Let $\pi: P \rightarrow M$ be a $G$-principal bundle. Define $\Omega^k_{\text{hor}}(P, \mathfrak{g})^{\text{Ad}}$ to be the set of $k$-forms $\omega$ taking values in the Lie algebra $\mathfrak{g}$ associated to $G$ such that it is horizontal and of type Ad, i.e. for all $p \in P$:
- $\omega_p(X_1, \ldots, X)k = 0$ whenever at least one of the vectors $X_i \in T_pP$ is vertical.
- $r^*_g \omega = \text{Ad}_{g^{-1}} \circ \omega$ for all $g \in G$.
I am studying the following theorem found in Hamilton's Mathematical Gauge Theory
The vector space $\Omega^k_{\text{hor}}(P, \mathfrak{g})^{\text{Ad}}$ is canonically isomorphic to the vector space $\Omega^k(M, \text{Ad}(P))$.
Here Ad$(P)$ is the adjoint bundle associated to $P$.
I have a simple question about this theorem. Here we are looking at vector-valued forms, however Ad$(P)$ is a vector bundle and not just a vector space. Thus at each point in $M$ the output of any such form $\omega$ takes values in the vector bundle Ad$(P)$.
How is this a vector valued form? Are we implicitly taking the output to be in the fiber in Ad$(P)$ over some point in $M$ (i.e. a vector space isomorphic to $\mathfrak{g}$)? If so, why not just write $\Omega^k(M, \mathfrak{g})$? What is the difference between $\Omega^k(M, \text{Ad}(P))$ and $\Omega^k(M, \mathfrak{g})$? Furthermore, if this is the case, how does this form retain any information about the adjoint representation as opposed to just $\mathfrak{g}$?
Why not just write $\Omega^k(M,\mathfrak{g})$ instead of $\Omega^k(M,\text{Ad}(P))$? Because they are not the same. Try to come up with an example where this is the case (of course, $G$ should be non-abelian, otherwise they are the same for sure!). For example, work out the case of the $\text{SU}(2)$ Hopf fibration.
And why are sections of $\Omega^k(M,\text{Ad}(P))$ still vector-valued $k$-forms? Well, $\text{Ad}(P)$ is locally trivial and so these $k$-forms obey similar rules to ordinary differential forms. However, they transform differently under coordinate changes than elements of $\Omega^k(M,V)$ with $V$ a vector space. And this is important to make the isomorphism $\Omega^k(M,\text{Ad}(P))\cong \Omega^k_\text{hor}(P,\mathfrak{g})^\text{Ad}$ work out.