equiv \hat{X}_{\alpha_1\beta_1}
\hat{X}_{\alpha_2\beta_2}-\hat{X}_{\alpha_1\beta_2}\hat{X}_{\alpha_2\beta_1}\label{reord}\ee
generate ideals of the enveloping algebra ${\cal U}(u(N_+,N_-))$.
\end{lem}
\noindent {\it Proof:} it suffices to realize that:
\bea
\left[\hat{X}_{\alpha\beta},\hat{X}_{\alpha_1\beta_1\alpha_2\beta_2}\right]&=&
\Lambda_{\alpha_1\beta}\hat{X}_{\alpha\beta_1\alpha_2\beta_2}-
\Lambda_{\alpha\beta_1}\hat{X}_{\alpha_1\beta\alpha_2\beta_2}+\nn\\
& &\Lambda_{\alpha_2\beta}\hat{X}_{\alpha_1\beta_1\alpha\beta_2}-
\Lambda_{\alpha\beta_2}\hat{X}_{\alpha_1\beta_1\alpha_2\beta} \blacksquare\eea
\begin{thm}\label{idealordering} Let 
${\cal 
I}_{\alpha_1\beta_1\alpha_2\beta_2}=x_{\alpha_1\beta_1\alpha_2\beta_2}C^\infty(\cG^*)$
represent ideals of the algebra $C^\infty(\cG^*)$ of smooth functions $L$ on the coalgebra $\cG^*$, with 
Poisson-Lie bracket (\ref{poissonlie}), generated by polynomials like 
(\ref{reord}) in the coordinates $x_{\alpha\beta}$.  Then, the Poisson 
algebra (\ref{nPoisson}) is 
equivalent to the quotient $C^\infty(\cG^*)/{\cal 
I}_{\alpha_1\beta_1\alpha_2\beta_2}$.
\end{thm}
\noindent In other words, the Poisson algebra (\ref{nPoisson}) and the Poisson-Lie 
algebra (\ref{poissonlie}) are equivalent up to ideals generated by polynomials arising 
out of ordering ambiguities. 

Before discussing quantum deformations of 
(\ref{nPoisson}), let us comment on the structure of some of its subalgebras as such. 



There are many possible ways of embedding the 
$u(N_+,N_-)$ generators (\ref{bosreal}) inside (\ref{nPoisson}), as there are also 
many possible choices of $su(1,1)$ inside (\ref{auaralcom}). However, 
a ``canonical'' choice is: 
\be
\hat{X}_{\alpha\beta}\equiv -i\hbar
L^{\delta_\alpha}_{e_{\alpha\beta}}\,, \;\;\;
e_{\alpha\beta}\equiv {\rm sign}(\beta-\alpha)
\sum_{\sigma=\min(\alpha,\beta)}^{\max(\alpha,\beta)-1} e_{\sigma,\sigma+1},\label{embedding}
\ee
where $e_{\sigma,\sigma+1}$ 
denotes an upper-triangular matrix with 
the $(\sigma,\sigma+1)$-entry equal to one and zero elsewhere, that is 
$(e_{\sigma,\sigma+1})_{\mu\nu}=\delta_{\sigma,\mu}\delta_{\sigma+1,\nu}$ 
(we set $e_{\alpha\alpha}\equiv 0$). 
For example, the $u(1,1)$ Lie-algebra generators correspond to:
\be\ba{ll}
\hat{X}_{12}=-i\hbar L^{(1,0)}_{\left(\begin{array}{cc} 0 &1\\ 0&0\ea\right)},& 
\hat{X}_{21}=-i\hbar L^{(0,1)}_{\left(\begin{array}{cc} 0 &-1\\ 0&0\ea\right)},\\
\hat{X}_{11}=-i\hbar L^{(1,0)}_{\left(\begin{array}{cc} 0 &0\\ 0&0\ea\right)},&
\hat{X}_{22}=-i\hbar L^{(0,1)}_{\left(\begin{array}{cc} 0 &0\\ 0&0\ea\right)}.\ea
\ee
Letting the lower-index $m=e_{\alpha\beta}$ in (\ref{embedding}) run to  
over arbitrary integral upper-triangular matrices $m$, we arrive to the  
following infinite-dimensional algebra (as can be seen from 
(\ref{nPoisson})):
\be
\left\{L^{\delta_\alpha}_{m}, 
L^{\delta_\beta}_{n}\right\}_P=-i(m^\beta L^{\delta_\alpha}_{m+n}-n^\alpha 
L^{\delta_\beta}_{m+n})
\,,\label{difeounm}
\ee
which we shall denote by $\ell_{\infty}^{(1)}(N_+,N_-)$. Reference \cite{Fradkin2} also considered 
infinite continuations of the finite-dimensional symmetries $SO(1,2)$ and $SO(3,2)$ as an  
``analytic continuation", i.e. an extension (``revocation") of the region of definition of the Lie-algebra 
generators' labels.  
It is easy to see that, for $u(1,1)$, the ``analytic continuation'' (\ref{difeounm}) 
leads to two Virasoro sectors: $L_{m_{12}}\equiv L^{(1,0)}_m,\,
\bar{L}_{m_{12}}\equiv L^{(0,1)}_m$. Its $3+1$ dimensional counterpart 
$\ell_{\infty}^{(1)}(2,2)$ contains four non-commuting Virasoro-like sectors 
$\ell_{\infty}^{(1_\alpha)}(2,2)=\{L^{\delta_\alpha}_{m}\}
,\,\alpha=1,\dots,4$ which, in their turn, 
hold three genuine Virasoro sectors for $m=k u_{\alpha\beta},\, 
k\in \mathbb Z,\, \alpha<\beta=2,\dots,4$, where $u_{\alpha\beta}$ denotes an 
upper-triangular matrix with components 
$(u_{\alpha\beta})_{\mu\nu}=\delta_{\alpha,\mu}\delta_{\beta,\nu}$. 
In general, $\ell_{\infty}^{(1)}(N_+,N_-)$ 
contains $N(N-1)$ distinct and non-commuting 
Virasoro sectors, 
\be \left\{V_k^{(\alpha\beta)}, V_l^{(\alpha\beta)}\right\}_P=-i
\Lambda^{\alpha\alpha}{\rm sign}(\beta-\alpha)\, (k-l)V_{k+l}^{(\alpha\beta)}\,,
\;\;\;\;\;V_k^{(\alpha\beta)}\equiv
L^{\delta_\alpha}_{k u_{\alpha\beta}}
\ee
and holds $u(N_+,N_-)$ as the {\it maximal 
finite-dimensional subalgebra}. 


The algebra $\ell_{\infty}^{(1)}(N_+,N_-)$ can be seen as the {\it minimal} 
infinite continuation of $u(N_+,N_-)$ representing the diffeomorphism 
algebra  diff$(N)$ of the $N$-torus $U(1)^N$. Indeed, the algebra (\ref{difeounm}) 
formally coincides with the algebra of vector fields 
$L^\mu_{f(y)}=f(y)\frac{\partial}{\partial y_\mu}$, where 
$y=(y_1,\dots,y_N)$ denotes a local system of coordinates and $f(y)$ 
can be expanded in a plane wave basis, such that 
$L^\mu_{\vec{m}}=e^{im^\alpha y_\alpha}
\frac{\partial}{\partial y_\mu}$ 
constitutes a basis of vector fields for  the so called 
generalized Witt algebra \cite{Ree}, 
\be \left[L^\alpha_{\vec{m}},L^\beta_{\vec{n}}\right]=-i(m^\beta L^{\alpha}_{\vec{m}+\vec{n}}-
n^\alpha L^{\beta}_{\vec{m}+\vec{n}}),\label{witt}\ee
of which there are studies about its representations (see e.g. 
\cite{ramostorus}). Note that, for us, the $N$-dimensional 
lattice vector $\vec{m}=(m_1,\dots,m_N)$ in (\ref{vecupper}) is, by 
construction, constrained to 
$\sum_{\alpha=1}^N m_\alpha=0$ (i.e. $L^\mu_{\vec{m}}$ is ``divergence free"), which 
introduces some novelties in (\ref{difeounm}) as regards the Witt algebra (\ref{witt}). 
In fact, the algebra (\ref{difeounm}) can be split into one ``temporal'' piece, constituted by 
an Abelian ideal generated by $\check{L}^N_m\equiv \Lambda_{\alpha\alpha} 
L^{\delta_\alpha}_{m}$, and a ``residual'' symmetry generated by the 
spatial diffeomorphisms 
\be
\tilde{L}^j_m\equiv\Lambda_{jj} 
L^{\delta_j}_{m}-\Lambda_{j+1,j+1} L^{\delta_{j+1}}_{m},\,j=1,\dots,N-1\,\, 
({\rm no \ sum \ on \ } j)\,,
\ee
which act semi-directly on the temporal part. More precisely, the 
commutation relations (\ref{difeounm}) in this new basis adopt the following 
form:
\bea
\left\{ \tilde{L}^j_m,\tilde{L}^k_n\right\}_P &=&-i(\tilde{m}^k \tilde{L}^j_{m+n} -
\tilde{n}^j \tilde{L}^k_{m+n})\,,\nn\\
\left\{ \tilde{L}^j_m,\tilde{L}^N_n\right\}_P &=&  i\tilde{n}^j \tilde{L}^N_{m+n}\,,
\label{inftempesp}\\
\left\{ \tilde{L}^N_m,\tilde{L}^N_n\right\}_P &=& 0\,,\nn
\eea
where $\tilde{m}_k\equiv m_k-m_{k+1}$. Only for $N=2$, the last 
commutator admits a central extension of the form  
$\sim n_{12}\delta_{m+n,0}$ 
compatible with the rest of commutation relations 
(\ref{inftempesp}). This result amounts to the fact that the 
(unconstrained) diffeomorphism algebra diff$(N)$ does not admit any 
non-trivial central extension except when $N=1$ \cite{nocentral}.

Another important point is in order here. We saw in Sec. \ref{nctsec} that the 
transformation (\ref{trigbasis}) in the Lie-algebra of $U(q)$ step-operators provided an 
``egalitarian" basis \cite{Fairlie} with trigonometric structure constants (\ref{Moyaleq}) 
and revealed a connection between the algebra of symplectic diffeomorphisms on the 
torus sdiff$(T^2)$ and $su(\infty)$. Here the expression (\ref{embedding}) reveals 
an embedding of the Lie algebra $u(N_+,N_-)$ inside the diffeomorphism algebra 
diff$(N_+,N_-)$ with commutation relations (\ref{difeounm}). That is, this new way 
of labelling $u(N_+,N_-)$ generators provides a direct ``analytic 
continuation" to diff$(N_+,N_-)$.

As well as  the diffeomorphisms ---or``$U(N_+,N_-)$-spin $I=\delta_\mu$ currents"---    
$L^{\delta_\mu}_{m}$ in (\ref{difeounm}), one can also introduce 
``higher-$U(N_+,N_-)$-spin $I$ currents" $L^I_m$ (in a sense 
similar to  that of Ref. \cite{Fradkin}) by letting the upper-index $I$ run to over an arbitrary 
half-integral $N$-dimensional lattice. Diffeomorphisms 
$L^{\delta_\mu}_{m}$ act semi-directly on ``$u(N_+,N_-)$-spin $J$ currents" $L^J_n$ as 
follows (see (\ref{nPoisson})):
\be
\left\{L^{\delta_\mu}_{m},L^J_n\right\}_P=-i\Lambda^{\alpha\beta}J_\alpha m_\beta 
L^{J+\delta_\mu-\delta_\alpha}_{m+n}+in^\mu L^J_{m+n}.\label{difaction}
\ee
Note that this action leaves stable Casimir quantum numbers like the trace 
$\sum_{\alpha=1}^NJ_\alpha$ [Casimir $C_1$ eigenvalue (\ref{Casimir})]. 

\subsection{Quantum (Moyal) deformations\label{qnPoissonsec}}

As happens to $w_\infty$-algebras, 
the quantization procedure, which entails unavoidable 
renormalizations (mainly due to ordering problems), must deform the classical ($\hbar\to 0$) 
``generalized $w_\infty$" algebra $\ell_\infty(N_+,N_-)$ in (\ref{nPoisson}) to a quantum 
algebra $\cL_\infty(N_+,N_-)$, by adding 
higher-order (Moyal-type) terms and central extensions like in 
(\ref{infq}). There is basically only one possible
deformation $\cL_\infty(N_+,N_-)$ of the bracket (\ref{nPoisson}) ---corresponding to a
full symmetrization--- that fulfils the Jacobi
identities \cite{Bayen}, which is the Moyal bracket (\ref{Moyal},\ref{star}), where now 
\[\Upsilon\equiv
\left(\begin{array}{cc} 0 & \Lambda \\ -\Lambda &0\ea\right)\] is a $2N\times 2N$ symplectic 
matrix. The calculation of higher-order terms in (\ref{Moyal}) is an 
arduous task, but the result can be summed up as follows:
\be
\left\{L^{I}_{m},L^{J}_{n}\right\}_{{\rm M}}=
\sum_{r=0}^{\infty}2(\frac{\hbar}{2})^{2r+1}f^{\alpha_1\dots\alpha_{2r+1}}_{\alpha_1\dots\alpha_{2r+1}}(I,m;J,n)
L^{I+J-\sum_{j=1}^{2r+1} \delta_{\alpha_j}}_{m+n},\label{qnPoisson}
\ee
where the higher-order structure constants
\be
f^{\alpha_1\dots\alpha_{2r+1}}_{\alpha_1\dots\alpha_{2r+1}}(I,m;J,n)\equiv\sum_{\ell=0}^{2r+1}
\frac{(-1)^\ell}{(2r+1-\ell)!\ell!}\prod_{s=1}^{2r+1}\Lambda^{\alpha_s\beta_s}
\Gamma_{\alpha_s}^\ell(I,-m)\Gamma_{\beta_s}^\ell(J,n)
\ee
are expressed in terms of the factors
\be
\Gamma_{\alpha_s}^\ell(I,m)\equiv I^{(s)}_{\alpha_s}+
(-1)^{\theta(\ell-s)}m_{\alpha_s}/2,
\ee
which are defined through the vectors (\ref{vecupper}) and  $U(N_+,N_-)$-spins
\be
I^{(s)}_{\alpha_s}= I_{\alpha_s}-\sum_{t=\theta(s-\ell-1)\ell+1}^{s-1} \delta_{\alpha_s}^{\alpha_t}\,,\;\;\;
I^{(0)}=I^{(\ell+1)}\equiv I,
\ee
with
\be
\theta(\ell-s)=\left\{\begin{array}{l} 0\,\;\;{\rm if}\;\;\ell<s
\\ 1\,\;\;{\rm if}\;\;\ell\geq s\ea\right.
\ee
the Heaviside function. For example, for $r=0$, the leading order (classical, $\hbar\to 0$) structure 
constants are:
\bea
f^{\alpha}_{\alpha}(I,m;J,n)=\Lambda^{\alpha\beta}(\Gamma_{\alpha}^0(I,-m)\Gamma_{\beta}^0(J,n)-
\Gamma_{\alpha}^1(I,-m)\Gamma_{\beta}^1(J,n))\nn\\
=\Lambda^{\alpha\beta}((I_{\alpha}-m_\alpha/2)(J_{\beta}+n_\beta/2)-
(I_{\alpha}+m_\alpha/2)(J_{\beta}-n_\beta/2)),
\label{leading}
\eea
which, after simplification, coincides with (\ref{nPoisson}). 


We have rephrased our previous (hard) problem of computing the commutators (\ref{commu}) of 
the tensor operators  (\ref{auarop2}) in terms of (more easy) Moyal brackets of 
functions on the coalgebra $u(N_+,N_-)^*$ (up to quotients by the ideals ${\cal 
I}_{\alpha_1\beta_1\alpha_2\beta_2}$ of theorem \ref{idealordering}). 
However, Moyal bracket captures the 
essence of more general deformations, which may include central extensions like  
\begin{eqnarray}
\left[\hat{L}^I_m, \hat{L}^J_n\right]=&&
\hbar\Lambda^{\alpha\beta}(J_\alpha m_\beta-
I_\alpha n_\beta)\hat{L}^{I+J-\delta_\alpha}_{m+n}+ {\rm O}(\hbar^3) \nn\\ &+&
\hbar^{(\sum_{\alpha=1}^N{I_\alpha+J_\alpha})}Q_I(m)\delta^{I,J}\delta_{m+n,0}
\mathbb I,
\end{eqnarray}
with central charges $Q_I(m)$ for all $U(N_+,N_-)$-spin $I$ currents 
$\hat{L}^I_m$. Note that, the structure of this central extension implies that the modes 
$\hat{L}^I_m$ and $\hat{L}^I_{-m}$ are conjugated, a fact inherited from the conjugation relation 
$\hat{X}_{\alpha\beta}^\dag=\hat{X}_{\beta\alpha}$ after (\ref{conjrel}) and the definition (\ref{auarop2}) of 
$\hat{L}^I_m$. An exhaustive study of this central extensions is in progress. 
Note that the diffeomorphisms subalgebra $\ell^{(1)}_\infty(N_+,N_-)$ 
remains unaltered by Moyal deformations.

\subsection{Volume-preserving diffeomorphisms and higher-extended objets 
(symplectic p-branes)\label{generalization}}

We showed in Sec. \ref{su2largeN} that the low-energy limit of
the $SU(\infty)$ Yang-Mills action (\ref{suinftyaction}), 
described by space-constant (vacuum configurations) $SU(\infty)$ vector potentials
$X_\mu(\tau;\vartheta,\varphi)\equiv A_\mu(\tau,\vec{0};\vartheta,\varphi)$,
turns out to reproduce the dynamics of the relativistic spherical
membrane. This view can be extended to arbitrary ``symplectic $p$-branes" (coadjoint orbits) $\cO_{C}$ of 
unitary groups just replacing the Poisson bracket on the sphere 
(\ref{spherepb}) by (\ref{poiscoad}) and using 
$\langle\psi^c_m|\psi^c_n\rangle$, with integration measure 
(\ref{intmeasleaf}), as a scalar product on a $p$-(symplectic)brane 
[e.g. $p=N(N-1)$ for maximal orbits ${\cO}_{N(N-1)}=U(N)/U(1)^N$, or $p=N-1$ for the projective space 
$CP^{N-1}=SU(N)/U(N-1)$]. 
Symplectic (volume-preserving) diffeomorphisms 
$L_\psi\equiv\{\psi,\cdot\}_P$ on the $p$-brane ${\cO}$ are the residual symmetry of the 
corresponding extended object in the light-cone gauge, and act 
as gauge transformations. 

We can also think of a $\cL_{\infty}(N_+,N_-)$-invariant [see (\ref{qnPoisson})] Yang-Mills 
gauge theory in $D$ dimensions with action functional: 
\bea
S&=&\int {\rm d}^Dx \langle F_{\nu\gamma}|
F^{\nu\gamma}\rangle\,,\nn\\
F_{\nu\gamma}&=&\partial_\nu A_\gamma-\partial_\gamma A_\nu +
\left\{A_\nu,A_\gamma\right\}_M\,,\\
A_\nu(x;z,\bar{z})&=&\sum_{\{I,m\}}A_{\nu I}^m(x) L^I_m(z,\bar{z})\,,\;\;\nu,\gamma=1,\dots,D\,,\nn
\eea
by setting, for example, $\langle L^I_m| L^J_n\rangle 
\propto \delta^{I,J}\delta_{m+n,0}$, where by $L^I_m(z,\bar{z})$ we denote the operator symbol of $\hat{L}^I_m$. 
Thus, all (infinite) higher-$U(N_+,N_-)$-spin 
vector fields $A_{\nu I}^m(x)$ on $\mathbb{R}^D$ are combined into a single field 
$A_\nu(x;z,\bar{z})$ on the extended manifold $\mathbb{R}^D\times \cO$, i.e. 
$A_{\nu I}^m(x)$ can be considered as a particular ``vibration mode" of the $p$-brane 
$\cO$.

In the same way, a 2+1-dimensional Chern-Simons $\cL_{\infty}(N_+,N_-)$-invariant gauge theory 
can be formulated with action:
\be
S=\int_{\mathbb{R}^3\times \cO} (A\wedge d A+\frac{1}{3}\{A,A\}\wedge 
A),\;\;A=A_\mu dx^\mu,
\ee
and equations of motion: $F=0$.



\subsection{Higher-dimensional non-commutative coadjoint orbits\label{ncam}}

Finally, let us comment on the potential relevance of the factor algebras  
$\cL_{\cas}({\cal G})$ on tractable 
non-commutative versions \cite{Connes} of 
algebraic (flag) manifolds $\cO_C$ of $\cal G^*$ like the 
maximal orbits $\cO_N=U(N)/U(1)^N$, where the notion of a pure state 
$\psi^I_m$ replaces that of a point. The possibility of describing 
phase-space physics in terms of the quantum (non-commutative) analog of the 
algebra of functions: the operator symbols (\ref{covsymb}), 
and the absence of localization 
expressed by the Heisenberg uncertainty relation, was noticed a long time 
ago by Dirac \cite{Dirac}. Just as the standard differential 
geometry of $\cO_C$ can be described by using the algebra $C^\infty(\cO_C)$ 
of smooth complex functions 
$\psi$ on $\cO_C$ (that is, $\lim_{\stackrel{\cas\to\infty}{\hbar\to 0}}
\cL_{\cas}({\cal G})$, when considered 
as an associative, commutative algebra), 
a non-commutative geometry for $\cO_C$ can be described by using 
the algebra $\cL_{\cas}({\cal G})$, seen as an associative algebra with 
a  non-commutative $*$-product like 
(\ref{symbolprod}). The appealing feature 
of a non-commutative space $\cO_C$ is that a $G$-invariant `lattice structure' 
can be constructed in a natural way, a desirable property as regards 
finite models of quantum gravity (see e.g. \cite{Madore} and Refs. therein). 
Indeed, as already mentioned in Theorem \ref{Burnside}, $\cL_{\cas}({\cal G})$ collapses to 
${\rm Mat}_{d_{\cas}}({\mathbb C})$ (the full matrix algebra of $d_{\cas}\times d_{\cas}$ 
complex matrices) whenever $\cas_\alpha$ coincides with the Casimir $\hat{C}_\alpha$ eigenvalue 
in a $d_\cas$-dimensional irrep $D_{\cas}$ of 
$G$. This fact provides a finite ($d_\cas$-points) `fuzzy' or `cellular' 
description of the non-commutative space $\cO_C$, 
the classical (commutative) case being 
recovered in the limit $\cas_\alpha\to\infty, \hbar\to 0$, so that $C_\alpha=\hbar^\alpha c_\alpha<\infty$. 
The notion of quantum space itself 
could be the ensemble (\ref{P-W}) of all of them, enclosed in a single structure (e.g. the group algebra), with different 
multiplicities. The multiplicity increases with $\cas$ 
(``the density of points"), so that classical-like spaces ($\cas\to\infty$) are ``more abundant". 
It is also a very important feature of 
$\cL_{\cas}(N_+,N_-)$ that the quantization deformation scheme 
(\ref{qnPoisson}) does not affect the the diffeomorphism subalgebra $\cL^{(1)}_\infty(N_+,N_-)$ and, 
accordingly, its maximal finite-dimensional subalgebra 
$u(N_+,N_-)$ (``good observables" or preferred coordinates \cite{Bayen}) of 
non-commuting ``position operators" 
\bea
&\hat{y}_{\alpha\beta}=\frac{\kbar}{2\hbar}(\hat{X}_{\alpha\beta}
+\hat{X}_{\beta\alpha})\,,\;\; 
\hat{y}_{\beta\alpha}=\frac{i\kbar}{2\hbar}(\hat{X}_{\alpha\beta}
-\hat{X}_{\beta\alpha})\,,\;\;\;\;\alpha<\beta\,,&\nn\\
&\hat{y}_\alpha=\frac{\kbar}{\hbar}(\Lambda_{\alpha\alpha}\hat{X}_{\alpha\alpha}-
\Lambda_{\alpha+1,\alpha+1}\hat{X}_{\alpha+1,\alpha+1})\,,&
\eea
where $\kbar$ denotes a parameter that 
gives $y$ dimensions of length 
(e.g., the Planck length $\kbar=\sqrt{\hbar G/c^3}$). 
The ``volume" $V_\alpha=\kbar^\alpha c_\alpha$ of each algebraic (flag) manifold  
$\cO_C$ (see \cite{harmanal} and \cite{Fulton} for a definition 
of flag manifolds) increases with $c_\alpha$. Thus, large volumes $V_\alpha$ (flat-like spaces) correspond to 
a high density of quantum points (large $\cas_\alpha$). In the classical limit 
$\kbar\to 0$ (large scales) and $\cas_\alpha\to \infty$, the coordinates $y$ commute.

\section*{Conclusions and outlook}

We have discussed the structure of new infinite-dimensional $\W$-like Lie algebras in a 
group theoretical framework as algebras of tensor operators. The (hard) problem of computing commutators 
of tensor operators has been rephrased in terms of (more easy) Moyal brackets of 
(polynomial) functions on the coalgebra $u(N_+,N_-)^*$ (up to quotients by the ideals ${\cal 
I}_{\alpha_1\beta_1\alpha_2\beta_2}$ of theorem \ref{idealordering}). That is, we intend to 
recover quantum commutators from quantum (Moyal) deformations of the classical limit. Moyal bracket captures the 
essence of more general deformations, and makes use of the standard 
oscillator realization of the basic $u(N_+,N_-)$-Lie algebra generators. 
These infinite-dimensional generalized $\W$-algebras can be seen as: 
1) analytic continuations of the finite-dimensional symmetries 
$u(N_+,N_-)$, or 2) higher-$U(N_+,N_-)$-spin extensions of the 
diffeomorphism algebra diff$(N_+,N_-)$ of a $N$-dimensional manifold (e.g. 
a $N$-torus). They provide a new arena for integrable field models in 
higher dimensions, of which we have mentioned gauge dynamics of 
higher-extended objects. An exhaustive study of central extensions should give us an 
important new ingredient regarding the constructions of unirreps and 
invariant geometric action functionals, just as central extensions of $\W$ 
algebras encode essential information. This should be our next step. 

An exhaustive study of the unirreps of $U(N_+,N_-)$, coherent states,  
$U(N_+,N_-)$-operator symbols and their classical limit is in progress \cite{harmanal}. 

\section*{Acknowledgment}

Work partially supported by the DGICYT under project BFM 2002-00778.


\begin{thebibliography}{9}
\bibitem{infdimal} M. Calixto: Structure constants for new infinite-dimensional Lie algebras of 
$U(N_+,N_-)$ tensor operators and applications, J. Phys. {\bf A33}, L69 (2000).
\bibitem{Zamolodchikov} A.B. Zamolodchikov: 
Infinite additional symmetries in two-dimensional conformal quantum-field theory, Theor. Math. Phys. {\bf 65},
1205 (1985);\\
 V.A. Fateev and A.B. Zamolodchikov: Conformal quantum-field theory models in 2 
 dimensions having z3 symmetry, Nucl. Phys. {\bf B280},
644 (1987).
\bibitem{Poly}  A.M. Polyakov: Quantum gravity in two-dimensions, 
 Mod. Phys. Lett.  {\bf A2}, 893 (1987).
 \bibitem{Bergshoeff} E. Bergshoeff, P.S. Howe, C.N. Pope, E. Sezgin, X. Shen and
K.S. Stelle: Quantisation deforms $w_\infty$ to $W_\infty$ gravity, Nucl. Phys. {\bf B363}, 163 (1991).
\bibitem{Shen} X. Shen: W Infinity and String Theory, Int. J. Mod. Phys. {\bf A7}, 6953 (1992).
\bibitem{Hull} C.M. Hull: $\W$ Geometry, Commun. Math. Phys. {\bf 156}, 245 (1993). 
\bibitem{Pope2} C.N. Pope, L.J. Romans and X. Shen: A new higher-spin algebra and the lone-star product, 
Phys. Lett. {\bf B242}, 401 (1990).
\bibitem{Capelli} A. Cappelli and G.R. Zemba: Modular invariant partition functions in the quantum Hall effect, 
Nucl. Phys. {\bf B490}, 595 (1997).
\bibitem{Bakas}I. Bakas: The large-$N$ limit of extended conformal symmetries, Phys. Lett. {\bf B228}, 57 (1989).
\bibitem{Witten} E. Witten: Ground Ring Of Two Dimensional String Theory, Nucl. Phys. {\bf B373}, 187 (1992).
\bibitem{Pope} C.N. Pope, X. Shen and L.J. Romans: $W_\infty$ and the Racah-Wigner algebra, 
Nucl. Phys. {\bf B339}, 191 (1990).
\bibitem{Hoppe2} M. Bordemann, J. Hoppe and P. Schaller: Infinite-dimensional matrix algebras, 
Phys. Lett. {\bf B232}, 199 (1989).
\bibitem{Fradkin2} E.S. Fradkin and V.Y. Linetsky: Infinite-dimensional generalizations 
of finite-dimensional symmetries, J. Math. Phys. {\bf 32}, 1218 (1991).
\bibitem{Fradkin} E.S. Fradkin and M.A. Vasiliev: Candidate for the role of higher-spin symmetry,  
Ann. Phys. (NY) {\bf 177}, 63 (1987).
\bibitem{Vasiliev} M.A. Vasiliev: Extended Higher-Spin Superalgebras and Their Realizations in Terms of Quantum 
Operators, Fortchr. Phys. {\bf 36}, 32 (1988).
\bibitem{vp} M. Calixto: Promoting finite to infinite symmetries: 
the (3+1)-dimensional analogue of the Virasoro algebra and higher-spin fields, Mod. Phys. Lett. {\bf A15}, 939 
(2000).
\bibitem{qg} M. Calixto: Higher-$U(2,2)$-spin fields and higher-dimensional $\W$ gravities: quantum AdS space 
and radiation phenomena, Class. Quant. Grav. {\bf 18}, 3857 (2001).
\bibitem{Bakas2} I. Bakas, B. Khesin and E Kiritsis: The Logarithm of the 
Derivative Operator and Higher Spin Algebras of ${\W}_\infty$ type, 
Commun. Math. Phys. {\bf 151}, 233 (1993).
\bibitem{Ramosh} F. Martinez-Moras and E. Ramos: Higher Dimensional 
Classical W-Algebras, Commun. Math. Phys. {\bf 157}, 573 (1993).
\bibitem{Connes} A. Connes: Noncommutative Geometry, Academic Press (1994).
\bibitem{Madore} J. Madore: An introduction to Noncommutative differential geometry
and its physical applications, London Mathematical Society, Lecture Note Series \textbf{257},
Cambridge University Press (1999).
\bibitem{Kirillov} A. A. Kirillov: Elements of the theory of 
representations, Springer, Berlin (1976).
\bibitem{Landsman} N. P. Landsman: Mathematical Topics Between Classical and
Quantum Mechanics, Springer-Verlag New York (1998).
\bibitem{harmanal} M. Calixto: Harmonic Analysis on $SU(N_+,N_-)$: 
coherent states and applications, in preparation.
\bibitem{Bergshoeff2} E. Bergshoeff, M.P. Blencowe and K.S. 
Stelle: Area-Preserving Diffeomorphisms and Higher-Spin Algebras,  
Commun. Math. Phys.{\bf 128}, 213 (1990).
\bibitem{glinfty} M. Bordemann, J. Hoppe, P.Schaler, M. Schlichenmaier: 
$gl(\infty)$ and Geometric Quantization, Commun. Math. Phys. \textbf{138}, 209 
(1991).
\bibitem{Berezin} F.A. Berezin: General concept of quantization, Commun. Math. Phys. {\bf 40}, 153 (1975) 
\bibitem{Biedenharn} L.C. Biedenharn and J.D. Louck: Angular momentum 
in quantum physics, Addison-Wesley, Reading, MA (1981).\\
L.C. Biedenharn and J.D. Louck: The Racah-Wigner algebra in quantum theory, 
Addison-Wesley, New York, MA (1981).\\
L.C. Biedenharn and M.A. Lohe: Quantum group symmetry and q-tensor 
algebras, World Scientific, Singapore (1995).
\bibitem{Hoppe} J. Hoppe, MIT Ph.D. Thesis (1982); 
J. Hoppe: Diffeomorphisms groups, quantization and $SU(\infty)$, Int. J. Mod. Phys. {\bf A4}, 5235 (1989).
\bibitem{Floratos} E.G. Floratos, J. Iliopoulos and G. Tiktopoulos: A 
note on $SU(\infty)$ classical Yang-Mills theories, 
Phys. Lett. {\bf B217}, 285 (1989).
\bibitem{Schild} A. Schild: Classical null strings,  
Phys. Rev. {\bf D16} 1722 (1977). 
\bibitem{Gervais} J.L. Gervais and A. Neveu: String structure of the master field on $U(\infty)$ Yang-Mills, 
Nucl. Phys. {\bf B192}, 463 (1981).
\bibitem{simplin} M. Calixto, V. Aldaya and J. Guerrero: Generalized conformal symmetry and extended 
objects from the free particle, Int. J.
Mod. Phys. {\bf A13}, 4889 (1998).
\bibitem{HoppeDS} Hoppe J., Olshanetsky M. and Theisen S.: Dynamical Systems on Quantum Tori Lie Algebras, 
Commun. Math. Phys. {\bf 155}, 429 (1993).
\bibitem{Nissimov} E. Nissimov, S. Pacheva and I. Vaysburd: Induced 
$W_\infty$ gravity as a WZNW model, 
Phys. Lett. {\bf B288}, 254 (1992).
\bibitem{Popehidden} C.N. Pope, X. Shen, K.W. Xu and Kajia Yuan, Nucl. Phys. {\bf B376}, 52 (1992).
\bibitem{Fairlie} D.B. Fairlie and J. Nuyts: Deformations and renormalizations of $W_\infty$, 
Commun. Math. Phys. {\bf 134}, 413 (1990).
\bibitem{nctorusref} M.A. Rieffel, Pac. J. Math. {\bf 93} (1981) 415;\\
A. Connes, C. R. Acad. Sci. Paris \textbf{A290} (1980) 599;\\
A. Connes and M.A. Rieffel, Contemp.
Math. \textbf{62} (1987) 237;\\ 
M.A. Rieffel, Can. J. Math. \textbf{40} (1988) 257.
\bibitem{Fairlie2} D.B. Fairlie, P. Fletcher and C.K. Zachos: Infinite-dimensional algebras and 
trigonometric basis for the classical Lie algebras, J. Math. Phys. {\bf 31}, 1088
(1990).
\bibitem{ConnesMT}  A. Connes, M.R. Douglas and A. Schwarz: Noncommutative geometry and matrix theory, 
J. High Energy Phys. \textbf{9802} (1998) 003.
\bibitem{Bayen} F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz and 
D. Sternheimer: Deformation theory and quantization, Ann. Phys.(NY) {\bf 111}, 61 (1978).
\bibitem{Ree} R. Ree: On generalized Witt algebras, Trans. Amer. Math. Soc. {\bf 83}, 510 (1956).
\bibitem{ramostorus} F. Figueirido and E. Ramos: Fock space representations of 
the algebra of diffeomorphisms of the N-torus, Int. J. Mod. Phys. 
\textbf{A6}, 771 (1991).
\bibitem{nocentral} E. Ramos, C.H. Sah and R.E. Shrock: Algebras of diffeomorphisms of the 
$N$-torus, J. Math. Phys. {\bf 31}, 1805 (1990).
\bibitem{Dirac} P.A.M. Dirac, Proc. Roy. Soc. {\bf A109}, 642 (1926);
 Proc. Camb. Phil. Soc. {\bf 23}, 412 (1926).
\bibitem{Fulton} W. Fulton and J. Harris: Representation Theory, 
Springer-Verlag, New York (1991).


\end{thebibliography}
\end{document}
Oscillator Realization of Higher-U(m,n)-Spin 
Algebras of W(infinity)-type and Quantized Simplectic Diffeomorphisms

This article is a further contribution to our research [M.Calixto J.Phys.A33(2000)L69] into a class of infinite-dimensional Lie algebras L_\infty(m,n)generalizing the standard W_\infty algebra, viewed as a tensor operator algebra of SU(1,1) in a group-theoretic framework. Here we interpret L_\infty(m,n)either as infinite continuations of pseudo-unitary symmetries or as 
``higher-U(m,n)-spin extensions"  of the diffeomorphism algebra diff(m,n). We also provide a deeper mathematical interconnection between Poisson (and 
symplectic diffeomorphism) algebras of functions on coadjoint 
orbits of pseudo-unitary groups U(m,n) and the classical limit of 
the corresponding tensor operator (and group) algebras. As potential applications we comment on the formulation of integrable higher-dimensional dynamical (field) systems and gauge theories of higher-extended objects. Some 
remarks on non-commutative geometry are also made.

