John McKay discovered a deep correspondence between the affine Coxeter-Dynkin diagrams of the A-D-E Lie algebras and the irreducible representations of the finite subgroups of SU(2) = Spin(3) = Sp(1) = S3.

John Baez, in his week 182, refers to "... Joris van Hoboken, Platonic solids, binary polyhedral groups, Kleinian singularities and Lie algebras of type A,D,E, Master's Thesis, University of Amsterdam, 2002, available at http://www.science.uva.nl/research/math/examen/2002/scriptiejoris.ps ...", which thesis John Baez has described in a sci.physics.research post as "... The most detailed online explanation ...[of the]... mysterious relationship between ADE and Platonic solids ...".

Jean-Luc Brylinski has described a correspondence dual to McKay's, involving such things as matrix-valued Fourier transforms.

The E-series of Lie Algebras ends at E8, and del Pezzo Surfaces also end at dimension 8.

Here is a copy of a Usenet Sci.Math paper of John McKay, MCKAY@Vax2.Concordia.CA, sent from John McKay to Tony Smith by e-mail in June 1993:

Kronecker: (in German!) The natural numbers are the work of God, everything else is the work of man.

I show here how to derive the Dynkin diagrams and all the semi-simple Lie algebras from the natural numbers:

Let us start with the semi-infinite graph:

1-2-3-4-5-...

Each edge is to be regarded as a pair of opposing directed edges.

Each node is an irreducible representation of SU2(C). The numbers are the degrees of these representations. I call this graph the representation graph of SU2(C) because it is built from the fundamental (natural) representation, R of SU2(C) and the equations:

(*) RxR[i] = + m[i,j]R[j] summed over j,

giving the decomposition of the tensor product of R (=R[2]) with the unique irreducible of dimension i. We start with the trivial representation of dimension = 1. m[i,j] is the multiplicity of the directed edge from node i to node j.

For each finite subgroup G in SU2(C), we restrict the representations R[i] to G. This then splits the representations of SU2(C) into irreducibles of G.

Example: G = quaternions. 1a / We get 1-2-1b from the initial segment of 1-2-3-... \ 1c

which is the affine Dynkin diagram of type D4. (We break off as soon as representations repeat.)

For each finite subgroup of SU2, we get an affine Dynkin diagram in this way. Affine means adding an extra node corresponding to the negative of the highest root. The correspondence is:

A[r] degenerate Cyclic[r+1] D[r] {2,2,r-2} Generalized quaternion[r-2] E[6] {2,3,3} 2.Alt[4] binary tetrahedral E[7] {2,3,4} 2.Symm[4] binary octahedral E[8] {2,3,5} 2.Alt[5]=SL(2,5) binary icosahedral where {a,b,c} =<x,y,z: x^a=y^b=z^c=xyz> (=-1). The non-ADE types correspond to certain pairs (G,H), H { G in SU2(C). The ADE Dynkin graphs are: 1 A[r] / \ An (r+1) -gon / \ 1-...-1 1 1 \ / D[r] 2-2...2-2 There are r+1 nodes. The sum / \ of the numbers = h = Coxeter 1 1 number. The sum of the squares 1 is the order of G. The numbers | are the degrees of irreducible 2 representations of G. They are | first Chern numbers (singularities). E[6] 1-2-3-2-1 They are the periods of products of pairs of Fischer involutions 2 mod centre (E[8]{=}Monster,E[7]{=} | 2.Baby,E[6]{=}3.F_24). Fundamental E[7] 1-2-3-4-3-2-1 group (Lie) = Schur multiplier (sporadics). 3 | E[8] 1-2-3-4-5-6-4-2

To get the other Dynkin diagrams, we "fold" these ones by replacing nodes by the equivalence classes of nodes under orbits of graph automorphisms. There are several ways of doing this. Repeated folding and reversing arrows and unfolding takes, for example: D4 - G2 - E6 - F4 - E7. One can interpret the reversals in terms of Frobenius reciprocity.

By taking traces of (*) on the identity, we see that the adjacency matrices m[i,j] of all these Dynkin graphs have a maximal eigenvalue of 2. This is the crucial property of affine Dynkin graphs.

The corresponding eigenvector has irreducible degrees for its components. The columns of the character table of G are eigenvectors and the row of the 2-dimensional representation is the row of eigenvalues.

The connection with platonic solids is as follows: See L.E. Dickson Algebraic Theories, Chapter 13. Project from the North pole of the sphere escribed to the Platonic solid, through each vertex on to the equatorial plane (which we interpret as the complex plane). Thus we may identify each vertex with a complex number,v[i]; we form the (homogeneous) polynomial V(x,y) = prod(x-v[i]y).

Similarly we form E(x,y) from the midpoints of the edges, and F(x,y) from the normals through the centre of the faces. These are three functions in two variables and so there is a relation f(V,E,F) = 0.

This is a singularity of the simplest kind which can be "desingularized" into a set of exceptional fibres which are complex projective lines intersecting as the dual of the Dynkin diagram of finite type.

The intersection matrix = M-2I = -C where M is the matrix m[i,j] which we started with (without the affine node), and C is a Cartan matrix from which we can derive the generators and relations for a Lie algebra (Serre's relations). Including the affine node yields Kac-Moody generators and relations and relates to elliptic singularities.

Caveat: for the E8 - icosahedral = <2,3,5> case, the singularity is x^2+y^3+z^5=0 (see exercise in Hartshorne's book on Algebraic Geometry) but it is NOT x^a+y^b+z^c generically.

This correspondence between the platonic groups and the Lie algebras of type A,D,E is described by Slodowy in Springer Yellow Series No. 815 . By using a subgroup to get the equivalence classes, we get the F,G series too.

There have been many applications of these ideas in many contexts. Let me cite two: Peter Kronheimer has used them in his paper on asymptotically locally flat and asymptotically locally euclidean spaces in connection with cosmological geometry. They are also being used in understanding the spectral lines associated with the newly discovered C60 molecules of fullerenes. This provides a good example of the unity of mathematics and the nonsense of the distinction between pure and applied.

This June, in London, Walter Feit is giving the London Math. Soc. Hardy Lecture on these ideas in connection with the representations of quivers.

This all generalises further. (May get published one day!)

References: Graphs, singularities and finite groups. Proc. Symp. Pure Math. vol 37. Amer. Math. Soc. (1980). Pages 183- and 265-. Representations and Coxeter Graphs. In "The Geometric Vein" Coxeter Festschrift (1982) Springer-Verlag. Pages 549-.

Moral: Read the original - Plato (ca. 430-350 B.C.) Timaeus.

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