ade-classificationsrt.representation-theory
The ADE type Dynkin diagrams seem to come up in seemingly different areas of math. Two places they come up are:
(1) Classification of simply laced complex simple lie algebras.
(2) Finite subgroups of $Sl_2 (\mathbb{C})$
Are there any other objects that they classify?
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 × 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?
I will first address the string theory part of the question.
String theory provides examples of physical systems admitting several descriptions that provide natural bridges between Kleinian singularities (and therefore Platonic solids), ALE spaces, quiver diagrams, ADE diagrams and two dimensional Conformal Field Theories.
The scene is given by compactifications of string theory on Kleinian orbifolds $M_\Gamma=\mathbb{C}^2/\Gamma$ where $\Gamma$ is a discrete subgroup of $SU(2)$. The space $M_\Gamma$ admits a Kleinian singularity at the origin. After studying this physical system, one is less surprised to see that Kleinian singularities, quiver diagrams, ALE spaces, ADE diagrams and 2 dimensional Conformal Field Theories all admit the same ADE classifications since they provide different descriptions of the same underlying physical system.
Michael Douglas and Gregory Moore have studied the compactification of string theory on Kleinian orbifold $M_\Gamma$ using D-branes as probes of the geometry. D-branes are extended objects on which strings can end. D-branes provide a physical description of the geometry in terms of supersymmetric gauge theories. Such supersymmetric gauge theories are efficiently summarized by a quiver diagram with a very natural physical interpretation: the nodes correspond to D-branes with specific gauge groups on them and the links between the nodes are open strings ending on the branes.
The minimal energy configurations (the vacua) of these supersymmetric gauge theories are obtained finding the extrema of a potential whose construction is equivalent to the hyperkhäler quotient construction of Asymptotic Locally Euclidian Spaces (ALE spaces) first obtained by Kronheimer. ALE spaces are HyperKähler four dimensional real manifolds whose anti-self-dual metrics are asymptotic to a Kleinian orbifold $M_\Gamma=\mathbb{C}^4/ \Gamma$. Physically ALE spaces described gravitational instantons. ALE spaces provide small resolutions of the Kleinian singularities where the singular point is replaced by a system of spheres whose intersection matrix is equivalent to the Cartan matrix of an ADE Dynkin diagram. One can also consider Yang-Mills instantons on such spaces. The gauge group associated with the Yang-Mills instantons is given by the type of ADE diagram obtained by the resolution of the singularity. This was analyzed in the math literature by Kronheimer and Nakajima. Physically the ALE instantons moduli space is equivalent to the vacua of the gauge theory description of D-branes located at the singularities.
The link between D-branes on ALE spaces (or equivalently Kleinian singularities) and the ADE classification of two dimensional Conformal Field Theories (CFT) was studied by Lershe, Lutken and Schweigert. Although the geometry is singular, the CFT description is smooth. The 2 dimensional CFT is coming directly from the string description: as a string evolves it described a 2 dimensional surface called the string worldsheet. D-branes enter the CFT as boundary states. In the description of the CFT, one recovers Arnold's ADE list of simple isolated singularities.
Updates
I would like to comment on the non-stringy part of the question. This is motivated by the comments of Victor Protsak.
If one removes all the string theory interpretation in the discussion above. What is left is Kronheimer's description of ALE spaces. Kronheimer's construction provides a beautiful realization of McKay's correspondence between Kleinian singularities, their crepant resolutions and ADE diagrams. This is reviewed in chapter 7 of Dominic Joyce's book "Compact Manifolds with Special Holonomy". From that perspective, the string theory description provides a physical interpretation of Kronheimer's construction and adds a natural link with quiver diagrams and 2 dimensional Conformal Field Theories.
Best Answer
They classify quivers for which the path algebra is of finite representation type., acording to a famous theorem of P. Gabriel.