To fix notations : let G be simply connected simple compact group, and $U_q(\mathcal{G})$ the DrinfeldJimbo universal algebra quantization of its complexified algebra defined as usual, with q not root of unity. Recall that it's a *algebra, and I denote $G_q$ the unital $C^*$algebra dual to it as usual (thus completion of matrix coefficient of representations...).
My question is very naive and loose, so all suggestions are welcome. Is there a general notion of fundamental group (e.g. using like étale), such that $G_q$ has trivial fundamental group?

$\begingroup$ Presumably you mean that the "noncommutative space" associated to the algebra $G_q$ should have trivial fundamental group? (Algebras are not spaces, to repeat one of my usual bits of nitpicking.) $\endgroup$– Yemon ChoiJan 1 '12 at 19:48

$\begingroup$ Yes either the $C*$algebra or the dense algebra of matrix elements, your choice ;) $\endgroup$– AminJan 1 '12 at 20:09
This is a negative answer to your question, or at least a partially negative one. I don't know if you've thought about things in this way, but there is a naive way to formulate the idea of the fundamental group of a space in terms of its $C^*$algebra of continuous functions. The problem is that, at least according to my quick calculations, the noncommutative space associated to the quantized function algebra $\mathcal{O}_q(SU_2)$ does not have trivial fundamental group. This may mean that this notion of fundamental group doesn't make much sense for noncommutative spaces, or that quantum phenomena arise which just make these quantum groups not simply connected, or something else. My suspicion is that the first is true, my reasoning being that if there was a good notion of fundamental group for noncommutative spaces, we would have heard of it already. Perhaps somebody can chime in on that?
Anyway, let me tell you what I've been thinking about. The fundamental group of a space $X$ is defined as homotopy classes of based continuous maps $S^1 \to X$. Continuous maps $S^1 \to X$ are in bijection with unital $*$homomorphisms $C(X) \to C(S^1)$, at least for $X$ compact. So for a compact noncommutative space with corresponding unital $C^*$algebra $A$, loops should be defined as morphisms (of unital $C^*$algebras) $A \to C(S^1)$.
I believe that the correct notion of homotopy of morphisms is the following: two morphisms $\Phi_0, \Phi_1 : A \to C(S^1)$ are homotopic if there is a morphism $\Psi : A \to C([0,1]) \otimes C(S^1)$ such that $\mathrm{ev}_0 \circ \Psi = \Phi_0$ and $\mathrm{ev}_1 \circ \Psi = \Phi_1$. This is spelled out in Chapter 1 of Elements of Noncommutative Geometry, by Varilly, GraciaBondia, and Figuera.
To say that a loop takes basepoint to basepoint is a little trickier for noncommutative spaces. However, for the quantum groups this is ok because there is still one classical point, namely the identity element, which corresponds to the counit $\varepsilon : \mathcal{O}_q(G) \to \mathbb{C}$. For a loop/morphism $\Phi : \mathcal{O}_q(G) \to C(S^1)$, then, we can say that $\Phi$ preserves basepoints if $\Phi(x)(1) = \varepsilon(x)$ for all $x \in \mathcal{O}_q(G)$. Or in more fancy terms we could say that $\mathrm{ev}_1 \circ \Phi = \varepsilon$.
Finally, we have to define the correct analogue of the constant loop. Again this may be difficult in general, but for the quantum groups we have the counit, so the constant loop is the morphism $x \mapsto \varepsilon(x)1$ of $\mathcal{O}_q(G)$ to $C(S^1)$.
So this gives us most of the elements to define the fundamental group of a noncommutative space with associated $C^*$algebra $A$: we take based morphisms $A \to C(S^1)$ modulo homotopy, and say that a loop is trivial if it is homotopic (through based morphisms) to the constant loop. I haven't said how you're supposed to compose loops, but this at least defines the fundamental group as a set.
Now the bad news: for $\mathcal{O}_q(SU_2)$, this doesn't give the right answer. For the sake of completeness, $\mathcal{O}_q(SU_2)$ (for $0 < q < 1$, say) is the unital $C^*$algebra generated by elements $a,b$ with relations $$ a^*a + b^*b =1, \quad aa^* + q^2 bb^* = 1, $$
$$ bb^* = b^*b, \quad ab = qba, \quad ab^* = qb^*a. $$ The first two relations show that if $\Phi : \mathcal{O}_q(SU_2) \to A$ is a morphism to a commutative $C^*$algebra $A$, then $(1q^2)\Phi(b^*b) = 0$, so $\Phi(b) = 0$. (Thus we really need $q \neq \pm 1$ for this to work.)
Now $C(S^1)$ is the universal $C^*$algebra generated by a single unitary element $u$. This is a quantum subgroup of $\mathcal{O}_q(SU_2)$ via the map $\pi : \mathcal{O}_q(SU_2) \to C(S^1)$ defined by $$ \pi(a) = u, \quad \pi(b) = 0. $$ Combining this with the preceding statement, any morphism $\Phi : \mathcal{O}_q(SU_2) \to C(S^1)$ must factor through the quotient map $\pi$, so homotopy classes of morphisms $\mathcal{O}_q(SU_2) \to C(S^1)$ are in bijection with homotopy classes of morphisms $C(S^1) \to C(S^1)$. These clases correspond to homotopy classes of maps $S^1 \to S^1$, so this means that the fundamental group is $\mathbb{Z}$, rather than the trivial group.
Again, my guess is that this notion of fundamental group is just too rigid to make sense for all noncommutative spaces. One needs at least one classical point in order for this to make sense, which excludes any simple algebras, such as noncommutative tori. The question is interesting, though, and I would be glad to know if anybody else has thought about other notions of a fundamental group for noncommutative spaces.

$\begingroup$ Hello, thanks for your thoughts, but you should know that already in abelian algebraic geometry, using $S^1$ is proscribed, this is the origin of étale theory, so I wasn't even thinking of such downtoearth approach, and you showed that it isn't a good idea. Also, you should know that for $C*$algebras, people don't really use $S^1$ to study coverings, but rather use extension theory, and that's what seem to be missing for quantum groups, as far as I know :). $\endgroup$– AminJan 3 '12 at 21:28

$\begingroup$ Yes, I thought that this approach was too simpleminded. I guess you are saying that rather than define the fundamental group directly, one can take the point of view of coverings, and then the fundamental group is something like a limit of groups of transformations that preserve the fibers of coverings? Do you know a good reference where extension theory for $C^*$algebras is discussed from the point of view of coverings? $\endgroup$– MTSJan 3 '12 at 22:13

1$\begingroup$ Yes that's exactly the kind of things I had in mind. For extension of abelian $C^*$algebras, I only know of the paper of "On covering spaces of $C^*$algebras and Galois extensions",Childs, but to quote it, most of the ideas were known to Grothendieck. $\endgroup$– AminJan 3 '12 at 22:33