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?




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**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.



What's the state of affairs concerning the identification between quantum group reps at root of unity, and positive energy affine Lie algebra reps?
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. 


