Adjoint bundle Ad(P)

Let $\pi:P\to M$ be a principal G-bundle for a Lie group $G$ . Let $G$ act on itself on the left by conjugation: $g\cdot h := ghg^{-1}$ .

Construction (adjoint bundle). The adjoint bundle is the associated bundle

\mathrm{Ad}(P) := P\times_G G,

formed using the conjugation action. Each fiber $\mathrm{Ad}(P)_x$ is canonically a group (isomorphic to $G$ , but not canonically identified without choosing a point in $P_x$ ), and the group law is defined fiberwise by

[p,h_1]\cdot [p,h_2] := [p,h_1h_2],

which is well-defined because conjugation is by group automorphisms.

Remarks

A choice of local section $s:U\to P$ identifies $\mathrm{Ad}(P)|_U$ with $U\times G$ , and changes of section act by conjugation.

Examples

If $P$ is trivial, $P\cong M\times G$ , then $\mathrm{Ad}(P)\cong M\times G$ as a bundle of groups.
If $G$ is abelian, conjugation is trivial, so $\mathrm{Ad}(P)\cong M\times G$ for every principal $G$ -bundle.
For the frame bundle of a vector bundle with structure group $\mathrm{GL}(n)$ , $\mathrm{Ad}(P)$ encodes the bundle of change-of-frame maps, with fibers identified (after choosing a frame) with $\mathrm{GL}(n)$ .