When studying quantum groups, in particular quantized universal enveloping algebras, people will tell you that such a quantization is in some sense unique. More specifically, you might hear that a quantization is a deformation and this deformation is unique up to something. As the second Hochschild cohomology of $U(sl_2)$ is zero, these deformations are clearly not simply algebra deformations.
In the already famous book 'tensor categories' by Etingof et al. they explain some links with the Davydov-Yetter cohomology. There other ways of looking at this uniqueness though (not from a homological point of view). I was wondering whether anyone knows a reference specifically towards understanding this uniqueness of quantizations.
Secondly, do these results still hold in an infinite-dimensional setting? (or is there hope that such results hold?)