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Lie algebras  One commutator set to rule them all?
Posted:
Dec 25, 2013 2:17 PM


The generator set of a Lie algebra is, so to say, a rank 3 tensor with 1 upper and 1 lower index running in the defining irrep and one lower running in the adjoint irrep. The structure constants C are a rank 3 tensor with indices only running in the adjoint irrep. (This is spoken very graphicoriented, replace with clean vectorspace notation if it makes you feel better :) What happens if one replaces the two instances of "defining" in the above sentence with two instances of "any irrep" R? The dimensions change, but the structure constants stay the same (or at least, you can choose a gauge where they do)? So still [Ri,Rj]=sumCijk*Rk?
At least the standard SU2 example affirms my hunch  the rules are always [Jx,Jy]=Jz regardless of the spin (= the irrep), and graphically, this means that the adjoint J=1 is in any Clebsch j(x)j (except J=0 of course), which is obviously true. (Taking R to be the adjoint even forces more cool stuff, IHX rule and so, but enough for now.)
Is my intution right?  Hauke Reddmann <:EX8 fc3a501@unihamburg.de Hund frißt Hund jeden Tag  Pal jetzt NEU mit Menschgeschmack Hund frißt Hund heißt der Sport  hoff', du stehst auf Völkermord (Der Nachwuchs)



