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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. 25

[8] D. Kazhdan and G. Lusztig, Tensor structures arising from affine Lie algebras, IV, J. Amer. Math. Soc. 7 (1994), 383–453.

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  • $\begingroup$ Can you be more precise about the constraints placed on the level $k$ in this literature? It gets a bit complicated. $\endgroup$
    – Jim Humphreys
    Aug 8, 2014 at 17:56
  • $\begingroup$ On the affine Lie algebra side, the level $k$ is any positive integer: $k\in\mathbb Z_{>0}$. On the quantum group side, this choice corresponds to $q=e^{\pi i/m(k+h^\vee)}$, where $h^\vee$ is the dual coxeter number, and $m\in\{1,2,3\}$ is the ratio between square-lengths of long roots and short roots. As far as I understand, $U_q\mathfrak g$ at other roots of unity is not directly related to affine Lie algebras. My basic reference for representations of quantum groups is math.tamu.edu/~rowell/umtcs.pdf $\endgroup$
    – André Henriques
    Aug 8, 2014 at 18:53
  • $\begingroup$ Great question! I have nothing useful to contribute towards an answer. $\endgroup$
    – Theo Johnson-Freyd
    Aug 9, 2014 at 1:24

3 Answers 3

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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.

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  • $\begingroup$ Are you saying that for categories with the same fusion rules as $E_8$ level 2, there is only one possible associator, braiding, and ribbon structure up to equivalence? Is that why the Finkelberg tensor functor (does his approach give a tensor functor?) is an equivalence of modular tensor categories. $\endgroup$
    – André Henriques
    Aug 11, 2014 at 22:15
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    $\begingroup$ @AndréHenriques: for fusion rules of the Ising category the associator, braiding, and ribbon structure are determined by the twist and sign of dimension of the non-invertible simple object (this is known at least since Moore and Seiberg paper on conformal field theory; you can $\endgroup$
    – Victor Ostrik
    Aug 12, 2014 at 4:59
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    $\begingroup$ find exposition in Appendix B of arxiv 0906.0620). These quantities are computable in this case which gives an equivalence. On the other hand it is not clear to me whether Finkelberg's functor is even defined in this case.. $\endgroup$
    – Victor Ostrik
    Aug 12, 2014 at 5:09
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    $\begingroup$ For Ising Fusion rules, there are 2 inequivalent Unitary Fusion categories and each can have 4 different braidings: the first $h=1/16,15/16,7/16,9/16$ and the other $h=3/16,13/16,5/16,11/16$, where $\exp(2\pi ih)$ is the the twist of the simple object with $d=\sqrt{2}$. One can check this using the Cuntz algebra approach. $\endgroup$
    – Marcel Bischoff
    Sep 15, 2014 at 16:38
  • $\begingroup$ @Victor: Where is it proven that the fusion rules of the affine Lie algebra version of $(E_8)_2$ are Ising? More generally, where is it proven that the fusion rules of affine Lie algebras are given by the quantum Racah formula? I have only been able to find references that prove the quantum Racah formula in the quantum groups approach. $\endgroup$
    – André Henriques
    Jan 14, 2015 at 10:25
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I take the freedom to copy here an email of Finkelberg to myself:

Dear Andre,

  1. The braidings and ribbon structures do also match. The braiding is reconstructed from the local systems; the ribbon structure (balance) comes from the action of $L_0$ and can be explicitly computed on any irreducible. Let me know if you need details on this.

  2. This is a question to Huang and Lepowsky. As far as I remember, they know it for all algebras and levels without exceptions (see the references in my Erratum). My argument needed the rigidity of KL categories which was only established with the above excetpions.

All the best, Michael.

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The following answer is from Yi-Zhi Huang. I post it here with his permission.

Finkelberg's approach cannot be used to give a tensor functor in the exceptional case ($\mathfrak g=E_6$, $k=1$, $\mathfrak g=E_7$, $k=1$ and $\mathfrak g=E_8$, $k=1$ or $2$). Finkelberg's approach can be sketched as follows:

Kazhdan-Lusztig constructed an equivalence between a quantum group module category at a root of unity and an affine Lie algebra module category of a negative integral level. On the quantum group side, one can take a subquotient to obtain a semisimple rigid braided tensor category (in fact a modular tensor category). The same procedure certainly works on the affine Lie algebra side. So one also has a semisimple subquotient of the affine Lie algebra module category of the negative integral level. The main idea of Finkelberg is to use the contragredient functor for affine Lie algebra modules. This functor sends modules of negative levels to those of positive levels (since the functor is the Lie algebra contragredient functor, not the vertex algebra contragredient functor). The crucial step is to prove that this functor is a tensor functor. This is the place where Finkelberg had a gap. One needs the Verlinde formula to fill the gap. From this description, you can see that Finkelberg's work has nothing to do with quantum groups. It is purely a result on the affine Lie algebra side. It is the work of Kazhdan-Lusztig that gives the connection with the quantum group.

Now since Kazhdan-Lusztig's work does not cover the exceptional cases, Finkelberg's approach fails completely. To construct a tensor functor, one has to do this directly. The abelian categories are obviously equivalent and the fusion rules are known to be the same. But these are far from a construction of a tensor category equivalence.

One possibility is to see whether Kazhdan-Lusztig's method can be adapted. But in this case one has to work directly with the semisimple subquotient of the quantum group category, since on the affine Lie algebra side the category is not a subquotient of a nonsemisimple rigid braided tensor category. It does not seem to be easy to directly adapt the method of Kazhdan-Lusztig since the representation theory of affine Lie algebras at positive integral levels are very different from the representation theory of affine Lie algebras at negative integral levels.

I think that if there is a construction in the exceptional cases (mainly the case $E_8$, level $2$), it should also work in the general positive level case. Finkelberg's construction is natural as a construction of an equivalence between a positive integral level category and a negative integral level category. But it is not natural as a construction of an equivalence between a quantum group category and a positive integral level category. The failure in the exceptional cases is an indication. There should be a direct and natural construction that works in all the cases and provides a true understanding of the connections between the modular tensor categories from quantum groups and from affine Lie algebras.

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