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\centerline {\bf TOPOLOGICAL QUANTUM DOUBLE}
\gsaut
\hskip6cm Philippe BONNEAU.
\gsaut
\gsaut
\saut
{\bf Abstract :} Following a preceding paper showing how the introduction of a
t.v.s. topology on quantum groups leads to a remarkable {\it unification} and
{\it rigidification} of the different definitions, we adapt here, in the same
way, the definition of quantum double. This topological double is {\it dualizable}
and {\it reflexive} (even for {\it infinite dimensional} algebras).
In a simple case we show, considering the double as the "zero class" of an extension
theory, the uniqueness of the double structure as a quasi-Hopf algebra.
\gsaut
\gsaut
{\bf R\'esum\'e :} A la suite d'un pr\'ec\'edent article montrant comment l'introduction
d'une topologie d'e.v.t. sur les groupes quantiques permet une {\it unification}
et une {\it rigidification} remarquables des diff\'erentes d\'efinitions, on adapte
ici de la m\^eme mani\`ere la d\'efinition du double quantique. Ce double topologique
est alors {\it dualisable} et {\it reflexif} (m\^eme pour des alg\`ebres de {\it
dimension infinie}).
Dans un cas simple on montre, en consid\'erant le double comme la "classe z\'ero" d'une th\'eorie
d'exten-
\noindent
sions, l'unicit\'e de cette structure comme alg\`ebre quasi-Hopf.
\gsaut
\gsaut
\gsaut
\gsaut
{\bf Address:} Universit\'e de Bourgogne - Laboratoire de Physique-Math\'ematique
\hskip1.6cm B.P. 138 - 21004 DIJON Cedex - FRANCE
\hskip1.6cm {\it T\'el.} : (33) 80 39 58 50 - {\it Fax} : (33) 80 39 58 69
\hskip1.6cm e-mail : flato@satie.u-bourgogne.fr
\gsaut
\saut
PREPRINT DIJON \hskip2cm {\it Juillet 1993}
Accepted for publication in Reviews in Mathematical Physics.
\vfill\eject
0. {\bf INTRODUCTION.}
\psaut
The topological model of quantum groups (abbreviated QG) we introduced in [BFGP]
is very useful: dualizing is straightforward and thus everything we show on
it is valid for both the Drinfeld [D1] and Faddeev-Reshetikhin-Takhatajan [FRT] versions
of QG. Moreover most of the works made in the generic case (not root of unity)
seem to be easily adaptable to the new model.
In this paper we do this explicitely for the quantum double. Here again the
notion of duality is primordial: the duality in the "QUE-dual" [D1] is very specific,
and defines the double in a particular context. But the double can be defined
more generally, for example in terms of the "double cross-product" of Majid [Ma1]. It is
also of interest in the theory of monoidal categories. But in these works, because
of the problem of dualizing an infinite dimensional Hopf algebra, it is supposed
most of the time that the constructions are done for finite dimensional Hopf algebras
(group algebras of finite groups for example).
Fr\'echet nuclear or dual of Fr\'echet nuclear topologies and strong duality get rid
of these difficulties in the infinite dimensional case (see [BFGP]). Therefore, we can
adapt the definition of Majid for finite groups to a quantum double structure on
${\cal H}_t (G) \bar{\otimes} {\cal A}_t (G)$ and ${\cal C}^\infty_t (G) \bar{\otimes}
{\cal D}_t (G)$ either for $t=0$ or $t$ generic parameter of the QG deformation
(see [BFGP]) and for every compact Lie group $G$ (${\cal H} (G)$ is the coefficient
algebra of $G,$ ${\cal A} (G)$ its strong dual, ${\cal C}^\infty (G)$ the $C^\infty$
complex functions on $G$ and ${\cal D} (G)$ the distributions on $G$ ; $\bar{\otimes}$
stands for the completed injective topological tensor product [Gr]).
We have, as topological vector spaces, $({\cal H}_t (G) \bar{\otimes} {\cal A}_t (G))^* =
{\cal A}_t (G) \hat{\otimes} {\cal H}_t (G)$ (projective topological tensor product)
and thus, we can define a dual double structure. {\it The double defined here is, of
course, reflexive.}
We were also interested in the notion of extensions for quantum groups. Two possibilities
arise :
1) to adapt the Hochschild theory of associative algebras to the Hopf case and then the
underlying vector space of the extension is a direct sum.
2) Since ${\cal U} ({\cal G}_1 \oplus {\cal G}_2) = {\cal U} {\cal G}_1
\otimes {\cal U} {\cal G}_2$ and $\hskip.1cm \Crm (G_1 \times G_2) = \hskip.1cm
\Crm G_1 \otimes \hskip.1cm \Crm G_2,$ for ${\cal G}_1, {\cal G}_2$ Lie algebras
and $G_1, G_2$ groups, we have to find, if we want to pass from the
level of Lie algebras or groups to that of their associated Hopf algebras and then to their
QG deformations, an extension theory for Hopf algebras compatible with an underlying
structure of tensor product of vector spaces.
This was done by Sweedler [Sw2] and Singer [Si] (with restrictions on the (co)-commutativity).
We remark that the double, in the non deformed case of ${\cal A} (G)$ and ${\cal D} (G),$
is a smash-product (see [Sw1] or [A]), i.e., the "zero class" of the extension theory.
Then, adapting the theory for quasi-Hopf algebras, we show with cohomological methods, that in the case of ${\cal D} (G)$
any quasi-Hopf extension of ${\cal D} (G)$ by ${\cal C}^\infty (G)$ is the double.
This is a kind of uniqueness result.
We were also motivated by an article of Dijkgraaf-Pasquier-Roche [DPR] who find
apparently non equivalent quasi-Hopf analogues of the double for finite groups. Here,
with the extension theory we take, this is not possible.
\psaut
The article is organized as follows :
We first recall what we need about the quantum double, then construct topological
quantum doubles and study their properties.
Next, we recall the extension theory we use and the classifying cohomology.
We adapt it for quasi-Hopf algebras in the way of [Bo] and prove the uniqueness property
of $D ({\cal D} (G))$ by showing the nullity of a second cohomology.
Finally, we discuss what is left to be done about the double and more generally about the
problem of extensions for QG.
\vfill\eject
1. {\bf BACKGROUND.}
\psaut
Let $(A, \mu, \Delta, \varepsilon, S)$ be a Hopf algebra, $A^*$ a Hopf algebra "in duality"
(to be defined) with $A,$ and $A^0$ be $A^*$ with the {\it opposite} coproduct.
The antipode on $A^0$ is then $^t S',$ the transpose of the skew antipode of
$A$ (i.e. the antipode of the coalgebra $A$ equipped with the opposite product
as an algebra).
If we consider the vector space $A^* \otimes A,$ Drinfeld [D1] defines the
quantum double as follows :
i) $D(A) \simeq A^0 \otimes A$ as coalgebras,
ii) $(f \otimes Id_A) . (Id_{A^0} \otimes b) = f \otimes b,$
iii) $(Id_{A^0} \otimes e_s) . (e^t \otimes Id_A) = \Delta^{kjn}_s \hskip.1cm
\mu^t_{plk} \hskip.1cm S'^p_n \hskip.1cm (e^l \otimes Id_A) \hskip.1cm (Id_{A^0} \otimes
e_j),$ where $\{ e_s \}$ is a basis of $A$ and $\hskip.1cm \{ e^t \}$ the dual basis.
This can be formulated in an other way ([Ma1]).
\saut
{\bf DEFINITION 1.}
\it Let $(A, \mu, \Delta, \varepsilon, S)$ be a Hopf algebra, $A^*$ its dual for a given
duality notion. We denote by $A^0$ the Hopf algebra $A^*$ with the opposite coproduct, $^t\mu', \hskip.1cm
S'$ the skew antipode of $A$ and $<.,.>$ the duality pairing.
We call {\bf quantum double of $A,$} $D(A),$ the Hopf algebra
$(A^0 \otimes A, \mu_D, \Delta_D = ^t\mu' \otimes \Delta, S_D = ^tS'
\otimes S)$ with
$$\mu_D \hskip.1cm ((f \otimes a) \otimes (g \otimes b)) = \displaystyle{\sum_{(a)}} \hskip.1cm
f \hskip.1cm \otimes \hskip.1cm a_{(2)}
\hskip.1cm b$$
where ? is a variable of $A,$ using the notation of Sweedler [Sw1] for the coproduct.
Equivalently, \rm
$$\mu_D ((f \otimes a) \otimes (g \otimes b)) = \sum_{(a) (g)}
\hskip.1cm ~~ \hskip.1cm fg_{(2)} \otimes a_{(2)} b$$
\psaut
{\bf LEMMA 1.} \it If $A$ is cocommutative,
$$\mu_D ((f \otimes a) \otimes (f \otimes b)) = \displaystyle{\sum_{(a)}} \hskip.1cm f(a_{(1)}
\bullet g) \otimes a_{(2)} b$$
with $x \bullet h$ the coadjoint action of $A$ on $A^0$ (i.e. $x \bullet h(y) =
\sum_{(x)} \hskip.1cm )$ \rm
\saut
{\it Proof :}
\psaut
(i) $S = S'$
(ii) we have $\sum_{(a)} \hskip.1cm f(a_{(1)} \bullet g) \otimes a_{(2)} b =
\sum_{(a) (a_{(1)})} \hskip.1cm f
\otimes a_{(2)} b$
\hskip2.8cm $= \sum_{(a)} \hskip.1cm f
\otimes a_{(3)} b$ \hskip1cm (coassociativity)
\hskip2.8cm $= \sum_{(a)} \hskip.1cm f
\otimes a_{(2)} b$ \hskip1cm (cocommutativity)
\square
The product in this cocommutative case is known as the {\it smash product} of
$A^0$ by $A$ via the coadjoint action [Sw2].
In the general case, we have the "double cross product" structure of Majid [Ma1] :
the product of the double can be written as follows :
\noindent
$(f \otimes a) . (g \otimes b) = \displaystyle{\sum_{(a) (g)}} \hskip.1cm f \hskip.1cm \alpha
(a_{(1)} \otimes g_{(1)}) \otimes \beta (a_{(2)} \otimes g_{(2)})b \hskip.3cm$
where $\alpha$ defines an $A$-module structure on $A^0$ by
$$\alpha (c \otimes h) = \sum_{(c)} \hskip.1cm $$
and $\beta$ defines an $A^0$-module structure on $A$ by
$$\beta (c \otimes h) = \sum_{(c)} \hskip.1cm c_{(2)} \hskip.1cm .$$
\vfill\eject
2. {\bf TOPOLOGICAL QUANTUM DOUBLE.}
\psaut
As in [BFGP] we would like, with the help of topology, to find a notion of
quantum double which is reflexive and more natural (in the definition of
duality for example).
I think that we can find general enough conditions on a topological Hopf algebra
$A$ so that the construction described in the following is always possible. This
would certainly require quite technical notions of topological vector
spaces. For example, the dual of the topological tensor product of a
Fr\'echet space with its dual, is, in the general case, especially pathological
(see [Gr, Chap. II, paragraph 4]). And this would lead us too far
from our main subject : the quantum groups.
Therefore, I shall restrict myself here to the two cases in which we were interested
in [BFGP], when $G$ is a compact, semi-simple, connected Lie group :
1) ${\cal A} (G) = \displaystyle{\Pi_{\pi \in \hat{G}}} \hskip.1cm End (V_\pi)
\hskip.5cm ({\cal A} (G)^* = {\cal H} (G) = \displaystyle{\oplus_{\pi \in \hat{G}}}
\hskip.1cm {\cal C}_\pi)$
\noindent
with $\hat{G} = \{$ non equivalent irreducible finite dimensional (f.d.) representations
of $G \}$
${\cal C}_\pi = \{$ coefficients of the f.d. representation $(\pi, V_\pi) \}$
\noindent
and their "quantum" deformations.
2) ${\cal D} (G) = \{$ distributions on $G \}$ $\hskip.5cm ({\cal D} (G)^* = C^\infty (G)$
$= \{ C^\infty$ complex functions on $G \})$
\noindent
and their "quantum deformations" (for details see [BFGP]).
\saut
{\bf DEFINITION 2.}
\it We call {\bf topological Hopf algebra,} a Hopf algebra $(A, \mu, \Delta, \varepsilon, S)$
such that
$\mu : A \tilde{\otimes} A \fl A$
$\Delta : A \fl A \tilde{\otimes} A$ \hskip1cm are continuous
$\varepsilon : A \fl \hskip.1cm \Crm$
$S : A \fl A$
\noindent
where $\tilde{\otimes}$ stands either for the completed projective topological
tensor product (noted $\hat{\otimes}$), or for the injective one (noted $\bar{\otimes}$)
(see {\rm [Gr]}). \rm
\psaut
Now we have
\psaut
{\bf PROPOSITION 1.} \it Let $A$ be a topological Hopf algebra (for $\bar{\otimes}$)
and $A^*$ its strong dual.
\psaut
Then Definition 1 gives a topological Hopf algebra
structure (for $\bar{\otimes}$) on $A^0 \bar{\otimes} A.$ \rm
\saut
The proof is straightforward :
\psaut
the elements $f \otimes a, \hskip.1cm f \in A^0, \hskip.1cm a \in A,$
generate topologically $A^0 \bar{\otimes} A,$
$\Delta_A, S'_A, \mu_A$ are (by definition) continuous and the duality
pairing is a separately continuous form on $A^0 \times A,$ thus a continuous
linear form on $A^0 \bar{\otimes} A.$
\square
\psaut
Therefore, we can define the double for every (injective) topological Hopf algebra. However the
problems mentioned above occur for the dualization. That is why, from now on, \it we
shall restrict ourselves to the cases ${\cal A} = {\cal A} (G)$
and ${\cal D} = {\cal D} (G)$ $({\cal A}^* = {\cal H}$ and ${\cal D}^* = {\cal C}),$
and to their QG deformations \rm (which have the same vector space structure).
\saut
{\bf PROPOSITION 2.} \it As vector spaces, we have
$({\cal A} \hat{\otimes} {\cal H})^* = {\cal H} \bar{\otimes} {\cal A}
\hbox{ and } ({\cal H} \bar{\otimes} {\cal A})^* = {\cal A} \hat{\otimes} {\cal H},$
since $A^* = {\cal H}$ and $A^{**} = A,$ and the same for ${\cal D}$ and ${\cal C}.$ \rm
\vfill\eject
{\it Proof :}
\psaut
1) {\it for ${\cal A}$ :}
$$\eqalign {
({\cal A} \hat{\otimes} {\cal A}^*)^* &= \biggl[\biggl(\prod_{\pi \in \hat{G}} \hskip.1cm {\cal L}
(V_\pi)\biggr) \hat{\otimes} \biggl(\bigoplus_{\pi' \in \hat{G}} \hskip.1cm {\cal L} (V_{\pi'})\biggr)\biggr]^* \cr
&\simeq \biggl[\prod_{\pi \in \hat{G}} \hskip.1cm \biggl({\cal L} (V_\pi) \hat{\otimes} (\bigoplus_{\pi' \in \hat{G}}
\hskip.1cm {\cal L} (V_{\pi'}))\biggr)\biggr]^* \cr
&\simeq \bigoplus_{\pi \in \hat{G}} \hskip.1cm \biggl[{\cal L} (V_\pi) \hat{\otimes}
(\bigoplus_{\pi' \in \hat{G}} \hskip.1cm {\cal L} (V_{\pi'}))\biggr]^* \cr
&\simeq \bigoplus_{\pi \in \hat{G}} \hskip.1cm \biggl[\bigoplus_{\pi' \in \hat{G}}
\hskip.1cm [{\cal L} (V_\pi) \otimes {\cal L} (V_{\pi'})]\biggr]^* \hskip.3cm
({\cal L} (V_\pi) \hbox{ normable }) \cr
&\simeq \bigoplus_{\pi \in \hat{G}} \hskip.1cm \prod_{\pi' \in \hat{G}} \hskip.1cm
[{\cal L} (V_\pi) \otimes {\cal L} (V_{\pi'})]^* \cr
&\simeq \bigoplus_{\pi \in \hat{G}} \hskip.1cm \prod_{\pi' \in \hat{G}} \hskip.1cm
({\cal L} (V_\pi) \otimes {\cal L} (V_{\pi'})) \cr
&\simeq \bigoplus_{\pi \in \hat{G}} \hskip.1cm \biggl[{\cal L} (V_\pi) \bar{\otimes}
(\prod_{\pi' \in \hat{G}} {\cal L} (V_{\pi'}))\biggr] \hskip.3cm
({\cal L} (V_\pi) \hbox{ Fr\'echet }) \cr
&\simeq \biggl(\bigoplus_{\pi \in \hat{G}} \hskip.1cm {\cal L} (V_\pi)\biggr) \bar{\otimes}
\biggl(\prod_{\pi' \in \hat{G}} {\cal L} (V_{\pi'})\biggr) \simeq {\cal A}^* \bar{\otimes}
{\cal A} \cr
}$$
Thus we obtain $({\cal A} \hat{\otimes} {\cal A}^*)^* \simeq {\cal A}^* \bar{\otimes}
{\cal A}$ and by the same arguments we have $({\cal A}^* \bar{\otimes} {\cal A})^* =
{\cal A} \hat{\otimes} {\cal A}^*.$
\psaut
2) {\it for ${\cal D} (G)$ :} in [Gr], chap. II, we see p. 129 that ${\cal D} (G)
\hat{\otimes} {\cal C}^\infty (G)$ is bornological, thus (theorem 14.1) its strong
dual is complete and therefore (corollary of lemma 9) $({\cal D} (G) \hat{\otimes}
{\cal C}^\infty (G))^* \simeq {\cal D} (G)^* \bar{\otimes} C^\infty (G)^*$
\noindent
$(= C^\infty (G)
\bar{\otimes} {\cal D} (G) \hskip.1cm )$ and ${\cal D} (G) \hat{\otimes} {\cal C}^\infty (G)$
is reflexive.
\square
\psaut
We now have to dualize the Hopf algebra structure : for the coproduct, this is
straightforward.
For the product we consider the "double cross-product" form :
$$(f \otimes a) . (g \otimes b) = \sum_{(a) (g)} f \hskip.1cm \alpha(a_{(1)} \otimes
g_{(1)}) \otimes \beta(a_{(2)} \otimes g_{(2)}) b$$
with $\alpha : A \bar{\otimes} A^* \fl A^*$ and $\beta : A \bar{\otimes} A^* \fl
A$ continuous (see definition of $\alpha$ and $\beta$).
An equivalent form (without Sweedler notation, denoting (34) the permutation of two
middle factors), is
$$\mu_D = (\mu_* \otimes \mu) \hskip.1cm (Id_* \otimes \alpha \otimes \beta \otimes
Id) \hskip.1cm (34) \hskip.1cm (Id_* \otimes \Delta_* \otimes \Delta \otimes
Id)$$
which shows by transposition that $^t\mu_D$ define a continuous coproduct
on $(A \bar{\otimes} A^*)^* \simeq A^* \hat{\otimes} A$ and thus make the dual
of the topological quantum double of $A$ a topological Hopf algebra (for $\hat{\otimes}$).
Because of the reflexivity of the vector spaces involved, we have in the same
way $D(A)^{**} = D(A).$
\psaut
All these arguments work for $A$ considered as a topological
$\hskip.1cm \Crm[[t]]$-Hopf algebra thanks to the material developped in [BFGP].
Therefore we can construct and dualize the double of the "topological quantum groups"
studied in [BFGP] $({\cal A} (G)$ and ${\cal D} (G)$ with deformed coproduct
on $\hskip.1cm \Crm[[t]]).$
We can formulate all of these into a theorem :
\saut
{\bf THEOREM 1.} \it Let $A$ be either ${\cal A} (G), \hskip.1cm {\cal D} (G),$ or
their deformed versions, $H$ be ${\cal H} (G), \hskip.1cm {\cal C}^\infty (G)$
or their deformed versions. Then :
$D(A) = A^* \bar{\otimes} A = H \bar{\otimes} A$ with the classical Hopf structure
of double
$D(A)^* = A \hat{\otimes} A^* = A \hat{\otimes} H$ with the dualized structure
and $D(A)^{**} = D(A).$ \rm
\saut
{\bf Remark :} $D ({\cal D} (G))^* = {\cal D} (G) \hat{\otimes} {\cal C}^\infty (G)
\simeq {\cal C}^\infty (G, {\cal D} (G)).$
$\hskip6.1cm \simeq {\cal D} (G, C^\infty (G))$
This can lead to explicit and interesting expressions for the coproduct on the
dual double [P].
\saut
3. {\bf THE DOUBLE AS TRIVIAL EXTENSION OF BIALGEBRAS.}
\psaut
In this section {\it we show a uniqueness property of the topological double}
in the non-deformed case of the double, $D({\cal D} (G)),$ of ${\cal D} (G).$
But first we have to fix the framework of this result :
\saut
{\bf Notations.}
Let $(A, \Delta_A, \varepsilon_A)$ be a
coalgebra, $(\Lambda, \mu_\Lambda, \eta_\Lambda)$ an algebra :
On $Hom_{\Crm} (A, \Lambda)$ we put the algebra structure defined
by the product (often called "convolution")
$$f * g = \mu_\Lambda \circ (f \otimes g) \Delta_A$$
The unit of $Hom_{\Crm} \hskip.1cm (A , \Lambda)$ for $*$ is $\eta_\Lambda \circ
\varepsilon_A.$
If $A$ is cocommutative and $\Lambda$ commutative, $*$ is commutative.
We note $Reg (A, \Lambda)$ the invertible elements of $Hom_{\Crm} \hskip.1cm
(A, \Lambda)$ for $*,$ and $f^{-1}$ the inverse of $f \in Reg (A, \Lambda).$
For the notions of module algebra, comodule algebra,...we refer to
the book of Abe [A].
For $\Psi$ a (right) $A$-comodule structure on $X,$ for $X$ a coalgebra,
we note $\Psi (x) = \displaystyle{\sum_{(x)}} x_X \otimes x_A.$
For $A$ a Hopf algebra, $\varepsilon_A$ its counit, we note $A^+ = Ker
\hskip.1cm \varepsilon_A.$
\saut
{\bf Extensions} :
\saut
DEFINITION 3 [Sw2]. \it Let $A$ be a {\it cocommutative} Hopf algebra,
$\Lambda$ a {\it commutative} $A$-module algebra.
We call $(B, \Psi)$ an {\bf algebra tensorial extension of $\Lambda$ by} $A$ if $B$ is an
algebra containing $\Lambda$ and
\noindent
$\Psi : B \fl B \otimes \Lambda$ is an algebra homomorphism verifying
(i) $A = \Psi^{-1} (B \otimes 1) = \{ b \in B \vert \Psi (b) = b \otimes 1 \}.$
(ii) $\Psi$ gives to $B$ the structure of a right $A$-comodule (with respect to the
underlying coalgebra structure of $A$).
(iii) $b \lambda = \displaystyle{\sum_{(b)}} (b_A . \lambda) \hskip.1cm b_B
\hskip.5cm Jb \in B, \hskip.1cm \lambda \in \Lambda.$
\noindent
We say that {\it the extension $(\Psi , B)$ is} {\bf cleft} if there is a comodule morphism in
$Reg (A,B).$ \rm
\saut
PROPOSITION 3 [By]. \it Let $A$ be a cocommutative Hopf algebra, $\Lambda$
a commutative one. Then
$(i, B, \rho)$ with $B$ a Hopf algebra, $i$ an injective Hopf morphism from $\Lambda$ to
$B, \rho$ a surjective one from $B$ to $A$ such that
1) $Ker \hskip.1cm \rho = \Lambda^+ B$ (ideal generated by $\Lambda^+$ in $B$)
2) $\exists \hskip.1cm q : B \fl \Lambda$ left $\Lambda$-modules map with $(i \circ q) *
(j \circ \rho) = 1$
3) $\exists \hskip.1cm j : A \fl B$ right $A$-comodules map with $q \circ j = y \circ
\varepsilon : A \fl \Lambda$
\noindent
defines a cleft algebra tensorial extension (with $\psi = (1 \otimes \rho) \circ
\Delta).$ \rm
\psaut
Proposition 3 can be taken as an alternative (and more natural) definition of
cleft algebra extension.
In the terminology of Byott, $(i, B, \rho)$ and $(i', B', \rho')$ are {\it
equivalent} if $\exists f : B \fl B'$ Hopf algebras isomorphism
$s.t. \hskip.5cm f \circ i = i$ \hskip.3cm and \hskip.3cm $\rho \circ f = \rho.$
This notion of extension is classified by a cohomological theory that I shall
briefly recall now, {\it the Sweedler cohomology} ([Sw2]) :
For $A$ a cocommutative Hopf algebra, $\Lambda$ a commutative A-module algebra,
$Reg (A^{\otimes n}, \Lambda)$ are n-cochains.
Let $D^n : Reg (A^{\otimes n}, \Lambda) \rightarrow Reg (A^{\otimes
n+1}, \Lambda)$
$$\eqalign {
D^n f = [\tau \circ (Id_A \otimes f)] &* [f^{-1} \circ (m_A \otimes Id_A \otimes...
\otimes Id_A)] \cr
&* [f \circ (Id_A \otimes m_A \otimes Id_A \otimes...\otimes Id_A] \cr
&* ... \cr
&* [f^{\pm 1} \circ (Id_A \otimes...\otimes Id_A \otimes m_A)] * [f^{\mp 1}
\otimes \varepsilon_A] \cr
}$$
where $m_A$ and $\varepsilon_A$ are the product and counit on $A$ and $\tau : A
\otimes \Lambda \fl \Lambda$ defines the $A-$module structure on
$\Lambda.$
{\it Normalized $n$-cochains} :
\noindent
$Reg_+ (A^{\otimes n}, \Lambda) = \{ f \in Reg (A^{\otimes n}, \Lambda) \vert f (a_1
\otimes... \otimes a_n) = \varepsilon_\Lambda (a_1)... \varepsilon_\Lambda (a_2)$
if $a_i \in \hskip.1cm \Crm,$ for at least one $i \}.$
\psaut
As always, we define $Z^n_{s\omega} (A, \Lambda), \hskip.1cm B^n_{s\omega} (A, \Lambda)$
and $H^n_{sw} (A, \Lambda)$ for $n$-cocycles, $n$-coboundaries, and $n$-cohomology
of the Sweedler complex $\{ Reg^n (A, \Lambda) = Reg (A^{\otimes n}, \Lambda)
\hskip.1cm ; \hskip.1cm D^n \}.$
\psaut
We have $Reg^0 (A, \Lambda) = A^{inv}$ and $H^0_{sw} (A, \Lambda) = \{ \lambda \in
\Lambda^{inv} / a \bullet \lambda = \varepsilon (a) \lambda \hskip.5cm \forall a \in A \}$
$$\eqalign {
f \in Z^1_{sw} (A, \Lambda) &\Leftrightarrow D^1 f = [\tau \circ (Id \otimes f)]*
[f^{-1} \circ m_A]*[f \otimes \varepsilon_A] = \eta_\Lambda \circ (\varepsilon_A
\otimes \varepsilon_A) \cr
&\Leftrightarrow f(ab) = \sum_{(a)} \hskip.1cm (a_{(1)} \bullet f(b)) \hskip.1cm f(a_{(2)}),
\hskip.5cm \forall a,b \in A \cr
f \in B^1_{sw} (A, \Lambda) &\Leftrightarrow \exists \lambda \in \Lambda^{inv}
\hbox{ s.t. } f = D^0 \hskip.1cm \lambda \hskip.5cm (\Leftrightarrow f(a) = (a \bullet \lambda)
\lambda \hskip.1cm ) \cr
}$$
Then we have :
\saut
{\bf PROPOSITION 4.} [Sw2]. \it If $B$ is a cleft algebra extension of
$\Lambda$ by $A,$ there exists $\tau \in Z^2_{sw} (A, \Lambda)$ s.t. the product on
$B \simeq \Lambda \otimes A$ (as vector spaces) is
$$(\lambda \sharp_\tau a) (\gamma \sharp_\tau x) = \sum_{(a) (x)} \hskip.1cm \lambda
(a_{(1)} \bullet \gamma) \hskip.1cm \tau (a_{(2)} \otimes x_{(1)}) \hskip.1cm \sharp_\tau
\hskip.1cm a_{(3)} \hskip.1cm x_{(2)}.$$
If $\tau \in B^2_{sw} (A, \Lambda),$ then the product on $B$ is equivalent to the
smash-product : \rm
$$(\lambda \sharp a).(\gamma \sharp x) = \sum_{(a)} \hskip.1cm \lambda (a_{(1)} \bullet \gamma)
\hskip.1cm \sharp \hskip.1cm a_{(2)} x.$$
As wanted $H^2_{sw} (A, \Lambda)$ classifies the extensions. Of course, we can
dualize Sweedler cohomology : this is the cohomology the complex $\{ Reg(A, \Lambda^{\otimes m}),
\hskip.1cm D^m_* \}$ with
$$\eqalign {
D^m_* f = [(Id_\Lambda \otimes f) \circ \varphi]&*[(\Delta_\Lambda \otimes Id_\Lambda \otimes
... \otimes Id_\Lambda) \circ f^{-1}] \cr
&* [(Id_\Lambda \otimes \Delta_\Lambda \otimes Id_\Lambda \otimes ... \otimes
Id_\Lambda) \circ f] \cr
&*... \cr
&* [(Id_\Lambda \otimes ... \otimes Id_\Lambda \otimes \Delta_\Lambda) \circ f^{\pm 1}]*
[f^{\mp 1} \otimes \eta_\Lambda] \cr
}$$
for $\Lambda$ a commutative Hopf algebra and $A$ a cocommutative $\Lambda$-comodule
coalgebra with $\varphi$ the structure map.
We will note, "canonically", $Z^n_{cosw} (A, \Lambda), \hskip.1cm
B^n_{cosw} (A, \Lambda)$ and $H^n_{cosw} (A, \Lambda)$ for the Sweedler complex
in the coalgebra case.
As before, we can define {\it cleft coalgebra tensorial extension of $A$ by $\Lambda$}
and prove that the underlying vector space is always $\Lambda \otimes A$ and that for each of
them, there exists $\theta \in Z^2_{cosw} (A, \Lambda)$ s.t. the coproduct on the
extension is
$$\Delta_\theta (\lambda \sharp^\theta a) = \sum_{(\lambda) (a) (a_{(1)})} (\lambda_{(1)}
\hskip.1cm a^I_{(1)} \hskip.1cm \sharp^\theta \hskip.1cm a_{(2)A}) \otimes (\lambda_{(2)}
\hskip.1cm a^{II}_{(1)} \hskip.1cm a_{(2) \Lambda} \hskip.1cm \sharp^\theta \hskip.1cm a_{(3)})$$
with $\theta (a) = \sum_{(a)} \hskip.1cm a^I \otimes a^{II}.$
The structures of algebra tensorial extension and coalgebra tensorial extension (from
now on we omit "cleft" but it will always be implicit) can be glued together if the
$\Lambda$-comodule Hopf algebra $A$ and the $A$-module Hopf algebra $\Lambda$
verify some conditions called "matched pair" conditions (for details see [Si]).
Then the compatibility condition between $\mu_\sigma$ and $\Delta_\theta$ is expressed
$D^{12} \theta = D^{21}_* \sigma.$
We call these extensions : {\it Hopf algebra tensorial extensions}. They are classified
by the $2$-cohomology of the following bicomplex [Si] :
$$\diagram{
Reg (A, \Lambda) & \flq{D^{11}} & Reg (A^{\otimes 2}, \Lambda) & \flq{D^{21}} &
Reg (A^{\otimes 3}, \Lambda) & \flq{D^{31}}... \cr
\downarrow D^{11}_* && \downarrow D^{21}_* && \downarrow D^{31}_* \cr
Reg (A, \Lambda^{\otimes 2}) & \flq{D^{12}} & Reg (A^{\otimes 2}, \Lambda^{\otimes 2}) &
\flq{D^{22}} & Reg (A^{\otimes 3}, \Lambda^{\otimes 2}) & \flq{D^{32}}... \cr
\downarrow D^{12}_* && \downarrow D^{22}_* && \downarrow D^{32}_* \cr
Reg (A, \Lambda^{\otimes 3}) & \flq{D^{13}} & Reg (A^{\otimes 2}, \Lambda^{\otimes 3}) &
\flq{D^{23}} & Reg (A^{\otimes 3}, \Lambda^{\otimes 3}) & \flq{D^{33}}... \cr
\downarrow D^{13}_* && \downarrow D^{23}_* && \downarrow D^{33}_* \cr
... && ... && ... \cr
}$$
with cochains $C^n (A, \Lambda) = \displaystyle{\bigoplus_{p+q=n+2, \hskip.1cm
p \geq 1, q \geq 1}} \hskip.1cm Reg (A^{\otimes p}, \Lambda^{\otimes q}).$
\saut
{\bf Remark :}
During the colloquium "Enveloping algebras and quantum groups" in Paris, July 92,
M. Rosso exposed the same kind of extension theory about {\it quasi-Hopf algebras}
(see [D2] for definition) and claimed that it is classified by the same bicomplex connected with the complex
$\{ Reg (\hskip.1cm \Crm, \Lambda^{\otimes n}) \simeq \Lambda^{\otimes n}, \hskip.1cm n \geq 1 \}$
as first column, the cochains becoming $\displaystyle{\bigoplus_{p+q=n+1, \hskip.1cm
p \geq 0, q \geq 1}} \hskip.1cm Reg (A^{\otimes p}, \Lambda^{\otimes q}).$
\psaut
What I shall expose now should be part of the proof of this result, but since
I never saw it anywhere and the point of view and the concepts here are a bit
different, I shall do it in details. The idea is to adapt the Hopf algebra case
to the bialgebra (not necessarily coassociative) one, exactly in the same way that
I did in [Bo].
\saut
{\bf DEFINITION 4.} \it Let $A$ be a cocommutative Hopf algebra, $\Lambda$ a commutative
one. $(A, \Lambda)$ form a matched pair {\rm [Si]}. We call {\bf bialgebra tensorial extension
of $\Lambda$ and} $A$ a cleft algebra extension of $\Lambda$ by $A$ (defined
by $\sigma \in Z^2_{sw} (A, \Lambda)$) with an extension coproduct defined by a
compatible $\theta \in Z^2_{cosw} (A, \Lambda).$
We say that two extensions $(B, \mu_\sigma, \Delta_\theta)$ and $(B', \mu_{\sigma'},
\Delta_{\theta'})$ are equivalent if $(B, \mu_\sigma) \buildrel\hbox{alg.} \over \simeq
(B', \mu_{\sigma'})$ and $\exists \hskip.1cm P_\Lambda \in (\Lambda \otimes \Lambda)^{inv}$
s.t. $\Delta_\theta = \tilde{P}_\Lambda \hskip.1cm \Delta_{\theta'} \hskip.1cm \tilde{P}^{-1}_\Lambda$
where $\tilde{P}_\Lambda = (23) \hskip.1cm (P_\Lambda \otimes 1_{A^{\otimes 2}}).$ \rm
\saut
{\bf THEOREM 2.} \it The $2$-cohomology of the complex
$$\diagram{
\Lambda^{inv} \simeq Reg \hskip.1cm (\Crm, \Lambda) & \fl & Reg (A, \Lambda) & \fl &
Reg (A^{\otimes 2}, \Lambda) & \fl & Reg (A^{\otimes 3}, \Lambda) & \fl ... \cr
\downarrow && \downarrow && \downarrow && \downarrow \cr
(\Lambda \otimes \Lambda)^{inv} \simeq Reg \hskip.1cm (\Crm, \Lambda^{\otimes 2}) & \fl &
Reg (A, \Lambda^{\otimes 2}) & \fl & Reg (A^{\otimes 2}, \Lambda^{\otimes 2}) &
\fl & Reg (A^{\otimes 3}, \Lambda^{\otimes 2}) & \fl ... \cr
}$$
with "Sweedler" bialgebra cochains $C^{n}_{bis \omega} (A, \Lambda) = \displaystyle{\bigoplus_{p+q=n+1,
\hskip.1cm p \geq 0, q=1,2}} Reg (A^{\otimes p}, \Lambda^{\otimes q})$
classifies the bialgebra tensorial extensions for the trivial $\Lambda$-comodule
structure on $A$ (ie : $a \mapsto 1_\Lambda \otimes a).$ \rm
\vfill\eject
We note $Z^n_{bis \omega} (A, \Lambda), \hskip.1cm B^n_{bis \omega} (A, \Lambda)$
and $H^n_{bis \omega} (A, \Lambda)$ the $n$-cocycles, $n$-coboundaries and $n$-cohomology
of the above complex.
\saut
{\it Proof :}
The "matched pair" property is reduced, for a trivial $\varphi,$ to the fact that
$A$ must be a $\Lambda$-comodule Hopf algebra, which follows from [Ma1].
For the product see [Sw2]. For the compatibility between product and coproduct
in the case of "an abelian matched pair" see [Si].
We consider the extension defined by $(\sigma, \theta) \in Z^2_{bisw} (A, \Lambda).$
We have $(\lambda \sharp^\theta_\sigma a) {\bf .}_\sigma (\gamma
\sharp^\theta_\sigma b) = \displaystyle{\sum_{(a)}} \lambda (a_{(1)} \bullet \gamma) \hskip.1cm
\sigma (a_{(2)} \otimes b_{(1)}) \hskip.1cm \sharp^\theta_\sigma \hskip.1cm
a_{(3)} b_{(2)}$
$\hskip1.5cm \Delta_\theta (\lambda \sharp^\theta_\sigma a) = \displaystyle{\sum_{
(\lambda)(a)(a_{(1)})}} (\lambda_{(1)} \hskip.1cm a^I_{(1)} \hskip.1cm \sharp^\theta_\sigma
\hskip.1cm a_{(2)}) \otimes (\lambda_{(2)} \hskip.1cm a^{II}_{(1)} \hskip.1cm
\sharp^\theta_\sigma \hskip.1cm a_{(3)})$
We have to show : $\Delta_\omega = \tilde{P}_\Lambda \hskip.1cm {\bf .}_\sigma \hskip.1cm
\Delta_\theta \hskip.1cm {\bf .}_\sigma \hskip.1cm \tilde{P}^{-1}_\Lambda$ \hskip.2cm iff
\hskip.2cm $\theta^{-1} * \omega$ is a coboundary
with $\tilde{P}_\Lambda = \displaystyle{\sum_{(P_\Lambda)}}
(p^{(1)} \hskip.1cm \sharp^\theta_\sigma \hskip.1cm 1_A) \otimes (p^{(2)} \hskip.1cm
\sharp^\theta_\sigma \hskip.1cm 1_A).$
To simplify notations, we shall write now $\sharp$ for $\sharp^\theta_\sigma,$ ${\bf .}$ for
${\bf .}_\sigma$ and omit the "$\sum$".
$$\eqalign {
\tilde{P}_\Lambda \hskip.1cm {\bf .} \hskip.1cm \Delta_\theta (\lambda \sharp a) &=
[(p^{(1)} \sharp 1_A) \otimes (p^{(2)} \sharp 1_A)] {\bf .} [(\lambda_{(1)} \hskip.1cm
a^I_{(1)} \hskip.1cm \sharp \hskip.1cm a_{(2)}) \otimes (\lambda_{(2)} \hskip.1cm
a^{II}_{(1)} \hskip.1cm \sharp \hskip.1cm a_{(3)})] \cr
&= [(p^{(1)} \sharp 1_A) {\bf .} (\lambda_{(1)} \hskip.1cm a^I_{(1)} \hskip.1cm \sharp \hskip.1cm a_{(2)})]
\otimes [(p^{(2)} \sharp 1_A) {\bf .} (\lambda_{(2)} \hskip.1cm a^{II}_{(1)} \hskip.1cm \sharp \hskip.1cm
a_{(3)})] \cr
&= [p^{(1)} (1_A \bullet (\lambda, a^I_{(1)})) \hskip.1cm \tau (1_A \otimes a_{(2)(1)})
\hskip.1cm \sharp \hskip.1cm a_{(2)(2)}] \cr
&\hskip1cm \otimes [p^{(2)} (1_A \bullet (\lambda_{(2)} \hskip.1cm a^{II}_{(1)}))
\hskip.1cm \tau (1_A \otimes a_{(3)(1)}) \hskip.1cm \sharp \hskip.1cm a_{(3)(2)}] \cr
}$$
We use {\it normalized} cochains then $\tau (a \otimes 1_A) = \tau (1_A \otimes a) =
\varepsilon (a), \hskip.1cm \forall a \in A.$
\psaut
So : $\tilde{P}_\Lambda \hskip.1cm {\bf .} \hskip.1cm \Delta_\theta (\lambda
\sharp a) = (p^{(1)} \hskip.1cm \lambda_{(1)} \hskip.1cm a^I_{(1)} \hskip.1cm \sharp \hskip.1cm a_{(2)})
\otimes (p^{(2)} \hskip.1cm \lambda_{(2)} \hskip.1cm a^{II}_{(1)} \hskip.1cm \sharp \hskip.1cm a_{(3)})$
applying the classical properties of counit.
We write $\tilde{P}^{-1}_\Lambda = \displaystyle{\sum_{(P^{-1}_\Lambda)}} (\bar{p}^{(1)}
\sharp 1_A) \otimes (\bar{p}^{(2)} \sharp 1_A)$
\psaut
$\tilde{P}_\Lambda \hskip.1cm {\bf .} \hskip.1cm \Delta_\theta (\lambda \sharp^\theta_\sigma a)
\hskip.1cm {\bf .} \hskip.1cm \tilde{P}^{-1}_\Lambda$
$$\eqalign {
&= [p^{(1)} \hskip.1cm \lambda_{(1)} \hskip.1cm a^I_{(1)} (a_{(2)(1)} \bullet
\bar{p}^{(1)}) \hskip.1cm \tau(a_{(2)(2)} \otimes 1_A) \hskip.1cm \sharp \hskip.1cm a_{(2)(3)}] \cr
&\hskip5cm \otimes
[p^{(2)} \hskip.1cm \lambda_{(2)} \hskip.1cm a^{II}_{(1)} \hskip.1cm (a_{(3)(1)}
\bullet \bar{p}^{(2)}) \hskip.1cm \tau(a_{(3)(2)} \otimes 1_A) \hskip.1cm \sharp \hskip.1cm a_{(3)(3)}] \cr
&= [p^{(1)} \hskip.1cm \lambda_{(1)} \hskip.1cm a^I_{(1)} (a_{(2)(1)} \bullet
\bar{p}^{(1)}) \hskip.1cm \sharp \hskip.1cm a_{(2)(2)}] \otimes [p^{(2)} \hskip.1cm \lambda_{(2)}
\hskip.1cm a^{II}_{(1)} \hskip.1cm (a_{(3)(1)} \bullet \bar{p}^{(2)}) \hskip.1cm \sharp \hskip.1cm a_{(3)(2)}] \cr
&= [p^{(1)} \hskip.1cm \lambda_{(1)} \hskip.1cm a^I_{(1)} (a_{(2)(1)} \bullet
\bar{p}^{(1)}) \hskip.1cm \sharp \hskip.1cm a_{(3)}] \otimes [p^{(2)} \hskip.1cm \lambda_{(2)}
\hskip.1cm a^{I}_{(1)} \hskip.1cm (a_{(1)} \bullet \bar{p}^{(2)}) \hskip.1cm \sharp \hskip.1cm a_{(3)}] \cr
}$$
from coassociativity we have
\noindent
$\displaystyle{\sum_{(a)}} a_{(1)} \otimes a_{(2)} \otimes a_{(3)} \otimes a_{(4)} \otimes a_{(5)} =
\displaystyle{\sum_{(a)}} a_{(1)(1)} \otimes a_{(1)(2)(1)} \otimes a_{(1)(2)(2)} \otimes a_{(2)}
\otimes a_{(3)}$
$\hskip4.5cm = \displaystyle{\sum_{(a)}} a_{(1)(1)} \otimes a_{(1)(2)(1)}
\otimes a_{(2)} \otimes a_{(1)(2)(2)} \otimes a_{(3)}$ (by cocommutativity), so
$$\tilde{P}_\Lambda {\bf .}_\sigma \hskip.1cm \Delta_\theta (\lambda
\sharp^\theta_\sigma a) {\bf .}_\sigma \tilde{P}^{-1}_\Lambda = \displaystyle{\sum_{(a)(a_{(1)}(\lambda)}}
[p^{(1)} \lambda_{(1)} a^I_{(1)(2)} (a_{(1)(2)(1)} \bullet
\bar{p}^{(1)}) \hskip.1cm \sharp \hskip.1cm a_{(2)}] \otimes [p^{(2)} \lambda_{(2)}
a^I_{(1)(1)} (a_{(1)(2)(2)} \bullet \bar{p}^{(2)}) \hskip.1cm \sharp \hskip.1cm a_{(3)}]$$
\noindent
if we write $\Delta_\omega (\lambda \sharp^\omega_\sigma a) = \displaystyle{\sum_{(a)(a_{(1)})(\lambda)}}
[\lambda_{(1)} \hskip.1cm a^{\tilde{I}}_{(1)} \sharp a_{(2)}] \otimes [\lambda_{(2)}
\hskip.1cm a^{\tilde{II}}_{(1)} \sharp a_{(3)}]$ \hskip.5cm where
$\omega (a) = \displaystyle{\sum_{(a)}} a^{\tilde{I}} \otimes a^{\tilde{II}}$
\noindent
we get $\Delta_\omega (\lambda \sharp^\omega_\sigma a) = \tilde{P}_\Lambda \hskip.1cm
{\bf .}_\sigma \hskip.1cm \Delta_\theta (\lambda \sharp^\theta_\sigma a) \hskip.1cm
{\bf .}_\sigma \hskip.1cm \tilde{P}^{-1}_\Lambda$ \hskip.2cm if and only if \hskip.2cm $\theta^{-1} * \omega =
D^{02} \hskip.1cm (P^{-1}_\Lambda).$
Now, as $\Delta_\omega = (\phi \otimes \phi) \circ \Delta_\theta \circ \phi^{-1}
\hskip.5cm \Leftrightarrow \hskip.5cm \theta^{-1} * \omega = D^{11}_* \hskip.1cm
\xi_\phi$ [Si] for $\phi : B_\theta \fl B_\omega$ isomorphism and $\xi_\phi \in
Reg (A, \Lambda)$ attached to $\phi,$ we have the result.
\square
\vfill\eject
{\bf Application to $D({\cal D} (G))$} :
Let $A$ be either ${\cal A} (G)$ or ${\cal D} (G),$ $\Lambda$ be ${\cal H} (G)$ or
${\cal C}^\infty (G).$ Theorem 2 could have been written with continuous
cochains. Then the continuous bialgebra tensorial extensions of $\Lambda (= A^*)$
and $A,$ via
$$\eqalign {
\varphi \hskip.1cm : \hskip.1cm &\Lambda \fl \Lambda \bar{\otimes} A \hskip.5cm
\hbox{ and } \hskip1cm \tau : A \bar{\otimes} \Lambda \fl \Lambda \hskip3cm , \cr
&\lambda \mapsto \lambda \otimes 1_A \hskip2.6cm a \otimes \lambda \mapsto \lambda
\circ ad (a) \cr
}$$
are classified by the continuous version of the "$bis \omega$" $2$-cohomology
{\it with respect to the injective topological tensor product.} $D(A)$ is the trivial
extension of $\Lambda$ and $A$ in this case.
(For details about topologization of algebraic cohomological theory see [Ta]).
\saut
{\bf THEOREM 3.} \it Let $G$ a simply connected, connected, compact Lie group. Then
the "biSweedler" continuous $2$-cohomology vanishes :
$$\hskip2.5cm H^2_{bis \omega, c} \hskip.1cm ({\cal D} (G), {\cal C}^\infty (G)) = \{ 0 \}$$
\rm
If we extend the name "double" to all the extensions for which the "classical"
quantum double is the zero class, then we show that $D({\cal D} (G))$
is the only bialgebra and thus the only quasi-Hopf double structure on ${\cal D} (G).$
It is interesting (and is one of the motivations of this work) to see
in [DPR] that for finite groups they find quasi-Hopf double structures apparently
not equivalent to the classical one. Their notion of extension is however apparently
somewhat different
\saut
\hskip1cm {\it Proof :}
\psaut
{\bf LEMMA 2.} \it We have the following long exact sequence
$$... \fl H^{n-1}_{sw, c} (A, \Lambda^{\otimes 2}) \fl H^n_{bis \omega, c} (A, \Lambda)
\fl H^n_{sw, c} (A, \Lambda) \fl H^n_{sw, c} (A, \Lambda^{\otimes 2}) \fl ...$$
\rm
\psaut
The proof is exactly the same as for the analoguous result in [Bo], and we
have the straightforward corollary.
\saut
{\bf COROLLARY 1.} \it If $H^1_{sw, c} (A, \Lambda^{\otimes 2}) = \{ 0 \} = H^2_{sw, c}
(A, \Lambda)$ \hskip.3cm then \hskip.3cm $H^2_{bis \omega, c} (A, \Lambda) = \{ 0 \}.$ \rm
\saut
{\bf LEMMA 3.} \it $H^n_{sw, c} ({\cal D} (G), \Lambda) \simeq H^n_{gr, c} (G,
\Lambda^{inv}),$ the continuous group cohomology. \rm
\saut
{\bf Proof :} Sweedler shows in [Sw2] that $H^n_{sw} (\hskip.1cm \Crm G, \Lambda) =
H^n_{gr} (G, \Lambda^{inv}).$ We can adapt the proof to have
$$H^n_{sw,c} (\hskip.1cm \Crm G, \Lambda) = H^n_{gr,c} (G, \Lambda^{inv})
\hskip1cm (1)$$
Indeed, if $f \in Reg_{c} (\hskip.1cm \Crm G^{\otimes n}, \Lambda)$ is a continuous
invertible element, its restriction to $G^n$ takes values in $\Lambda^{inv}$ and
is continuous.
$f^{-1}$ (on $G$) maps $g$ into $[f(g)]^{-1}$ and is continuous.
Conversely if $f$ is a continuous map : $G \fl \Lambda^{inv},$ one can
extend it by linearity to an element of $Reg_{c} (\hskip.1cm \Crm G^{\otimes n},
\Lambda).$ Since $G^n$ is a basis of $\hskip.1cm \Crm G^{\otimes n}$ we obtain
an isomorphism of complexes.
\psaut
Now, as $f^{-1}$ (for $f \in Reg_{c} (\hskip.1cm \Crm G^{\otimes n}, \Lambda)$)
is continous, we can extend $f$ and $f^{-1}$ to $\hskip.1cm \overline{\Crm G^{\otimes n}} =
{\cal D} (G)^{\otimes n}$ and they stay inverse to each other for $*.$ Thus
$$Reg_{c} (\hskip.1cm \Crm G^{\otimes n}, \Lambda) \simeq Reg_{c} ({\cal D}^{\otimes n},
\Lambda) \hskip1cm (2)$$
and this is an isomorphism of complexes. By (1) and (2) we have the result.
\square
Finally, we have to calculate $H^2_{gr, c} (G, {\cal C}^\infty (G)^{inv})$ and
$H^1_{gr, c} (G, [{\cal C}^\infty (G)^{\otimes 2}]^{inv}).$
We have the following exact sequence :
$$0 \fl \Zrm \fl {\cal C}^\infty (G) \flq{exp} {\cal C}^\infty (G)^{inv} \fl 0$$
where $exp$ is the exponential.
Then by the classical long exact sequence of cohomology and the fact that
\noindent
$H^2_{gr,c} (G, \Zrm) = \{ 0 \} = H^2_{gr,c} (G, {\cal C}^\infty (G))$ for $G$
compact, we have $H^2_{gr,c} (G, {\cal C}^\infty (G)^{inv}) = \{ 0 \}.$
In the same way, knowing that ${\cal C}^\infty (G) \hat{\otimes} {\cal C}^\infty
(G) = {\cal C}^\infty (G) \bar{\otimes} {\cal C}^\infty (G) = {\cal C}^\infty (G \times G)$
we have
\noindent
$0 \fl \Zrm \times \Zrm \fl {\cal C}^\infty (G \times G) \fl {\cal C}^\infty
(G \times G)^{inv} \fl 0$ and therefore,
$H^1_{gr,c} (G, [{\cal C}^\infty (G) \bar{\otimes} {\cal C}^\infty (G)]^{inv}) =
\{ 0 \}.$
Theorem 3 then follows from Lemma 3 and Corollary 1.
\square
\saut
4. {\bf CONCLUSION.}
\psaut
The paper can be completed in several directions. First it is interesting to look more in details at the explicit structure
of the dual of the double (namely at its coproduct). Using
${\cal D} (G) \hat{\otimes} {\cal C}^\infty (G) \simeq {\cal C}^\infty (G, {\cal D}
(G)) \simeq {\cal D} (G, {\cal C}^\infty (G)),$ this should be achieved by an
adaptation of the formulas given by Majid
[Ma1] for finite groups.
\psaut
Secondly, it should be possible to take a general coaction $\psi$ and not just the
trivial one in Theorem 2.
\psaut
Third, Theorem 3 should also be true for ${\cal A} (G)$. For this one needs to
prove
$$H^1_{gr,c} (G, [{\cal H} (G) \bar{\otimes} {\cal H} (G)]^{inv}) = \{ 0 \} =
H^2_{gr,c} (G, {\cal H} (G)^{inv}).$$
Finally this extension theory should be developed in the quantum groups domain.
Indeed paragraph 3 is just a first step, an example of the techniques and the concepts
to use.
But to look at extensions of a Hopf algebra $A$ and an arbitrary $A$-module algebra $M$ we must
have a theory which works with a non commutative $M$ and a non-cocommutative $A$
(or to begin with a, may be, easier problem, extensions of deformations of a cocommutative
$A$ and a $M$ ; this will be enough for the quantum group cases of ${\cal A}_t (G)$ or
${\cal D}_t (G)$). This seems not straightforward
but there are some interesting examples which might be useful : Majid gives some
in [Ma2] and Truini and Varadarajan try in [TV], to define a "quantum extension"
of two algebras of coefficients of (compact) semi-simple groups. This is very interesting
because extending two semi-simple Lie algebras, one by the other, is always trivial
in the category of Lie algebras. But is it the same in the category of Hopf algebras,
or bialgebras (non necessarily coassociative) ? And for two "quantum groups" ?
The [TV] example is not trivial in the sense that it is not
a smash-product. Moreover, this extension seems not to be one in the sense of
Molnar [Mo], who gives a very natural and interesting concept of extensions of
Hopf algebras since it is valid in every case, even for non (co)-commutative ones.
But he does not give a cohomological classification for these. It is the difficult
point because we shall certainly have to go to the domain of non commutative cohomology
theory.
\gsaut
\gsaut
{\bf Acknowledgments :} I want to thank G. PINCZON for the very interesting discussions
we had on the subject of this paper, W. SCHMID for arguments on group
cohomology and D. STERNHEIMER for a careful reading and many corrections. And,
of course, I thank M. FLATO for many reasons, but here, especially
for his disponibility, even in holidays.
\vfill\eject
\centerline {\bf BIBLIOGRAPHY}
\gsaut
[A] E. ABE : "Hopf algebras". Cambridge University Press (1980).
\psaut
[Bo]JP. BONNEAU : "Cohomology and associated deformations for not necessarily
coassociative bialgebras". {\it Lett. Math. Phys.} {\bf 26} (1992), 277-283.
\psaut
[By] N.P. BYOTT : "Cleft extensions of Hopf algebras". {\it J. of alg.} {\bf 157}
(1993), 405-429.
\psaut
[BFGP] P. BONNEAU, M. FLATO, M. GERSTENHABER, G. PINCZON : "Hidden group structure
of quantum groups : strong duality, rigidity and preferred deformations". To appear
in {\it Comm. Math. Phys.}
\psaut
[DPR] R. DIJKGRAAF, V. PASQUIER, P. ROCHE : "Quasi-Hopf algebras, group cohomology
and orbifold models". {\it Nucl. Phys. B (proc. suppl.)} {\bf 18B} (1990), 60-72.
\psaut
[D1] V.G. DRINFELD : "Quantum groups" in {\it Proc.} ICM 1986, AMS (ed. A.M. Gleason)
Providence (1987), {\bf vol. 1}, 798-820.
\psaut
[D2] V.G. DRINFELD : "Quasi-Hopf algebras". {\it Leningrad Math. J.} {\bf 1} (1990),
1419-1457.
\psaut
[Gr] A. GROTHENDIECK : "Produits tensoriels topologiques et espaces nucl\'eaires".
{\it Mem. Amer. Math. Soc.} {\bf 16} (1955).
\psaut
[Ma1] S. MAJID : "Physics for algebraists : non commutative and non cocommutative
Hopf algebras by a bicross-product construction". {\it J. of alg.} {\bf 130} (1990),
17-64.
\psaut
[Ma2] S. MAJID : "More examples of bicross-product and double cross product Hopf
algebras". {\it Isra\"el J. Math.} {\bf 72} (1990), 133-148.
\psaut
[Mo] R. MOLNAR : "Semi-direct products of Hopf algebras". {\it J. of alg.} {\bf 47}
(1977), 29-51.
\psaut
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\psaut
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\end
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