From: jamesrheckman@gateway.netnospam (Jim Heckman)
Subject: Re: Exceptional Lie algebras
Date: 03 Mar 2001 09:55:32 GMT
Newsgroups: sci.math
Summary: Correspondence between Dynkin diagrams and polyhedra

>From: "Marco" mdein@katamail.com 
>Date: 3/2/01 4:20 AM Pacific Standard Time
>Message-id: <3a9f900d$1@spamkiller.newsfeeds.com>
>
>jamesrheckman@gateway.netnospam (Jim Heckman) wrote:
>
>>I don't know what to say, beyond "it's the geometry". Almost
>>any book on Lie algebras and/or groups will derive all possible
>>Dynkin diagrams, subject to the relevant geometrical constraints,
>>and them construct the root systems in explicit detail.
>
>Thanks for the references. The reason why I asked was that I 
>noticed that if you take the platonic solids and order them by 
>the number of faces, then you take V+E+F for each one and 
>multiply it by the place in this sequence, then you get 14, 52, 
>78, 248, 310. Pity it doesn't work for E7, which has dimension
>133. I know there's a correspondence between the Platonic 
>solids and the ADE algebras but AFAIK it doesn't extend to G2
>or F4. It's probably just a coincidence.

Yeah, the ADE classification crops up all over the place in
math, and it wouldn't surprise me at all if what you've
discovered has some "deeper" meaning vis-a-vis it -- as another
poster pointed out.

BTW, there's at least one other way in which the Platonic solids
correspond very definitely to Coxeter graphs. The solids are
the 3-d versions of what are called regular convex polytopes,
and it turns out every RCP has as its symmetry group the Weyl
group of a 'non-branched' (i.e., not D or E) graph.

The A_n graphs go with the n-d 'simplexes': the triangle in 2-d;
the tetrahedron in 3-d; ... The n-d simplex is 'made' from (n+1)
copies of the (n-1)-d simplex.

The B/C_n graphs go with the n-d 'cross polytopes' and their
duals the 'measure polytopes' (or 'hypercubes'). The n-d cross
polytope is made from 2^n copies of the (n-1)-d simplex, so
in 3-d it's the octahedron. The n-d hypercube is made from
2n copies of the (n-1)-d hypercube, so in 3-d it's the cube.

F_4 goes with a very unique 4-d RCP made of 24 octahedrons,
which is the *only* non-simplex self-dual RCP in *any*
number of dimensions.

H_3 (the H's are Coxeter graphs, but not Dynkin diagrams,
since they don't satisfy the 'crystallographic' condition) goes
with the 3-d dodecahedron and its dual the icosahedron.

H_4 goes with the 4-d RCP made of 120 dodecahedrons, and
its dual made of 600 tetrahedrons.

And that's all there are. Beyond 4-d, the only RCPs are the
simplex, and the cross and measure polytopes, 'because' there
are no non-branched Coxeter graphs except for A and B/C.

-- 
 
Jim Heckman