# How do gauge fields transform with an extra inhomogeneous term even though they are Lie Algebra valued in Non-Abelian gauge theories?

I am trying to work out Non-Abelian gauge theories but I couldn't get my head around the fact that gauge fields transform with an extra inhomogeneous term under the adjoint action of a group $G$, that is

$A_{\mu} \rightarrow gA_{\mu}g^{-1} - \partial_{\mu}g g^{-1}$

where $g \in G$. As far as I understand, the adjoint action of a group on a Lie Algebra valued object is given as

$Ad(g) T = gTg^{-1}$

where $T \in Lie(G)$. So how do (can) gauge fields transform differently even though the gauge fields themselves are Lie Algebra valued ? I understand we want to keep covariant derivatives gauge covariant but that doesn't make anything clear about this behavior of gauge fields.

• The notation you use is not familiar to me. So when you refer to a Lie-Algebra valued gauge field, I guess that means that you are contracting the non-Abelian gauge field with its associated generator, but still I then would expect the first expression to contain a term that represents some derivative of a scalar field also contracted with the generators. Guess that is the second term, but with some notation that I don't understand. Then because it is non-Abelian theory, I'm also expecting to see a third term that contains the fabc's. – flippiefanus Sep 15 '16 at 4:24
• physics.stackexchange.com/questions/191981/… – Elliot Schneider Sep 15 '16 at 14:49
• – Cosmas Zachos Nov 10 '18 at 16:05