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 $U_q(\mathfrak{su}(2))$?