This is a generic question, a good answer to it may be a reference to a corresponding paper\textbook, but any useful comments would be okay too.


Let $\mathfrak{g}$ be a (simple) Lie algebra and $U\_q(\mathfrak{g})$ be its q-deformation of its universal enveloping algebra. For example, for $\mathfrak{g} = \mathfrak{su}(2)$ with positive $\mathfrak{e}$, negative $\mathfrak{f}$ roots and Cartan element $\mathfrak{h}$ the commutation relation read


$[\mathfrak{e},\mathfrak{f}] =\mathfrak{h},\quad [\mathfrak{h},\mathfrak{e}] = 2\mathfrak{e},\quad [\mathfrak{h},\mathfrak{e}] = -2\mathfrak{f}.$ 


After q-deformation the first relationship above turns into


$[\mathfrak{e},\mathfrak{f}] =[\mathfrak{h}]\_q :=\frac{q^\mathfrak{h}-q^{-\mathfrak{h}}}{q-q^{-1}}$,


where $q \in\mathbb{C}^\ast$.


Let us now consider Lie group $G$ such that $\mathfrak{g}$ is its Lie algebra. Thinking in terms of differential geometry, Lie group can be seen as a smooth (real) manifold. Let some coordinate patch be parametrized by local coordinates $x\_1,\dots x\_n$. Here we can implement a noncommutative deformation of the coordinates. There are two examples I'm aware of: $xy-yx=\hbar$ and $xy = t yx$ for some nonzero $t$.


My goal is to understand relationship (if any) between the above mentioned deformations: q-deformation of the Lie algebra and noncommutative deformation of the coordinates on the corresponding Lie group. Returning back to $\mathfrak{su}(2)$, we have $SU(2)=S^3$ as its Lie group. The question is, does the noncommutative $S^3$ have anything to do with $U\_q(\mathfrak{su}(2))$?



Relationship between "different" quantum deformations
There is certainly a way to quantize the algebra of functions on a Lie group in a way that is compatible with the $q$-deformation of the universal enveloping algebra of its Lie algebra. The standard way to do it is this: let $G$ be a simple, connected, simply connected complex Lie group with Lie algebra $\mathfrak{g}$. Construct $U\_q(\mathfrak{g})$ as you have done in your example, using the Cartan data of $\mathfrak{g}$.


Then it is a theorem that, up to some characters acting on the Cartan part of $U\_q(\mathfrak{g})$ (i.e. on the $K\_i$'s) the finite-dimensional representation theory of $U\_q(\mathfrak{g})$ is the same as that of $\mathfrak{g}$. That is, the representations have the same dimensions, the same weight structures, the same decompositions of tensor products, etc.


For any finite-dimensional representation $V$ of $U\_q(\mathfrak{g})$, one can define its matrix coefficients: for $v \in V$ and $f \in V^\*$, the matrix coefficient is the linear functional
$$ c^V\_{f,v} : U\_q(\mathfrak{g}) \to \mathbb{C}$$
given by
$$ c^V\_{f,v}(a) = f (a\cdot v).$$


Denote by $\mathcal{O}\_q(G)$ the span of all of the $c^V\_{f,v}$ inside the dual of $U\_q(\mathfrak{g})$. With the algebraic operations defined by duality from those in $U\_q(\mathfrak{g})$, this is a Hopf subalgebra of the finite dual $U\_q(\mathfrak{g})^{\circ}$ which separates points of $U\_q(\mathfrak{g})$. This is the analogue of the algebra of polynomial functions on $G$. For some values of $q$, you can introduce a $\*$-structure and a norm and take a completion to get the analogue of the algebra of continuous functions.


Basically this is just mimicking the Peter-Weyl decomposition of the algebra of functions on a compact group. You can do lots of stuff with this, for instance look at quantum homogeneous spaces, building analogues of the de Rham complex, etc.


A good reference is Klimyk and Schmudgen, Quantum Groups and Their Representations.


