Classification of quantum Lie groups Obviously, I tumbled over
Classification of (compact) Lie groups
- are the quantum Lie groups (or make that: algebras) easier to classify?
Or does the whole q-deformation thingie make it even more complicated?
(The classification scheme from the linked post is already to technical
for me, although I understand the basic idea. If an exhaustive list of
quantum Lie groups exists, I would already be happy with that for now.)
 A: What Scott's comment is getting at is that you need to have an abstract definition of "quantum Lie group" if you want to have a classification result.  As the theory of quantized enveloping algebras and quantized coordinate algebras is currently formulated, this is not really how it works.
Rather, you start with a (finite-dimensional, semisimple, complex) Lie algebra $\mathfrak{g}$, and from its Cartan data you write down the Hopf algebra $U_q(\mathfrak{g})$.  I assume here that $q$ is not a root of unity.  This is functorial in the somewhat limited sense that if the Dynkin diagram for $\mathfrak{g}_1$ includes into the Dynkin diagram for $\mathfrak{g}_2$ then there is a corresponding map of Hopf algebras $U_q(\mathfrak{g}_1) \to U_q(\mathfrak{g}_2)$ (so diagram automorphisms give rise to Hopf algebra automorphisms).  However, this is different from defining a class of Hopf algebras satisfying some properties and then showing that they must all be of the form $U_q(\mathfrak{g})$ for some $\mathfrak{g}$.
If you take $U_q(\mathfrak{g})$ as given, however, then you can construct quantized coordinate algebras corresponding to all of the connected groups $G$ which have Lie algebra $\mathfrak{g}$.
First, let $G$ be the connected, simply connected Lie group with Lie algebra $\mathfrak{g}$.  Now $U_q(\mathfrak{g})$ has (morally) the same finite-dimensional representation theory as $U(\mathfrak{g})$ does.  If we restrict to Type 1 representations, i.e. those in which the Cartan generators $K_i$ act as powers of $q$ on weight vectors (as opposed to acting as negative powers of $q$, which is well-defined since $q$ isn't a root of unity), then the finite-dimensional representations are in 1-1 correspondence.
For each of these Type 1 representations, we can define matrix coefficients.  Say $V$ is a finite-dimensional representation of $U_q(\mathfrak{g})$, and let $v \in V$ and $\phi \in V^*$.  Then the associated matrix coefficient is $ c^V_{\phi,v} : U_q(\mathfrak{g}) \to \mathbb{C}, $ defined by
$$ c^V_{\phi,v}(X) = \phi(X\cdot v).  $$
These matrix coefficients actually live in the finite dual $U_q(\mathfrak{g})^\circ$, and you define $\mathcal{O}_q(G)$ to be the (Hopf) subalgebra of $U_q(\mathfrak{g})^\circ$ generated by all of the matrix coefficients of all of the finite-dimensional representations.  (Of course it suffices just to take irreducibles.)  This gives you the quantized analogue of the Peter-Weyl decomposition of the algebra of functions on $G$.
Now there are other connected, but not simply connected, groups which have the same Lie algebra.  Let's pick one and call it $G'$.  Since $G'$ is a quotient of $G$, then (classically) the coordinate algebra $\mathcal{O}(G')$ is a subalgebra of $\mathcal{O}(G)$.  In terms of the Peter-Weyl decomposition, $\mathcal{O}(G')$ is generated by matrix coefficients of all of the finite-dimensional representations of $\mathfrak{g}$ which integrate to give a representation of $G'$.  Then you can define $\mathcal{O}_q(G')$ to be the subalgebra of $\mathcal{O}_q(G)$ generated by only those matrix coefficients.
Anyway, I don't know if that's what you were looking for exactly.  Hope it's helpful.  If you think of a more precise question to ask then you may find more answers forthcoming.
