From Osher Doctorow
John Baez in his recent "Message in the sky" posting which cited S. Hsu
and A. Zee's "Message in the sky," physics/0510102 v1 11 Oct 2005,
definitely in my opinion did us all a service mathematically in
remindng us or teaching us about Cartan matrices. I suspect that he
and Hsu and Zee don't really care about who if anybody created the
Universe as we know it, but Cartan matrices are quite fascinating. See
"All simple Lie algebras, over the complex field: due to Cartan with
detailed notes on those of ranks 1 and 2,"
http://panda.unm.edu/Courses/finley/p500Fall05/Rank2Lie.pdf, which is
by Daniel Finley of U. New Mexico, and also his "Structure of the root
spaces for simple Lie algebras," same http except that instead of
Rank2Lie.pdf it ends in roots.pdf.
Needless to say, the presence of only 2's on the main diagonal and -1s
and -2s and 0s on subdiagonals or sub...subdiagonals for all simple,
complex Lie algebras except for G2 which has -3 as the row 1 column 2
element of its 2 x 2 matrix, is quite interesting. In fact, it's
somewhat obsessed me for about a day and a half, and I've barely
scratched the surface.
Osher Doctorow
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