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 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.
Means that as a $\mathbb C[[\hbar]]$ associative algebra $U_\hbar(\mathfrak g)$ is isomorphic to $U(\mathfrak g)[[\hbar]]$. Each element can be understood as a formal power series $\sum_{n=0}^\infty \hbar^n X_n$ where each $X_n\in U(\mathfrak g)$ and the product is given by the usual product formula of formal power series $$\sum_{n=0}^\infty \hbar^n X_n \cdot \sum_{n=0}^\infty \hbar^n Y_n=\sum_{n=0}^\infty \hbar^n \sum_{j=0}^n X_j\cdot Y_{n-j}$$
Means that $U_\hbar(\mathfrak g)$ is a $\mathbb C[[\hbar]]$ Hopf algebra such that $U_\hbar(\mathfrak g)/\hbar U_\hbar(\mathfrak g)$ is again a Hopf algebra isomorphic (as Hopf algebra) to $U(\mathfrak g)$ . To see this as a deformation problem in a cohomological manner you should look for what is called Gerstenhaber-Shack cohomology. You can have a look at the original papers on the subject:
- 1 Gerstenhaber and Shack, Bialgebra cohomology, deformations and quantum groups, Proc. Nat. Acad. Sci. USA 87 (1990).
- [2] Gerstenhaber and Shack, Algebras, bialgebras, quantum groups, and algebraic deformations, Contemp. Math. 134 (1992).
I have the impression (I ask for specialists to correct me on this point) that in general computing GS cohomology is very hard and very few general results are known and that is why these works for quite some time were somewhat not really considered in the qg community.
As an aside, difficulties in finding the English version of the paper by Drinfel'd were already addressed here: English version of “Quasi-Hopf Algebras”. It is often difficult to find the very short lived Leningrad Journal of Math but with help from you library you should succeed.