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The ordinary McKay correspondence relates the subgroups of SU(2) to the affine ADE Dynkin diagrams. The correspondence is that the vertices correspond to irreducible representations of the subgroup, and edges correspond to tensor product with the 2-dimensional representation (which is given by the inclusion of G in SU(2)).

The quantum McKay correspondence does the same thing for "quantum subgroups" of the quantum group U_q(su(2)) for q a root of unity and the ordinary ADE Dynkin diagrams. One way to make this precise is that "quantum subgroups" are module categories over semisimplified categories of modules for U_q(su(2)), so the simple objects in the module category are the vertices, and the edges come from the tensor product with the defining rep of U_q(su(2)). In this language the result is due to Kirillov-Ostrik, but it goes back further in both the subfactor literature (due to Ocneanu and others) and the conformal field theory literature (which I don't know as well). These quantum subgroups come in two sorts, type 1 and type 2, and the type 1 ones can also be realized as fusion categories where the module structure comes from "restrict and tensor." The A_n, the D_2n, E_6 and E_8 are Type 1.

Ok, so now for the question, does the E_8 "quantum subgroup of quantum su(2)" have anything whatesoever to do with Lie algebra E_8?

I think the answer should be "no." The main reason I'm guessing this is that although the D_2n fusion categories are related to certain SO quantum groups under level-rank duality, the level-rank dual fusion category is not related to the quantum group for the Lie algebra D_2n it's instead related to the Lie algebra D_n-1 (see section 4.3 of my paper with Scott and Emily and the references therein).

The reason that I ask is in reference to Borcherds's question about detecting E_8 experimentally. I don't totally understand Will Orrick's answer but it sure sounds to me like he's saying that they've detected *the quantum subgroup E_8*, not the Lie algebra E_8. In particular its the quantum subgroup E_8 that has 8 objects ("particle types" in physics speak) one of which (all the way at the far end) has dimension the golden ratio (and so the golden ratio should come up in comparing it with the trivial object in some experiment).

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+1. I think this question is great, and one I've been wondering about. We've closed some questions recently for being too "bloggy", whereas this is a nice example of a question with good motivation, refers to other questions, all sorts of things that are nice on a blog, but still asks a well-posed and narrow question. –  Theo Johnson-Freyd Jul 19 '10 at 4:55
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It does indeed contain "all sorts of things that are nice on a blog" but I don't see what the precise question is at all! "Does X have anything whatsoever to do with Y", where X has not been defined, and with motivation that something else named Y has recently occurred in an entirely different context that the author couldn't understand does $\textit{not}$ strike me as particularly suitable for MO. Also, is it intentional that the Lie $\textrm{group}$ $SU(2)$ has been mislabeled as $su(2),$ the notation which is normally reserved for the corresponding Lie $\textit{algebra}$? –  Victor Protsak Jul 19 '10 at 6:55
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But if you really want one: Let Rep^ss(U_q(su_2)) be semisimplification (that is the quotient by all negligible morphisms) of the category of tilting modules for the Lusztig form of the Drinfel'd-Jimbo quantum group restricted to q a 60th root of unity. This is a braided fusion category. There is a unique (up to complex conjugation) algebra object structure on V_0+V_10+V_18+V_28 which we call A. This algebra object is commutative with respect to the braiding on C, so the category of A-module objects in C is itself a (non-braided) fusion category. Its fusion graph is E_8. –  Noah Snyder Jul 19 '10 at 14:43
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Good point Theo, all the categories I'm talking about involve finite dimensional representations anyway though you're right that I should have emphasized that when making the algebra vs. group point. Also, for non simply connected groups there's also a difference, I think the right notation is that quantum SO(3) should refer to only the reps of U_q(su_2) which come from deformations of representations of su_2 whose highest weight is in the root lattice. However, this notation is not standard. People typically use quantum SO to refer to what I'd prefer be called quantum Spin. –  Noah Snyder Jul 19 '10 at 19:19
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I added the tag "lie-algebras", which however may be just what you want to remove. This entire area of questioning may be hard to pin down rigorously, but I too was told some months ago by a local physicist about some experimental confirmation that I still can't relate coherently to familiar mathematics. –  Jim Humphreys Jul 21 '10 at 13:44
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1 Answer

This is not really an answer. It's more like a stub.
I hope that by making it wiki, I'll encourage others to contribute.


So there are plenty of things classified by A-D-E (or variants thereof, such as A-B-C-D-E-F-G, or A-D-E-T, or A-Deven-Eeven). In particular, there are plently of things called "the E8 ...".

Instead of making a complete graph, and showing that for any X and Y, "the E8 X" is directly related to "the E8 Y", one should maybe be less ambitious, and only construct a connected graph.

So, our task is to connect "the E8 Lie group" to "the E8 quantum subgroup of SU(2)". I suggest the following chain:

E8 Lie group --- E8 Lie algebra --- E8 surface singularity --- E8 subgroup of SU(2) --- E8 quantum subgroup of SU(2).

(1) E8 Lie group --- E8 Lie algebra
No comment.

(2) E8 Lie algebra --- E8 surface singularity
Look at the nilpotent cone $C$ inside $\mathfrak g_{E_8}$. That's a singular algebraic variety with a singular stratum in codimension 2. The transverse geometry of that singular stratum yields a surface singularity.

(3) E8 surface singularity --- E8 subgroup of SU(2)
Given a finite subgroup $\Gamma\subset SU(2)$, the surface singularity is $X:=\mathbb C^1/\Gamma$.
Conversely, $\Gamma$ is the fundamental group of $X\setminus \{0\}$.

(4) E8 subgroup of SU(2) --- E8 quantum subgroup of SU(2)
Is this quantization?!?

So let's recall what one really means when one talks about the "E8 quantum subgroup of SU(2)". We start with the fusion category Rep(SU(2))28, which one can realize either using quantum groups, or loop groups, or vertex algebras. That category is a truncated version of Rep(SU(2)): whereas Rep(SU(2)) has infinitely many simple objects, Rep(SU(2))28 has only finitely many, 29 to be precise.

Now this is what the "E8 quantum subgroup of SU(2)" really is: it's a module category for Rep(SU(2))28. In other words, it's category M equipped with a functor Rep(SU(2))28 × MM, etc. etc. That's where one sees that "quantum subgroup of SU(2)" is really a big abuse of language.

...so I don't know how to relate subgroups of SU(2) with the corresponding "quantum subgroups".
Can anybody help?

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Something I've learned since asking this question. Ocneanu has a somewhat mysterious theory he's been working on for a decade that supposedly gives a direct relationship between the quantum subgroups of SU(2) and the semisimple Lie algebras, and then gives constructions of "higher dimensional analogues of Lie algebras" coming from quantum subgroups of other Lie groups. I don't understand any of it, but there are some pictures in this video: msri.org/web/msri/online-videos/-/video/showVideo/2239 –  Noah Snyder Dec 13 '11 at 17:37
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Wow! There's soooo much stuff in this video! –  André Henriques Dec 15 '11 at 20:40
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