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I'm looking for references regarding an unpublished Deligne's manuscript "Une descrption de catégorie tressée (inspiré par Drinfeld)" and the subject it touches, that is described in the post title. If it's not available online maybe someone can orient me to some other references regarding the same connection or to the connection itself for me to try to develop the details.

Thanks,

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Welcome to MO. One place where this idea is somewhat explained is Bezrukavnikov-Finkelberg- Schechtman, Factorizable sheaves and quantum groups (https://arxiv.org/abs/q-alg/9712001) but maybe this is where you heard about it.

I'd say nowadays this idea might be best understood as a particular case of the identifications between:

  • braided monoidal categories, and algebras over the little disks operad in Cat
  • algebras over the little disks operad and locally constant factorization algebras on $\mathbb{R}^2$.
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  • $\begingroup$ Thank you, the reference is good, for the place I've read about this is "Kahzdan, Lusztig - structures arising from affine lie algebras" i'm currently reading it so I'll use both sources. $\endgroup$
    – Rebour
    Commented Mar 31, 2021 at 14:24
  • $\begingroup$ Indeed, one of there motivation is to give a RIemann-Hilbert proof of the Kazhdan--Lusztig equivalence. I'm not entirely sure that proof is written in full, though. You can also look at papers by Gaitsgory explaining and making precise similar ideas, in a somewhat more technical language. $\endgroup$
    – Adrien
    Commented Mar 31, 2021 at 14:31

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