Convention: Ad(P) uses the conjugation action on G

Let $\pi:P\to M$ be a principal G-bundle . In this project we reserve the notation $\operatorname{Ad}(P)$ for the associated bundle with fiber the group $G$ itself, where $G$ acts on $G$ by conjugation.

Convention

We define the adjoint bundle

using the conjugation action of $G$ on itself:

together with the general convention that associated bundles use a left G-action on the fiber while $P$ carries a right principal action (see right action convention ).

This is compatible with the construction in construction of the adjoint bundle , and it is distinct from the adjoint Lie algebra bundle

which uses the adjoint representation on the Lie algebra ; see construction of ad(P) .

Because conjugation is by group automorphisms, $\operatorname{Ad}(P)$ is a bundle of Lie groups (fiberwise multiplication is well-defined). In particular, the gauge group can be identified with the smooth sections of $\operatorname{Ad}(P)$ once this convention is fixed.

Examples

1. Trivial principal bundle.
   If $P\cong M\times G$, then $\operatorname{Ad}(P)\cong M\times G$ as a bundle of groups (choosing the obvious global section produces the identification).

2. Frame bundle interpretation.
   If $E\to M$ is a rank $n$ vector bundle and $P=\operatorname{Fr}(E)$ is its frame bundle , then

   $$\operatorname{Ad}(P)=\operatorname{Fr}(E)\times_{GL(n)}GL(n)$$

   can be identified with the bundle $\operatorname{Aut}(E)\to M$ of fiberwise linear automorphisms of $E$ (conjugation is exactly change-of-basis for matrices).

3. Abelian structure group.
   If $G$ is abelian, conjugation is trivial, so $\operatorname{Ad}(P)\cong M\times G$ canonically. For instance, for the Hopf fibration as a principal $U(1)$-bundle, the adjoint bundle is $S^2\times U(1)$.