Hi! I'm trying to show that the differential from equation
$$D \star F = 0$$ transform homogeneously under the adjoint action ##F \mapsto gFg^{-1}## of the lie group ##G##, where ##D## denotes the covariant exterior derivative ##D\alpha = d \alpha + A \wedge \alpha## for some lie algebra valued form ##\alpha##. Since
$$A \mapsto gAg^{-1} + gdg^{-1}$$
we get
$$D\star F \mapsto d(g\star Fg^{-1}) + (A \mapsto gAg^{-1} + gdg^{-1})\wedge \star F = dg \wedge \star F g^{-1} + g d\star F g^{-1} + g \star F \wedge dg^{-1} + g A \wedge \star F g^{-1} + g dg^{-1} \wedge g^{-1} \star F g^{-1}$$
where if we use that ##gdg^{-1} = - (dg) g^{-1}##, then the last term seem to cancel the first. However, if this is correct, then the third term has to be zero.
Is the third term zero, and if so why?
Am I going wrong somewhere else?
$$D \star F = 0$$ transform homogeneously under the adjoint action ##F \mapsto gFg^{-1}## of the lie group ##G##, where ##D## denotes the covariant exterior derivative ##D\alpha = d \alpha + A \wedge \alpha## for some lie algebra valued form ##\alpha##. Since
$$A \mapsto gAg^{-1} + gdg^{-1}$$
we get
$$D\star F \mapsto d(g\star Fg^{-1}) + (A \mapsto gAg^{-1} + gdg^{-1})\wedge \star F = dg \wedge \star F g^{-1} + g d\star F g^{-1} + g \star F \wedge dg^{-1} + g A \wedge \star F g^{-1} + g dg^{-1} \wedge g^{-1} \star F g^{-1}$$
where if we use that ##gdg^{-1} = - (dg) g^{-1}##, then the last term seem to cancel the first. However, if this is correct, then the third term has to be zero.
Is the third term zero, and if so why?
Am I going wrong somewhere else?
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