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 email@example.com Hund frißt Hund jeden Tag - Pal jetzt NEU mit Menschgeschmack Hund frißt Hund heißt der Sport - hoff', du stehst auf Völkermord (Der Nachwuchs)