[Math] Does the quantum subgroup of quantum su_2 called E_8 have anything at all to do with the Lie algebra E_8

Question

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).

Answer 1

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.

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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-D_even-E_even). 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 × M → M, 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?