A question on algebras and multipletsby metroplex021 Tags: lie algebra, lie group, multiplet, symmetry 

#1
Aug3111, 09:35 AM

P: 115

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 lowestdimensional nontrivial 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! 



#2
Aug3111, 10:48 AM

Sci Advisor
Thanks
P: 3,853

metroplex021, I don't have the paper to refer to, but the author should have stated the theorem a little more carefully.




#3
Aug3111, 03:31 PM

P: 115

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 lowestdimensional, nontrivial *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 multivalued rep. Are there any general arguments as why we should not expect to find multivalued reps in nature? Any references or thoughts on this would be massively appreciated! Thanks again! 


Register to reply 
Related Discussions  
Algebras generated as commutative algebras.  Calculus & Beyond Homework  2  
SUSY multiplets  simple question  High Energy, Nuclear, Particle Physics  3  
Simple question on SUSY multiplets  High Energy, Nuclear, Particle Physics  15  
Prove there are exactly 4 nonisomorphic algebras among algebras Af  Calculus & Beyond Homework  0  
Currents / Multiplets  High Energy, Nuclear, Particle Physics  1 