Help with Derivation: u^-1(\partial_\mu u)u^-1 = \partial_\mu(u^-1)

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SUMMARY

The equality -u^{-1}(\partial_\mu u)u^{-1} = \partial_\mu(u^{-1}) holds true in the context of gauge theory, specifically when u represents a NxN representation of a continuous gauge group. This relationship is derived from the properties of the gauge group and the behavior of the derivative operator in spacetime. Understanding this equality is crucial for grasping the mathematical framework of the standard model of particle physics.

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  • Understanding of gauge theory and continuous gauge groups
  • Familiarity with differential operators in the context of physics
  • Knowledge of the standard model of particle physics
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Kontilera
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Hello there!
I'm trying to read a set of notes on the standard model and the author states the following equality without any explanation:
-u^{-1}(\partial_\mu u)u^{-1} = \partial_\mu(u^{-1}),
u is here a NxN representation of some continuous gauge group and is a function over spacetime.
Does this equality hold and in that case why?

Thanks!
// Kontilera
 
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