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Topic: Lie algebras - One commutator set to rule them all?
Replies: 0

 Hauke Reddmann Posts: 545 Registered: 12/13/04
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
graphic-oriented, replace with clean vector-space
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@uni-hamburg.de
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