Weight diagrams and Lie algebras

[metroplex021]

Very often we've identified sets of particles with the weights of a semi-simple Lie algebra - for example, the 8 particles of the baryon octet with the weights of the 8 representation of SU(3) global flavour symmetry (in the old days of the Eightfold Way), or the 3 weak bosons with the weights of the triplet representation of local SU(2) weak isospin symmetry. Does anybody know if it's possible for a single weight diagram to belong to two semi-simple Lie algebras? Or does the structure of a weight diagram determine the associated algebra uniquely? Any thoughts would be really appreciated.

 

[Bill_K]

A mathematician immediately thinks of the trivial case -- the singlet representation.

 

[metroplex021]

OK - then let *me* specify non-trivial cases! And - as I should have said before - the weight diagrams in question correspond to *irreducible* representations (since these are what we usually deal with in particle physics anyway). Thanks.