I have read in [these lecture notes](http://webpages.math.luc.edu/~ptingley/oldseminars/QuantumGroupsSpring2011/lecture1.pdf) that every deformation $U\_h(\mathfrak{g})$ of $U(\mathfrak{g})$ is trivial, i.e. isomorphic to $U(\mathfrak{g})[[h]]$ as associative $\mathbb{C}[[h]]$-algebras. Why is this true? The reason they cite is that $HH^2(U(\mathfrak{g}),U(\mathfrak{g}))=0$. Now, I know how to compute Hochschild cohomology of simpler algebras, but several problems arise for me in this setup: 


1. I am unsure how to proceed in the computation of $HH^2(U(\mathfrak{g}),U(\mathfrak{g}))$ (or more generally, replacing $U(\mathfrak{g})$ with $T(V)$, the tensor algebra of a vector space). How can I conclude that this is $0$?
2. Why does the "trivial" deformation look like a power series ring? They say that the multiplication in $U\_h(\mathfrak{g})$ looks like "multiplication modulo $h$" in $U(\mathfrak{g})$; what does this mean precisely?
3. The authors instead deform $U(\mathfrak{g})$ as a Hopf algebra, but don't elaborate on what such a deformation should look like, what cohomology theory classifies such deformations, etc. Where can I find a resource that concretely answers these questions? An English translation of Drinfel'd's paper "Quasi-Hopf algebras" would be a good starting point, but I haven't been able to find one.



Why is every deformation of the universal enveloping algebra of a complex semisimple Lie algebra trivial?
The article [Deformation par quantification et rigidite des
algebres enveloppantes](https://arxiv.org/abs/math/0211416v1) by M. Bordemann, A. Makhlouf, T. Petit addresses these questions. They call Lie algebras $\mathfrak{g}$ with $HH^2(U(\mathfrak{g}),U(\mathfrak{g}))=0$ *strongly rigid*, and show that then
every formal associative deformation is equivalent to the trivial deformation. For semisimple Lie algebras over an algebraically closed field of characteristic zero one has $HH^2(U(\mathfrak{g}),U(\mathfrak{g}))=H^2(\mathfrak{g},S\mathfrak{g})$, which is zero by Whitehead's second Lemma for Lie algebra cohomology.


