Adjoint representations and Lie Algebras

AI Thread Summary
The adjoint representation of a Lie Group G involves elements of the Lie Algebra, where the transformation g^{-1} A_\mu g results in a new element of the Lie Algebra. This is due to the properties of Lie groups and their corresponding algebras, which ensure that the commutation relations, such as [A_\mu, A_\nu] = if_{ijk} A_\sigma, hold true. The structure constants f_{ijk} arise from the algebra's basis and define how the algebra elements interact under the adjoint action. Understanding this relationship is crucial for grasping the underlying structure of Lie groups and algebras. The discussion highlights the fundamental connection between the group elements and their algebraic representations.
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I have a very superficial understanding of this subject so apologies in advance for what's probably a stupid question.

Can someone please explain to me why if we have a Lie Group, G with elements g, the adjoint representation of something, eg g^{-1} A_\mu g takes values in the Lie Algebra of G?

Ie. Why does it necessarily mean that [A_\mu,A_\nu]=if_{ijk}A_\sigma where f is the structure constant

Thanks.
 
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