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Topic: "Mini" Lie algebras
Replies: 3   Last Post: Dec 24, 2013 9:15 AM

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Hauke Reddmann

Posts: 545
Registered: 12/13/04
"Mini" Lie algebras
Posted: Dec 23, 2013 1:02 PM
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Just for fun (if you call nervous index breakdown fun) I computed
some Lie algebras by simply taking 2*2 matrices as generators.
(If I'm through with that I go to 3*3 :-) Assume a complex field.

The simplest rule would be [M1,M2]=M1. But there already the
troubles begin - the metric tensor is singular!

Is this because I'm too stupid to compute, or because this doesn't define
a Lie algebra, or the Lie algebra is not foo/bar/insert favorite
Lie algebra adjective here and this can happen?

Which Lie algebras can be done with 2*2 matrices anyway? The
abelian (2*2=1+1+1+1#), the SU2 (2*2=1+3#), the OSP(2,1) (whatever
that is - 2*2=1+2+1#), # marking the adjoint irrep, have I botched
it up again and are the more? What about 2*2=4#, again with
singular metric tensor? (Where's the 1 then?)

Please assume I don't know anything about Lie algebras except
the Wikipedia matrix generator definition :-)
Hauke Reddmann <:-EX8
Hund frißt Hund jeden Tag - Pal jetzt NEU mit Menschgeschmack
Hund frißt Hund heißt der Sport - hoff', du stehst auf Völkermord
(Der Nachwuchs)

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