Let $R$ be an associative unital $k$-algebra. If $\alpha \in End_k(R)$ and $\delta$ is a $\alpha$-derivation, then one can define the skew polynomial algebra $R[x; \alpha,\delta]$ by letting $ax = x \alpha(a) + \delta(a)$. For the algebra $\mathscr{U}(\mathfrak{sl_2})$ applying this skew polynomial "functor" 2 times we can get the respective quantum algebra, however not every quantum group can be realized this way. Because of the fact above, it looks like that the skew polynomial algebra is a kind of deformation.
Is, in fact, the polynomial algebra a deformation along a cocycle of the Hochschild cochain complex?
Thanks in advance.