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\centerline {\bf THE HIDDEN GROUP STRUCTURE OF
QUANTUM GROUPS:}
\psaut
\centerline {\bf STRONG DUALITY, RIGIDITY AND
PREFERRED
DEFORMATIONS.}
\gsaut
\saut
\gsaut
\saut
\hskip.3cm P. BONNEAU \footnote{${}^1$}{Universit\'e de
Bourgogne
- Laboratoire de Physique Math\'ematique
B.P. 138, 21004~DIJON Cedex - FRANCE,
e-mail: flato@satie.u-bourgogne.fr }
M. FLATO${}^1$, M. GERSTENHABER
\footnote{${}^2$}{Department of
Mathematics, University
of Pennsylvania, Philadelphia,
PA 19104-6395~U.S.A.
e-mail: mgersten@mail.sas.upenn.edu and murray@math.upenn.edu }
and G. PINCZON ${}^1$
\gsaut
\gsaut
\gsaut
{\bf Abstract:}
\psaut
A notion of well-behaved Hopf algebra is introduced;
reflexivity
(for strong duality)
between Hopf algebras of Drinfeld-type and their duals,
algebras of coefficients
of compact semi-simple groups, is proved.
A hidden classical group structure is clearly indicated for
all generic models of quantum groups.
Moyal-product-like deformations are naturally found for all
FRT-models on coefficients
and $C^\infty$-functions. Strong rigidity ($H^2_{bi} = \{ 0 \}$)
under deformations
in the category of bialgebras is proved and consequences
are deduced.
\gsaut
\gsaut
\psaut
\noindent
AMS classification: Primary 17B37, 16W30, 22C05
\hskip3.5cm 46H99, 81R50.
\saut
{\sl Running title :} Topological quantum groups.
(In press in Communications in Mathematical Physics,
end of 1993)
\vfill\eject
\centerline {\bf Introduction.}
\saut
There presently exist at least four models of quantum
groups, introduced
respectively by Drinfeld ("D" model) [6], Jimbo ("J" model)
[16], Faddeev-Reshetikhin-Takhtajan
("FRT" model) [9] and Woronowicz ("W" model) [23]. We
apologize for missing other models
or authors. All these models are Hopf algebras that intend
to be ``deformations",
in the following sense: they depend on a parameter, say
$q$ (or
$e^{it}),$ and
when $q=1$ (or $t=0),$ one finds a very classical and well
known Hopf algebra, such as, e.g.: the
enveloping algebra of a simple Lie algebra (D), the algebra
of coefficients
on an algebraic reductive group (FRT), an algebra of
continuous functions
on a compact group (W). It is often claimed that the
classical limit of the
J model is the enveloping algebra of the corresponding
simple Lie algebra;
however this claim is not quite correct (see eg. [5] or [3]):
in fact, the classical limit of the J model is an extension of
${\cal U} (g)$ by $r$
parities ($r = rank \hskip.1cm g$). As a matter of fact this
should have
been obvious even a priori, since all deformations of the
(multiplicative structure of the) enveloping algebra of a
simple algebra are trivial [8], while the J model is a
non-trivial deformation
of its classical limit (see the end of this introduction). It is
also often asserted that the D and
FRT models are mutually dual. Although this claim seems
quite reasonable because
generators of D models can be found in the dual of FRT
models [9], a canonical
duality $*$ such that D$^*$ = FRT and FRT$^*$ = D has not
yet been constructed.
Here is a short summary of some puzzling problems
concerning quantum groups:
{\bf a)} A first problem is the meaning that one should give
to the word ``deformation". A very good
discussion is given in [11]. If we take the usual notion of the
third author
then the D model is a deformation of ${\cal U} (g);$
Drinfeld has
shown that this deformation is trivial from the algebra
viewpoint, so that the
only thing which is really deformed is the coproduct. We
shall discuss later the
J model. For the FRT model, [11] shows that defining
relations of the type ${\cal R}
\hskip.1cm T_1 \hskip.1cm T_2 = T_2 \hskip.1cm T_1
\hskip.1cm {\cal R}$ do not necessarily
define a deformation, even if ${\cal R}$ is Yang-Baxter;
however, one shows [11] that the FRT
model is a deformation if $G = SU(n).$ The problem of
showing that the FRT model
is a deformation for other $G,$ such as $SO(n),$ $Sp(n),$
has not been solved up to now.
{\bf b)} A second problem is related to duality. It is not only
a problem concerning quantum
groups, but really a problem concerning Hopf algebras in
general, when not
finite dimensional. For such Hopf algebras $A$, the
algebraic dual $A'$ is not
a Hopf algebra, and $A \subset_{\neq} A''.$
Therefore if one starts with the naive but suggestive idea
saying that if the D model is a
deformation then the FRT model, being its dual, should also
be a deformation, one
gets stuck in trying to prove it rigorously.
What is necessary, as noted in [9] (remark 23), is a theory
of reflexive
Hopf algebras which avoids the technical difficulties
coming from the lack of reflexivity
of the usual algebraic theory. Obviously, such a theory has
to be a topological one!
{\bf c)} A third problem is to give an interpretation of the
Tanaka-Krein philosophy in the case
of quantum groups: it has often been noticed that, in the
generic case, finite
dimensional representations of a quantum group are
(essentially) representations
of its classical limit. So the algebras involved should be the
same, and this is
justified by the abovementioned rigidity result of Drinfeld.
Such a remark shows
that the initial classical group is still there, acting as some
``hidden variables"
of this quantum group theory, and it is an interesting
challenge to discover these hidden variables.
{\bf d)} Related to the third problem is the result of the third
author and S. D. Schack on preferred presentations [12].
Assuming that some duality does exist for which the FRT
model is dual
to the D model, it will follow that the FRT model is a
deformation of the algebra
of coefficients of $G$. As the algebra structure of the D
model remains the initial
one, the coalgebra structure of the FRT model remains also
the initial one, so one
should be able to realize the FRT model as a deformation of
the algebra of coefficients
of $G,$ with unchanged coproduct (a preferred deformation).
{\bf e)} Finally (once more if a convenient duality does
exist) the FRT model should
extend to $C^\infty$-functions and provide a deformation of
the usual product
with the Poisson bracket as leading term. This picture looks
very much like the Moyal
product of quantum mechanics, and is given heuristically
in [6] as a justification
for the name quantum groups (see also [1]).
The goal of the present paper is to give an answer to the
above questions. We shall:
{\bf i)} recover FRT-models from D models by a suitable
duality argument.
{\bf ii)} show that there exists a preferred deformation (in the
Gerstenhaber-Schack sense)
of the Hopf algebra of coefficient functions on $G$ ($G$ a
compact connected Lie
group) satisfying the FRT relation ${\cal R} \hskip.1cm T_1
\hskip.1cm T_2 = T_2 \hskip.1cm
T_1 \hskip.1cm {\cal R},$ with ${\cal R}$ Yang-Baxter as in [9],
and that this deformation extends to a deformation of the algebra
$C^\infty (G).$
{\bf iii)} discuss properties of D models, FRT models and J
models in the framework of deformation theory.
In order to get these results, we introduce several new
concepts, such as those of well-behaved
Hopf algebras and deformation theory of topological
algebras, which are of
interest by themselves; for instance, the (strong) dual of a
well-behaved Hopf
algebra is a well behaved Hopf algebra, and the
(topological) deformation theory of
a well-behaved Hopf algebra is equivalent to the
(topological) deformation theory of
its (strong) dual. Let us give a brief survey of the principal
results of the paper.
First, we introduce a category of topological Hopf algebras
for which duality
works as if the algebra was finite dimensional. For this
reason, we call them
``well-behaved" Hopf algebras. If $A$ is such an algebra,
then its (strong) dual $A^*$
is also a well-behaved Hopf algebra, with Hopf structure
obtained by transposition
of the Hopf structure of $A,$ and $A^{**} = A$ as Hopf
algebras. We show that any
countable-dimension Hopf algebra can be given a natural
topology for which it is
a well-behaved Hopf algebra. Moreover any $H(G) =
C^\infty (G),$ $G$ a compact
connected Lie group, is a well-behaved Hopf algebra, and
so is its dual $A(G),$
the convolution algebra of distributions on $G.$ There are
many other examples, essentially all Hopf algebras of interest.
While FRT models are usually constructed for complex
reductive algebraic groups this context can be
replaced by that of compact connected Lie groups. We
prefer to deal with the second case,
since this brings us back to the usual harmonic analysis on
(compact) Lie groups,
and also because we believe in the power and
simplicity of Weyl's unitary trick. In section 2, related to the
Tanaka-Krein philosophy,
we give a complete description of the well-behaved Hopf
algebra ${\cal H} (G)$
($G$ a compact connected Lie group) of coefficients of
$G,$ and of its dual ${\cal A} (G)$;
our description is in fact an algebraic version of the Fourier
transform on compact groups.
In the third section, since the algebras to be deformed are
topological, we have to
present a topological version of the Gerstenhaber theory of
deformations. This construction
is parallel to the usual one (needed technical results are
given in Appendix
3). The main result is that to deform (in the topological
sense) a well-behaved
Hopf algebra is tantamount to deforming its strong dual.
Note, however, that in
the case of countable dimension Hopf algebras, the purely
algebraic deformation theory and
topological deformation theory do coincide.
In the fourth section, we study deformations of the
well-behaved algebras ${\cal H} (G),$
$H(G),$ and their duals ${\cal A} (G), A(G),$ previously
introduced. Just like
Drinfeld's result for ${\cal U} (g),$ we show that $A(G)$ and
${\cal A} (G)$ are
rigid in the associative category, and the new coproduct is
obtained from the initial
one by a twist, so it is still quasi-coassociative and
quasi-cocommutative (see
[2] for a rigidity interpretation of this result, and a
discussion of cohomology
and deformations of quasi-coassociative bialgebras). It
should nevertheless be
remarked at this point that one of the basic results of the
present paper (generalizing a similar result of strong rigidity
for the case of
$SU(2)$ [3] to the general case of a compact connected Lie
group) is as
follows: While the J model (as will be explained later) does
not exhibit any rigidity, and
the D model, as shown by Drinfeld is rigid under global
deformations, what we show for $A(G)$ and ${\cal A} (G)$
is strong rigidity. That is, they are
rigid under both infinitesimal and {\it a fortiori} global
deformations in the category
of bialgebras because the corresponding second
cohomology space, as defined in [3], vanishes identically.
We thus have a kind of Whitehead lemma for our version of
quantum groups. By duality, any coassociative
deformation of ${\cal H} (G),$ or $H(G),$ has a preferred
presentation. For ${\cal H} (G),$
this corresponds to the Gerstenhaber-Schack result, while
for $H(G)$ it is
new. Moreover, we show that in such a deformation of
${\cal H} (G),$ or $H(G),$ the product
of central functions (e.g., characters) is unchanged,
something which was never
noticed. Finally, we show that preferred deformations of
$H(G)$ do restrict to ${\cal H} (G).$
In the fifth section, we introduce a notion of quotient
deformation, by showing
that preferred deformations of ${\cal H} (G),$ or $H(G),$ do
produce preferred
deformations of ${\cal H} (G / \Gamma),$ or $H(G /
\Gamma),$ if $\Gamma$ is a
closed normal subgroup of $G.$ This result is interesting
even if $G$ is simple
(it can be used e.g. for $G = SU(2)$ and $G / \Gamma =
SO(3)$).
This extends naturally to cases where $\Gamma$ is
closed but not necessarily normal.
There one can still define ${\cal H} (G / \Gamma),$
respectively
$H(G / \Gamma),$ which now, however, are comodule
algebras over
${\cal H} (G)$, respectively $H(G).$ For a comodule
algebra over a bialgebra the notion of a
preferred deformation (relative to a preferred deformation of
the bialgebra) is still defined: we require that the
bialgebra and the comodule coalgebra structure over it
simultaneously so deform that we continue to have a
comodule algebra over the bialgebra, with the comultiplications
of both the bialgebra and comodule algebra remaining unchanged.
In this way it is meaningful to speak, in particular, of preferred
deformations of homogeneous spaces.
In section 6, we achieve another important goal of this
paper: we justify duality
between D models and FRT models. Here is a brief sketch:
Denote by ${\cal U}_t$
the D model deformation of ${\cal U} (g)$. We make a
``good choice" of a
Drinfeld isomorphism $\varphi: {\cal U}_t \simeq {\cal U} (g)
[[t]].$ Using
$\varphi,$ we imbed ${\cal U}_t$ as a dense subalgebra of
${\cal A} (G) [[t]]$ or as a subalgebra
of $A(G) [[t]]$ ($G$ a compact connected semi-simple Lie
group with complexified Lie algebra
$g$). We then show that the coproduct $\tilde{\Delta}$ of
${\cal U}_t$ extends to
${\cal A} (G) [[t]]$ (or $A(G) [[t]]$), and so does the antipode
and counit. Therefore
we get a Hopf deformation of topological Hopf algebras
${\cal A} (G) [[t]]$ and
$A(G) [[t]],$ and (using the main result of section 3) a
preferred Hopf deformation
of ${\cal H} (G)$ and $H(G),$ which satisfies the FRT
relation ${\cal R} \hskip.1cm
T_1 \hskip.1cm T_2 = T_2 \hskip.1cm T_1 \hskip.1cm {\cal
R}$ with ${\cal R}$
Yang-Baxter. Let us emphasize that this works not only for
$G = SU(n), SO(n)$ or
$Sp(n),$ as in [9], but also for $G = Spin (n),$ and for
exceptional $G.$
For instance, in the case of $Spin (2p),$ it predicts the
existence of a preferred
deformation of type ${\cal R} \hskip.1cm T_1 \hskip.1cm T_2 =
\hskip.1cm T_2 \hskip.1cm T_1
\hskip.1cm {\cal R},$ ${\cal R}$ Yang-Baxter, based on the
direct sum of the spinor-representations
(see also remark 23 in [9]); such a model has never been
explicitely described
up to now. Note that our result of realization of quantum
groups as preferred
deformations of ${\cal H} (G)$ and $H(G)$ (instead of
algebras defined by generators
and relations) gives a complete answer to a question of [11]
(solved for $SU(n)$
in [11]). On the other hand, the abovementioned ``good
choice" of the Drinfeld
isomorphism $\varphi$ is far from being explicit; on the
contrary, it seems to be
an incredibly complicated problem to give an explicit
$\varphi.$ This can be said
as follows: hidden variables (actually the initial group $G$)
do exist for quantum
groups, but it is a non trivial problem to give an explicit
realization!
In section 7, we discuss the J model, in the case $g = sl(2),
G = SU(2).$ It is usually
thought that the J model is a clever, but equivalent,
notation for the D model. We show
that though clever, it is not at all equivalent. Denote by
$A_t$ the J model and
$\tilde{A}_t$ the same with $t$ a formal parameter. It is an
easy matter to specialize (or ``contract")
$A_t$ to $t = 0.$ The $A_0$ obtained is not ${\cal U} (g),$
but an extension of ${\cal U} (g)$
by a parity $C$ $(C^2 = 1).$ This constitutes a first
difference.
So $A_0$ is not a domain. Since $\tilde{A}_t$
is a domain, $\tilde{A}_t$ is a non-trivial deformation of
$A_0,$ a second
difference with the D model. Moreover, we can consider
$\tilde{A}_{t_0+t}$ as a deformation
of $A_{t_0},$ and we show that it is a non-trivial one, a third
difference.
We have shown in [3] that $A_t,$ for generic $t,$ can be
realized as a dense
subalgebra of ${\cal A} (G)$. Actually, ${\cal A} (G)$ has
several nice properties:
it is rigid as an algebra, the D model can be recovered as a
dense subalgebra of a
deformation of the Hopf algebra ${\cal A} (G),$ duality
provides the FRT model
(since ${\cal A} (G)^* = {\cal H} (G)$), finite dimensional
representations of
${\cal A} (G)$ and $G$ are the same, etc.. So, in our
opinion, ${\cal A} (G)$
is exactly the model of hidden variables of quantum groups,
in that case.
The paper contains four appendices, which introduce
needed technical material and
present the terminology and notations we use. In particular,
Appendix 2 is devoted
to the natural ${\cal L} {\cal F}$-nuclear
topology on countable dimension spaces, a very simple,
but fundamental
notion, since it provides any countable dimension Hopf
algebra with a well-behaved Hopf structure.
Many ideas of the present paper originate in [3], where the
case $G = SU(2)$ is explicitely treated.
Finally, any reader will notice that this paper is a
discussion of the generic
case of quantum groups. We do not discuss the ``roots of
unity" case, because we are
dealing with deformation theory. Some explicit deformation
formulas, e.g., the Moyal product remain well-defined
when $\hbar$ is a root of unity, but the latter seems never
to have been discussed. Nevertheless we believe that the
really interesting
applications of quantum groups should come from the root
of unity case, a case
that should be compared with representation theory on
finite fields rather than with
real (or complex) Lie group theory.
\gsaut
{\bf Acknowledgements:}
\psaut
The authors are very happy to thank Wilfried Schmid for
inspiring discussions,
and Daniel Sternheimer for helpful comments and a very
careful reading of the manuscript.
\saut
\centerline {\bf 1. Well-behaved topological bialgebras.}
\psaut
We refer to Appendix 4 for notions of topological algebras,
bialgebras, etc...
{\bf (1.1) The framework}: Let us assume that $A$ is a
topological bialgebra, and moreover, that as
a complete topological vector space (c.t.v.s.) $A$ is
nuclear, and Fr\'echet or dual of Fr\'echet. Then $A$
is Montel [21], so, by (A.1.3), $A$ is reflexive. Using
(A.1.5), transposition of
the product and coproduct of $A$ defines a coproduct and
a product on $A^*$, so $A^*$
becomes a topological bialgebra. Now $A^*$, as a t.v.s.,
satisfies exactly the same conditions as $A,$ so we can
repeat the transposition operation, and since
$A$ is reflexive, we recover the initial bialgebra
structure of $A.$
If we assume that $A$ is associative (with unit), then $A^*$
will be coassociative with counit,
and if we assume that $A$ is a topological Hopf algebra,
then the counit
of $A$ will define the unit of $A^*$; the counit of $A^*$ is
the evaluation on the unit of $A,$
and the antipode of $A^*$ is the transpose of the antipode
of $A,$ so $A^*$ is also a topological
Hopf algebra. Therefore, we introduce the following
definition:
\psaut
{\bf (1.2) Definition}:
\it A topological algebra (resp: bialgebra, Hopf algebra) is
{\bf well-behaved} if (as
a c.t.v.s.) it is nuclear and Fr\'echet, or nuclear and dual of
Fr\'echet. \rm
\psaut
Now, we summarize the results of this section:
\psaut
{\bf (1.3) Proposition}:
\it When $A$ is a well-behaved topological bialgebra (resp:
a Hopf algebra) then
the transposition defines on $A^*$ a well-behaved
topological bialgebra (resp: a Hopf algebra)
structure, and the initial structure of $A = A^{**}$ is
recovered by transposition of the structure of $A^*.$ \rm
\psaut
{\bf (1.4) Remarks}: the classical algebraic theory of Hopf
algebras suffers from an obvious lack of reflexivity,
as soon as the algebra is not finite
dimensional, and (1.3) shows that the category of
well-behaved algebras is really
of interest, because reflexivity now holds and everything
works almost
as in finite dimensional case. While the condition of being
well-behaved
may seem unnatural, in the rest of this section, we
show that almost all bialgebras or Hopf algebras of interest
are in fact well-behaved.
\psaut
{\bf (1.5) Natural topology}: First, we note that bialgebras or
Hopf algebras used in algebraic theories
always have countable dimension. In this case, Appendix 2
provides a natural topology
on $A,$ and the following result shows that it is the good
one:
\psaut
{\bf (1.5.1) Proposition}:
\it If $A$ is a countable dimension bialgebra (resp: Hopf
algebra), and if $A$
is given its natural topology (see $A2$), then $A$ is a
well-behaved topological
bialgebra (resp: Hopf algebra). \rm
\psaut
{\bf Proof}:
By (A.2.4), the natural topology is nuclear and complete,
and $A$ is reflexive.
By (A.2.5), $A^*$ is Fr\'echet, so we have only to prove
that $A$ is a topological
bialgebra (resp: Hopf algebra), but it is an obvious
consequence of (A.2.8) and
(A.2.2) \hskip2cm Q.E.D.
\psaut
{\bf (1.6) Example}: Let $\Crm [t]$ be the
space of polynomials, with its natural topology, and $\Crm
[[t]]$ its dual (see
(A.2.10)). We define a well-behaved topological Hopf
algebra structure on $\Crm [t]$ by:
$t^n \times t^p = \pmatrix{ n+p \cr p \cr }
\hskip.1cm t^{n+p},$ coproduct $\delta (t^n) = \sum^n_{i=0}
\hskip.1cm t^i \otimes t^{n-i},$ counit $\varepsilon (t^n) =
\delta_{n0},$ antipode
$S (t^n) = (-1)^n \hskip.1cm t^n.$ Using (1.3), we get a
well-behaved topological
Hopf algebra structure on $\Crm [[t]].$
It is easily seen that the product is the usual product of
$\Crm [[t]],$ and the coproduct is given by:
$$f(t) \in \Crm [[t]], \hskip.1cm \Delta (f) = f(t+t') \in
\hskip.1cm \Crm [[t,t']] \simeq
\hskip.1cm \Crm [[t]] \hat{\otimes} \hskip.1cm \Crm [[t]].$$
{\bf (1.7) The algebras $H$ and $A$}: Let $G$ be a compact
connected Lie group, and $H(G) = C^\infty (G).$ Here
the topology of $H$ is the usual Fr\'echet topology, which
is nuclear [13]. Now, the product
on $H$ is the pointwise abelian product of functions, the
coproduct is defined by $\delta (f)
(x,y) = f(xy),$ using the standard isomorphism $H(G)
\hat{\otimes} H(G) \simeq H(G \times G),$
the counit is the Dirac distribution at the unity of $G,$ and
the antipode is $\Delta (f) (x) =
f(x^{-1}),$ so we get a well-behaved topological Hopf
algebra. Using (1.3), $H(G)^* = A(G)$
is also a well-behaved topological Hopf algebra, but
$H(G)^*$ is
the space of distributions on $G,$ and it is easy to check
that the product so obtained
is the usual convolution product of distributions; identifying
$G$ and Dirac
distributions, one gets from the compactness of $G$ a
topological inclusion $G \subset A(G).$
Since $G^\perp = \{ 0 \}, \hskip.1cm \overline{Vect (G)} =
A(G),$ where $ {Vect(G)} $ is the linear span of $G.$
Then, denoting
by $\Delta$ the transpose of the product of $H(G),$ one has
$\Delta (x) = x \otimes x,
\hskip.1cm x \in G.$ Finally, $1_{A(G)} = 1_G,$ and the
counit of $A(G)$ is the trivial
representation of G. Summarizing:
\psaut
{\bf (1.7.1) Proposition}:
\it $H(G)$ and $A(G)$ are well-behaved topological Hopf
algebras. The product on $A(G)$
is the convolution product of distributions, the coproduct is
$\Delta (x) = x \otimes x,
\hskip.1cm x \in G,$ and the counit is the trivial
representation of $G.$ \rm
\saut
\centerline {\bf 2. The Hopf algebra of coefficients of $G,$
and its dual.}
\psaut
Let $G$ be a compact connected Lie group and $g$ be the
complexification of the Lie
algebra of $G.$ We have already introduced in (1.7) the
well-behaved topological
Hopf algebras $H = H(G) = C^\infty (G),$ and its dual $A =
A(G),$ the well-behaved Hopf
algebra of distributions. We shall now present another
model of well-behaved
topological Hopf algebra naturally associated to $G$: the
algebra of coefficients
${\cal H},$ and its dual ${\cal A},$ which can be treated as
an algebra of formal
distributions on $G.$
We denote by $\hat{G}$ a fixed set of irreducible
inequivalent unitary finite
dimensional representations of $G,$ such that $\hat{G}$
contains one and only one element of each equivalence class.
Let $\rho$ be any finite dimensional representation of $G,$
acting on $V_\rho.$
\psaut
{\bf (2.1) Definition}:
\it Given $M \in {\cal L} (V_\rho),$ the coefficient of $\rho$
associated with $M$
is the function $C^\rho_M \in C^\infty (G)$ defined by:
$x \in G, \hskip.1cm C^\rho_M (x) = Tr (M \rho (x)).$ We
denote by ${\cal C}_\rho$ the
space of coefficients of $\rho.$ \rm
\psaut
We note that the algebra structure of the bicommutant of $\rho$
induces an associative
algebra structure on ${\cal C}_\rho$ with
unit element is $\xi^\rho = C^\rho_{Id},$ i.e.
{} From the Burnside-Schur theorem, ${\cal C}_\rho \simeq
{\cal L} (V_\rho)$
if and only if $\rho$ is irreducible.\psaut
{\bf (2.2) Proposition}:
\it Let $\rho$ and $\rho'$ be finite dimensional
representations of $G,$ then
$\rho \simeq \rho'$ if and only if ${\cal C}_\rho = {\cal
C}_{\rho'}$ (as algebras),
which occurs if and only if $\xi^\rho = \xi^{\rho'}.$ \rm
\psaut
{\bf Proof}:
Assume $\rho \simeq \rho'$ by $f: V_\rho \fl V_{\rho'},$
then $C^{\rho'}_M = C^\rho_{f^{-1}Mf},$
so ${\cal C}_\rho = {\cal C}_{\rho'},$ and $\xi^\rho =
\xi^{\rho'}.$ On the other hand, if
${\cal C}_\rho = {\cal C}_{\rho'}$ as algebras, then their
units do coincide, so $\xi^\rho =
\xi^{\rho'}$ which is equivalent to $\rho \simeq \rho'$ by
e.g. the Peter-Weyl theorem.
\hskip2cm Q.E.D.
By (2.2) and the Peter Weyl theorem, the subspace of
$C^\infty (G)$ generated by
coefficients functions of finite dimensional representations
of $G$ is exactly
${\cal H} = {\cal H}(G) = \oplus_{\pi \in \hat{G}} \hskip.1cm
{\cal C}_\pi$; note that ${\cal C}_\pi
\simeq {\cal L} (V_\pi)$ as algebras, when $\pi \in \hat{G}.$
Now ${\cal H}$ has another algebra structure, coming from
the usual (commutative)
multiplication "${\times}$" in $C^\infty (G)$ :
\psaut
{\bf (2.3) Proposition}:
\it ${\cal H}$ is a subalgebra of $C^\infty (G).$ One has: \rm
$$M \in {\cal L} (V_\rho), \hskip.1cm M' \in {\cal L}
(V_{\rho'}), \hskip.3cm C^\rho_M
\times C^{\rho'}_{M'} = C^{\rho \otimes \rho'}_{M \otimes M'}$$
{\bf Proof}:
We use the usual identification ${\cal L} (V_\rho) \otimes
{\cal L} (V_{\rho'}) \simeq
{\cal L} (V_\rho \otimes V_{\rho'});$ then the formula in (2.3)
is an obvious application
of the properties of the trace. \hskip2cm Q.E.D.
\psaut
{\bf (2.4) The algebra ${\cal H}$}:
\psaut
{\bf (2.4.1)} In the foregoing, ${\cal H} = {\cal H}(G)$ with its
commutative algebra structure
inherited from $C^\infty (G)$ will be called {\it the algebra of
coefficients of
$G.$} Since $G$ is a compact connected Lie group, ${\cal
H}$ is of countable
dimension (actually, it is known that ${\cal H}$ is a finitely
generated algebra [4]),
and has an obvious well-behaved topological Hopf algebra
structure: topology is
the natural one (see Appendix 2) and:
(1) Let $\delta$ be the coproduct of $C^\infty (G)$ (see
section 1). Set $C^\pi_{ij} =
C^\pi_{E_{ij}},$ where $E_{ij} = e^*_i \otimes e_j,$ $\{ e_i
\}$ basis of $V_\pi.$
Then $\delta (C^\pi_{ij}) = \sum_k \hskip.1cm C^\pi_{ik}
\otimes C^\pi_{kj} \in
{\cal H} \otimes {\cal H},$ so the restriction of $\delta$ to
${\cal H}$ defines a
coproduct on ${\cal H}.$
%----------------------------------------------------------------
%The \check on top of a superscript will be practically
%invisible in print.
%In this case you might prefer to set it on line using
%\spcheck. %If \widecheck
%exists then that might be readable. If you agree, do a
%search and replace.
%----------------------------------------------------------------
(2) Let $S$ be the antipode of $C^\infty (G)$ (see section
1), then $S (C^\pi_{ij}) =
C^{\check{\pi}}_{ji} \in {\cal H}.$
(3) The counit of $C^\infty (G)$ is a counit for ${\cal H}.$
{} From the definition of the topology of ${\cal H},$ the results
of Appendix 2 do apply: ${\cal H}$
is nuclear, and therefore reflexive (which answers a
question of [9]),
${\cal H}$ is well-behaved, and the dual ${\cal H}^*$ is also
a well-behaved
topological Hopf algebra. We shall next give a complete
description of ${\cal H}^*.$
For the time being, let us give another realization of ${\cal
H}$: consider the (right
or left) regular representation of $G$ in ${\cal L}^2 (G)$;
%--------------------------------
%Do you really want \cal rather than plain roman here?
%---------------------------------
then, from the Peter
Weyl theorem, ${\cal H}$ is exactly the space of
$G$-finite vectors of the regular
representation. ${\cal C}_\pi, \pi \in \hat{G},$ is an
isotypical component (of
type $\check{\pi}$ for the left regular, of type $\pi$ for the
right regular). So
${\cal H}$ is closely related to harmonic analysis on $G,$
and the construction
of ${\cal H}^*,$ that we shall describe in (2.5), is nothing
but the Fourier transform
on $G$ (up to normalization coefficients).
{\bf (2.4.2)} We note that any coefficient is an analytic
function on $G,$ so any element
of ${\cal H}$ is an analytic function on $G,$ and therefore
{\it ${\cal H}$
is a domain.}
{\bf (2.4.3)} Since $G$ is compact, there exist finite
dimensional irreducible representations
$\pi_1,...,\pi_k,$ such that any element $\pi$ of $\hat{G}$
can be obtained as a
subrepresentation of tensor products of $\pi_1,...,\pi_k.$ From
(2.3) and the
proof of (2.2), it follows that the coefficients of $\pi$ are
polynomials in the
coefficients of $\pi_1,...,\pi_k.$ So ${\cal H}$ is a finitely
generated algebra.
We shall give more details about the choice of $\pi_1 ...
\pi_k$ in section 7.
\psaut
{\bf (2.5) The algebra ${\cal A}$}: Given $\pi \in \hat{G},$
we identify ${\cal L} (V_\pi)$ and ${\cal L} (V_\pi)^*$
by the following duality:
%------------You need a space in the following formula
$$M,M' \in {\cal L} (V_\pi), = Tr (MM')
\leqno{(2.5.1)}$$
We introduce ${\cal A} = {\cal A}(G) = \Pi_{\pi \in \hat{G}}
\hskip.1cm {\cal A}_\pi,$ where
${\cal A}_\pi = {\cal L} (V_\pi),$ with product topology and
associative structure
of algebra defined by:
%-----------------------------------------------------------
%The use of ``o" in the following formula and elsewhere
%looks
%strange because of its size. If you have \circ that would
%probably look much better, but you would have to make
%the %changes by hand sine you can't use an ordinary
%search and
%replace (unless you use the spaces on both sides of the
%``o" as
%part of what you search for.
%------------------------------------------------------------
$$a = (a_\pi), b = (b_\pi), c = (c_\pi) = a*b, \hbox{ with }
c_\pi = a_\pi o b_\pi,
\hskip.1cm \forall \pi \in \hat{G} \leqno{(2.5.2)}$$
We get a topological algebra. As a vector space, we have
${\cal A}(G) = {\cal A} = {\cal H}^*,$
if we define the duality by:
$$a = (a_\pi) \in {\cal A}, \pi_0 \in \hat{G}, M \in {\cal L}
(V_{\pi_0}), = . \leqno{(2.5.3)}$$
We define a map $i: G \fl {\cal A}$ by $i(x) = (\pi(x)) \in {\cal
A}, x \in G.$
%-----------------------------------
%You might want to add "and denote by $overline...$ the
%linear %subspace spanned by the image." Also, to avoid
%beginning a %sentence with a symbol, I have added a few
%words/
%-------------------------------------------
\psaut
{\bf (2.5.4) Lemma}:
\it The mapping $i$ is one to one, bicontinuous from $G$
onto $i(G),$ and $\overline{Vect (i(G))} = {\cal A}.$ \rm
\psaut
{\bf Proof}:
Obviously $i$ is continuous, and bicontinuous since $G$ is
compact. It is one to
one because $\hat{G}$ is a complete set of representations
of $G$ (Peter-Weyl).
Since $i(G)^\perp = \{ 0 \},$ one has $\overline{Vect (i(G))}
= {\cal A}.$ \hskip2cm Q.E.D.
Henceforth, we identify $G$ and $i(G),$ so we consider
that $G \subset {\cal A}.$
\psaut
{\bf (2.5.5) Lemma}:
$$\forall x, x' \in G, h \in {\cal H}, = $$
\psaut
{\bf Proof}:
We can restrict to $h = C^\pi_M, \pi \in \hat{G}, M \in {\cal
L} (V_\pi).$ Then
$$ = Tr (\pi(x) \pi(x') M) = Tr (M \pi
(xx')) = C^\pi_M (xx'). \hskip.5cm Q.E.D.$$
(2.5.4) and (2.5.5) prove that our product $*$ on ${\cal A}$
is exactly the transpose $^T \delta$ of $\delta.$
So we achieve the definition of a topological Hopf structure
on ${\cal A}$ using
(1.3). Let us the check coproduct, counit and antipode:
(1) Let $\Delta_0 = ^T\times,$ where $\times$ is the product
on ${\cal H}.$ Then:
$$x \in G, <\Delta_0 (x) \vert h \otimes h'> = =
= $$
so $\Delta_0 (x) = x \otimes x, x \in G.$
(2) $1_{{\cal H}}$ is exactly the trivial representation
$\varepsilon$ of $G$ (or ${\cal A}$).
(3) The antipode $S$ of ${\cal A}$ is defined as the
mapping $S: {\cal A}_\pi \fl
{\cal A}_{\check{\pi}}, \pi \in \hat{G},$ given by:
$$a \in {\cal A}_\pi = {\cal L} (V_\pi), S(a) = ^Ta \in {\cal L}
(V^*_\pi) = {\cal A}_{\check{\pi}}$$
So one has: $a = (a_\pi) \in {\cal A}, S(a) = ((S(a))_\pi)$ and
$S(a)_\pi = ^Ta_
{\check{\pi}}.$ When $x \in G,$ one finds $S(x) = x^{-1}.$
The topological Hopf structure on ${\cal A} = {\cal H}^*$ is
exactly
the transpose of the Hopf structure of ${\cal H},$ as defined
in (1.3). ${\cal H},$
and ${\cal A},$ being well-behaved by applying
transposition to the Hopf structure
of ${\cal A},$ one recovers the Hopf structure of ${\cal H}$
(see (1.3)).
\psaut
{\bf (2.6) Further properties of ${\cal A}$}: So far we have
an explicit form for $\Delta_0$ when restricted to $G.$
It is of interest to know how $\Delta_0 (a)$ can be
computed for any $a \in {\cal A}.$
For this purpose, we consider
${\cal A} \hat{\otimes} {\cal A} = \Pi_{\pi,\pi' \in \hat{G}}
{\cal A}_\pi \otimes
{\cal A}_{\pi'}$ ((1.6.3)) and identify ${\cal A}_\pi \otimes
{\cal A}_{\pi'} =
{\cal L} (V_\pi) \otimes {\cal L} (V_{\pi'}) = {\cal L} (V_\pi
\otimes V_{\pi'}).$
Then ${\cal A} \hat{\otimes} {\cal A} = \Pi_{\pi,\pi' \in
\hat{G}} {\cal A}_{\pi,\pi'},$
with ${\cal A}_{\pi,\pi'} = {\cal L} (V_\pi \otimes V_{\pi'}),$
and product:
$a = (a_{\pi,\pi'}), b = (b_{\pi,\pi'}) \in {\cal A}
\hat{\otimes} {\cal A},$
then $a * b = (C_{\pi,\pi'}),$ with $C_{\pi,\pi'} = a_{\pi,\pi'}
\hskip.1cm o \hskip.1cm b_{\pi,\pi'}$
in ${\cal L} (V_\pi \otimes V_{\pi'}).$
As usual, given $\pi, \pi' \in \hat{G},$ the tensor product
representation $\rho_{\pi,\pi'} =
\pi \otimes \pi'$ is the representation of $G$ in $V_\pi
\otimes V_{\pi'}$ defined
by: $\rho_{\pi,\pi'} (x) = \pi \otimes \pi' (\Delta_0 (x)), x \in
G,$ where $\pi
\otimes \pi': {\cal A} \hat{\otimes} {\cal A} \fl {\cal
A}_{\pi,\pi'} = {\cal L}
(V_\pi \otimes V_{\pi'})$ is the canonical projection: $\pi
\otimes \pi' (a \otimes b) =
\pi (a) \otimes \pi' (b), a,b \in {\cal A}.$
Now $\rho_{\pi,\pi'}$ extends to a representation of ${\cal
A}$ on $V_\pi \otimes V_{\pi'},$
say $\rho_{\pi,\pi'},$ defined exactly by the same formula.
So we have:
$$a \in {\cal A}, \hskip.1cm \Delta_0 (a)_{\pi,\pi'} = (\pi
\otimes \pi') (\Delta_0 (a)) = \rho_{\pi,\pi'} (a).$$
This gives the component of $\Delta_0 (a)$ on ${\cal
A}_{\pi,\pi'}.$ Precisely, since
$G$ is compact, we have $V_\pi \otimes V_{\pi'} =
\sum_{\rho \in t(\pi, \pi')} W_\rho,$
with $t(\pi, \pi')$ a finite subset of $\hat{G},$ and $\rho_{\pi
\pi'} \vert_{W_\rho}
\simeq n_\rho.\rho, \hskip.1cm n_\rho \in \Nrm.$ Therefore
there exists $\alpha_\rho: W_\rho
\fl V_\rho \hskip.1cm \underbrace{\oplus...\oplus}_{n_\rho}
\hskip.1cm V_\rho = n_\rho.V_\rho$ such that
$\rho_{\pi \pi'} \vert_{W_\rho} = \alpha^{-1}_\rho \hskip.1cm
o \hskip.1cm n_\rho \hskip.1cm \rho \hskip.1cm o \hskip.1cm
\alpha_\rho$; defining
$C_{\pi,\pi'} = \sum_{\rho \in t(\pi,\pi')} \alpha_\rho,$ and
the representation
$r_{\pi, \pi'} = \sum_{\rho \in t(\pi,\pi')} n_\rho.\rho,$ on
the space
%\noindent
$\oplus_{\rho \in t(\pi,\pi')} \hskip.1cm n_\rho.V_\rho,$ we
have $\rho_{\pi,\pi'}
= C^{-1}_{\pi,\pi'} o \hskip.1cm r_{\pi,\pi'} o \hskip.1cm
C_{\pi,\pi'},$ and
$\Delta_0 (a)_{\pi,\pi'}$ is explicitely computed.
\psaut
{\bf (2.6.1) Lemma}:
\it Let $i: {\cal U} (g) \fl {\cal A}$ be the linear map defined
by $i(u) = (\pi(u)),
\hskip.1cm u \in {\cal U} (g).$ Then $i$ is one to one, and $i
\overline {({\cal U} (g)} = {\cal A}.$ \rm
\psaut
As a consequence, we can consider that ${\cal U} (g)
\subset {\cal A}$ as Hopf algebras (by (2.6)). Another
expression of (2.6.1) is the following: $\hat{G}$ is a
complete set of (irreducible)
finite dimensional representations of ${\cal U} (g)$ (this is
not completely obvious
a priori, since $\hat{G}$ is generally not identical with the
set of all finite
dimensional irreducible representations of ${\cal U} (g)).$
\psaut
{\bf Proof}:
Assume that $u \in {\cal U} (g),$ and $\pi(u) = 0, \forall \pi
\in \hat{G}.$ Let $\rho$
be the (left) regular representation space of $G$ on
$C^\infty
(G),$ which is a $C^\infty$
representation of $G$; then $\rho (u)$ is a differential
operator on $G,$ and it
is well-known that $\rho: {\cal U} (g) \fl \hbox{ Diff } (G)$ is
one to one ([15]).
The space of $G$-finite vectors of $\rho$ is ${\cal H},$ and
the restriction of
$\rho$ to ${\cal H}$ reduces as $\rho \vert_{\cal H} =
\sum_{\pi \in \hat{G}}
\hskip.1cm (dim \hskip.1cm \pi).\pi$ (Peter Weyl); so $\rho
(u) \vert_{\cal H} = 0,$
and since $\overline{{\cal H}} = C^\infty (G),$ we deduce
that $\rho (u) = 0,$ and
then that $u = 0.$ For the density, we consider $V =
\sum_{\pi \in \hat{G}} V_\pi,$
endowed with the representation $\oplus_{\pi \in \hat{G}}
\pi$; this is a semi-simple
${\cal U} (g)$-module, with irreducible isotypical
components $V_\pi.$ Its bicommutant
is ${\cal A},$ so, by Jacobson's density theorem, given
$v_1,...,v_n \in V, \hskip.1cm
a \in {\cal A},$ we can find $u \in {\cal U}$ such that
$\oplus_{\pi \in \hat{G}}
\pi (a) \hskip.1cm (v_i) = \oplus_{\pi \in \hat{G}} \hskip.1cm
\pi (u) (v_i),$ which proves the result. \hskip2cm Q.E.D.
\psaut
{\bf (2.6.2) Remark}:
\psaut
(1) One has ${\cal H} \subset C^\infty (G),$ this injection
is continuous and
$\overline{\cal H} = C^\infty (G).$ By transposition, we get
that $A \subset {\cal A},$
and $\bar{A} = {\cal A}$ (another proof of (2.5.4)). All these
inclusions are compatible with the Hopf structures. So ${\cal
A}$ is a completion of
the Hopf algebra $A$ of distributions on $G$; this is the
reason why we call
${\cal A}$ the algebra of formal distributions on $G.$
(2) The map $i$ of (2.6.1) is valued in $A,$ so ${\cal U} (g) \subset A,$
as Hopf algebras. However, $\overline{{\cal U} (g)} \neq A.$
\psaut
{\bf (2.7) Ideals and representations of ${\cal A}.$}
{\bf (2.7.1) Proposition}:
\it (1) Let $I$ be a left (resp: right) closed ideal of ${\cal
A},$ then there
exists a left (resp: right) closed ideal $J$ such that ${\cal A}
= I \oplus J$ (topological direct sum).
(2) Let $I$ be a left (resp: right) closed ideal of ${\cal A},$
then $I = \Pi_{\pi \in \hat{G}} (I \cap {\cal A}_\pi).$
(3) Let $I$ be a two sided closed ideal of ${\cal A},$ then
there exists a subset
$\hat{G}_I \subset \hat{G}$ such that:
$$I = \Pi_{\pi \in \hat{G}_I} \hskip.1cm {\cal A}_\pi.$$
(4) Let $\rho$ be a finite dimensional ${\cal A}$-module,
then there exists $G_\rho
\subset \hat{G}$ such that
\noindent
$\rho \simeq \sum_{\pi \in G_\rho} \hskip.1cm n_\pi.\pi,$
with $n_\pi
\in \Nrm$; as a consequence $\rho$ is a continuous ${\cal
A}$-module. \rm
\psaut
{\bf Proof}:
(1) Using (A.2.9), $I$ has a topological supplement $V$ in
${\cal A}.$ So, if
we set $\rho (x) (a) = xa, x \in G, a \in {\cal A}, \bar{\rho}
(x) (\bar{a}) =
\overline{xa}, \bar{a} \in {\cal A} / I,$ the obtained
representation $\rho$ of
$G$ is an extension of $\bar{\rho}$ in the sense of [18].
Since the canonical
map ${\cal A} \fl {\cal A} / I$ has a continuous section, such
an extension is
related to a cohomology, namely to $H^1 (G, {\cal L}_c (A /
I, I))$ (see [18]).
But $G$ being compact, $H^1$ vanishes ([14]), so the
extension splits, i.e. $I$
has a $\rho$-stable topological supplementary, which is the
wanted ideal $J.$
(2) First, since $I$ is closed, if $i_\pi \in I, \forall \pi \in
\hat{G},$ then
$(i_\pi) \in I,$ so $\Pi_{\pi \in \hat{G}} (I \cap {\cal A}_\pi)
\subset I.$ Then,
using (1), we write ${\cal A} = I \oplus J,$ where $J$ is a
closed left ideal,
and deduce that $1 = 1_I + 1_J$; if $i \in I,$ then $i = i.1_I.$
Let $i = (i_\pi),$
$1_I = (1_{I \pi}),$ one has $i = (i_\pi.1_{I \pi}),$ but
$i_\pi.1_{I \pi} = i_\pi.1_\pi
\in I,$ so the result is proved.
(3) Using (2), $I = \Pi_{\pi \in \hat{G}} (I \cap {\cal A}_\pi),$
and since
${\cal A}_\pi = {\cal L} (V_\pi)$ is a simple algebra, one has
$I \cap {\cal A}_\pi =
\{ 0 \}$ or ${\cal A}_\pi.$ \hskip.2cm Q.E.D.
(4) Using (2.5.4), the result is trivial if $\rho$ is assumed continuous. To
carry out a proof
without this assumption, we have to introduce the center
$Z({\cal A}).$ It is clear
that $Z({\cal A}) = \Pi_{\pi \in \hat{G}} \hskip.1cm \Crm
\hskip.1cm 1_\pi.$ Now $\hat{G}$
is countable, so we can choose a bijection between
$\hat{G}$ and ${\bf Z};$ let us
denote by $n_\pi$ the integer which corresponds to $\pi \in
\hat{G},$ and introduce
$Q = (n_\pi 1_\pi) \in Z({\cal A}).$
Now, let $\lambda_1,...,\lambda_k$ be the eigenvalues of $\rho (Q),$ and $V(\lambda_1),...,V (\lambda_k)$ the corresponding generalized eigenspaces ; then $V_\rho = \oplus^k_{i=1} \hskip.1cm V (\lambda_i),$ and each $V (\lambda_i)$ is a sub-${\cal A}$-module, so we can restrict to the case $V_\rho = V (\lambda), \hskip.1cm \lambda \in \Crm.$ Setting $\alpha = dim \hskip.1cm V_\rho,$ one has $(Q - \lambda)^\alpha \in Ker \hskip.1cm \rho$; if $\lambda \neq n_\pi, \hskip.1cm \forall \pi \in \hat{G},$ then $(Q - \lambda)^\alpha$ has an inverse in ${\cal A},$ so ${\cal A} = ker \hskip.1cm \rho$; if $\lambda = n_{\pi_0}, \hskip.1cm \pi_0 \in \hat{G},$ then $(Q - n_{\pi_0})^\alpha \hskip.1cm {\cal A} = \Pi_{\pi \neq \pi_0} \hskip.1cm {\cal A}_\pi \subset ker \hskip.1cm \rho,$ so $\rho$ is actually a representation of the simple algebra ${\cal A}_{\pi_0} = {\cal A} / \Pi_{\pi \neq \pi_0} \hskip.1cm {\cal A}_\pi,$ and this completes the proof of (4). \hskip2cm Q.E.D.
\psaut
{\bf (2.7.2) Remarks.}
Let us come back to the center $Z({\cal A}),$ and explain
some properties of the
very special element $Q$ introduced in the proof of (2.7.1)
(4). (Actually, $Q$
can be seen as some kind of generalized Casimir, with very
nice properties):
(1) First, from the definition of $Q,$ a finite dimensional irreducible
representation $\rho$ is
characterized, up to equivalence, by the number $\rho
(Q).$
(2) $Q$ generates $Z({\cal A})$ in the following sense:
given $C = (\lambda_\pi
1_\pi) \in {\cal A}, \lambda_\pi \in \Crm,$ there exists an
entire function $f(z) =
\sum_\rho \hskip.1cm f_\rho \hskip.1cm z^\rho, z \in \Crm,$
such that $f(n_\pi) =
\lambda_\pi.$ Now, the series $f(Q) = \sum_\rho \hskip.1cm
f_\rho
\hskip.1cm Q^\rho$ converges in ${\cal A},$ and $C =
f(Q).$ So, if we denote by
${\cal E}$ the algebra of complex entire functions, by ${\cal
I}$ the closed ideal generated by
%----------------------------------------------
%The following looks funny in print.
% Also, you now have a conflict in the meaning
%of \pi (but the reader will understand).
%-----------------------------------------------
$\hbox{ sin} \hskip.1cm J \pi z,$ one has
$Z({\cal A})
\simeq {\cal E} / {\cal I}.$
\saut
\centerline {\bf 3. Deformation theory of topological
algebras.}
\psaut
Let us first extend the notion of topological algebras,
bialgebras, etc...to
the $\hskip.1cm \Crm [[t]]$ case.
\psaut
{\bf (3.1) Definition}:
\it A topological $\hskip.1cm \Crm [[t]]$-algebra $\tilde{A}$
is a topologically free $\hskip.1cm \Crm [[t]]$-module
$\tilde{A} \simeq A[[t]],$ where $A$ is a c.t.v.s., with a
$\hskip.1cm \Crm [[t]]$-bilinear
continuous product $\tilde{\mu}.$ \rm
\psaut
Obviously, $\tilde{\mu}$ induces a product $\mu$
on $A,$ endowing $A$ with a topological algebra structure,
as defined in (A,4,1).
We call $A$ the {\it classical algebra} associated to
$\tilde{A}.$
\psaut
{\bf (3.2) Definition}:
\it Given a topological algebra $A,$ a deformation of $A$ is
a topologically free
$\hskip.1cm \Crm [[t]]$-algebra $\tilde{A}$ (see (A.3.2)),
such that $\tilde{A} / t \tilde{A} \simeq A.$ \rm
\psaut
The simplest example is the {\it trivial deformation}: let
$\mu$ be the product
of $A,$ then $\mu \in {\cal L} (A \hat{\otimes} A, A) \subset
{\cal L} (A \hat{\otimes} A,A)
[[t]] \simeq {\cal L}_t (A[[t]] \hat{\otimes}_t A[[t]], A[[t]])$
(A(3.1) and (A(3,2)),
so $\mu$ extends to a continuous $\Crm [[t]]$-bilinear
product on $A[[t]].$
\psaut
{\bf (3.3) Equivalence}: As usual, deformations are {\it
equivalent} if they are isomorphic as topological
$\Crm[[t]]$-algebras, the isomorphism reducing to the
identity modulo $t$, and a deformation is {\it trivial} if it is
equivalent to
the trivial deformation. A deformation $\tilde{A}$ with
product $\tilde{\mu}$ of
an algebra $A$ with product $\mu$ is completely specified
when one knows:
$$a, b \in A, \tilde{\mu} (a,b) \buildrel def \over{=} a
\times_t b = a \times b + t C_1 (a,b) +
t^2 C_2 (a,b) + ..., \hbox{ with } C_i \in {\cal L} (A
\hat{\otimes} A,A). \leqno{(3,3,1)}$$
This looks exactly like the usual algebraic theory [10],
except that the cochains
$C_i$ involved have to be continuous. Note that if $A$ is
an algebra of countable
dimension, then by (A.2.9), the topological deformation
theory of the topological
algebra $A$ (see (1.5.1)), and the usual algebraic theory
are the same.
\psaut
{\bf (3.4) Topological deformations and cohomology}: If $A$
is a (general) associative topological algebra, one can restrict
to associative deformations, and the usual cohomological
machinery can be used, noticing
that cohomology in that case is {\it continuous
cohomology.} So, from (3.2.1),
the {\it leading term} $C_1 \in Z^2_c (A,A),$ obstructions
are in $Z^3_c (A,A),$
etc... Then, by standard arguments [10] one has:
\psaut
{\bf (3.4.1) Lemma}:
\it (1) If $H^2_c (A,A) = \{ 0 \},$ any deformation of $A$ is
trivial ($A$ is rigid).
(2) If $H^3_c (A,A) = \{ 0 \},$ any cocycle in $Z^2_c (A,A)$
is the leading term of at least one deformation. \rm
\psaut
{\bf (3.5) Unit}: Let us assume that $\tilde{A}$ is a
deformation with associative
product of a topological asociative algebra $A.$ We recall
that in our terminology,
associative means: associative product and existence of
unit. Exactly as in the
algebraic case (see [10]), it can be shown that $\tilde{A}$
has a unit, so
$\tilde{A}$ is an associative deformation, and passing, if
necessary, to an equivalent deformation, one may even
assume that $1_{\tilde{A}} = 1_A.$
\psaut
{\bf (3.6) Formal series}: We need to extend our topological
notions of bialgebras, Hopf algebras,
etc... to $\Crm [[t]]$-algebras (3.1). This is straightforward,
so we give less
details: actually in the axioms of (A,4,1), in order to define
topological $\Crm
[[t]]$-bialgebras, Hopf algebras, etc..., one has to assume:
(1) that $\tilde{A}$ is a topologically free $\Crm [[t]]$-module,
$\tilde{A} \simeq A[[t]],$ where $A$ is a c.t.v.s.
(2) that the mappings involved, such as coproduct, counit,
antipode, are $\hskip.1cm \Crm [[t]]$-linear
and continuous.
(3) and to replace $\hat{\otimes}$ by $\hat{\otimes}_t$ (see
(A.3.5.1)).
Given a topologically free $\hskip.1cm \Crm [[t]]$-bialgebra
$\tilde{A},$ then $A = \tilde{A} / t \tilde{A}$
is a topological bialgebra: actually we can assume that
$\tilde{A} = A[[t]],$ as
$\hskip.1cm \Crm[[t]]$-module, so
$\tilde{A} \widehat{\otimes}_t \tilde{A} = (A
\widehat{\otimes} A) [[t]]$ then ${\cal L}_t
(\tilde{A}, \tilde{A} \hat{\otimes}_t \tilde{A}) \simeq {\cal L}
(A, A \hat{\otimes} A)
[[t]],$ therefore the coproduct $\tilde{\Delta}$ is completely
specified by:
$a \in A, \tilde{\Delta} (a) = \Delta (a) + t D_1 (a) + t^2 D_2
(a) + ...,$ where
$D_i \in {\cal L} (A, A \hat{\otimes} A)$ and $\Delta$ is the
coproduct for $A.$
Similar arguments show that when $\tilde{A}$ is a
topological $\Crm [[t]]$-Hopf
algebra, then $A = \tilde{A} / t \tilde{A}$ is a Hopf algebra.
Now the notion of deformation is clear:
\psaut
{\bf (3.7) Definition}:
\it Given a topological bialgebra (resp: Hopf algebra) $A,$
a deformation of $A$
is a topological $\hskip.1cm \Crm [[t]]$-bialgebra (resp:
Hopf algebra) $\tilde{A},$ such
that $\tilde{A} / t \tilde{A} \simeq A.$ \rm
\psaut
Equivalence is defined as in (3.2), and has to relate
respective coproducts, antipode,
etc..., as usual.
(Up to equivalence one may assume that
The unit and counit of the deformed bialgebra are identical
with those of the original. However, while any deformation
of a Hopf algebra continues to be Hopf, i.e., will continue to
have an antipode, it will generally not be the same as the
original.
\psaut
{\bf (3.8) Proposition}:
\it Let $\tilde{A}$ be a bialgebra (resp: Hopf) deformation of
a well-behaved
topological bialgebra (resp: Hopf) algebra $A.$ Then the
$\hskip.1cm \Crm [[t]]$-dual
$\tilde{A}^*_t$ (see (A.3.3.2)) is a deformation of the
topological Hopf algebra
$A^*.$ Deformations $\tilde{A}$ and $\tilde{A}'$ of $A$ are
equivalent if and
only if $\tilde{A}^*_t$ and $\tilde{A}'^{*}_t$ are equivalent
deformations of $A^*.$ \rm
\psaut
{\bf Proof}:
We prove (3.8) in the Hopf case.
The Hopf structure of $A^*$ was defined in (1.3). We can
assume that $\tilde{A} =
A[[t]],$ and $\tilde{A}^*_t$ stands for $A [[t]]^*_t \simeq A^*
[[t]],$ as $\hskip.1cm \Crm[[t]]$-modules (A.3.3.3).
Let $\tilde{\mu}, \tilde{\Delta}, \tilde{\varepsilon}$ and
$\tilde{S}$ be the respective
product, coproduct, counit and antipode on $A[[t]],$ which
deform the corresponding
objects $\mu, \Delta, \varepsilon$ and $S$; we can assume
that $1_{\tilde{A}} = 1_A$
(see (3.5)). Using $\hskip.1cm \Crm [[t]]$-transposition, as
defined in (A.3.4), and $(\tilde{A}
\hat{\otimes}_t \tilde{A})^*_t = (A \hat{\otimes} A[[t]])^*_t
\simeq (A \hat{\otimes} A)^*
[[t]] = A^* \hat{\otimes} A^* [[t]] = \tilde{A}^*_t \otimes_t
\tilde{A}^*_t$
(cf. appendices $A_1$ and $A_3$), we obtain a
$\hskip.1cm \Crm [[t]]$-Hopf
structure on $\tilde{A}^*_t,$
with respective product, coproduct, counit and antipode
defined by: $^T \tilde{\Delta},
^T \tilde{\mu}, ^T 1_A,$ and $^T \tilde{S}$; the unit of
$\tilde{A}^*_t$ is
$^T \tilde{\varepsilon}.$
Now, the Hopf structure of $A^*$ is defined exactly in the
same way, using usual
transposition; so $\tilde{A}^*_t$ deforms $A^*.$
The same transposition argument and $A^{**} = A$ proves
the last claim. \hskip1cm
Q.E.D.
\psaut
{\bf (3.9) Example}:
Let $A$ be a countable dimensional Hopf algebra. Then, as
shown in (1.5.1) $A$ is
a well-behaved topological algebra for its natural topology.
So (3.8) applies
to this case. As noticed in (3.3.1), algebraic deformation
theory [10] of $A,$ or
(topological) deformation theory are the same. Using (3.8),
deformation theory of
$A$ is the same as (topological) deformation of $A^*$;
unless $dim \hskip.1cm A < \infty,$
this is generally not identical to algebraic deformation
theory of $A^*.$ This is
a rather striking example of how continuity can be hidden in
an, a priori, purely algebraic problem !
\psaut
{\bf (3.10) Remark}:
Let $A$ be a Hopf algebra ; it is proved in [12] that any
associative and
coassociative bialgebra algebraic deformation of $A$ is
actually a Hopf algebra.
The same results holds in the topological case, and the
proof is essentially the same.
\saut
{\bf 4. Deformations of the topological Hopf algebras of a
compact Lie group.}
\psaut
In this section, we study the properties of deformations, as
defined in section 3,
of the Hopf algebras ${\cal H} = {\cal H} (G),$ or $H = H(G)
= C^\infty (G),$ or, equivalently,
(see (3.8)), of their respective dual algebras ${\cal H}^* =
{\cal A} (G) = {\cal A},$
or $H^* = A(G) = A.$ From (3.9), deformation theory of
${\cal H}$ is equivalent to
algebraic deformation theory. Our notion of (topological)
deformations makes it
possible to study deformations of $H$ or $A$; we have not
heard that any algebraic theory was tried in that case.
\psaut
{\bf (4.1) Deformations of representations}: We shall need
some results about
deformations of representations of $G,$ that we now
introduce (see [17] for more details):
Given a t.v.s. $V,$ ${\cal L}_t (V[[t]])$ is an algebra, so the
isomorphism
of (A.3.3.1) defines an algebra structure on ${\cal L} (V)
[[t]].$ More precisely, one has:
$$\sum_n \hskip.1cm t^n \hskip.1cm T_n, \hskip.1cm \sum_n \hskip.1cm t^n \hskip.1cm U_n \in {\cal L}(V) [[t]], (\sum_n t^n T_n) o (\sum_n
t^n U_n) \buildrel def \over{=} \hskip.1cm
\sum_n \hskip.1cm t^n (\sum^n_{i=0} T_i o U_{n-i}).$$
Now, it is obvious that $\sum_n \hskip.1cm t^n \hskip.1cm
T_n$ has an inverse
if an only if $T_0$ has an inverse in ${\cal L} (V).$
\psaut
{\bf (4.1.1) Definition}:
\it Given a continuous representation $\pi$ of $G$ in $V,$ a
deformation (or formal
representation) $\tilde{\pi}$ of $\pi$ is a morphism
$\tilde{\pi}: G \fl {\cal L}(V)
[[t]]$ such that
(1) $\tilde{\pi} = \pi + \sum_{n \geq 1} t^n \hskip.1cm \pi_n$
(2) $(g,v) \fl \pi_n (g)(v)$ is continuous from $G \times V$
into
$V, \forall n.$
Deformations $\tilde{\pi}$ and $\tilde{\pi}'$ of $\pi$ are
equivalent if there exists
$\phi = Id + \sum_{n \geq 1} t^n \hskip.1cm \phi_n \in {\cal
L}(V) [[t]]$ such
that $\tilde{\pi}' = \phi o \tilde{\pi} o \phi^{-1}.$ A
deformation $\tilde{\pi}$
of $\pi$ is trivial if it is equivalent to $\pi.$ \rm
\psaut
Assume that $dim \hskip.1cm V < \infty.$ We extend the
trace map
$Tr: {\cal L}
(V) \fl \Crm$ to a $\hskip.1cm \Crm [[t]]$-linear trace map
$\tilde{Tr}: {\cal L} (V) [[t]]
\fl \Crm [[t]],$ defined by $\tilde{Tr} = Id_{\Crm [[t]]} \otimes
Tr.$ Formula
$\widetilde{Tr} (\tilde{A} o \tilde{B}) = \tilde{Tr} (\tilde{B} o
\tilde{A})$ is still valid
for $\tilde{A}, \tilde{B} \in {\cal L} (V) [[t]].$
Given a continuous representation $\pi$ of $G$ in $V,$ and
a deformation $\tilde{\pi}$
of $\pi,$ we can still define coefficients of $\tilde{\pi}$ by:
(4.1.2) $\tilde{M} \in {\cal L} (V) [[t]],
C^{\tilde{\pi}}_{\tilde{M}} (x) = \widetilde{Tr}
(\tilde{M} \hskip.1cm o \hskip.1cm \tilde{\pi} (x)) \in \Crm
[[t]].$
The formal character of $\tilde{\pi}$ is:
(4.1.3) $\xi^{\tilde{\pi}} = C^{\tilde{\pi}}_{Id}$
Now $G$ is compact, so representations are rigid:
\psaut
{\bf (4.1.4) Lemma}:
\it If $\tilde{\pi}$ is a deformation of a continuous
representation $\pi$ of $G$
in a t.v.s. $V,$ then $\tilde{\pi}$ is a trivial deformation. \rm
\psaut
{\bf Proof} (see [17]).
We define $\phi \in {\cal L} (V) [[t]]$ by $\phi = \int_G
\hskip.1cm \tilde{\pi} (x) \hskip.1cm \pi (x^{-1}) dx$;
one has $\phi_0 = Id_V,$ and due to the invariance of the
Haar measure $\tilde{\pi}
\hskip.1cm o \hskip.1cm \phi = \phi \hskip.1cm o
\hskip.1cm \pi.$ \hskip2cm Q.E.D.
Assuming once more that $dim \hskip.1cm V < \infty,$ and
setting $\tilde{\pi} =
\phi \hskip.1cm o \hskip.1cm \pi \hskip.1cm o \hskip.1cm
\phi^{-1},$ $\phi \in
{\cal L} (V) [[t]],$ which is, from (6.1.4), the general case of
deformations of $\pi,$ one has:
given $\tilde{M} \in {\cal L} (V) [[t]],$
\noindent
$C^{\tilde{\pi}}_{\tilde{M}} =
C^\pi_{\phi^{-1} o \tilde{M} o \phi}$; let $\tilde{N} = \phi^{-1}
\hskip.1cm o \hskip.1cm
\widetilde{M} \hskip.1cm o \hskip.1cm \phi = \sum
\hskip.1cm t^n \hskip.1cm N_n,$ one has
$C^{\tilde{\pi}}_{\tilde{M}} = \sum \hskip.1cm t^n \hskip.1cm
C^\pi_{N_n},$ so (see section 2):
(4.1.5) $C^{\tilde{\pi}}_{\tilde{M}} \in {\cal H} [[t]],$ and
(4.1.6) $\xi^{\tilde{\pi}} = \xi^\pi.$
\psaut
{\bf (4.2) Deformations of the Hopf algebras ${\cal A}$ and
$A.$}:
\psaut
{\bf (4.2.1) Proposition}:
\it Any associative deformation of the algebras ${\cal A}$ or
$A$ is trivial. \rm
\psaut
{\bf Proof}:
By (3.4.1) the result is contained in the following lemma:
\psaut
{\bf (4.2.2) Lemma}: $H^n_c ({\cal A}, {\cal A}) = H^n_c
(A,A) = \{ 0 \}, \forall \hskip.1cm n \geq 1.$
\psaut
{\bf Proof for ${\cal A}$}:
Let $f = \Pi_k \hskip.1cm f_k \in Z^n_c ({\cal A}, {\cal A}),$
then each $f_k \in
Z^n_c ({\cal A}, {\cal A})$; for fixed $k,$ we set ${\cal A} =
{\cal A}' \oplus
{\cal A}_k,$ $1_{\cal A} = 1' + 1_{{\cal A}_k}, \hskip.1cm a
= a' +a_k,$ for $a \in {\cal A}$ and we get:
$$f(a_1,...,a_n) = f(a'_1,a_2,...,a_n) +
f(a_{1k},a'_2,a_3,...,a_n) + f(a_{1k},
a_{2k},a'_3,a_4,...,a_n) + ... +$$
$$+ f(a_{1k},...,a_{n-1 \hskip.1cm k},a'_n) + f
(a_{1k},...,a_{nk}).$$
First $f_k \vert \otimes_n \hskip.1cm \vert_{{\cal A}_k} \hskip.1cm = d(\varphi_k),$ with
$\varphi_k \in
{\cal L}^{n-1} ({\cal A}_k, {\cal A}_k),$ since $H^n ({\cal
A}_k, {\cal A}_k) =
\{ 0 \}.$ Then, on ${\cal A}_k \otimes ... \otimes {\cal A}_k
\otimes {\cal A}'_s
\otimes {\cal A} \otimes ... \otimes {\cal A},$ we introduce
$\psi^s_k (x_{1k},...,
x_{s-1 \hskip.1cm k}, x'_s, x_{s+1},..., x_{n-1}) = - (-1)^s
\hskip.1cm f_k (x_{1k},...,
x_{s-2 \hskip.1cm k}, 1', x_{s-1 \hskip.1cm k}, x'_s,
x_{s+1},...,x_{n-1}), \theta_k =
\sum^{n-1}_{s=1} \hskip.1cm \psi^s_k + \varphi_k,$ and
using cocycle relation, we
see that $f_k = d (\theta_k)$; then, if $\xi = \Pi_k \hskip.1cm
\theta_k,$ one
has $f = d \xi.$ Moreover, from the definition of $\xi,$ and
the continuity of $f, \xi$ is continuous. \hskip2cm Q.E.D.
\psaut
{\bf (4.2.3) Remarks}:
\psaut
(1) By the same proof, $H^n ({\cal A}, {\cal A}) = \{ 0 \},
\hskip.1cm n \geq 1,$
so, as a consequence, ${\cal A}$ is also rigid in the
algebraic [10] sense.
(2) By arguments similar to the proof of (4.2.2), one has:
$$H^1 ({\cal A}, {\cal A} \hat{\otimes} {\cal A}) = H^1_c ({\cal A}, {\cal A} \hat{\otimes}{\cal A}) = \{ 0 \}.$$
It results from [2], that for the cohomology of bialgebras $H_{bi}$ defined in [2], one has $H^2_{bi} \hskip.1cm ({\cal A}, {\cal A}) = H^2_{bi, c} \hskip.1cm ({\cal A}, {\cal A}) = \{ 0 \},$ so ${\cal A}$ is rigid in the category of bialgebras, in the sense defined in [2].
\psaut
{\bf Proof for $ A$}:
First, we note that ${\cal L} (\hat{\otimes}_n A,A) \simeq
(\hat{\otimes}_n A)^*
\hskip.1cm \hat{\otimes} A = A (G^n)^* \hskip.1cm
\hat{\otimes} A \simeq
C^\infty (G^n) \hat{\otimes} A = C^\infty (G^n, A),$ so given
$c \in {\cal L}
(\hat{\otimes}_n A,A),$ the restriction $c \vert_{G^n} \in
C^\infty (G^n, A),$ and
conversely, given $c \in C^\infty (G^n, A),$ then $c$ is the
restriction of an
element of ${\cal L} (\hat{\otimes}_n A,A).$
Now, given $\xi \in Z^n_c (A,A),$ then we set
$\theta (x_1,...,x_n) = \xi (x_1,...,x_n) x^{-1}_n ... x^{-1}_1,$
and get $\theta
\in Z^n_\infty (G,A)$ (see [14]), where $G$ acts on $A$ by
the representation: $T_x (a) = x
\hskip.1cm a \hskip.1cm x^{-1}, \hskip.1cm x \in G,
\hskip.1cm a \in A.$ Since
$G$ is compact, one has $H^n_\infty (G,A) = \{ 0 \}$ [14],
so there exists $L \in C^\infty
(G^{n-1}, A)$ such that $\theta = d_G (L)$; but $L \in {\cal
L} (\hat{\otimes}_{n-1}
A,A),$ so $\xi \in B^n_c \hskip.1cm (A,A).$ \hskip1cm
Q.E.D.
\psaut
{\bf (4.2.4) Proposition}:
\it Let $({\cal A} [[t]], \tilde{\Delta})$ (resp: $(A[[t]],
\tilde{\Delta})$)
be an associative deformation of the bialgebra ${\cal A}$
(resp: $A$); using
(4.2.1), we assume that the product is unchanged. Then
there exists $\tilde{P}
\in {\cal A} \hat{\otimes} {\cal A} [[t]]$ (resp: $A
\hat{\otimes} A[[t]]$) such
that $\tilde{\Delta} = \tilde{P} \hskip.1cm \Delta_0
\hskip.1cm \tilde{P}^{-1}.$ \rm
\psaut
{\bf Proof}:
$\tilde{\Delta} \in {\cal L}_t ({\cal A} [[t]], {\cal A} [[t]]
\hat{\otimes}_t
{\cal A} [[t]]) \simeq {\cal L} ({\cal A}, {\cal A} \hat{\otimes}
{\cal A}) [[t]]$
(see (A.3.3.1) and (A.3.5.1)); write $\tilde{\Delta} = \Delta_0
+ \sum_{n \geq 1} \hskip.1cm
t^n \hskip.1cm \Delta_n,$ and introduce $\tilde{\Delta} (x) =
\sum_n \hskip.1cm
t^n \hskip.1cm \Delta_n (x) \in {\cal A} \hat{\otimes} {\cal A}
[[t]], x \in G,$
and $\tilde{P} = \int_G \hskip.1cm \tilde{\Delta} (x)
\hskip.1cm
\Delta_0 (x)^{-1}
\hskip.1cm dx \in {\cal A} \hat{\otimes} {\cal A} [[t]];$ due to
the invariance of the Haar measure, one has:
$\tilde{P} \hskip.1cm \Delta_0 (y) = \tilde{\Delta} (y)
\hskip.1cm \tilde{P}, \hskip.1cm \forall y \in G,$ and since
$\overline{Vect (G)} = {\cal A},$
one gets $\Delta = \tilde{P} \hskip.1cm \Delta_0 \hskip.1cm
\tilde{P}^{-1}$ (note
that $\tilde{P}^{-1}$ exists since $\tilde{P}_0 = 1 \otimes 1$).
The proof is the same in the case of $A.$ \hskip1cm Q.E.D.
\psaut
{\bf Remark}: The explicit integral formula for the twist $\tilde{P} = \int_G
\hskip.1cm \tilde{\Delta} (x) \hskip.1cm \Delta_0 (x)^{-1} dx$ is of interest.
Note that for any $\tilde{H} \in {\cal A} \hat{\otimes} {\cal A} [[t]],$ or
$A \hat{\otimes} A [[t]],$ with $\tilde{H}_0 = 1 \otimes 1, \hskip.1cm \tilde{Q} =
\int_G \hskip.1cm \tilde{\Delta} (x) \hskip.1cm \tilde{H} \hskip.1cm \Delta_0
(x)^{-1} dx$ also defines a twist between $\tilde{\Delta}$ and $\Delta_0.$
\psaut
{\bf (4.2.5) Corollary}:
\it Any associative deformation of the bialgebra ${\cal A}$
(resp: ${\cal A}$)
is quasicocommutative and quasicoassociative. \rm
\psaut
{\bf Proof}:
In the terminology of [7], $\tilde{\Delta}$ is obtained from
$\Delta_0$ from
twisting by $\tilde{P},$ so, using [7] and the fact that
$\Delta_0$ is cocommutative
and coassociative, $\tilde{\Delta}$ is quasicocommutative
and quasicoassociative. \hskip1cm Q.E.D.
\psaut
{\bf (4.2.6) Proposition}:
\it Let $({\cal A} [[t]], \tilde{\Delta})$ (resp: $(A[[t]],
\tilde{\Delta})$) be
an associative and coassociative deformation of the
algebra ${\cal A}$ (resp: $A$),
with unchanged product. Then the counit $\varepsilon$ of
${\cal A}$ (resp: $A$)
is still a counit of ${\cal A} [[t]]$ (resp: $A[[t]]$), and there
exists an antipode
$\tilde{S},$ so ${\cal A} [[t]]$ (resp: $A [[t]]$) is a $\Crm [[t]]$-Hopf
algebra. Let $S_0$
be the antipode of ${\cal A}$ (resp: $A$), then there exists
$\tilde{a} \in {\cal A} [[t]]$
(resp: $A[[t]]$) such that $\tilde{S} = \tilde{a} \hskip.1cm S_0
\hskip.1cm \tilde{a}^{-1}.$ \rm
\psaut
{\bf Proof}:
As noted in (3.5), the corresponding deformation ${\cal H}
[[t]]$ of ${\cal H}$
(see (3.8)) has a unit $\tilde{\varepsilon},$ which satisfies
$\delta (\tilde{\varepsilon}) =
\tilde{\varepsilon} \otimes \tilde{\varepsilon}$; So
$\tilde{\varepsilon} (xy) =
\tilde{\varepsilon} (x) \hskip.1cm \tilde{\varepsilon} (y),
\hskip.1cm \forall
x,y \in G,$ and this proves that $\tilde{\varepsilon} \vert_G$
is
a deformation
of the trivial representation $\varepsilon \vert_G.$ By
(4.1.4),
$\tilde{\varepsilon} \vert_G =
\varepsilon \vert_G,$ so $\tilde{\varepsilon} = \varepsilon.$
It was noted in (3.10) that ${\cal A} [[t]]$ has an antipode
$\tilde{S},$ which
%------------------------------------------
%We come again to a notation which is very hard to read.
%---------------------------------------------------
deforms $S_0.$ For any $\pi \in \hat{G},$
$\tilde{\check{\pi}} = ^T\pi o \tilde{S}$
is a deformation of $\check{\pi}$; using (4.1.4), there exists
$\tilde{a}_\pi \in
{\cal L} (V^*_\pi) [[t]]$ such that $\tilde{\check{\pi}} =
\tilde{a}_\pi \hskip.1cm
\check{\pi} \hskip.1cm \tilde{a}^{-1}_\pi,$ so $^T
\tilde{\check{\pi}} = \pi \hskip.1cm o
\hskip.1cm \tilde{S} = (^T \tilde{a}_\pi)^{-1} (\pi \hskip.1cm o
\hskip.1cm S)
^T \tilde{a}_\pi$; let $\tilde{a} = (^T a_\pi)^{-1} \in {\cal A}
[[t]],$ then $\tilde{S} = \tilde{a} \hskip.1cm S_0 \hskip.1cm
\tilde{a}^{-1}.$
Similarly, by (3.10), $A[[t]]$ has an antipode $\tilde{S}$
which deforms $S_0.$
Let $\tilde{a} = \int_G \hskip.1cm \tilde{S} (x) \hskip.1cm
S_0 (x^{-1}) dx \in
A[[t]];$ using the (right) invariance of Haar measure, and
the antihomomorphism
property of antipodes, one deduces that $\tilde{a} S_0 (y) =
\tilde{S} (y) \tilde{a},
\hskip.1cm \forall y \in G$; moreover $\tilde{a} (0) = 1,$ so
$\tilde{a}$ has an
inverse in $A[[t]],$ and using the density of $G$ in $A,$ we
get:
$$\tilde{S} (a) = \tilde{a} S_0 (a) \hskip.1cm \tilde{a}^{-1},
\hskip.1cm \forall a \in A. \hskip1cm \hbox{Q.E.D.}$$
{\bf (4.3) Dualization to ${\cal H}$}: The results of (4.2)
translate by $\Crm [[t]]$-duality (3.8) as follows:
\psaut
{\bf (4.3.1) Proposition}:
\it Let $({\cal H} [[t]], \tilde{\times}, \tilde{\delta})$ (resp:
$(H[[t]], \tilde{\times},
\tilde{\delta})$) be a coassociative deformation of the
bialgebra ${\cal H}$ (resp:
$H$); then, up to equivalence, it can be assumed that
$\tilde{\delta} = \delta.$
The product is quasi-commutative, and quasi-associative,
the counit unchanged. If
the product is associative, then ${\cal H} [[t]]$ (resp:
$H[[t]]$) is a $\Crm [[t]]$-Hopf algebra,
with the same unit and counit than ${\cal H}$ (resp: $H$).
\rm
\psaut
{\bf (4.3.1.1) Definition}:
\it A deformation of the bialgebra ${\cal H}$ (resp: $H$)
with unchanged coproduct will be called a preferred
deformation of ${\cal H}$ (resp: $H$). \rm
Let us now develop consequences of (4.2.4). We need
some notations. Given $\pi$
and $\pi' \in \hat{G},$ we denote by $\rho_{\pi \pi'}$ their
tensor product
$\rho_{\pi \pi'} (x) = \pi \otimes \pi' (\Delta_0 (x)), x \in G,$
and by $\tilde{\rho}_{\pi \pi'}$
their new tensor product given by $\tilde{\rho}_{\pi \pi'} (x) =
\pi \otimes \pi'
(\tilde{\Delta} (x)).$
Using (4.2.4), one has:
$$\tilde{\rho}_{\pi \pi'} = \pi \otimes \pi' (\tilde{P})
\hskip.1cm \rho_{\pi \pi'}
\hskip.1cm \pi \otimes \pi' (\tilde{P})^{-1}.$$
We write ${\cal A} \hat{\otimes} {\cal A} = \Pi_{\pi, \pi' \in
\hat{G}} {\cal A}_\pi
\otimes {\cal A}_{\pi'},$ with projections $\pi \otimes \pi':
{\cal A} \hat{\otimes}
{\cal A} \fl {\cal A}_\pi \otimes {\cal A}_{\pi'},$ and ${\cal
A}[[t]] \hat{\otimes}_t
{\cal A}[[t]] = ({\cal A} \hat{\otimes} {\cal A}) [[t]] =
\Pi_{\pi, \pi' \in \hat{G}}
({\cal A}_\pi \otimes {\cal A}_{\pi'}) [[t]],$ with the same
projections.
Likewise, for elements $P \in {\cal A} \hat{\otimes} {\cal A}$
(resp:
$\tilde{P} \in {\cal A}
[[t]] \hat{\otimes}_t {\cal A} [[t]])$ we write $P = (P_{\pi
\pi'}),$ with $P_{\pi \pi'} =
\pi \otimes \pi' (P) \in {\cal A}_\pi \otimes {\cal A}_{\pi'}$
(resp: $\tilde{P} =
(\tilde{P}_{\pi \pi'}),$ with $\tilde{P}_{\pi \pi'} = \pi \otimes
\pi' (P) \in
({\cal A}_\pi \otimes {\cal A}_{\pi'}) [[t]]).$
With these notations, we have:
$$\rho_{\pi \pi'} (x) = \Delta_0 (x)_{\pi \pi'}, \hskip.3cm
\tilde{\rho}_{\pi \pi'} (x) = \tilde{\Delta}
(x)_{\pi \pi'} = \tilde{P}_{\pi \pi'} \hskip.1cm \Delta_0
(x)_{\pi \pi'} \hskip.1cm
\tilde{P}^{-1}_{\pi \pi'}.$$
Then, given $M \in {\cal L} (V_\pi),$ and $M' \in {\cal L}
(V_{\pi'}),$ we compute,
using the trace (as defined in (4.1.1)):
$$\eqalign{
x \in G, \hskip.3cm C^\pi_M \tilde{\times} C^{\pi'}_{M'} (x)
&= < C^\pi_M \otimes C^{\pi'}_{M'}
\vert \tilde{\Delta} (x)> \in \Crm [[t]] \cr
&= < C^\pi_M \otimes C^{\pi'}_{M'} \vert \tilde{\Delta}
(x)_{\pi \pi'}> \cr
&= < C^\pi_M \otimes C^{\pi'}_{M'} \vert \tilde{\rho}_{\pi
\pi'} (x)> \cr
&= \widetilde{Tr} (M \otimes M' \hskip.1cm o \hskip.1cm
\tilde{\rho}_{\pi \pi'} (x)> = C^{\tilde{\rho}_{\pi \pi'}}_
{M \otimes M'} (x). \cr }$$
So we have proved that (2.3) is valid for preferred
deformations, namely:
\psaut
{\bf (4.3.2) Proposition}:
\it Let $({\cal H} [[t]], \tilde{\times})$ be a preferred
deformation of ${\cal H},$ then
if $\pi, \pi' \in \hat{G}, \hskip.1cm M \in {\cal L} (V_\pi),
\hskip.1cm M' \in {\cal L} (V_{\pi'}),
\hskip.1cm C^\pi_M \hskip.1cm \tilde{\times} \hskip.1cm
C^{\pi'}_{M'} = C^{\pi \tilde{\otimes} \pi'}_
{M \otimes M'},$ where $\pi \tilde{\otimes} \pi' =
\tilde{\rho}_{\pi \pi'}$ stands
for the new tensor product of the representations $\pi$ and
$\pi'$ of $G.$ \rm
\psaut
{\bf (4.3.3) Corollary}:
\it Under the same assumptions, let ${\cal H}_c$ be the
subalgebra of central
functions; then the product on ${\cal H}_c$ is unchanged.
\rm
\psaut
{\bf Proof}:
${\cal H}_c$ is linearly generated by the characters
$\xi^\pi$ of the elements of $\hat{G},$ so we compute:
$$\xi^\pi \hskip.1cm \tilde{\times} \hskip.1cm \xi^{\pi'} =
C^\pi_{Id V_\pi} \hskip.1cm
\tilde{\times} \hskip.1cm C^{\pi'}_{Id V_{\pi'}} = C^{\pi
\tilde{\otimes} \pi'}_{Id V_\pi \otimes Id V_{\pi'}} =
C^{\pi \tilde{\otimes} \pi'}_{Id (V_\pi \otimes V_{\pi'})} =
\xi^{\pi \tilde{\otimes} \pi'} =
\xi^{\pi \otimes \pi'}$$
$$= \xi^\pi \times \xi^{\pi'}. \hskip2cm \hbox{Q.E.D.}$$
This raises a natural question: do (4.3.2) and (4.3.3) hold if
${\cal H}$
is replaced by $H$? Actually, the answer is yes, and this
will be a consequence of the following proposition:
\psaut
{\bf (4.3.4) Proposition}:
\it If $(H, \tilde{\times})$ is a preferred deformation of $H,$
then it defines, by
restriction, a preferred deformation of ${\cal H},$ i.e.: \rm
$$\forall h,h' \in {\cal H}, \hskip.1cm h \hskip.1cm
\tilde{\times} \hskip.1cm h' \in {\cal H} [[t]].$$
\psaut
{\bf Proof}:
The inclusion ${\cal H} \subset H$ is a (continuous)
injective morphism, with
dense range; moreover, the coproduct of ${\cal H}$ is the
restriction of the
coproduct of $H.$ Therefore, using transposition we obtain
a continuous injective
morphism $A = H^* \mapsto {\cal A} = {\cal H}^*,$ with
dense range. So we can
consider that $A$ is a subalgebra of ${\cal A};$ moreover,
since the product of
${\cal H}$ is the restriction of the product of $H,$ the
coproduct of $A$ is the
restriction of the coproduct $\Delta_0$ of ${\cal A}.$ Using
(3,8), the given
preferred deformation of $H$ induces a corresponding
deformation of $A,$ with
unchanged product, and coproduct $\tilde{\Delta} =
\tilde{P} \hskip.1cm \Delta_0
\hskip.1cm \tilde{P}^{-1}, \hskip.1cm \tilde{P} \in A
\hat{\otimes} A[[t]],$ by
(4.2.4). Since $\tilde{P} \in {\cal A} \hat{\otimes} {\cal A}
[[t]],$
$\tilde{\Delta}$ is actually a coproduct on ${\cal A}.$
Now, given $h, h' \in {\cal H},$ we write $\tilde{\Delta} =
\sum \hskip.1cm t^n \hskip.1cm \Delta_n,$ and:
$$d \in A, \hskip.1cm = = \sum \hskip.1cm t^n =
\sum \hskip.1cm t^n \hskip.1cm C_n (h,h') (d)$$
Actually, $C_n (h,h')$ is an element of $H,$ completely
determined by its restriction
to $G \subset A.$ On the other hand, the formula $D_n
(h,h') (a) = $ defines an element of ${\cal H},$ also
completely determined by
its restriction to $G.$ Since $C_n (h,h')$ and $D_n (h,h')$
coincide on $G,$ one
has $C_n (h,h') = D_n (h,h') \in {\cal H},$ so:
$$h \hskip.1cm \tilde{\times} \hskip.1cm h' = \sum_n
\hskip.1cm t^n \hskip.1cm C_n (h,h')
\in {\cal H} [[t]]. \hskip2cm \hbox{Q.E.D.}$$
\psaut
{\bf (4.3.5) Corollary}: (4.3.2) {\it and} (4.3.3) {\it hold
with} $H$ {\it replacing} ${\cal H}.$
\psaut
{\bf Proof}:
Let $H_c$ be the subalgebra (in $H$) of central functions,
then $\overline{{\cal H}}_c = H_c,$
so using (4.3.4), (4.3.3) and the continuity of $C_n$ in the
development of the product
$$h \hskip.1cm \tilde{\times} \hskip.1cm h' = \sum
\hskip.1cm t^n \hskip.1cm C_n (h,h'),
\hskip.1cm h,h' \in H,$$ we get (4.3.3) for $H.$ \hskip2cm Q.E.D.
\psaut
{\bf (4.3.6) Proposition}:
\it Let $\tilde{\times}$ be a preferred associative
deformation of $H.$ By (4.3.1), it
is a Hopf deformation, with antipode $\tilde{S}.$ Then
$\tilde{S}$ restricts to ${\cal H},$ i.e.:
$$\forall h \in {\cal H}, \hskip.1cm \tilde{S} (h) \in {\cal H}
[[t]],$$
so the restriction of a Hopf deformation of $H$ defines a
Hopf deformation of ${\cal H}.$ \rm
\psaut
{\bf Proof}:
Using (4.3.4), the restriction defines a preferred associative
deformation of the
bialgebra ${\cal H}.$ From (4.3.1), the unit and counit of
$H$ are unchanged, so do
restrict to ${\cal H}.$ Again by (4.3.1), the obtained
deformation of ${\cal H},$
being associative, is a Hopf deformation, so it has an
antipode. The problem is
to show that this antipode is the restriction of the antipode
of $H,$ and this will
be done (using unicity of antipode) by showing that
$\tilde{S}$ restricts to ${\cal H}.$
We use $\hskip.1cm \Crm [[t]]$-duality (3.8): we have $A =
H^* \subset {\cal A} = {\cal H}^*,$
and our preferred deformation leads to deformations of $A$
and ${\cal A},$ with
unchanged product, counit and same coproduct (see
(4.3.4)). Both have antipode,
say $\tilde{s}$ and $\tilde{{\cal S}},$ and we want to show
that $\tilde{{\cal S}}
\vert_A = \tilde{s}.$ By (4.2.6), $\tilde{s} = \tilde{a}
\hskip.1cm S_0 \hskip.1cm
\tilde{a}^{-1},$ with $\tilde{a} \in A[[t]],$ so $\tilde{s}$
extends to an antipode
on ${\cal A}[[t]],$ defined by the same formula, and the
unicity of the antipode proves
that this extension is exactly $\tilde{{\cal S}}.$ \hskip2cm
Q.E.D.
\psaut
\centerline {\bf 5. Quotient deformations.}
\psaut
Given a compact connected Lie group $G,$ we continue to
denote by ${\cal H} (G)$ the algebra
of coefficients of $G,$ by $H(G) = C^\infty (G)$ the algebra
of $C^\infty$ functions
on $G,$ and by ${\cal A} (G)$ and $A(G)$ their respective
duals. We recall that
${\cal H} (G \times G) \simeq {\cal H} (G) \otimes {\cal H}
(G),$ and $H(G \times
G) \simeq H(G) \hat{\otimes} H(G)$; using (A.1.5), ${\cal A}
(G \times G) \simeq
{\cal A} (G) \hat{\otimes} {\cal A} (G),$ and $A(G \times G)
\simeq A(G) \hat{\otimes} A(G).$
Let $\Gamma$ be a normal subgroup of $G$; we introduce
$H(G)^{\Gamma} = \{ f \in H(G) / f(x \gamma) = f(x),
\hskip.1cm \forall x \in G,
\hskip.1cm \gamma \in \Gamma \},$ and ${\cal H}
(G)^\Gamma = {\cal H}(G) \cap
H(G)^\Gamma,$
\noindent
which are subalgebras respectively of $H(G)$ and ${\cal H}
(G),$ stable by the antipode since $\Gamma$ is normal.
Now, there is an obvious isomorphism $\phi: H(G)^\Gamma
\simeq H(G / \Gamma),$
which induces on $H(G)^\Gamma$ the coproduct of $H (G
/ \Gamma)$; one has $H(G)^\Gamma
\hat{\otimes} H(G)^\Gamma \simeq H(G / \Gamma \times G
/ \Gamma) \simeq H(G \times G /
\Gamma \times \Gamma),$ and since $\Gamma$ is normal,
this shows that the restriction
of the coproduct of $H(G)$ to $H(G)^\Gamma$ is exactly
the coproduct of $H(G)^\Gamma,$
so that $H(G / \Gamma) \simeq H(G)^\Gamma$ is
a Hopf subalgebra of $H(G).$
Let us show that the same result holds when $H$ is
replaced by ${\cal H}.$ First
we note that any element $\pi$ of $\widehat{G / \Gamma}$
is actually an element of
$\hat{G}$ satisfying $\pi \vert \Gamma = Id_{V_\pi}.$
So $\widehat{G / \Gamma} = \{ \pi \in \hat{G} / \pi
\vert_\Gamma = Id_{V_\pi} \}
\subset \hat{G}.$ We introduce the notations ${\cal C} (G /
\Gamma)_\pi,$ and ${\cal C} (G)_\pi$ as in (2,1).
\psaut
{\bf (5.1) Lemma}:
\it The restriction of $\phi$ to ${\cal H} (G)^\Gamma$ is an
isomorphism onto ${\cal H} (G / \Gamma).$
One has: \rm
$${\cal H} (G)^\Gamma = \sum_{\pi \in \widehat{G /
\Gamma}}
\hskip.1cm {\cal C} (G)_\pi.$$
{\bf Proof}:
Consider $\psi = \phi^{-1} \vert_{{\cal H} (G / \Gamma)}.$ It
is clear that
$\psi ({\cal C} (G / \Gamma)_\pi) \subset {\cal C} (G)_\pi,$ if
$\pi \in \widehat{G / \Gamma},$
and the Peter Weyl theorem gives $dim \hskip.1cm {\cal C}
(G / \Gamma)_\pi = dim \hskip.1cm
{\cal C} (G)_\pi = (dim \hskip.1cm \pi)^2,$ if $\pi \in
\widehat{G / \Gamma},$ so
\noindent
$\psi ({\cal C} (G / \Gamma)) = \sum_{\pi \in \hat{G /
\Gamma}} \hskip.1cm {\cal C} (G)_\pi.$
${\cal H} (G)^\Gamma$ is a sub $G$-module of ${\cal H}$
for the left regular
representation $L,$ so it reduces on isotypical components:
${\cal H} (G)^\Gamma =
\sum_{\pi \in \hat{G}} \hskip.1cm {\cal H} (G)^\Gamma \cap
{\cal C} (G)_\pi.$ Given
$f \in {\cal H} (G)^\Gamma, \hskip.1cm \gamma \in
\Gamma,$ one has $L_\gamma (f) = f$
since $\Gamma$ is normal, so $L_\gamma \vert_{{\cal H}
(G)^\Gamma \cap {\cal C} (G)_\pi} = Id.$
But $L$ acts on ${\cal C} (G)_\pi$ as a sum of $dim
\hskip.1cm \pi$ representations
isomorphic to $\check{\pi},$ so there are two cases:
${\cal H} (G)^\Gamma \cap {\cal C} (G)_\pi \neq \{ 0 \},$
and then $\pi \in \widehat{G / \Gamma},$
or ${\cal H} (G) \cap {\cal C} (G)_\pi = \{ 0 \}.$ When $\pi \in
\widehat{G / \Gamma},$
any element of ${\cal C} (G)_\pi$ is in ${\cal H}
(G)^\Gamma$ ($\Gamma$ is normal), so we conclude that
${\cal H} (G)^\Gamma = \sum_{\pi \in \widehat{G /
\Gamma}} \hskip.1cm
{\cal C} (G)_\pi = \psi ({\cal H} (G / \Gamma)).$ \hskip2cm
Q.E.D.
Now, the coproduct of ${\cal H} (G / \Gamma)$ induces a
coproduct on ${\cal H} (G)^\Gamma,$
and it is easy to check on coefficients (see (2.4.1)) that it is
exactly the restriction
of the coproduct of ${\cal H} (G).$ So ${\cal H} (G /
\Gamma) \simeq {\cal H} (G)^\Gamma$
is a Hopf subalgebra of ${\cal H} (G).$
\psaut
{\bf (5.2) Proposition}:
\it Let $\tilde{\times}$ be a preferred deformation of the
product of ${\cal H} (G)$
(resp.: $H(G)$), then $\tilde{\times}$ can be restricted to
${\cal H} (G)^\Gamma$ (resp: $H(G)^\Gamma$), i.e.:
$$\forall h,h' \in {\cal H} (G)^\Gamma (\hbox{resp.}:
H(G)^\Gamma), \hskip.1cm
h \tilde{\times} h' \in {\cal H} (G)^\Gamma [[t]] \hskip.1cm
(\hbox{resp.}: H(G)^\Gamma [[t]]).$$
and defines a preferred deformation of ${\cal H}
(G)^\Gamma$ (resp.: $H(G)^\Gamma$). \rm
\psaut
{\bf Proof}:
We begin by ${\cal H} (G)^\Gamma.$ We can restrict to $h
= C^\pi_M, \hskip.1cm
h' = C^{\pi'}_{M'}, \hskip.1cm \pi, \pi' \in \widehat{G /
\Gamma},$
$M \in {\cal L} (V_\pi), \hskip.1cm M' \in {\cal L} (V_{\pi'}).$
\noindent By (4.3.2):
$C^\pi_M \hskip.1cm \tilde{\times} \hskip.1cm C^{\pi'}_{M'}
= C^{\pi \tilde{\otimes} \pi'}_{M \otimes M'}$
\noindent
But $\pi \tilde{\otimes} \pi' = \tilde{P} \hskip.1cm \pi \otimes
\pi' \hskip.1cm
\tilde{P}^{-1},$ with $\tilde{P} \in {\cal L} (V_\pi \otimes
V_{\pi'}) [[t]]$ (4,2,4).
\noindent
So $C^\pi_M \hskip.1cm \tilde{\times} \hskip.1cm
C^{\pi'}_{M'} =
C^{\pi \otimes \pi'}_
{\tilde{P}^{-1} M \otimes M' \tilde{P}}.$ Using (4,1,2), we
compute:
$$\eqalign{
x \in G, \gamma \in \Gamma, C^\pi_M \hskip.1cm
\tilde{\times} \hskip.1cm C^{\pi'}_{M'}
(x \gamma) &= \widetilde{Tr} (\tilde{P}^{-1}.M \otimes M'.
\tilde{P}. \pi \otimes \pi' (x \gamma)) \cr
&= \widetilde{Tr} (\tilde{P}^{-1}.M \otimes M'. \tilde{P}. \pi
\otimes \pi' (x)) \hbox{ since }
\pi \otimes \pi' \in \widehat{G / \Gamma} \cr
&= C^\pi_M \hskip.1cm \tilde{\times} \hskip.1cm
C^{\pi'}_{M'} (x), \cr }$$
which proves that $C^\pi_M \hskip.1cm \tilde{\times}
\hskip.1cm C^{\pi'}_{M'}
\in {\cal H} (G)^\Gamma [[t]],$ as wanted.
Now, we use $\overline{{\cal H} (G / \Gamma)} = H (G /
\Gamma)$ to deduce $\overline{{\cal H} (G)^\Gamma} =
H(G)^\Gamma.$
We write, for $h, h' \in H(G)$:
$$h \hskip.1cm \tilde{\times} \hskip.1cm h' = \sum_n
\hskip.1cm t^n \hskip.1cm C_n (h,h').$$
By (4.3.4), $C_n (h,h') \in {\cal H} (G)$ if $h, h' \in {\cal H}
(G),$ by the beginning of the proof, $C_n (h,h') \in {\cal H}
(G)^\Gamma$ if $h,h' \in {\cal H} (G)^\Gamma.$
Given $h,h' \in H(G)^\Gamma,$ we choose sequences
$(h_p), (h'_p) \in {\cal H} (G)^\Gamma$
such that $\displaystyle{lim_p} \hskip.1cm h_p = h$ and
$\displaystyle{lim_p} \hskip.1cm h'_p = h'.$ From the
continuity of $C_n,$ the sequence $C_n (h_p, h'_p)$
converges in $H(G)$ to $C_n (h,h'),$
but since $C_n (h_p, h'_p) \in {\cal H} (G)^\Gamma \subset
H(G)^\Gamma, \hskip.1cm
\forall p,$ and since $H(G)^\Gamma$ is closed in $H(G),$ it
results that $C_n (h,h') \in
H(G)^\Gamma, \hskip.1cm \forall n,$ and, therefore, that $h
\hskip.1cm \tilde{\times}
\hskip.1cm h' \in H(G)^\Gamma [[t]].$
So any preferred deformation of $H(G)$ (resp: ${\cal H}
(G)$) restricts to $H(G)^\Gamma$
(resp: ${\cal H}(G)^\Gamma$); the obtained deformation of
$H(G)^\Gamma$ (resp: ${\cal H} (G)^\Gamma$)
is a preferred deformation because the coproduct of
$H(G)^\Gamma$ (resp: ${\cal H} (G)^\Gamma$)
is the restriction of the coproduct of $H(G)$ (resp: ${\cal H}
(G)$), as mentioned above. \hskip1cm Q.E.D.
\psaut
{\bf (5.3) Remark}:
Using the isomorphism of Hopf algebras ${\cal H}
(G)^\Gamma \simeq {\cal H} (G / \Gamma)$
(resp: $H(G)^\Gamma \simeq H(G / \Gamma)$) and (5.2),
we see that any preferred
deformation of ${\cal H} (G)$ (resp: $H(G)$) provides a
preferred deformation
of ${\cal H} (G / \Gamma)$ (resp: $H (G / \Gamma)$); this is
the justification
for the term quotient deformation. This result is useful, even
if $G$ is simple
(in which case $\Gamma$ is a finite subgroup of the center
of $G$), because if
one starts with a simple simply connected compact $G$
(this is exactly the general case
of universal covering of a simple compact group), from
deformations at the level
of $G,$ using (5.2), we get deformations at the level of any
group covered by
$G$ (e.g.: deformations for $SU(2)$ will provide
deformations for
$SO (3)$).
Also, as remarked in the Introduction, one can
One can even remove the condition that $\Gamma$ be normal by
considering comodule algebras.
However, if $\Gamma$ is not normal, there will be conditions on the deformed product of ${\cal H},$ or $H,$ if one wants to get directly "a quantized homogeneous space" $G/\Gamma.$ This will be treated elsewhere.
\saut
{\bf 6. Quantum groups and deformation theory.}
\psaut
{\bf (6.1) Generators}: It was mentioned in (2.4.2), (2.4.3)
that ${\cal H} = {\cal H} (G)$ is a domain, and
a finitely generated algebra. We need some facts about
generators of ${\cal H},$
when $G$ is simple. They are classical, but since we don't
know where they are completely explained,
we give some details. We follow [4], chap. VI.
\psaut
{\bf (6.1.1) Definition}:
\it A subset $\{ \pi_1 ... \pi_r \}$ of $\hat{G}$ is said
complete if the coefficients
of $\pi_1,...,\pi_r$ provide a generator system of ${\cal H}.$
\rm
\psaut
Using the Stone-Weierstrass and Peter-Weyl theorems, it is
easy to check that $\{ \pi_1 ... \pi_r \}$
is complete if:
(1) $\forall \hskip.1cm i = 1,...,r, \hskip.1cm \check{\pi}_i \in
\{ \pi_1,...,\pi_r \}$
(2) $\pi = \oplus^r_{i=1} \hskip.1cm \pi_i$ is a faithful
representation of $G.$
Faithful finite dimensional representations do exist for any
compact connected
Lie group, so complete sets do exist, and this proves that
${\cal H}$ is finitely generated.
\psaut
{\bf (6.1.2) Proposition}:
\it (1) If $G = SU(n),$ or $SO(n),$ or $Sp(n),$ and $\pi_s$ is
the standard natural
representation, then $\{ \pi_s \}$ is a complete set.
(2) If $G = Spin (n),$ there are two cases:
(i) $n = 2p + 1,$ the irreducible spin representation {\rm [4]}
is a complete set.
(ii) $n = 2p,$ $\pi_{\pm}$ the two irreducible spin
representations {\rm [4]},
then $\{ \pi_+, \pi_- \}$ is a complete set.
(3) We denote by $\tilde{G}_2, \tilde{F}_4, \tilde{E}_6,
\tilde{E}_7, \tilde{E}_8$
the compact simply connected Lie groups with respective
Lie algebra $G_2, F_4, E_6, E_7$ and $E_8,$ then:
(i) if $G = \tilde{G}_2, \tilde{F}_4$ or $\tilde{E}_8,$ any
irreducible $f.d.$ representation is a complete set.
(ii) if $G = \tilde{E}_7,$ there exists an irreducible $f.d.$
representation which is a complete set.
(iii) if $G = \tilde{E}_6,$ there exist two irreducible $f.d.$
representations such
that $\{ \pi_1, \pi_2 \}$ is a complete set. \rm
\psaut
{\bf Proof}:
In case (1), if $G = SO(n)$ or $Sp(n),$ $\pi_s$ is faithful
and self contragredient,
so we apply the above criteria. If $G = SU(n),$
$\check{\pi}_s =
%--------------------------------------------------
%The meaning of "Ext" in the following is obvious in
%context, but
%do you really want this notation (which conflicts with
%standard usage in homological algebras)?
%----------------------------------------------------
Ext_{n-1} (\pi_s),$ so (1) is true.
In case (2)(i), the irreducible spin representation is faithful
and self contragredient;
in case (2)(ii), $\{ \pi_+, \pi_- \}$ satisfies the conditions
of the criteria.
For exceptional $\tilde{G}_2, \tilde{F}_4$ or $\tilde{E}_8,$
the center $Z(G)$
is trivial [15], and any irreducible $f.d.$ representation is
self contragredient [20], therefore (3)(i) is true.
For $\tilde{E}_7,$ the center $Z(G)$ is $Z_2$ [15], any f.d.
irreducible is self contragredient [19], so any irreducible
subrepresentation of $Ind^G_{Z(G)}
\varepsilon, \hskip.1cm \varepsilon$ the alternate
character of $Z_2,$ will provide a complete set.
For $\tilde{E}_6$ the center is $Z_3$ [15] and we can find
faithful irreducible
representations by reduction of $Ind^G_{Z(G)} \xi,$ where
$\xi$ is a faithful
character of $Z(G);$ given such a $\pi,$ then $\{ \pi,
\check{\pi} \}$ is a complete
set; in that case, $\check{\pi} \not\simeq \pi$ if $\pi$ is
faithful, because $Z(G) = Z_3.$ \hskip1cm Q.E.D.
Generator systems will provide a description of
deformations of ${\cal H}$
which is quite similar to the FRT model of quantum groups:
\psaut
{\bf (6.1.3) Proposition}:
\it Assume that $G$ is one of the groups listed in (6.1.2). Let
$\pi_0$ be the direct
sum of the irreducible $f.d.$ representations of $G$
appearing in a complete set
as described in (6.1.2) and $\{ C_{ij} \}$ the coefficients of
$\pi_0$ in a fixed basis. If $\tilde{\times}$ is a preferred Hopf
deformation of ${\cal H},$
then $(C_{ij})$ is a (topological) generator system of the
$\hskip.1cm \Crm [[t]]$-algebra
(${\cal H}[[t]], \tilde{\times}$); if $T$ is the matrix
$[C_{ij}], \hskip.1cm T_1 =
T \otimes Id, \hskip.1cm T_2 = Id \otimes T,$ then there
exists an invertible
${\cal R}$ in ${\cal L} (V_{\pi_0} \otimes V_{\pi_0}) [[t]]$
such that:
$${\cal R}(T_1 \hskip.1cm \tilde{\times} \hskip.1cm T_2) =
(T_2 \hskip.1cm \tilde{\times} \hskip.1cm
T_1){\cal R}$$
(here, by topological generator system we mean that the
closure of the $\hskip.1cm \Crm [[t]]$-algebra
generated by $\{ C_{ij} \}$ is ${\cal H}[[t]]$) \rm
\psaut
{\bf Proof}:
First, we note that formula (4.3.2) can easily be generalized
to any formal representations
$\tilde{\pi}$ and $\tilde{\pi'}$ of $G,$ which are
deformations of f.d. representations
$\pi$ and $\pi'$ (see (4.1.1)):
{\bf (6.1.4)} $C^{\tilde{\pi}}_M \hskip.1cm \tilde{\times}
\hskip.1cm
C^{\tilde{\pi'}}_{M'} = C^{\tilde{\pi} \tilde{\otimes}
\tilde{\pi'}}_ {M \otimes M'},$
if $M \in {\cal L} (V_\pi), \hskip.1cm M' \in {\cal L}
(V_{\pi'})$
(see (4.1.2) for the definition of generalized coefficients).
>From the associativity
of $\tilde{\times},$ the new tensor product of
representations of $G$ is associative,
so we deduce:
$C^{\tilde{\pi}_1}_{M_1} \hskip.1cm
\tilde{\times}...\tilde{\times} \hskip.1cm C^{\tilde{\pi}_n}_
{M_n} = C^{\tilde{\pi}_1 \tilde{\otimes}...\tilde{\otimes}
\tilde{\pi}_n}_{M_1
\otimes...\otimes M_n},$ if $\tilde{\pi}_i$ deforms f.d.
$\pi_i,$ and $M_i \in
{\cal L} (V_\pi), \hskip.1cm i = 1...n.$
Now we take $\tilde{\pi}_i = \pi_0, \hskip.1cm \forall i$ and
set $\pi_0 \tilde{\otimes}...
\tilde{\otimes} \pi_0 = \displaystyle{\tilde{\otimes}_n}
\hskip.1cm \pi_0,$
$M_e = e^*_{i_e} \otimes e_{j_e}, \hskip.1cm \{ e_i \}$
being a basis of $V_{\pi_0},$ and
obtain:
$$C_{i_1 j_1} \tilde{\times}...\tilde{\times} \hskip.1cm C_{i_n
j_n} = C^{\displaystyle{\tilde{\otimes}_n}
\pi_0}_M, \hbox{ with } M = (e^*_{i_1} \otimes...\otimes
\hskip.1cm e^*_{i_n})
\otimes (e_{j_1} \otimes...\otimes \hskip.1cm e_{j_n})$$
Since $\displaystyle{\tilde{\otimes}_n} \hskip.1cm \pi_0$ is
a deformation of $\displaystyle{\otimes_n}
\pi_0,$ it results from (4.1.4) that we can find
$\tilde{P} \in {\cal L} (\displaystyle{\otimes_n}
\hskip.1cm V_{\pi_0}) [[t]]$ such that
$\displaystyle{\tilde{\otimes}_n} \pi_0 = \tilde{P}.
\displaystyle{\otimes_n} \pi_0. \tilde{P}^{-1}.$ Then, by
(2.3) and (4.1.2):
$$C_{i_1 j_1} \hskip.1cm \times...\times \hskip.1cm C_{i_n
j_n} = C^{\displaystyle{\otimes_n} \pi_0}_M =
C^{\displaystyle{\tilde{\otimes}_n} \pi_0}_{\tilde{P} M
\tilde{P}^{-1}}$$
But $\tilde{P} \hskip.1cm M \hskip.1cm \tilde{P}^{-1} =
\sum_{\alpha_1...\alpha_n,
\beta_1...\beta_n} \tilde{\lambda}_{\alpha_1...\alpha_n
\hskip.1cm \beta_1...\beta_n}
(e^*_{\alpha_1} \otimes...\otimes \hskip.1cm
e^*_{\alpha_n})
\otimes (e_{\beta_1}
\otimes...\otimes \hskip.1cm e_{\beta_n}),$ with
$\tilde{\lambda}_{\alpha_1...\alpha_n
\hskip.1cm \beta_1...\beta_n} \in \Crm [[t]],$ so:
$$C_{i_1 j_1} \hskip.1cm \times...\times \hskip.1cm C_{i_n
j_n} = \sum_{\alpha_1...\alpha_n,
\beta_1...\beta_n} \tilde{\lambda}_{\alpha_1...\alpha_n
\hskip.1cm \beta_1...\beta_n}
\hskip.1cm C_{\alpha_1 \beta_1} \hskip.1cm \tilde{\times}
\hskip.1cm C_{\alpha_2 \beta_2}
\hskip.1cm \tilde{\times}...\tilde{\times} \hskip.1cm
C_{\alpha_n \hskip.1cm \beta_n}$$
Therefore any polynomial in the $\{ C_{ij} \}$ for the initial
product, can be written
as a polynomial of $\{ C_{ij} \}$ for the new product. Note
that the degree does not
increase, and that, in the new product case, we are dealing
with non commutative
polynomials with coefficients in $\Crm [[t]].$ Now any $h$ in
${\cal H} [[t]]$
can be written $h = \sum_n \hskip.1cm t^n h_n, \hskip.1cm
h_n \in {\cal H},$
and any $h_n$ is a non-commutative $\Crm
[[t]]$-polynomial in $\{ C_{ij} \}$ for the
new product, so $\{ C_{ij} \}$ is a topological generator
system of $({\cal H}
[[t]], \tilde{\times}).$
As to relation ${\cal R}.T_1 \hskip.1cm \tilde{\times}
\hskip.1cm T_2 = T_1 \hskip.1cm
\tilde{\times} \hskip.1cm T_2. {\cal R},$ it is an equivalent
(but illuminating) way
to express the quasicommutativity (4.3.1) of $\tilde{\times},$
combined with formula (4,3,2), as we now show:
Let $\tilde{\Delta}' = \sigma.\tilde{\Delta}$ and denote by
$\tilde{\rho}$ the new tensor product
of $\pi_0$ by itself obtained from $\tilde{\Delta},$ and by
$\tilde{\rho'}$ the new tensor
product of $\pi_0$ by itself obtained from $\tilde{\Delta}'.$
Then, by the quasicocommutativity
of $\tilde{\Delta}$ (4.2.5), one has $\tilde{\Delta}' =
\tilde{{\cal R}}.\tilde{\Delta}.\tilde{{\cal R}}^{-1},$
so if we set ${\cal R} = \pi_0 \hskip.1cm \otimes \hskip.1cm
\pi_0 (\tilde{{\cal R}}),$
we get that $\tilde{\rho}' = {\cal R} \hskip.1cm \tilde{\rho}
\hskip.1cm {\cal R}^{-1}.$
Using now (6.1.4), the notation $E_{ij} = e^*_i \hskip.1cm
\otimes \hskip.1cm e_j,$
and $^T \tilde{\Delta}' (h,h') = h' \hskip.1cm \tilde{\times}
\hskip.1cm h,$ we deduce:
$C_{i'j'} \tilde{\times} \hskip.1cm C_{ij} =
C^{\tilde{\rho}'}_{E_{ij} \otimes E_{i' j'}} =
C^{{\cal R} \tilde{\rho} {\cal R}^{-1}}_{E_{ij} \otimes E_{i'
j'}} = C^{{\cal R} \tilde{\rho}
{\cal R}^{-1}}_{(i,i') (j,j')},$ and
$C_{ij} \tilde{\times} \hskip.1cm C_{i' j'} =
C^{\tilde{\rho}}_{E_{ij} \otimes E_{ij'}} =
C^{\tilde{\rho}}_{(i,i') (j,j')},$
so that from these two relations, one has:
$$T_2 \hskip.1cm \tilde{\times} \hskip.1cm T_1 = {\cal R}
(T_1 \hskip.1cm \tilde{\times}
\hskip.1cm T_2) {\cal R}^{-1}. \hskip2cm \hbox{Q.E.D.}$$
\psaut
{\bf (6.1.5) Remark}:
$\{ C_{ij} \}$ being a generator system of $({\cal H},
\times),$ ${\cal H}$
has a natural filtration coming from the one of the
polynomial algebra $\Crm [C_{ij}].$
Now, let us define a topological filtration in a topological
$\Crm [[t]]$-algebra
to be an increasing sequence of $\hskip.1cm \Crm
[[t]]$-submodules $A_n$ such that $A_n.A_p
\subset A_{n+p}, \hskip.1cm \forall n,p$ and
$\overline{\cup_n
\hskip.1cm A_n} = A.$
Then $({\cal H} [[t]], \times)$ has a topological filtration
inherited from $\Crm [C_{ij}],$
as said above, and, from the proof of (6.1.3), this filtration
also works
for $({\cal H} [[t]], \tilde{\times}).$ Moreover, from ${\cal R}
\hskip.1cm T_1 \hskip.1cm T_2 =
T_2 \hskip.1cm T_1 \hskip.1cm {\cal R},$ any monomial in
$\{ C_{ij} \}$ for the
new product can be reordered, so that elements of degree
at most $n$ are actually
$\hskip.1cm \Crm [[t]]$-linear combinations of ordered
monomials of degree at most $n$ in the
new product (roughly speaking, the graded associated
algebra is still $\hskip.1cm \Crm [C_{ij}]$).
\psaut
{\bf (6.2) The Drinfeld models}: We give a brief description
of the D models of quantum groups, following
[6], [8]: Given a simple complex f.d. Lie algebra $g,$ with
${\cal U}$ its enveloping
algebra, there exists a Hopf deformation ${\cal U}_t$ of
${\cal U}$ which is a
topologically free complete $\Crm [[t]]$-module (i.e ${\cal
U}_t \simeq {\cal U} [[t]]$ as
$\hskip.1cm \Crm [[t]]$-modules),
with coproduct $\tilde{\Delta}: {\cal U}_t \fl {\cal U}_t
\hskip.1cm \hat{\otimes}_t
\hskip.1cm {\cal U}_t$; therefore ${\cal U}_t$ is a
deformation of ${\cal U}$ in
the sense of (3.7), when ${\cal U}$ is given its natural
topology. It is shown
in [8] that ${\cal U}$ is (deformation) rigid, therefore ${\cal
U}_t \simeq {\cal U} [[t]]$
as an algebra, and it is also shown that $\tilde{\Delta}$ is
obtained from the
standard coproduct $\Delta_0$ by a twist, i.e. there exists
$\tilde{P} \in {\cal U}_t
\hskip.1cm \hat{\otimes}_t \hskip.1cm {\cal U}_t = ({\cal U}
\otimes {\cal U}) [[t]]$
such that $\tilde{\Delta} = \tilde{P} \hskip.1cm \Delta_0
\hskip.1cm \tilde{P}^{-1}.$
Let us note the analogy between these results and (4,2,1),
(4,2,4). Now, let $G$
be the compact simply connected Lie group with Lie
algebra $g_0$ such that $g_0
\otimes_{\Rrm} \hskip.1cm \Crm = g.$ We introduce $H =
H(G) = C^\infty (G),$ $A = A(G) = H(G)^*,
{\cal H} = {\cal H} (G)$ and ${\cal A} = {\cal A} (G)$ as in
section 2; using (2.6.2), ${\cal U} \subset
A \subset {\cal A},$ so ${\cal U} [[t]] \subset A[[t]] \subset {\cal A} [[t]].$
\psaut
{\bf (6.2.1) Proposition}:
\it The Hopf deformation ${\cal U}_t$ can be extended to a
Hopf deformation of
$A$ (resp.: ${\cal A}$) with unchanged product, unit and
counit. \rm
U{\rm sin}g $\hskip.1cm \Crm [[t]]$-duality (3.8), we deduce:
\psaut
{\bf (6.2.2) Corollary}:
\it The Hopf deformation ${\cal U}_t$ produces (by
$\hskip.1cm \Crm [[t]]$-duality) a
preferred Hopf deformation of ${\cal H}$ and $H.$ \rm
\psaut
{\bf (6.2.3) Remark}:
So our results, and especially (4.3.2), (4.3.3) and (6.1.3)
can be applied. Note
that the ${\cal R}$-matrix of (6.1.3) can be specified to be a
solution of the
Yang-Baxter equation, because in D model, the twist
between $\tilde{\Delta}$ and
$\tau. \tilde{\Delta}$ can be chosen with this property [6]. It
is interesting
that the D model will produce on ${\cal H}$ a preferred
deformation for any $G$
listed in (6.1.2), and not only for $SU(n), SO(n)$ and
$Sp(n)$ as the FRT-model does
(see also remark 23 in [9]). In the case of $Spin (2p)$ and
$\tilde{E}_6,$ the
coefficients of two (and not only one) irreducible f.d.
representations will be
needed to describe the product as in (6.1.3). We also note
that nobody has ever
found a preferred deformation on $H,$ which is however, in
our opinion, a very
good candidate for deformation of Poisson brackets !
\psaut
{\bf Proof of (6.2.1)}:
{\rm Since} $\tilde{\Delta} = \tilde{P} \hskip.1cm \Delta_0
\hskip.1cm \tilde{P}^{-1},$
with $\tilde{P} \in ({\cal U} \hskip.1cm \otimes \hskip.1cm
{\cal U}) [[t]],$
$\tilde{\Delta}$ extends to a coproduct on $A[[t]],$ and
${\cal A}[[t]].$
With this new coproduct and the standard product, $A[[t]]$
and ${\cal A}[[t]]$
are $\hskip.1cm \Crm [[t]]$-bialgebras. U{\rm sin}g (3.10)
there exists an antipode and a counit.
>From (4.2.6), this counit is the standard counit of $A,$ or
${\cal A};$
since it restricts to ${\cal U}_t = {\cal U} [[t]],$ by
unicity of the counit, the
restriction has to be the counit of ${\cal U}_t.$ Now let
$\tilde{S}$ be the
antipode of ${\cal U}_t,$ then given any irreducible
representation $\pi$ of $g,$
we define a deformation $\tilde{\rho} = ^T(-\pi) \hskip.1cm o
\hskip.1cm \tilde{S}$
of $\check{\pi}$ {\rm sin}ce $g$ is simple, $\check{\pi}$ is
rigid, so $\tilde{\rho} =
\alpha_\pi \hskip.1cm o \hskip.1cm \check{\pi} \hskip.1cm
o \hskip.1cm \alpha^{-1}_\pi,$
with $\alpha_\pi \in {\cal L} (V^*_\pi) [[t]].$ But $\check{\pi}
= ^T\pi \hskip.1cm
o \hskip.1cm S_0,$ with $S_0$ the standard antipode of
${\cal A},$ so $\pi (\tilde{S} (u)) =
a_\pi \hskip.1cm o \hskip.1cm \pi (S_0 (u)) \hskip.1cm o
\hskip.1cm a^{-1}_\pi, \hskip.1cm \forall u \in
{\cal U},$ with
%---------------------------------------------------
%The notation in the following is confusing.
%-----------------------------------------------------
$a_\pi = ^T \alpha^{-1}_\pi \in {\cal L} (V_\pi) [[t]]$; so we
can extend
$\tilde{S}$ to ${\cal A} [[t]]$ by the following formula:
$$b \in {\cal A}, \hskip.1cm \tilde{S} (b)_\pi = a_\pi
\hskip.1cm S_0
(b)_\pi \hskip.1cm a^{-1}_\pi = a_\pi \hskip.1cm ^T
b_{\check{\pi}} \hskip.1cm a^{-1}_\pi.$$
This defines a second antipode for the above mentioned
Hopf structure of ${\cal A} [[t]]$;
by unicity of the antipode, it must be the same.
By (3.10), $A[[t]],$ with its coproduct $\tilde{\Delta},$ has
an antipode
$\tilde{S}',$ and by (4.2.6), one has $\tilde{S}' = \tilde{a}
\hskip.1cm S_0
\hskip.1cm \tilde{a}^{-1},$ with $\tilde{a} \in A[[t]].$ But this
last formula
defines an antipode on ${\cal A} [[t]]$ with its coproduct
$\tilde{\Delta},$ so
by unicity of the antipode, $\tilde{S}' = \tilde{S}.$ So we
have proved that the coproduct,
counit and antipode of ${\cal U}_t = {\cal U} [[t]]$ extend to
$A[[t]]$ and ${\cal A} [[t]].$ \hskip1cm Q.E.D.
\psaut
{\bf (6.3) Drinfeld isomorphisms}: We continue with the
notations of (6.2) and discuss the following problem:
in (6.2.1) an isomorphism $\tilde{\varphi}: {\cal U}_t \simeq
{\cal U} [[t]]$
(called Drinfeld isomorphism in the following) is fixed. Now,
such a Drinfeld
isomorphism is certainly not unique; what happens if it is
changed?
\psaut
{\bf (6.3.1) Proposition}:
\it Let $\tilde{\varphi}$ and $\tilde{\psi}$ two Drinfeld
isomorphisms, then the
corresponding preferred deformations of ${\cal H} (G)$ are
equivalent. \rm
Before proving (6.3.1), let us note that a good choice of the
Drinfeld isomorphism
has still some importance, because it will simplify defining
relations of the
corresponding deformation of ${\cal H} (G).$ We shall come
back to this problem in next subsections.
\psaut
{\bf Proof of (6.3.1)}:
Drinfeld isomorphisms are constructed as follows: take any
section of the canonical
morphism ${\cal U}_t \fl {\cal U}$ (such sections do exist by
rigidity of ${\cal U},$
see [8]), and extend it to ${\cal U} [[t]]$ by $\Crm
[[t]]$-linearity. Therefore,
given $\tilde{\varphi}$ and $\tilde{\psi},$ and defining
$\tilde{\theta} = \tilde{\psi}
\hskip.1cm o \hskip.1cm \tilde{\varphi}^{-1}, \hskip.1cm
\tilde{\theta}$ is a
$\hskip.1cm \Crm [[t]]$-linear automorphism of ${\cal U}
[[t]],$ and $\tilde{\theta}_0 =
Id_{{\cal U}}.$ It results that $\pi \hskip.1cm o \hskip.1cm
\tilde{\theta},
\hskip.1cm \pi \in \hat{G},$ is a deformation of the
representation $\pi$ of $g,$
and {\rm sin}ce $g$ is simple, $\pi \hskip.1cm o \hskip.1cm
\tilde{\theta} = \tilde{\alpha}_\pi
\hskip.1cm o \hskip.1cm \pi \hskip.1cm o \hskip.1cm
\tilde{\alpha}^{-1}_\pi,$
with $\alpha_\pi \in {\cal L} (V_\pi) [[t]].$ Now let $\tilde{a}
= (\tilde{\alpha}_\pi)
\in {\cal A} [[t]] = \Pi_{\pi \in \hat{G}} \hskip.1cm {\cal L}
(V_\pi) [[t]];$ we
extend $\tilde{\theta}$ to (a continuous automorphism of)
${\cal A} [[t]]$ by:
$$\tilde{\theta} (\tilde{b}) = \tilde{a} \hskip.1cm \tilde{b}
\hskip.1cm \tilde{a}^{-1},
\tilde{b} \in {\cal A} [[t]].$$
The coproduct $\tilde{\Delta}$ of ${\cal U}_t$ induces
coproducts $\tilde{\Delta}_\varphi$
and $\tilde{\Delta}_\psi$ of ${\cal U} [[t]]$ by formula:
$$\tilde{\Delta}_\varphi = \tilde{\varphi} \hskip.1cm \otimes
\hskip.1cm \tilde{\varphi}
\hskip.1cm o \hskip.1cm \tilde{\Delta} \hskip.1cm o
\hskip.1cm \tilde{\varphi}^{-1}
\hskip.1cm (\hbox{resp.:} \tilde{\Delta}_\psi = \tilde{\psi}
\hskip.1cm \otimes
\hskip.1cm \tilde{\psi} \hskip.1cm o \hskip.1cm \tilde{\Delta}
\hskip.1cm o \hskip.1cm \tilde{\psi}^{-1})$$
Now, one has $\tilde{\Delta}_\psi = \tilde{\theta} \hskip.1cm
\otimes \hskip.1cm
\tilde{\theta} \hskip.1cm o \hskip.1cm \tilde{\Delta}_\varphi
\hskip.1cm o \hskip.1cm
\tilde{\theta}^{-1}.$ It was shown in (6.2.1) that
$\tilde{\Delta}_\varphi$ and
$\tilde{\Delta}_\psi$ extend to (continuous) coproducts on
${\cal A} [[t]].$
Since $\tilde{\theta}$ extends to ${\cal A} [[t]],$ and
{\rm sin}ce $\overline{{\cal U} [[t]]} =
{\cal A} [[t]]$ (2.6.1), the formula $\tilde{\Delta}_\psi = \tilde{\theta}
\hskip.1cm \otimes
\hskip.1cm \tilde{\theta} \hskip.1cm o \hskip.1cm
\tilde{\Delta}_\varphi \hskip.1cm
o \hskip.1cm \tilde{\theta}^{-1}$ is valid for the extensions of
$\tilde{\Delta}_\varphi$
and $\tilde{\Delta}_\psi$ to ${\cal A} [[t]].$ So $({\cal A}
[[t]], \tilde{\Delta}_\varphi)$
and $({\cal A} [[t]], \tilde{\Delta}_\psi)$ are equivalent
deformations of $({\cal A},
\Delta_0),$ and (3.8) finishes the proof of (6.3.1). \hskip1cm
Q.E.D.
\psaut
{\bf (6.4) FRT models}: We now show how FRT-models are
recovered by a good choice of the Drinfeld
isomorphism. First, we need the following lemma:
\psaut
{\bf (6.4.1) Lemma}:
\it Let $\tilde{\rho}$ be a representation of ${\cal U}_t,$ and
$\pi = \tilde{\rho}_0,$
$\pi \in \hat{G}.$ Then there exists a Drinfeld isomorphism
$\tilde{\varphi}$ such that $\tilde{\rho} = \pi \hskip.1cm o
\hskip.1cm \tilde{\varphi}.$ \rm
\psaut
{\bf Proof}: Fix any Drinfeld isomorphism $\tilde{\varphi}'.$
Now $\tilde{\rho} \hskip.1cm o \hskip.1cm
\tilde{\varphi}^{'-1}$ is a deformation of the
representation $\pi$ of $g,$ hence a trivial deformation. So
there exists $\tilde{\alpha}_\pi
\in {\cal L} (V_\pi) [[t]],$ such that $\tilde{\rho} \hskip.1cm o
\hskip.1cm \tilde{\varphi}^{'-1} =
\tilde{\alpha}_\pi \hskip.1cm o \hskip.1cm \pi \hskip.1cm o
\hskip.1cm
\tilde{\alpha}^{-1}_\pi.$ We write $\tilde{\alpha}_\pi =
\sum_n \hskip.1cm t^n
\hskip.1cm \beta_n, \hskip.1cm \beta_n \in {\cal L}
(V_\pi),$ and use the Jacobson
density theorem: there exists $u_n \in {\cal U}$ such that
$\beta_n = \pi (u_n),$
and therefore there exists $\tilde{u} = \sum_n \hskip.1cm t^n
\hskip.1cm u_n \in
{\cal U} [[t]]$ such that $\tilde{\alpha}_\pi = \pi (\tilde{u}).$
Now we obtain that
$\tilde{\rho} \hskip.1cm o \hskip.1cm \tilde{\varphi}^{'-1}
(\tilde{b}) = \pi (\tilde{u}
\hskip.1cm \tilde{b} \hskip.1cm \tilde{u}^{-1}), \hskip.1cm
\forall \hskip.1cm \tilde{b} \in {\cal U} [[t]],$ so:
$\tilde{\rho} (\tilde{c}) = \pi (\tilde{u} \hskip.1cm
\tilde{\varphi}' (\tilde{c})
\tilde{u}^{-1}), \hskip.1cm \forall \hskip.1cm \tilde{c} \in
{\cal U}_t.$
We define a Drinfeld isomorphism $\tilde{\varphi}$ by
$\tilde{\varphi} (\tilde{c}) = \tilde{u}
\hskip.1cm \tilde{\varphi}' (\tilde{c}) \tilde{u}^{-1},
\hskip.1cm \forall \hskip.1cm
\tilde{c} \in {\cal U}_t,$ and then $\tilde{\rho} = \pi
\hskip.1cm o \hskip.1cm
\tilde{\varphi}.$ \hskip1cm Q.E.D.
{\bf (6.4.2}) Let us apply (6.4.1) to the case $G = SU(n).$
We follow closely ([6], section 7), except
that we replace $h$ by $t,$ and we denote by $\tilde{\rho}$
the representation
of ${\cal U}_t$ denoted by $\rho$ in that paper. Then
$\tilde{\rho} (0) = \pi_s,$
the standard natural representation of $g = sl(n),$ which is
also the standard
natural representation of $G.$ We choose a Drinfeld
isomorphism $\tilde{\varphi}$
such that $\tilde{\rho} = \pi_s \hskip.1cm o \hskip.1cm
\tilde{\varphi}$ (6.4.1).
We denote by $C_{ij}$ the coefficients of $\pi_s$; they are
a generator system
of ${\cal H}.$ The deformation ${\cal U}_t$ (and the choice
of $\tilde{\varphi}$)
produces a deformation of ${\cal H}$ (and $H$), and we
want to compute the
new relations of the generator system $\{ C_{ij} \}$ (see
(6.1.3)). But this
computation is explicitely done in ([6], section 7): let
$\rho_{ij}$ be the coefficients
of $\tilde{\rho},$ they are elements of $({\cal U}_t)^*_t;$ by
the choice of
$\tilde{\varphi},$ one has $^T \tilde{\varphi} (C_{ij}) =
\rho_{ij}$; the Hopf
algebras $({\cal U}_t, \tilde{\Delta})$ and $({\cal U} [[t]],
\tilde{\Delta}_{\tilde{\varphi}} =
\tilde{\varphi} \otimes \tilde{\varphi} \hskip.1cm o
\hskip.1cm \tilde{\Delta}
\hskip.1cm o \hskip.1cm \tilde{\varphi}^{-1})$ are
isomorphic by $\tilde{\varphi},$
so $({\cal U}_t)^*_t \simeq ({\cal U} [[t]])^*_t = {\cal U}^*
[[t]]$ by $^T \tilde{\varphi}.$
So in formulas (16), (17), (18), (19) given in ([6], section 7)
for the products of
$\rho_{ij},$ one has only to replace $\rho_{ij}$ by $C_{ij}$
to get the relations
between the generators $\{ C_{ij} \}$ of ${\cal H}$ (or $H$),
for the deformed
product $\tilde{\times}$ provided by (6.2.2). A glance at [9]
is enough to be convinced
that we recover (a formal version of) the FRT quantization
of $SL(n).$ Note that
$({\cal H}, \tilde{\times})$ has an antipode (6.2.2), and so
does the FRT quantization;
by unicity, they have to be the same. Actually, this proves
that the FRT quantization
of $SL(n)$ can be seen as a preferred Hopf deformation of
${\cal H} (SU(n))$; this result
has been obtained in [11], [12], by different techniques.
Moreover, applying (6.2.2), this
deformation extends to a preferred Hopf deformation of
$H(G) = C^\infty (G),$
a result which is completely new.
{\bf (6.4.3)} FRT quantizations (or rather formal versions)
can be similarly recovered
from D. models in cases $G = SO(n),$ or $G = Sp(n),$ and
this proves that there
exists a preferred Hopf deformation of ${\cal H} (SO(n))$ or
${\cal H} (Sp(n))$
which satisfies the relations of FRT quantifications as given
in [9]. We insist
that this result is a justification of the terminology
"deformation", often
employed, but never justified in these cases (see e.g. [11],
where it is shown
that relations of type ${\cal R} \hskip.1cm T_1 \hskip.1cm
T_2 = T_2 \hskip.1cm T_1
\hskip.1cm {\cal R}$ need not define a deformation, even if
${\cal R}$ is Yang-Baxter). Now,
the proof looks very much like the case of $SU(n),$ but
cannot be so explicit,
the main task being to choose the representation
$\tilde{\rho}$ used
above. This can be done u{\rm sin}g the reconstruction
theorem 12 of [9], and the
fundamental corepresentation $\tau$ defined in ([9], Def.
20). Applying the method
developed in section (6.4), the D. model will produce by
(6.2.2) a preferred Hopf
deformation of ${\cal H} (G)$ satisfying FRT-relations. Note
that this deformation
extends to a deformation of $H(G) = C^\infty (G).$
{\bf (6.4.4)} D. models and (6.2.2) predict the existence of a
preferred Hopf deformation
of ${\cal H} (G)$ (or $H(G)$) for any $G$ listed in (6.1.2),
with generators $T =
(C_{ij})$ satisfying $T_1 \hskip.1cm \tilde{\times}
\hskip.1cm T_2 = {\cal R} \hskip.1cm T_2
\hskip.1cm \tilde{\times} \hskip.1cm T_1 \hskip.1cm {\cal
R}^{-1},$ (6.1.3), and ${\cal R}$ Yang-Baxter
(6.2.3). It would be of interest to describe such an FRT
model when e.g. $G = Spin (n)$
and especially $Spin (2p),$ where two irreducible
representations have to be used
(6.1.2). Such a model will induce a preferred Hopf
deformation of ${\cal H} (G / \Gamma),$
for any $\Gamma$ in the center of $G$ (5.3). The case of
exceptional $G$ is also
of interest, {\rm sin}ce very little seems to be known.
Finally, we mention that the Reshetikhin model [19] (see also [22]),
being obtained by twisting the
standard coproduct, is also in our deformation framework: it
will also produce
deformations of ${\cal H} (G)$ (or $H(G)$) of the above
type.
\saut
\centerline {\bf 7. J models.}
\psaut
For complex $t \notin 2 \pi {\bf Q}, \hskip.1cm k \in Z,$ we
get $q = e^{it},$
and define the J model $A_t (g), g = sl(2),$ as the algebra
generated by $\{ F,G, K,K^{-1} \}$ with relations:
$$[F,G] = {K^2 - K^{-2} \over q - q^{-1}}, \hskip.1cm FK =
q^{-1} \hskip.1cm KF, \hskip.1cm GK = q KG. \leqno{(7.1)}$$
Direct interpretation in deformation theory is not possible,
because (7,1) is not defined
at $t = 0.$ Nevertheless, as shown in [5] or [3], it is not
difficult to swallow this {\rm sin}gularity, and actually define
$A_t (g)$ for any $t \in \Crm,$
by introducing $S = {K - K^{-1} \over q - q^{-1}}, \hskip.1cm
C = {K + K^{-1} \over 2},$
so that $A_t (g)$ is the algebra generated by $\{ F,G,S,C
\}$ with relations:
$$[F,G] = 2SC, \hskip.1cm FS = (S {\rm {\rm cos}t} - C) F,
\hskip.1cm FC = (C {\rm {\rm cos}t} + S
{\rm {\rm sin}}^2t)F, \leqno{(7.2)}$$
$$GS = (S {\rm {\rm cos}t} + C) G, \hskip.1cm GC = (C {\rm
{\rm cos}t} - S {\rm {\rm sin}}^2t) G, \hskip.1cm
C^2 + S^2 {\rm {\rm sin}}^2t = 1, \hskip.1cm [S,C] = 0.$$
Though (7.2) is a rather lengthy definition, it does define
$A_t (g)$ for any
$t \in \Crm.$ Moreover, if $t$ is now a formal parameter, we
can define $\tilde{A}_t (g)$
(the formal J model) as the $\Crm [[t]]$-algebra generated
by $\{ S,C,K,K^{-1} \}$ with relations (7.2).
Now for $t = 0,$ (7.2) becomes:
$$[F,G] = 2SC, \hskip.1cm [C,F] = [C,G] = [C,S] = 0,
\hskip.1cm [S,F] = CF, \hskip.1cm
[S,G] = -CG, \hskip.1cm C^2 = 1.$$
Setting $\hat{Y} = SC, \hskip.1cm \hat{F} = FC, \hskip.1cm
\hat{G} = GC,$ one
finds the commutation rules of $g = sl(2).$ So $A_0 (g)
\simeq {\cal U} (g) \otimes P,$
with $P \simeq \Crm [x] / x^2 - 1,$ and this proves that the
classical limit of
the J model is not ${\cal U} (g),$ as often asserted, but an
extension of ${\cal U} (g)$
by a parity $C.$ Let us note that a similar result holds for
general simple $g,$
by similar arguments: the classical limit of the J model will
be an extension of
${\cal U} (g)$ by $r$ parities ($r = rank \hskip.1cm g$). Now
we come back to
$g = sl(2).$ From the Poincar\'e-Birkhoff-Witt theorem, we
deduce that $\tilde{A}_t (g)$
is a deformation of $A_0 (g).$ It is well-known that
$\tilde{A}_t (g)$ is a domain,
and it is obvious that $A_0 (g)$ is not, so the $\hskip.1cm
\Crm [[t]]$-algebras $\tilde{A}_t (g)$
and $A_0 (g) [[t]]$ cannot be isomorphic, which proves:
\psaut
{\bf (7.3) Proposition}:
\it $\tilde{A}_t (g)$ is a non trivial deformation of $A_0 (g)
\simeq {\cal U} (g)
\otimes \hskip.1cm \Crm [x] / x^2 - 1.$ \rm
\psaut
The same argument shows that the algebras $A_t (g)$ and
$A_0 (g)$ cannot be isomorphic
when $t \notin 2 \pi {\bf Q}.$ It was shown in [3] that $A_t
(g) \simeq A_{t'} (g)$
if and only if $t' = \pm t + 2 k \pi, \hskip.1cm k \in {\bf Z},$
and this has an intuitive
interpretation as non-rigidity of $A_t (g)$ (and not only of
$A_0 (g)$ as shown
in (7.3)). Let us now give a complete justification of this
interpretation:
We fix $t_0 \in \Crm,$ and define $\tilde{B}_t (g)$ as the
$\Crm [[t]]$-algebra
generated by $\{ F,G,S,C \}$ with relations:
$$[F,G] = 2SC, \hskip.1cm FS = (S {\rm cos} (t_0+t) - C)F,
\hskip.1cm FC = (C {\rm cos}(t_0+t) +
S {\rm sin}^2 (t_0+t) F, \leqno{(7.4)}$$
$$GS = (S {\rm cos} (t_0+t) + C) G, GC = (C {\rm cos}
(t_0+t) - S {\rm sin}^2t)G,
C^2 + S^2 {\rm sin}^2 (t_0+t) = 1, [S,C] = 0.$$
The classical limit $t = 0$ is $A_{t_0} (g),$ and u{\rm sin}g
once more the Poincar\'e-Birkhoff-Witt
theorem, $\tilde{B}_t (g)$ is a deformation of $A_{t_0} (g).$
We denote by $D_n$ the irreducible representation of $g$
of dimension $(2n+1),
\hskip.1cm n \in {1 \over 2} \Nrm,$ and also by $D_n$ its
extension to $A_{t_0} (g),$
$t_0 \notin 2 \pi {\bf Q},$ as defined in [3].
\psaut
{\bf (7.5) Lemma}:
\it Let $(\pi, W)$ be a finite dimensional representation of
$A_{t_0} (g), \hskip.1cm
t_0 \notin 2 \pi {\bf Q};$ then
$$H^1 (A_{t_0} (g), {\cal L} (W)) = \{ 0 \}.$$
\psaut
{\bf (7.5.1) Corollary}:
\it When $t_0 \notin 2 \pi {\bf Q},$ the finite dimensional
representations of $A_{t_0} (g)$ are rigid. \rm
\psaut
{\bf Proof}:
Given $\xi \in Z^1 (A_{t_0} (g), \hskip.1cm {\cal L} (W)),$
we define a new representation
$\rho$ of $A_{t_0} (g)$ on $W \times W$ by
$\rho = \pmatrix{ \pi \hskip.3cm \xi \cr
0 \hskip.3cm \pi \cr }.$
But $\rho$ is semi-simple, and {\rm sin}ce it is an extension
of $\pi$ by itself [18],
the extension cocycle $\xi$ is a coboundary.
Now $H^1 (A_{t_0} (g), {\cal L} (W)) = \{ 0 \}$ is the
standard sufficient condition for rigidity of $\pi$ [17].\hskip1cm Q.E.D.
\psaut
{\bf (7.5.2) Proposition}:
\it When $t_0 \notin 2 \pi {\bf Q},$ $\tilde{B}_t (g)$ is a non
trivial deformation of $A_{t_0} (g).$ \rm
\psaut
{\bf (7.5.3) Corollary}:
$$H^2 (A_{t_0} (g), \hskip.1cm A_{t_0} (g)) \neq \{ 0 \}.$$
{\bf Proof}:
Let us assume that the $\Crm [[t]]$-algebras $A_{t_0} (g)
[[t]]$ and $\tilde{B}_t (g)$
are isomorphic by $\phi.$ Let $\tilde{\pi}_i$ be the $(2i+1)
(\hskip.1cm \Crm [[t]])$-dimensional
representation of $\tilde{B}_t (g)$ as defined in [3] (on p.
79). Then the
value of the Casimir element
$Q_t = GF + SC + S^2 {\rm cos} (t_0 + t)$ of $\tilde{B}_t (g)$ is
${\rm sin} i (t_0 + t) \hskip.1cm {\rm sin} (i+1) (t_0+t) \hskip.1cm / \hskip.1cm {\rm sin}^2 (t_0 + t).$
At $t = 0, \hskip.1cm \tilde{B}_t (g)$ has classical limit
$A_{t_0} (g),$ and
$\tilde{\pi}_i$ has classical limit $\pi_i,$ the irreducible
$(2i+1)$-dimensional
representation $\pi_i$ of $A_{t_0} (g).$ Then setting
$\hat{\pi}_i = \tilde{\pi}_i
\hskip.1cm o \hskip.1cm \phi,$ we get a deformation of the
representation $\pi_i$
of $A_{t_0} (g),$ which has to be trivial by (7.5.1), so the
value of the Casimir
$Q_0$ of $A_{t_0} (g)$ in $\hat{\pi}_i$ is
${\rm sin} i t_0 \hskip.1cm {\rm sin} (i+1) t_0 \hskip.1cm / \hskip.1cm {\rm sin}^2 t_0.$
Now, the respective centers of $\tilde{B}_t (g)$ and
$A_{t_0} (g) [[t]]$ are $Z (\tilde{B}_t (g)) = \Crm [[t]] [Q_t],$
and $Z(A_{t_0} (g)
[[t]]) = \Crm [[t]] [Q_0].$ One has $Z(\tilde{B}_{t_0} (g)) =
\phi (Z(A_{t_0} (g)
[[t]])) = \Crm [[t]] [\phi (Q_0)],$ therefore $\phi (Q_0) =
\tilde{\alpha} \hskip.1cm
Q_t + \tilde{B},$ with $\tilde{\alpha}, \tilde{\beta} \in \Crm
[[t]],$ and $\hat{\pi}_i
(Q_0) = \tilde{\alpha} \hskip.1cm \tilde{\pi}_i (Q_t) +
\tilde{\beta}$ (7.6.1).
Taking $i=0,$ we get $\tilde{\beta} = 0;$ then $i = 1,2$ lead
to ${\rm cos} t_0 = \pm
\hskip.1cm {\rm cos} (t_0 + t),$ a contradiction. \hskip1cm
Q.E.D.
\psaut
\centerline {\bf Appendix 1: \hskip.3cm Topological Vector
Spaces.}
\psaut
We refer to [21] or [13] for topological tensor
products, nuclear spaces, etc... We only fix our notations,
and mention some results we need.
{\bf (A.1.1)} A t.v.s. is a complex vector space $V$ with a
locally convex Hausdorf vector
space topology; a c.t.v.s. is a complete t.v.s.. Given t.v.s.
$V_1$ and $V_2,$
$V_1 \simeq V_2$ means topological isomorphism, $V_1
\hat{\otimes} V_2$ is the
c.t.v.s. projective tensor product of $V_1$ and $V_2$;
given t.v.s. $V_i,$ $i \in I,$
we endow $\Pi_{i \in I} V_i$ with the t.v.s. product topology.
{\bf (A.1.2)} We denote by $L (V_1, V_2)$ (resp: ${\cal L}
(V_1, V_2)$) the space of
linear maps (resp: continuous linear maps) from $V_1$ into
$V_2.$ There are several
topologies for which ${\cal L} (V_1, V_2)$ is a t.v.s.: if
nothing is mentioned,
${\cal L} (V_1, V_2)$ has the topology of uniform convergence on bounded sets.
When $V_1$ is Montel (e.g.: $V_1$ quasi-complete nuclear and barreled) then
${\cal L} (V_1, V_2) = {\cal L}_c (V_1, V_2),$ where subscript $c$ means topology
of uniform convergence on compact sets. When $V_1 = V_2 = V,$ we use notations
$L(V)$ and ${\cal L} (V)$ for $L (V,V)$ and ${\cal L} (V,V).$
{\bf (A.1.3)} We denote by $V^*$ the t.v.s. ${\cal L} (V,
\Crm).$ When $V$ is Montel,
$V$ is reflexive, i.e. the canonical mapping from $V$ into
$V^{**}$ is an isomorphism.
{\bf (A.1.4)} Assume that $V_1$ and $V_2$ are c.t.v.s., that
$V_1$ is barreled, $V^*_1$
nuclear and complete, then ${\cal L} (V_1, V_2)$ is
complete, and ${\cal L} (V_1, V_2)
\simeq V^*_1 \hat{\otimes} V_2.$ These assumptions on
$V_1$ are satisfied e.g.
if $V_1$ is nuclear and Fr\'echet, or if $V_1$ is the dual of
a nuclear and Fr\'echet space.
{\bf (A.1.5)} Let $V_1$ and $V_2$ be two Fr\'echet (or dual
of Fr\'echet) spaces;
assume $V_1$ is nuclear, then $(V_1 \hat{\otimes} V_2)^*
\simeq V^*_1 \hat{\otimes} V^*_2.$
{\bf (A.1.6)} Let $V_i, \hskip.1cm i \in I,$ and $W$ be t.v.s.,
then:
$$(\prod_{i \in I} V_i) \hat{\otimes} W \simeq \prod_{i \in I}
(V_i \hat{\otimes} W)$$
\psaut
\centerline {\bf Appendix 2: \hskip.3cm Natural topology of
countable dimensional vector spaces.}
\psaut
We assume in Appendix 2 that $V$ is a countable
dimensional vector space.
\psaut
{\bf (A.2.1) Definition}:
\it An increasing sequence $(V_n)_{n \in \Nrm},$ of
finite-dimensional subspaces of $V$ is a
sequence of definition if $\displaystyle{\bigcup_{n \in
\Nrm}} \hskip.1cm V_n = V.$ \rm
\psaut
Given a sequence of definition, we endow $V$ with the
c.t.v.s. strict inductive
limit topology defined by $(V_n)$ [21].
\psaut
{\bf (A.2.2) Lemma} [21]:
\it Any linear mapping from $V$ into a t.v.s. is continuous.
\rm
\psaut
As a consequence of (A.2.2), the topology defined from
$(V_n)$ does not depend
on the choice of the sequence of definition, so we set:
\psaut
{\bf (A.2.3) Definition}:
\it The strict inductive limit topology defined on $V$ from
any sequence of definition
is called the natural topology of $V.$ \rm
\psaut
{\bf (A.2.4) Proposition} [21]:
\it The natural topology of $V$ is complete, nuclear, and
Montel. $V$ is reflexive. \rm
\psaut
Given a sequence of definition $(V_n),$ we write $V_n =
V_{n-1} \hskip.1cm \oplus
\hskip.1cm V^n,$ and get an isomorphism $V^* \simeq
\displaystyle{\prod_{n \in \Nrm}}
\hskip.1cm (V^n)^*,$ if duality is defined by:
$\varphi_n \in (V^n)^*, \hskip.1cm v_n \in V, \hskip.1cm
<\displaystyle{\prod_{n \in \Nrm}} \hskip.1cm
\varphi_n, \hskip.1cm \displaystyle{\sum_{\hbox{finite}}}
\hskip.1cm v_n> =
\displaystyle{\sum_{\hbox{finite}}} \hskip.1cm <\varphi_n
\vert v_n>.$
This proves:
\psaut
{\bf (A.2.5) Proposition}:
\it $V^* \simeq \displaystyle{\prod_{n \in \Nrm}} \hskip.1cm
(V^n)^*$ is a Fr\'echet space. \rm
\psaut
{\bf (A.2.6) Lemma}:
\it Let $(V_n)$ and $(V'_n)$ be sequences of $f.d.$ spaces,
and $X$ any c.t.v.s., then:
(i) $(\displaystyle{\prod_{n \in \Nrm}} \hskip.1cm V_n)
\hskip.1cm \hat{\otimes} \hskip.1cm X \simeq
\displaystyle{\prod_{n \in \Nrm}} \hskip.1cm (V_n
\hskip.1cm \otimes \hskip.1cm X)$
(ii) $(\displaystyle{\prod_{n \in \Nrm}} \hskip.1cm V_n)
\hat{\otimes} (\displaystyle{\prod_{n \in \Nrm}} \hskip.1cm
V'_n) \simeq
\displaystyle{\prod_{n,p}} \hskip.1cm (V_n \hskip.1cm
\otimes \hskip.1cm V_p).$ \rm
\psaut
{\bf Proof}:
(A.2.5) is a consequence of (A.1.6), noticing that, when
$W$ is a f.d. space,
then $W \hskip.1cm \hat{\otimes} \hskip.1cm X = {\cal L}
(W^*, X) = L (W^*, X) =
W \hskip.1cm \otimes \hskip.1cm X.$ \hskip1cm Q.E.D.
\psaut
{\bf (A.2.7) Proposition}:
\psaut
\it Let $V$ and $V'$ be two countable dimensional vector
spaces, endowed with the natural topology, then:
(i) $(V \hskip.1cm \hat{\otimes} \hskip.1cm V')^* \simeq V^*
\hskip.1cm \hat{\otimes}
\hskip.1cm V'^*$
(ii) $V \hskip.1cm \hat{\otimes} \hskip.1cm V' = V
\hskip.1cm \otimes \hskip.1cm
V',$ with natural topology. \rm
\psaut
{\bf Proof}:
(i) results from (A.2.4) and (A.1.5).
(ii) From (A.1.5) and (i), $V \hskip.1cm \hat{\otimes}
\hskip.1cm V' \simeq
(V^* \hskip.1cm \hat{\otimes} \hskip.1cm V'^*)^*,$ but
\noindent
$V^* = \displaystyle{\prod_{n \in \Nrm}} (V^n)^*,
V'^* = \displaystyle{\prod_{n \in \Nrm}} (V'_n)^*,$ so, from
(A.2.5), $V^* \hskip.1cm \hat{\otimes}
\hskip.1cm V'^* \simeq \displaystyle{\prod_{n,p}} (V^n)^*
\otimes (V^p)^*,$ so $V \hskip.1cm \hat{\otimes}
\hskip.1cm V' \simeq \sum_{n,p} \hskip.1cm V^n
\hskip.1cm \otimes \hskip.1cm V'^p$
(as t.v.s.), i.e. $V \hskip.1cm \hat{\otimes} \hskip.1cm V' =
V \hskip.1cm \otimes
\hskip.1cm V'$ with natural topology. \hskip1cm Q.E.D.
\psaut
{\bf (A.2.8) Corollary}:
\it Any bilinear mapping from $V \times V$ into a t.v.s. is
continuous. \rm
\psaut
{\bf (A.2.9)} U{\rm sin}g (A.2.2), any subspace of $V$ is
closed, and has a topological
supplementary. By standard orthogonality arguments, any
closed subspace of $V^*$
has a topological supplementary (namely: the orthogonal of
any supplementary subspace
in $V$ of the orthogonal of the given subspace of $V^*$).
{\bf (A.2.10)} We develop a very simple, but fundamental
example:
Let $V = \Crm [t]; \hskip.1cm V_n = \Crm [t]_n$ (the
subspace of polynomials of
degree at most $n$) is a sequence of definition for the
natural topology of $V.$
Defining duality between polynomials and formal series as
usual (cf. e.g. [21]), we
get $V^* = \Crm [[t]].$ $V$ and $V^*$ are nuclear, Montel,
reflexive, and $V^*$
is Fr\'echet.
\psaut
\centerline {\bf Appendix 3:\hskip.3cm Deformations of
t.v.s.}
\psaut
Given a vector space $V,$ and a commutative algebra
$A,$
$\tilde{V} = A \otimes V$
has a natural $A$-module structure. In deformation theory
[10], where $A = \Crm [[t]],$
one is rather interested in $V [[t]] = \{ \sum_n \hskip.1cm t^n
\hskip.1cm v_n,
\hskip.1cm v_n \in V \},$ which contains $\tilde{V},$ but is
generally different
(unless $V$ is f.d.). Usually, one introduces a closure with
respect to $t$-adic
topology; unfortunately, if $V$ is a t.v.s., a $t$-adic closure
will no longer
be a t.v.s., so one has to introduce a coarser topology on
$V [[t]].$ Since $V [[t]] =
L(\Crm [t], V) = {\cal L} \hskip.1cm (\Crm [t], V)$ (A.2.2), we define:
\psaut
{\bf (A.3.1) Definition}:
\it Given a t.v.s. $V,$ the associated formal space is the
t.v.s. $V[[t]] = {\cal L}
(\Crm [t], V).$ \rm
The topology on $V [[t]]$ is exactly the product topology
$\displaystyle{\prod_n} \hskip.1cm t^n
V,$ with $t^n V \simeq V.$ It results that
$\displaystyle{lim_{n
\fl \infty}} \hskip.1cm
t^n \hskip.1cm \tilde{v}_n = 0,$ for any sequence
$(\tilde{v}_n)$ of elements of
$V [[t]].$ If $V$ is complete, then $V [[t]] \simeq \hskip.1cm
\Crm [[t]] \hskip.1cm
\hat{\otimes} \hskip.1cm V$ (A,1,4); generally if
$\overline{V}$ is the completion of
$V,$ $V[[t]]$ is a dense subspace of $\overline{V} [[t]] =
\hskip.1cm \Crm [[t]] \hskip.1cm
\hat{\otimes} \hskip.1cm \overline{V} = \hskip.1cm \Crm [[t]]
\hskip.1cm \hat{\otimes}
\hskip.1cm V$ ((A,1,4) and $\hskip.1cm \Crm \hskip.1cm
\hat{\otimes} \hskip.1cm
V = \overline{V}$).
Given $f \in \hskip.1cm \Crm [[t]],$ and $\tilde{v} \in V [[t]] =
{\cal L} (\Crm [t], V),$ we define $f. \tilde{v}$ by
$(f.\tilde{v}) (P) = f(P) \hskip.1cm \tilde{v} (P),$
\noindent
$P \in \hskip.1cm
\Crm [t],$ u{\rm sin}g $\hskip.1cm \Crm [t]^* = \hskip.1cm
\Crm [[t]]$ (A,2,9).
Therefore $V [[t]]$ is a $\hskip.1cm \Crm [[t]]$-module; if $f
= \displaystyle{\sum_n} \hskip.1cm
f_n \hskip.1cm t^n, \hskip.1cm \tilde{v} =
\displaystyle{\sum_n} \hskip.1cm t^n
\hskip.1cm v_n,$ one has:
$f.\tilde{v} = \displaystyle{\sum_n} \hskip.1cm t^n
\hskip.1cm \displaystyle{\sum_{i+j=n}}
\hskip.1cm f_i \hskip.1cm v_j,$ so the map $(f, \tilde{v}) \fl
f.\tilde{v}$ is continuous.
\psaut
{\bf (A.3.2) Definition}:
\it A topologically free (t.f.) $\hskip.1cm \Crm[[t]]$-module is
a formal space $V[[t]]$
associated with a t.v.s. $V,$ with its natural $\hskip.1cm
\Crm [[t]]$-module structure. \rm
\psaut
We mention that we do not need to define general
topological $\hskip.1cm \Crm [[t]]$-modules,
but only free ones, {\rm sin}ce we are only interested in
topological version of deformation
theory. Nevertheless it might be of interest to work in full
generality, even
for deformation theory. For the time being, $t.f. \hskip.2cm
\Crm [[t]]$-modules are enough for our purpose.
{\bf (A.3.3)} Given $t.f. \hskip.1cm \Crm [[t]]$-modules
$V[[t]]$ and $W[[t]],$ we consider the space
${\cal L}_t (V [[t]],$ $W[[t]])$ of continuous
$\hskip.1cm
\Crm [[t]]$-linear maps from $V [[t]]$ into $W[[t]].$
\psaut
{\bf (A.3.3.1) Lemma}:
\it ${\cal L}_t (V[[t]], \hskip.1cm W[[t]]) \simeq {\cal L} (V,W)
[[t]],$ as
t.v.s. and $\hskip.1cm \Crm [[t]]$-modules. \rm
\psaut
{\bf Proof}:
${\cal L}_t (V[[t]], \hskip.1cm W[[t]])$ has the t.v.s. structure
induced by
${\cal L} (V [[t]], \hskip.1cm W[[t]]).$
Given $F \in {\cal L}_t (V[[t]], \hskip.1cm W[[t]]),$ let $F(v) =
\displaystyle{\sum_n}
\hskip.1cm t^n F_n (v), \hskip.1cm v \in V;$ then $F_n \in
{\cal L} (V,W),$
so $\tilde{F} = F \vert_V \in {\cal L} (V,W) [[t]].$ $F$ can be
reconstructed from $\tilde{F}$ by:
$$F (\displaystyle{\sum_n} \hskip.1cm t^n\hskip.1cm v_n) =
\displaystyle{\sum_n}
\hskip.1cm t^n \hskip.1cm \displaystyle{\sum_{i+j=n}}
\hskip.1cm F_i (v_j)$$
So $F \fl \tilde{F}$ will give the wanted isomorphism; it is
not difficult to
check that it is a topological one. Moreover, ${\cal L}_t
(V[[t]], \hskip.1cm W[[t]])$
has a natural $\hskip.1cm \Crm [[t]]$-module structure
defined by $(f.F) (\tilde{v}) =
f. (F(\tilde{v})),$ and $\widetilde{f.F} = f.\tilde{F}.$
\hskip1cm Q.E.D.
\psaut
{\bf (A.3.3.2) Definition}:
\it The $\hskip.1cm \Crm [[t]]$-dual of $V[[t]]$ is $V[[t]]^*_t =
{\cal L}_t (V[[t]],
\hskip.1cm \Crm [[t]]).$ \rm
\psaut
U{\rm sin}g (A.3.3.1):
\psaut
{\bf (A.3.3.3) Proposition}: $V[[t]]^*_t \simeq V^* [[t]].$
\psaut
{\bf (A.3.4)} We denote by $<\hskip.1cm, \hskip.1cm>_t$
the $\hskip.1cm \Crm [[t]]$-bilinear
duality between $V[[t]]$ and $V[[t]]^*_t.$
Given $\phi \in {\cal L}_t (V[[t]], \hskip.1cm W[[t]]) \simeq
{\cal L} (V,W) [[t]],$
we write $\phi = \displaystyle{\sum_n} \hskip.1cm t^n
\hskip.1cm \phi_n, \hskip.1cm
\phi_n \in {\cal L} (V,W)$ and define the transpose $^T \phi
= \displaystyle{\sum_n}
\hskip.1cm t^n \hskip.1cm ^T \phi_n \in {\cal L} (W^*, V^*)
[[t]] \simeq {\cal L}_t
(W [[t]]^*_t, \hskip.1cm V[[t]]^*_t).$ One has:
$$<\tilde{v} \vert ^T \phi (\tilde{w}^*)>_t = <\phi (\tilde{v})
\vert \tilde{w}^*>_t
\hskip.1cm \hbox{ if } v \in V[[t]], \hskip.1cm \tilde{w}^* \in
W[[t]]^*_t = W^* [[t]].$$
{\bf (A.3.5)} We now define the notion of (topological)
$\hskip.1cm \Crm [[t]]$-tensor product:
\psaut
{\bf (A.3.5.1) Definition}:
\it The (topological) $\hskip.1cm \Crm [[t]]$-tensor product
of $V[[t]]$ and $W[[t]]$ is \rm
$$V[[t]] \hskip.1cm \displaystyle{\hat{\otimes}_t} \hskip.1cm
W[[t]] = (V \hskip.1cm
\hat{\otimes} \hskip.1cm W) [[t]].$$
The characteristic property of the topological tensor
product of t.v.s. is the factorization
property of continuous bilinear mappings. Here, the same
will hold, if one replaces
bilinear by $\hskip.1cm \Crm [[t]]$-bilinear.
We need some notations. First we define a canonical
continuous $\hskip.1cm \Crm [[t]]$-bilinear
mapping $\tilde{i}: V[[t]] \times W[[t]]$ into $V[[t]] \hskip.1cm
\displaystyle{\hat{\otimes}_t}
\hskip.1cm W[[t]]$ by:
$$\tilde{i} (\displaystyle{\sum_n} \hskip.1cm t^n \hskip.1cm
v_n, \hskip.1cm
\displaystyle{\sum_n} \hskip.1cm t^n \hskip.1cm w_n) =
\displaystyle{\sum_n}
\hskip.1cm t^n \hskip.1cm \displaystyle{\sum_{i+j=n}}
\hskip.1cm v_i \hskip.1cm
\otimes \hskip.1cm w_j, \hskip.1cm v_n \in V, \hskip.1cm
w_n \in W.$$
Now, it is quite natural to use the notation:
$$\tilde{i} (\tilde{v}, \tilde{w}) = \tilde{v} \hskip.1cm
\displaystyle{\otimes_t}
\hskip.1cm \tilde{w} \hskip.3cm, \hskip.1cm \tilde{v} \in
V[[t]], \hskip.1cm \tilde{w} \in W[[t]].$$
\psaut
{\bf (A.3.5.2) Proposition}:
\it Given a continuous $\hskip.1cm \Crm [[t]]$-bilinear map
$\tilde{F}: V[[t]]
\times W[[t]] \fl X[[t]],$ where $X$ is a c.t.v.s., there exists
a continuous
$\hskip.1cm \Crm[[t]]$-linear map
$\tilde{G}: V[[t]]
\hskip.1cm \displaystyle{\otimes_t}
\hskip.1cm W[[t]] \fl X[[t]]$ such that: \rm
$$F (\tilde{v}, \tilde{w}) = \tilde{G} (\tilde{v}
\displaystyle{\otimes_t} \hskip.1cm
\tilde{w}).$$
\psaut
{\bf Proof}:
Given $v \in V, \hskip.1cm w \in W, \hskip.1cm \tilde{F}
(v,w) = \displaystyle{\sum_n}
\hskip.1cm t^n \hskip.1cm F_n (v,w), \hskip.1cm F_n \in
{\cal L}_2 (V,W;X),$ so
$F_n (v,w) = G_n (v \otimes w)$ with $G_n \in {\cal L} (V
\hat{\otimes} W, X).$ Now
$$F(\tilde{v}, \tilde{w}) = \tilde{F} (\displaystyle{\sum_n}
\hskip.1cm t^n \hskip.1cm
v_n, \displaystyle{\sum_n} \hskip.1cm t^n \hskip.1cm w_n)
= \displaystyle{\sum_n}
\hskip.1cm t^n \hskip.1cm \displaystyle{\sum_{i+j+k=n}}
\hskip.1cm F_k (v_i, w_j) =$$
$$= \displaystyle{\sum_n} \hskip.1cm t^n \hskip.1cm
\displaystyle{\sum_{i+j+k=n}}
\hskip.1cm G_k (v_i \otimes w_j) = (\displaystyle{\sum_n}
\hskip.1cm t^n \hskip.1cm
G_n) (\displaystyle{\sum_n} \hskip.1cm t^n \hskip.1cm
\displaystyle{\sum{i+j=n}}
\hskip.1cm v_i \otimes w_j) = \tilde{G} (\tilde{v} \hskip.1cm
\otimes_t \hskip.1cm
\tilde{w})$$ if one defines
$\tilde{G} = \displaystyle{\sum_n} \hskip.1cm t^n
\hskip.1cm G_n
\in {\cal L} (V \hat{\otimes} W, X) [[t]] = {\cal L}_t (V
\hat{\otimes} W [[t]],
X[[t]]) = {\cal L}_t (V[[t]] \displaystyle{\hat{\otimes}_t}
W[[t]], X[[t]]).$
\hskip7cm Q.E.D.
\psaut
\centerline {\bf Appendix 4: \hskip.3cm Vocabulary.}
\psaut
There are several, more or less restrictive, notions of
algebras, bialgebras,
Hopf algebras, etc... in the literature, so we give precise
definitions of the notions we use in this paper, including
corresponding topological definitions.
{\bf (A.4.1)} A vector space $A$ (resp: c.t.v.s.) is an algebra
(resp: a topological algebra)
if one has fixed a linear (resp: continuous linear) map $\mu:
A \otimes A$ (resp:
$A \hat{\otimes} A$) $\fl A.$ As usual, we note $\mu (a
\otimes a') = aa', \hskip.1cm a, a' \in A.$
{\bf (A.4.2)} An associative (resp: topological associative)
algebra is an algebra
(resp: topological algebra) with an associative product and
a unit element.
{\bf (A.4.3)} When $A$ is an algebra (resp: a topological
algebra) then $A \otimes A$
(resp: $A \hat{\otimes} A$) is also an algebra (resp: a
topological algebra),
the product being defined by:
$$(a \otimes b).(a' \otimes b') = (aa') \otimes
(bb')\hskip.1cm ,
\hskip.1cm a,a', b, b' \in A.$$
{\bf (A.4.4)} A bialgebra (resp.: a topological bialgebra) is
an algebra (resp.: a topological
algebra) with a morphism (resp: continuous morphism)
$\Delta: A \fl A \otimes A$
(resp: $A \hat{\otimes} A$).
{\bf (A.4.5)} Hopf algebras are defined as in [6]. Topological
Hopf algebras are defined
by adding the continuity condition of the antipode and
counit.
\vfill\eject
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\end
ENDBODY