Question on tensor notation in group theory

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SUMMARY

The discussion focuses on the commutation relations for the generators of the special orthogonal group SO(3) as presented in Zee's book on Group Theory. Specifically, the commutator [J^{ij}, J^{lk}] is defined as J^{ij}*J^{lk} - J^{lk}*J^{ij}, and the expression involving the Kronecker delta, i(δ^{ik}J^{jl}), raises questions about its significance. The Kronecker delta serves as an identity operator in this context, linking different indices while maintaining the structure of the algebra. Understanding these notations is crucial for grasping the underlying principles of group theory and its applications in physics.

PREREQUISITES
  • Familiarity with group theory concepts, particularly Lie algebras.
  • Understanding of the special orthogonal group SO(3) and its generators.
  • Knowledge of commutation relations in quantum mechanics.
  • Basic understanding of tensor notation and Kronecker delta notation.
NEXT STEPS
  • Study the properties of Lie algebras and their representations.
  • Explore the role of the Kronecker delta in tensor calculus.
  • Learn about the applications of SO(3) in quantum mechanics and classical mechanics.
  • Investigate the commutation relations in other groups, such as SU(2) and SU(3).
USEFUL FOR

This discussion is beneficial for physicists, mathematicians, and students studying advanced topics in group theory, particularly those interested in the mathematical foundations of quantum mechanics and theoretical physics.

BWV
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in the appendix on Group Theory in Zee's book there is a discussion of commutations for SO(3)

two questions

- does [J^{ij},J^{lk}] = J^{ij}*J^{lk}-J^{lk}*J^{ij}?

and there is an expression in the appendix that the commutator equals i(\delta^{ik}J^{jl} ...

i don't understand the why you are multiplying the matrix by the kronecker delta with different upstairs indexes, is not it simply an identity?
 
Last edited:
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bumping this - this apparently is some notation I don't understand, the Wiki entry on Special Unitary groups has this as well

304a3ac6552e3b2d08fc00e5bdb18736.png


I don't get the significance of the kronecker deltas with the different indexes
 

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