Its commonly said that gauge bosons transform according to the adjoint representation of the corresponding gauge group. For example, for SU(2) the gauge bosons live in the adjoint 3 dimensional representation and the gluons in the 8 dimensional adjoint of SU(3).
Nevertheless, they transform according to
Aμ→A′μ=UAμU†−ig(∂μU)U†,
which is not the transformation law for some object in the adjoint representation. For example the W bosons transform according to
(Wμ)i=(Wμ)i+∂μai(x)+ϵijkaj(x)(Wμ)k.
Answer
A gauge field transforms in the adjoint of the gauge group, but not in the adjoint (or any other) representation of the group of gauge transformations.
In detail:
Let G be the gauge group, and G={g:M→G|g smooth} the group of all gauge transformations.
A gauge field A is a connection form on a G-principal bundle over the spacetime M, which transforms as A↦g−1Ag+g−1dg
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