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Bundle valued forms

For \(E\searrow M\) a vector bundle, an \(E{\hbox{-}}\)valued differential form on \(M\) is \begin{align*} { { {\Omega}^{\scriptscriptstyle \bullet}} }_{M}(E) \coloneqq{{\Gamma}\qty{E\otimes { { { {\bigwedge}^{\scriptscriptstyle \bullet}} }^{\scriptscriptstyle \bullet}} {\mathbf{T}} {}^{ \vee }M} } \cong { { {\Omega}^{\scriptscriptstyle \bullet}} }_M \otimes_{C^\infty(M; {\mathbf{R}})} {{\Gamma}\qty{E} } \end{align*}

Lie algebra valued forms

\begin{align*} { {\Omega}^{\scriptscriptstyle \bullet}} _{X}(M, {\mathfrak{g}}) \coloneqq{{\Gamma}\qty{( {\mathfrak{g}}\times M) \otimes { { { {\bigwedge}^{\scriptscriptstyle \bullet}} }^{\scriptscriptstyle \bullet}} {\mathbf{T}} {}^{ \vee }M} } \end{align*} # The adjoint bundle

Commonly used to define the adjoint bundle: for \(P\) a principal bundle with structure group \(G\in \mathsf{Lie}{\mathsf{Grp}}\) and $\mathsf{Lie}(G) \coloneqq{\mathfrak{g}}\in \mathsf{Lie} \mathsf{Alg} $, since there is a big adjoint action \begin{align*} G\curvearrowright{\mathfrak{g}}: { \operatorname{Ad} }_g(M) = gMg^{-1} \end{align*} yielding a representation of \(G\). Thus one can form the associated bundle \begin{align*} { \operatorname{ad}} P \coloneqq P \overset{\scriptscriptstyle {{ \operatorname{Ad} }} }{\times}{\mathfrak{g}},\qquad (pg, x)\sim (p, { \operatorname{Ad} }_g(x)) \end{align*}