I wonder if anyone can help me with this question regarding algebras and multiplets. In a nice review paper (McVoy, Rev Mod Phys 37(1)) the author states the following theorem:
Given any set of operators which satisfy the [Lie algebra commutation relations], there exists a Lie group which has these operators as its generators. Its multiplets are uniquely determined by the structure constants.'' He then goes on to say that it follows thatgiven the generator commutators, and nothing else, we can directly work out the multiplets of the group and all their properties, with no further assistance.''
But I'm confused about this, for the following reason. As a general rule, there are a number of different groups corresponding to the same Lie algebra (which are identical locally but which may differ globally). SU(3) and SU(3)/Z3 is a case in point. But (as anyone who's familiar with the history of the Eightfold Way and subsequent quark model will be aware) the triplet irrep is an irrep of SU(3), but not of SU(3)/Z3. (The lowest-dimensional non-trivial representation of SU(3)/Z3 is the 8.) So how can the algebra determine the multiplets of a group corresponding to it, if different groups with the same algebra will in general have different multiplets?
Any help much appreciated!