If two \({G}\)-bundles \({(E,M,F)}\) and \({(E^{\prime},M,F^{\prime})}\), with the same base space and structure group, also share the same trivializing neighborhoods and transition functions, then they are each called an **associated bundle** with regard to the other. It is possible to construct (up to isomorphism) a unique principal \({G}\)-bundle associated to a given \({G}\)-bundle; going in the other direction, given a principal \({G}\)-bundle and a left action of \({G}\) on a fiber \({F}\), we can construct a unique associated \({G}\)-bundle with fiber \({F}\). In particular, given a principal bundle \({(P,M,G)}\), the rep of \({G}\) on itself by inner automorphisms defines an associated bundle \({(\mathrm{Inn}P,M,G)}\), and the adjoint rep of \({G}\) on \({\mathfrak{g}}\) defines an associated bundle \({(\mathrm{Ad}P,M,\mathfrak{g})}\). If \({G}\) has a linear rep on a vector space \({\mathbb{K}^{n}}\), this rep defines an assocated bundle \({(E,M,\mathbb{K}^{n})}\), which we explore next.

The above depicts how given a principal bundle, we can construct an associated bundle for the action of \({G}\) on a vector space \({\mathbb{K}^{n}}\) by a linear rep, on itself by inner automorphisms, and on its Lie algebra \({\mathfrak{g}}\) by the adjoint rep. The action of the structure group is shown in general and for the case in which \({G}\) is a matrix group, with matrix multiplication denoted as juxtaposition. Although denoted identically, the \({f_{i}}\) are those corresponding to each bundle.

Δ The \({G}\)-bundle \({E}\) with fiber \({F}\) associated to a principal bundle \({P}\) is sometimes written \({E=P\times_{G}F\equiv(P\times F)/G}\), where the quotient space collapses all points in the product space which are related by the right action of some \({g\in G}\) on \({P}\) and the right action of \({g^{-1}}\) on \({F}\). |