In his paper [1], Finkelberg used Kazhdan-Lusztig's massive work [4,5,6,7,8] to prove that $Rep^{ss}(U_q\mathfrak g)$ (the semisimplification of the category of finite dimensional reps of $U_q\mathfrak g$) is equivalent to the category $Rep_k(\widetilde{L\mathfrak g})$
of level $k$ integrable highest weight modules over the affine Lie algebra $\widetilde{L\mathfrak g}$.
But then, I recently learned (from Section 3 of [3]) that there was an erratum [2] where an error was discovered and corrected, and that there are cases (namely $E_6$, $E_7$, $E_8$ level 1, and $E_8$ level 2) where the Kazhdan-Lusztig story [4,5,6,7,8] cannot be applied…
Question 1: Is the equivalence $Rep^{ss}(U_q\mathfrak g)\cong Rep_k(\widetilde{L\mathfrak g})$ known for all $\mathfrak g$ and $k$, or are there exceptions?
Question 2: Is the equivalence $Rep^{ss}(U_q\mathfrak g)\cong Rep_k(\widetilde{L\mathfrak g})$ known just at the level of fusion categories, or have the braidings also been compared? How about the ribbon structures?
References:
[1] M. Finkelberg, An equivalence of fusion categories, Geom. Funct. Anal. 6 (1996),
249–267.
[2] M. Finkelberg, Erratum to: An equivalence of fusion categories, Geom. Funct.
Anal. 6 (1996), 249–267; Geom. Funct. Anal. 23 (2013), 810–811.
[3] Y.-Z. Huang and J. Lepowsky, Tensor categories and the mathematics of
rational and logarithmic conformal field theory, ArXiv:1304.7556
[4] D. Kazhdan and G. Lusztig, Affine Lie algebras and quantum groups, Duke Math.
J., IMRN 2 (1991), 21–29.
[5] D. Kazhdan and G. Lusztig, Tensor structures arising from affine Lie algebras, I,
J. Amer. Math. Soc. 6 (1993), 905–947.
[6] D. Kazhdan and G. Lusztig, Tensor structures arising from affine Lie algebras, II,
J. Amer. Math. Soc. 6 (1993), 949–1011.
[7] D. Kazhdan and G. Lusztig, Tensor structures arising from affine Lie algebras, III,
J. Amer. Math. Soc. 7 (1994), 335–381.
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[8] D. Kazhdan and G. Lusztig, Tensor structures arising from affine Lie algebras, IV,
J. Amer. Math. Soc. 7 (1994), 383–453.
Best Answer
Here is my understanding of the situation. Question 1: yes, the equivalence is known in all cases. Kazhdan-Lusztig work does have some limitations on the level, so Finkelberg's approach is not applicable. However the categories in question are easy enough to work with explicitly: for instance $E_8$ at level 1 has just 1 simple object and $E_8$ at level 2 has 3 simple objects and fusion rules of the Ising category. I should note that in these small level cases there is a problem with the loop group side; for example it is non-trivial to show that this category is rigid (Finkelberg deduces this from his equivalence, which is not available for small levels). This could be verified on a case by case basis, but currently we have Huang's proof of Verlinde conjecture which takes care of all cases and much more.
Question 2: this equivalence is an equivalence of modular tensor categories, so it includes both braided and ribbon structure. The terminology changed since the time of Finkelberg's paper; his "fusion categories" are what is called "ribbon fusion categories" or "premodular categories" nowadays.