The natural numbers are the work of God,
everything else is the work of man. -
Kronecker (translated from the German)
I show here how to derive the Dynkin graphs 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 SU(2). The numbers are the degrees of these representations. I call this graph the representation graph of SU(2) because it is built from the fundamental (natural) representation, R of SU(2) and the equations:
(*) R ⊗ R[i] = ⊕ m[i,j] R[j]
j
giving the decomposition of the tensor product of R ( = R[2] ) and
the unique irreducible of dimension i, R[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 SU(2), we restrict the representations R[i] to G. This then splits the representations of SU(2) into irreducibles of G.
Example: G = the quaternion group
1a
/
1-2-1b
\
1c
from the initial segment of 1-2-3-...
which is the affine Dynkin graph of type D4. (We break off as soon as representations repeat.)
For each finite subgroup of SU2, we get an affine Dynkin graph in this way. "Affine" means adding an extra node corresponding to the negative of the highest root. The correspondence is:
A[r]: degenerate Cyclic of order r+1
D[r]: <2,2,r-2> Generalized quaternion of order 4(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> is the group with generators x,y,z and relations
xa = yb = zc = xyz(In fact, xyz = -1 in SU(2).)
The non-ADE types correspond to certain pairs (G,H), H < G in SU(2). The ADE Dynkin graphs are:
1
A[r] / \ (an (r+1)-gon)
/ \
1-...-1
1 1 The sum of the numbers at nodes adjacent
D[r] \ / to node v is twice the value at v.
2-2...2-2 There are r+1 nodes. The sum
/ \ of the numbers = h = Coxeter
1 1 number. The sum of the squares
E[6] 1 is the order of G. The numbers
| are the degrees of irreducible
2 representations of G. They are
| first Chern numbers (singularities).
1-2-3-2-1 They are the periods of products
of pairs of Fischer involutions
E[7] 2 mod centre (E[8]<=>Monster,E[7]<=>
| 2.Baby,E[6]<=>3.F_24). Fundamental
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 graphs, 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, yields, for
example, the sequence: D4 - G2 - E6 - F4 - E7. One may 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 characterizing 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 described in
Caveat: for the E8 = icosahedral = <2,3,5> case, the singularity is x2+y3+z5 = 0 (see exercise in Hartshorne's book on Algebraic Geometry) but it is NOT xa+yb+zc = 0 in every case:
E8: x2+y3+z5 = 0
E7: x2+y3+yz3 = 0
E6: x2+y3+z4 = 0
This correspondence between the Platonic groups and the Lie algebras of type A,D,E is described in
We can solve the geometric eigenproblem for rational polynomial solutions in t, of:
t*n[i] = ∑ n[j]
{successor nodes, j, of i}
We find that t occurs naturally as 2cos(a), and so put t = q+1/q,
clear denominators, and normalize the affine node to n[0]= 1+qh,
where h is the Coxeter number of the corresponding Lie group by
my correspondence. This gives polynomial solutions in q. If we now
divide by (1-qa)(1-qb) where a + b = h + 2, and ab = 2|G|, then
we find a solution given by
X[i](g-1) n[i](q)
m[i](G) = (1/|G|) ∑ ____________ = _____________
g ∈ G det(I-gq) (1-qa)(1-qb)
where X[i] is the ith irreducible character of G, (X[0]=1), and
G < SU2(C), by my correpondence, is the generalised Molien
series.
There is a remarkable connection between the three groups: M(onster), 2.B(aby Monster) and 3.F24 and the graphs for E8, E7, and E6 respectively.
In each of the above groups there are involutions (elements of period two) such that the product of pairs of them lies in one of 9,8,7 conjugacy classes respectively. The periods of elements in these classes are exactly the numbers used to label the extended Dynkin graphs (the coefficients of the highest roots) as long as we read this period modulo the centre. We may alternatively work with the folded Dynkin graphs of type E8, F4, G2 and collapse the centre. By moonshine, this means that each node of these extended graphs can be labelled by a modular function given by the moonshine correspondence, with the elliptic modular function j(z) corresponding to the identity of M and so to the affine node of the E8 Dynkin graph. There is a similar interpretation for the other nodes. Adjacency is not understood but there should be a mechanism for obtaining the neighbours of a moonshine modular function occurring here.
There have been many applications of these ideas in various 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 finding 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 correspondence has appeared in the book Roots of Consciousness as a basis for an explanation of consciousness!
This all generalizes further. (May get published one day!)
Moral: Read the original - the Timaeus by Plato (ca. 430-350 B.C.).
For more elementary stuff about ADE theory, try these issues of This Week's Finds: