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A question on algebras and multiplets

by metroplex021
Tags: lie algebra, lie group, multiplet, symmetry
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Aug31-11, 09:35 AM
P: 127
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 that ``given 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!
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Aug31-11, 10:48 AM
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Bill_K's Avatar
P: 4,160
metroplex021, I don't have the paper to refer to, but the author should have stated the theorem a little more carefully.
SU(3) and SU(3)/Z3 is a case in point.
And an even more familiar case is SU(2)/Z2 = SO(3). Given a Lie algebra there exists a unique simply connected Lie group, the universal covering group, in which all the irreducible representations of the Lie algebra are single valued. They are still representations of the other groups like SO(3) if you allow multi-valued representations.
Aug31-11, 03:31 PM
P: 127
Thank you Bill_K, that is (again) really helpful. So just to be clear: the triplet rep of SU(3) *is* also a rep of SU(3)/Z3, but is a multivalued rep? Does that mean that the 8 is the lowest-dimensional, non-trivial *vector* representation of SU(3)/Z3?

It's funny - I thought the Eightfold Way originally chose the SU(3)/Z3 specifically to exclude (amongst others) the triplet rep; but now it sounds like SU(3)/Z3 does not after all exclude it but rather possesses it, albeit as a multi-valued rep. Are there any general arguments as why we should not expect to find multi-valued reps in nature?

Any references or thoughts on this would be massively appreciated! Thanks again!

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