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Roughly 1990, Lusztig wrote a series of papers on quantum groups. Perhaps the result that the braid groups acts on $U_q(\mathfrak{g})$ is the proof which is least conceptual.

The original paper contains a case by case check for $m=2,3,4,6$ as far as I understand.

If it is not a coincidence, then it needs an explanation. My question is

Any conceptual proof up to now is known?

At least not one by one.

  • For example, can we define quantum groups for the type $I_n$? (The naive version is to work in the field of Puiseux series.)

  • Perhaps we have very limited 'conceptual' understanding on quantum groups up to now. There is a lot on the negative part $\mathfrak{f}$ (geomatric realization, categorification, etc.). But the braid group action is on the complete quantum group. The classic one is the Hall algebra. For this are there any interpolation of the braid group actions?

  • The only other result I know is the work of Nakajima which showed that the whole algebra can be realized by the equivariant K-theory of Nakajima varieties. (any result about how this action reflecting the geometry side?)

  • For example, the simplest case, in the springer realization of $U_q(\mathfrak{sl}_3)$, does the braid group action have any geometric meaning?

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  • $\begingroup$ Have you seen mathoverflow.net/a/115252 ? In particular, Drinfeld's ICM talk has an explicit formula for the universal đť‘…-matrix. $\endgroup$
    – Paul Gustafson
    Dec 28, 2020 at 1:40
  • $\begingroup$ Do you need this for the Lusztig integral form? Or would the h-adic version be enough? In the latter case I thought you could get it pretty directly from the classical r-matrix following Drinfeld, as Paul has suggested. If you really want the integral form done carefully then you probably just need to do what Lusztig does, but the h-adic version tells you what formulas you should use. $\endgroup$
    – Noah Snyder
    Dec 28, 2020 at 4:19
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    $\begingroup$ I think the question is about the quantum Weyl group (denoted by $T_{i, e}', T_{i, e}''$ in Lusztig's book) rather than about the quantum $R$-matrix. $\endgroup$
    – Pavel Safronov
    Dec 28, 2020 at 11:08

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