Standard Model and Beyond · Problem 3.3

Gauge Theories

Physics — Particle Physics · jacobi-identity-adjoint-representation

Problem

Prove the Jacobi identity

[L^i, [L^j, L^k]] + [L^j, [L^k, L^i]] + [L^k, [L^i, L^j]] = 0

for an arbitrary representation and use it to prove that the adjoint representation matrices in (3.16) satisfy the commutation rules in (3.7).