Weight diagrams and Lie algebras

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SUMMARY

This discussion centers on the relationship between weight diagrams and semi-simple Lie algebras, specifically in the context of particle physics. It highlights examples such as the baryon octet corresponding to the 8 representation of SU(3) and the weak bosons associated with the triplet representation of SU(2). The main inquiry is whether a single weight diagram can represent two different semi-simple Lie algebras or if it uniquely determines the associated algebra. The focus is on irreducible representations, which are commonly utilized in particle physics.

PREREQUISITES
  • Understanding of semi-simple Lie algebras
  • Familiarity with weight diagrams and their significance
  • Knowledge of SU(3) and SU(2) representations
  • Basic concepts of particle physics and irreducible representations
NEXT STEPS
  • Research the properties of irreducible representations in Lie algebras
  • Explore the implications of weight diagrams in particle physics
  • Study the relationship between different semi-simple Lie algebras
  • Learn about the classification of representations in SU(3) and SU(2)
USEFUL FOR

Particle physicists, mathematicians specializing in algebra, and researchers interested in the representation theory of Lie algebras.

metroplex021
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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.
 
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A mathematician immediately thinks of the trivial case -- the singlet representation.
 
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.
 

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