Let $\mathfrak{g}$ be a semisimple Lie algebra. Let $U_h(\mathfrak{g})$ be the Drinfeld-Jimbo quantum group, i.e. the $\mathbb{C}[[h]]$-algebra topologically generated by $X_i,Y_i,H_i$ where $1\leq i\leq$rank of $\mathfrak{g}$ and the "usual" relations as in Kassel's book. Now, Kassel's book which I am reading is sketchy about two points I would like to know:

(1) $U_h(\mathfrak{g})$ is a topologically free $\mathbb{C}[[h]]$-module via PBW-argument.

(2) The universal $R$-matrix of $U_h(\mathfrak{g})$ is given by a complicated formula which should be written explicitly.

For both points, Kassel refers vaguely to Drinfeld with sentences as "Drinfeld proves" but I am unable to find anything by Drinfeld on quantum groups online. Whence, the question: Where can I find Drinfeld's original papers on quantum groups which should contain a detailed proof of point (1) and (2) above? PS. I don't read russian. It should be an English version.

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    $\begingroup$ Isn't it the paper with the long name about quasi triangular quasi Hopf quasi algebras and a group related to the Galois group (i.e. GT)? (I'm on my phone) $\endgroup$ Oct 11, 2020 at 10:15
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    $\begingroup$ Not saying this is what you're looking for, but one nice thing by Drinfeld about quantum groups is his 1986 ICM lecture, pg's 798-820 of: mathunion.org/fileadmin/ICM/Proceedings/ICM1986.1/… $\endgroup$ Oct 11, 2020 at 13:12
  • $\begingroup$ Sam gave the correct reference I think, an expanded version is here link.springer.com/article/10.1007%2FBF01247086. He gives the standard presentation of quantum groups, and the construction of the R matrix using quantum doubles. $\endgroup$
    – Adrien
    Oct 11, 2020 at 14:16
  • $\begingroup$ J. Soviet Math. 41 (1988), no. 2, 898–915. This is the English translation of his paper Quantum groups. (Russian. English summary) Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 155 (1986) which is the expanded version of his ICM talk on quantum groups. $\endgroup$ Oct 11, 2020 at 14:23
  • $\begingroup$ Okay, the amount of detail in this paper is even less than in Kassel's book: "one can show...". Thank you anyway. $\endgroup$
    – user66288
    Oct 12, 2020 at 7:09


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