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CRE5FF5.tmp
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\author{Simon-Raphael Fischer\footnote{Email: \href{mailto:sfischer@ncts.tw}{sfischer@ncts.tw}; ORCiD: \href{https://orcid.org/0000-0002-5859-2825}{0000-0002-5859-2825}} }
\title{Integrating curved Yang-Mills gauge theories}
\subtitle{Gauge theories related to principal bundles equipped with Lie group bundle actions}
\date{\today}
\maketitle
\thispagestyle{empty}
\begin{center}
National Center for Theoretical Sciences, Mathematics Division, National Taiwan University\\
No. 1, Sec. 4, Roosevelt Rd., Taipei City 106, Taiwan Room 503, Cosmology Building, Taiwan
\ \\
\ \\
\ \\
\textbf{Abstract}\footnote[2]{Abbreviations used in this paper: \textbf{LGB} for Lie group bundle, \textbf{LAB} for Lie algebra bundle.}
%\footnote[2]{Abbreviations used in this paper: \textbf{(C)YMH GT} for (curved) Yang-Mills-Higgs gauge theory.}
\begin{abstract}
\small{
In this paper we construct a gauge theory based on principal bundles $\mathcal{P}$ equipped with a Lie group bundle action instead of a Lie group action. Due to the fact that pushforwards of right-translation act now only on the vertical structure of $\mathcal{P}$ we fix a connection on $\mathcal{G}$, which modifies the pushforward via right translation by subtracting the fundamental vector field generated by the Darboux derivative of a section of $\mathcal{G}$.
}
\end{abstract}
\end{center}
\textit{2020 MSC:} Primary 53D17; Secondary 81T13, 17B99.
\textit{Keywords:} \texttt{Mathematical Gauge Theory}, Differential Geometry, High Energy Physics - Theory, Mathematical Physics
\end{titlepage}
%%%%%%%%%%%%%%%%%%%%%%%%%%%% Inhaltsverzeichnis %%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%% Hier beginnt der Hauptteil %%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Introduction}
\subsection{Basic notations and remarks}\label{BasicNotations}
\begin{itemize}
\item The appendix serves for providing extra information or background knowledge which did not fit in the flow of this paper's text. We will sometimes refer to the appendix, but the experienced reader may be able to ignore the appendix.
\item In the main text we usually repeat and reintroduce needed objects for the statements (like "\textbf{Let $M$ be a manifold [...]}" in every statement) (almost) allowing to just read the statements without having read the text introducing it, while the appendix is written as a continuous text which has to be read as a whole in order to understand the essential statements.
\item As usual, there will be definitions of certain objects depending on other elements, and for keeping notations simple we will not always explicitly denote all dependencies. It will be clear by context on which it is based on, that is for example, if we define an object $A$ using the notion of Lie algebra actions $\gamma$ and we write "Let $X$ be an object $A$", then it will be clear by context which Lie algebra action is going to be used, for example given in a previous sentence writing "Let $\gamma$ be a Lie algebra action".
\item Throughout this work we always use Einstein's sum convention if suitable.
\item With $f^*F$ we denote the pullback/pull-back of the fibre bundles $F \to M$ under a smooth map $f: N \to M$. Similarly we denote the pullbacks of sections of a fibre bundle.
%We will also have sections $F$ as an element of $\Gamma\left( \left(\bigotimes_{m=1}^{l} E_m^*\right) \otimes E_{l+1} \right)$, where $E_1, \dots E_{l+1}$ ($l \in \mathbb{N}$) are real vector bundles of finite rank over $N$. Those pull-back as section, denoted by $\Phi^*F$, we will view as an element of $\Gamma\left( \mleft(\bigotimes_{m=1}^{l} \mleft(\Phi^*E_m\mright)^*\mright) \otimes \Phi^*E_{l+1} \right)$, and it is essentially given by
%\bas
%(\Phi^*F)(\Phi^*\nu_1, \dotsc , \Phi^*\nu_l)
%&=
%\Phi^*\mleft( F\mleft( \nu_1, \dotsc, \nu_l \mright) \mright)
%\eas
%for all $\nu_1 \in \Gamma(E_1), \dotsc, \nu_l \in \Gamma(E_l)$, using that pullbacks of sections generate the sections of a pullback bundle.
\item For $V \to M$ a vector bundle over $M$ do not confuse the pull-back of sections with the pull-back of forms $\omega \in \Omega^l(M; V)$ ($l \in \mathbb{N}_0$), here denoted by $f^!\omega$, which is an element of $\Gamma\left( \mleft(\bigwedge_{m=1}^{l} \mathrm{T}^*M \mright) \otimes f^*V \right) \cong \Omega^l(M; f^*V)$, and not of $\Gamma\left( \mleft(\bigotimes_{m=1}^{l} \mleft(f^*\mathrm{T}N\mright)^*\mright) \otimes f^*V \right)$ like $f^*\omega$.
%$f^!\omega$ is defined by
%\ba
%\mleft.\mleft(f^!\omega\mright)(Y_1, \dots, Y_l)\mright|_p
%&\coloneqq
%\omega_{f(p)}\mleft(\mathrm{D}_pf\mleft(\mleft.Y_1\mright|_p\mright), \dots, \mathrm{D}_pf\mleft(\mleft.Y_l\mright|_p\mright)\mright)
%\ea
%for all $p \in M$ and $Y_1, \dots, Y_l \in \mathfrak{X}(M)$.
\item Let $F \stackrel{\pi_F}{\to} M$ and $G \stackrel{\pi_G}{\to} N$ be two fibre bundles over smooth manifolds $M$ and $N$, and let $\phi: N \to M$ be a smooth map. Furthermore, let us assume we have a morphism $\Phi: G \to F$ of fibre bundles over $\phi$, that is, $\Phi$ is a smooth map such that the following diagram commutes
\begin{center}
\begin{tikzcd}
G \arrow{d}{\pi_G} \arrow{r}{\Phi} & F \arrow{d}{\pi_F}\\
N \arrow{r}{\phi} & M
\end{tikzcd}
\end{center}
especially, $\pi_F \circ \Phi = \phi \circ \pi_G$.
We make often use of that such morphisms have a 1:1 correspondence to \textbf{base-preserving} fibre bundle morphisms $\widetilde{\Phi}: G \to \phi^*F$, \textit{i.e.}\ $\widetilde{\Phi}$ is a smooth map with $\phi^*\pi_F \circ \widetilde{\Phi} = \pi_G$. For $p \in N$ the morphism $\widetilde{\Phi}$ has the form
\bas
\widetilde{\Phi}_p
&=
(p, \Phi_p),
\eas
that is,
\bas
\widetilde{\Phi}_p(g)
&=
\bigl(p, \Phi_p(g) \bigr)
\eas
for all $g \in G_p$, which is well-defined since $\Phi_p(g) \in F_{\phi(p)}$. The map $\widetilde{\Phi} \mapsto \Phi \coloneqq \mathrm{pr}_2 \circ \widetilde{\Phi}$ is then a bijective map between base-preserving morphisms $G \to \phi^*F$ and morphisms $G \to F$ over $\phi$, where $\mathrm{pr}_2$ is the projection onto the second component.
In total, $\widetilde{\Phi}$ is a base-preserving morphism if and only if $\Phi$ is a morphism over $\phi$; in fact, one defines pullback bundles in such a way that this equivalence holds. Observe that $\widetilde{\Phi}$ is an isomorphism (diffeomorphism) if and only if $\Phi$ is a fibre-wise isomorphism (diffeomorphism).
One can extend all of this similarly for more specific types of morphisms like vector bundle-morphisms.
Most of the time we will not mention this 1:1 correspondence explicitly, it should be clear by context. Hence, we will also denote $\widetilde{\Phi}$ by $\Phi$. In fact, we usually calculate with $\widetilde{\Phi}$, while $\Phi$ and its diagram may only arise to give an illustration about the geometry.
\item When we differentiate maps $\gamma$ depending on just one parameter $t \in \mathbb{R}$, then we may shortly write
\bas
\frac{\mathrm{d}}{\mathrm{d}t} \gamma(t)
&\coloneqq
\mleft.\frac{\mathrm{d}}{\mathrm{d}t} [t \mapsto \gamma]\mright|_t.
\eas
\end{itemize}
\subsection{Assumed background knowledge}
It is highly recommended to have basic knowledge about differential geometry and gauge theory as presented in \cite[especially Chapter 1 to 5]{Hamilton}, and we will follow the style and labeling as in \cite{Hamilton} when we generalize certain notions; however, sometimes we will still give explicit references to help with more technical details. It can be useful to have knowledge about Lie algebra and Lie group bundles, and even Lie algebroids and Lie groupoids, but we will introduce their basic notions such that it is not necessarily needed to have knowledge about these upfront.
We also often give references about Lie group bundles (LGBs), but the given references are often about Lie groupoids. If the reader has no knowledge about Lie groupoids, then it is important to know that LGBs are a special example of Lie groupoids; Lie groupoids carry "two projections", called \textbf{source} and \textbf{target}. An LGB is a special example of a Lie groupoid whose source equals the target.\footnote{But not every Lie groupoid with equal source and target is an LGB, they're in general bundles of Lie groups which is not completely the same; this nuance will not be important here.} If you look into such a reference, then the source and target are often denoted by $\alpha$ and $\beta$, or by $s$ and $t$; simply put both to be the same and identify these with our bundle projection which we often denote by $\pi$. In that way it should be possible to read the references without the need to know Lie groupoids. However, we try to re-prove the needed statements such that these types of references could be avoided by the reader.
See also the previous subsection about notions we assume to be known.
%\twocolumn
%\section{Basic definitions}\label{BasicDefinitions}
%
%In the following, we denote with $V^*$ the dual of a vector bundle $V \to N$ over a smooth manifold $N$, and $\Phi^*V$ denotes the pull-back of $V$ by $\Phi: M \to N$, a smooth map from a smooth manifold $M$ to $N$. We have a similar notation for the pull-back of sections, especially we will have sections $F$ as an element of $\Gamma\left( \left(\bigotimes_{m=1}^{l} E_m^*\right) \otimes E_{l+1} \right)$, where $E_1, \dots, E_{l+1} \to N$ ($l \in \mathbb{N}$) are real vector bundles of finite rank over a smooth manifold $N$, and $\Gamma(\cdot)$ denotes the space of smooth sections. Then we view the pull-back $\Phi^*F$ as an element of $\Gamma\left( \mleft(\bigotimes_{m=1}^{l} \mleft(\Phi^*E_m\mright)^*\mright) \otimes \Phi^*E_{l+1} \right)$, and it is essentially given by
%\bas
%(\Phi^*F)(\Phi^*\nu_1, \dotsc , \Phi^*\nu_l)
%&=
%\Phi^*\mleft( F\mleft( \nu_1, \dotsc, \nu_l \mright) \mright)
%\eas
%for all $\nu_1 \in \Gamma(E_1), \dotsc, \nu_l \in \Gamma(E_l)$. In general we also make use of that sections of $\Phi^*E$ can be viewed as sections of $E$ along $\Phi$, where $E \stackrel{\pi}{\to} N$ is any vector bundle over $N$. Let $\mu \in \Gamma(\Phi^*E)$, then it has the form $\mu_p = (p, u_p)$ for all $p \in M$, where $u_p \in E_{\Phi(p)}$, the fibre of $E$ at $\Phi(p)$; and a section $\nu$ of $E$ along $\Phi$ is a smooth map $M \to E$ such that $\pi \circ \nu = \Phi$. Then on one hand $\mathrm{pr}_2 \circ \mu$ is a section along $\Phi$, where $\mathrm{pr}_2$ is the projection onto the second component, and on the other hand $M \ni p \mapsto (p, \nu_p)$ defines an element of $\Gamma(\Phi^*E)$. With that one can show that there is a 1:1 correspondence of $\Gamma(\Phi^*E)$ with sections along $\Phi$. Similarly, vector bundle morphisms $L: G \to E$ over $\Phi$ have 1:1 correspondences to base-preserving vector bundle morpishms $G \to \Phi^*E$, where $G \to M$ is a vector bundle over $M$. We do not necessarily mention it when we make use of such trivial identifications, it should be clear by the context. For example $\mathrm{D}\Phi$ denotes the total differential of $\Phi$ (also called tangent map). It can be viewed as a vector bundle morphism $\mathrm{T}M \to \mathrm{T}N$ over $\Phi$, and we often view it as an element of $\Omega^1(M; \Phi^*\mathrm{T}N)$ by $\mathfrak{X}(M) \ni Y \mapsto \mathrm{D}\Phi(Y)$, where $\mathrm{D}\Phi(Y) \in \Gamma(\Phi^*\mathrm{T}N), M \ni p \mapsto \mathrm{D}_p\Phi(Y_p)$.
%
%Additionally, with $\Omega^k(N; E)$ ($k \in \mathbb{N}_0$) we denote $k$-forms on $N$ with values in a vector bundle $E \to N$, and we always use the Einstein's sum convention. If one has a connection $\nabla$ on a vector bundle $V \to N$, then one has the notion of the exterior covariant derivative on $\Omega^p(M;E)$, denoted by $\mathrm{d}^\nabla$. In the case of a trivial vector bundle $V=N \times W \to N$, where $W$ is some vector space, we will often use the \textbf{canonical flat connection} for $\nabla$, defined by $\nabla \nu = 0$, where $\nu$ is a constant section of $N \times W$, see \textit{e.g.}~\cite[Example 5.1.7; page 260f.]{hamilton}\ for a geometric interpretation as horizontal distribution. The canonical flat connection is clearly uniquely defined (if a trivialization is given) because constant sections generate all sections and due to the Leibniz rule and linearity of $\nabla$. Let $\mleft( e_a \mright)_a$ be a constant global frame of $N \times W$, thence,
%\bas
%\mathrm{d}^\nabla \omega
%&=
%\mathrm{d} \omega^a \otimes e_a
%\eas
%for all $\omega \in \Omega^p(M; W)$, where we write $\omega= \omega^a \otimes e_a$. Hence, we define
%\ba
%\mathrm{d}\omega
%&\coloneqq
%\mathrm{d}^\nabla \omega,
%\ea
%when $\nabla$ is the canonical flat connection. $\mathrm{d}$ is clearly a differential.
%
%As usual, there will be definitions of certain objects depending on other elements, and for keeping notations simple we will not always explicitly denote all dependencies. It will be clear by context on which it is based on, that is, when we define an object $A$ using the notion of Lie algebra actions $\gamma$ and we write "Let $A$ be [as defined before]", then it will be clear by context which Lie algebra action is going to be used, for example given in a previous sentence writing "Let $\gamma$ be a Lie algebra action".
%%, and recall the following wedge product\footnote{As also defined in \cite[\S 5, third part of Exercise 5.15.12; page 316]{hamilton}.} of forms with values in a vector bundle $E$ and values in its space of endomorphisms $\mathrm{End}(E)$,
%%\bas
%%\wedge: \Omega^k(N; \mathrm{End}(E)) \times \Omega^l(N; E)
%%&\mapsto
%%\Omega^{k+l}(N; E) \\
%%(T, \omega) &\mapsto T \wedge \omega
%%\eas
%%for all $k, l \in \mathbb{N}_0$, given by
%%\ba\label{DefVonWedgedemitEnd}
%%\mleft( T \wedge \omega \mright) \mleft( Y_1, \dotsc, Y_{k+l} \mright)
%%&\coloneqq
%%\frac{1}{k! l!} \sum_{\sigma \in S_{k+l}} \mathrm{sgn}(\sigma) ~
%%T \mleft( Y_{\sigma(1)}, \dotsc, Y_{\sigma(k)} \mright)
%%\mleft( \omega\mleft( Y_{\sigma(k+1)}, \dotsc, Y_{\sigma(k+l)} \mright) \mright),
%%\ea
%%where $S_{k+l}$ is the group of permutations $\{1, \dotsc, k+l\}$. This is then locally given by, with respect to a frame $\mleft( e_a \mright)_a$ of $E$,
%%\bas
%%T \wedge \omega &= T(e_a) \wedge w^a,
%%\eas
%%where $T$ acts as an endomorphism on $e_a$, \textit{i.e.}~$T(e_a) \in \Omega^k(N; E)$, and $\omega = \omega^a \otimes e_a$. Also recall that there is the canonical extension of $\nabla$ on $\mathrm{End}(E)$ by forcing the Leibniz rule. We still denote this connection by $\nabla$, too.
%
%We also need the following definitions.
%
%\begin{definitions}{Graded extension of products, \newline \cite[generalization of Definition 5.5.3; page 275]{hamilton}}{GradingOfProducts}
%Let $l \in \mathbb{N}$ and $E_1, \dots E_{l+1} \to N$ be vector bundles over a smooth manifold $N$, and $F \in \Gamma\left( \left(\bigotimes_{m=1}^{l} E_m^*\right) \otimes E_{l+1} \right)$. Then we define the \textbf{graded extension of $F$} as
%\bas
%\Omega^{k_1}(N; E_1) \times \dots \times \Omega^{k_l}(N; E_l)
%&\to \Omega^{k}(N; E_{l+1}), \\
%(A_1, \dots, A_l)
%&\mapsto
%F\mleft(A_1\stackrel{\wedge}{,} \dotsc \stackrel{\wedge}{,} A_l\mright),
%\eas
%where $k := k_1+\dots k_l$ and $k_i \in \mathbb{N}_0$ for all $i\in \{1, \dots, l\}$. $F\mleft(A_1\stackrel{\wedge}{,} \dotsc \stackrel{\wedge}{,} A_l\mright)$ is defined as an element of $\Omega^{k}(N; E_{l+1})$ by
%\bas
%&F\mleft(A_1\stackrel{\wedge}{,} \dotsc \stackrel{\wedge}{,} A_l\mright)\mleft(Y_1, \dots, Y_{k}\mright)
%\coloneqq \\
%&\frac{1}{k_1! \cdot \dots \cdot k_l!} \sum_{\sigma \in S_{k}} \mathrm{sgn}(\sigma) ~ F\left( A_1\left( Y_{\sigma(1)}, \dots, Y_{\sigma(k_1)} \right), \dots, A_l\left( Y_{\sigma(k-k_l+1)}, \dots, Y_{\sigma(k)} \right) \right)
%\eas
%for all $Y_1, \dots, Y_{k} \in \mathfrak{X}(N)$, where $S_{k}$ is the group of permutations of $\{1, \dots, k\}$ and $\mathrm{sgn}(\sigma)$ the signature of a given permutation $\sigma$.
%
%$\stackrel{\wedge}{,}$ may be written just as a comma when a zero-form is involved.
%
%Locally, with respect to given frames $\mleft( e^{(i)}_{a_i} \mright)_{a_i}$ of $E_i$, this definition has the form
%\ba\label{CoordExprOfGradedExtension}
%F\mleft(A_1\stackrel{\wedge}{,} \dotsc \stackrel{\wedge}{,} A_l\mright)
%&=
%F\mleft(e^{(1)}_{a_1}, \dotsc, e^{(l)}_{a_l}\mright) \otimes A_1^{a_1} \wedge \dotsc \wedge A_l^{a_l}
%\ea
%for all $A_i = A_i^{a_i} \otimes e^{(i)}_{a_i}$, where $A_i^{a_i}$ are $k_i$-forms on $N$.
%\end{definitions}
%
%\begin{remark}
%\leavevmode\newline
%Assume $F \in \Gamma\left( \mleft(\bigwedge_{m=1}^{l} \mathrm{T}^*N \mright) \otimes E \right) \cong \Omega^l(N; E)$ for some vector bundle $E$, \textit{i.e.}~$F$ is an $l$-form on $N$ with values in $E$. The pull-back $\Phi^*F$ by $\Phi$ can be then viewed as an element of $\Gamma\left( \bigwedge_{m=1}^{l} \mleft(\Phi^*\mathrm{T}N\mright)^* \otimes \Phi^*E \right)$.
%
%Do not confuse this pull-back with the pull-back of forms, here denoted by $\Phi^!F$, which is an element of $\Gamma\left( \mleft(\bigwedge_{m=1}^{l} \mathrm{T}^*M \mright) \otimes \Phi^*E \right) \cong \Omega^l(M; \Phi^*E)$ defined by
%\ba
%\mleft.\mleft(\Phi^!F\mright)(Y_1, \dots, Y_l)\mright|_p
%&\coloneqq
%F_{\Phi(p)}\mleft(\mathrm{D}_p\Phi\mleft(\mleft.Y_1\mright|_p\mright), \dots, \mathrm{D}_p\Phi\mleft(\mleft.Y_l\mright|_p\mright)\mright)
%\ea
%for all $p \in M$ and $Y_1, \dots, Y_l \in \mathfrak{X}(M)$. Then
%\ba\label{EqPullBackFormelFuerVerschiedeneDefinitionen}
%\Phi^!F
%&=
%\frac{1}{l!}~
%\mleft(\Phi^*F\mright) ( \underbrace{\mathrm{D}\Phi \stackrel{\wedge}{,} \dotsc \stackrel{\wedge}{,} \mathrm{D}\Phi}_{l \text{ times}} )
%\ea
%by using the anti-symmetry of $F$ and Def.~\ref{def:GradingOfProducts}, \textit{i.e.}
%\bas
%&\mleft.\frac{1}{l!}~
%\Big(\mleft(\Phi^*F\mright) ( \mathrm{D}\Phi \stackrel{\wedge}{,} \dotsc \stackrel{\wedge}{,} \mathrm{D}\Phi ) \Big) (Y_1, \dots, Y_l)\mright|_p \\
%&\hspace{1cm}
%=
%\frac{1}{l!}~
%\sum_{\sigma \in S_{l}} \mathrm{sgn}(\sigma) ~ \underbrace{(\Phi^*F)\mleft(\mathrm{D}\Phi\mleft(Y_{\sigma(1)}\mright), \dots, \mathrm{D}\Phi\mleft(Y_{\sigma(l)}\mright)\mright)}_{\mathclap{= \mathrm{sgn}(\sigma) ~ (\Phi^*F)\mleft(\mathrm{D}\Phi\mleft(Y_{1}\mright), \dots, \mathrm{D}\Phi\mleft(Y_{l}\mright)\mright)}}\Big|_p \\
%&\hspace{1cm}
%=
%\frac{1}{l!}~ \underbrace{\mleft( \sum_{\sigma \in S_{l}} 1 \mright)}_{= l!} ~
%F_{\Phi(p)}\mleft(\mathrm{D}_p\Phi\mleft(\mleft.Y_{1}\mright|_p\mright), \dots, \mathrm{D}_p\Phi\mleft(\mleft.Y_{l}\mright|_{p}\mright)\mright) \\
%&\hspace{1cm}
%= \mleft.\mleft(\Phi^!F\mright)(Y_1, \dots, Y_l)\mright|_p
%\eas
%for all $p \in M$ and $Y_1, \dots, Y_l \in \mathfrak{X}(M)$.
%\end{remark}
%
%In case of antisymmetric tensors we of course preserve that.
%
%\begin{propositions}{Graded extensions of antisymmetric tensors}{GradedExtensionPlusAntiSymm}
%Let $E_1, E_2 \to N$ be real vector bundles of finite rank over a smooth manifold $N$, $F \in \Omega^2(E_1; E_2)$. Then
%\ba
%F \mleft( A \stackrel{\wedge}{,} B \mright)
%&=
%-\mleft( -1 \mright)^{km}
%F \mleft( B \stackrel{\wedge}{,} A \mright)
%\ea
%for all $A \in \Omega^k(N; E_1)$ and $B \in \Omega^m(N; E_2)$ ($k,m \in \mathbb{N}_0$). Similarly extended to all $F \in \Omega^l(E_1; E_2)$.
%\end{propositions}
%
%\begin{remark}
%\leavevmode\newline
%This is a generalization of similar relations just using the Lie algebra bracket $\mleft[ \cdot, \cdot\mright]_{\mathfrak{g}}$ of a Lie algebra $\mathfrak{g}$, see \cite[\S 5, first statement of Exercise 5.15.14; page 316]{hamilton}.
%\end{remark}
%
%\begin{proof}
%\leavevmode\newline
%Trivial by using Eq.~\eqref{CoordExprOfGradedExtension}.
%\end{proof}
%
%We also need to know what a Lie algebroid is, a generalization of both, tangent bundles and Lie algebras; this concept will just be defined, refer to the references for thorough discussions of these definitions, especially \cite{mackenzieGeneralTheory} and \cite[\S VII; page 113ff.]{DaSilva}.
%
%\begin{definitions}{Lie algebroid, \newline \cite[\S 3.3, first part of Definition 3.3.1; page 100]{mackenzieGeneralTheory}}{test}
%%\leavevmode\newline
%Let $E \to N$ be a real vector bundle of finite rank. Then $E$ is a smooth Lie algebroid if there is a bundle map $\rho: E \to \mathrm{T}N$, called the \textbf{anchor}, and a Lie algebra structure on $\Gamma(E)$ with Lie bracket $\mleft[ \cdot, \cdot \mright]_E$ satisfying
%\ba
%\mleft[\mu, f \nu\mright]_E = f \mleft[\mu, \nu\mright]_E + \mathcal{L}_{\rho(\mu)}(f) ~ \nu
%\label{eq:E-Leibniz}
%\ea
%for all $f \in C^\infty(N)$ and $\mu, \nu \in \Gamma(E)$, where $\mathcal{L}_{\rho(\mu)}(f)$ is the action of the vector field $\rho(\mu)$ on the function $f$ by derivation. We will sometimes denote a Lie algebroid by $\mleft( E, \rho, \mleft[ \cdot, \cdot \mright]_E \mright)$.
%%We will sometimes denote a Lie algebroid by $\mleft( E, \rho, \mleft[ \cdot, \cdot \mright]_E \mright)$.
%\end{definitions}
%
%%Tangent bundles are a canonical example of Lie algebroids, their anchor is the identity with which we also equip them; another canonical example with zero anchor are the Lie algebra bundles:
%Tangent bundles and bundles of Lie algebras are canonical examples of Lie algebroids, their anchor is the identity and zero, respectively. The important example for us is a mixture of those examples:
%
%\begin{propositions}{Action Lie algebroids, \cite[\S 16.2, Example 5; page 114]{DaSilva}}{ActionLieoidsAreOids}
%Let $\mleft(\mathfrak{g}, \mleft[\cdot, \cdot \mright]_{\mathfrak{g}}\mright)$ be some Lie algebra equipped with a Lie algebra action $\gamma: \mathfrak{g} \to \mathfrak{X}(N)$ on a smooth manifold $N$. Then there is a unique Lie algebroid structure on $E = N \times \mathfrak{g}$ such that we have
%\ba
%\rho(\nu)
%&=
%\gamma(\nu),
%\\
%\mleft[\mu, \nu\mright]_E
%&=
%\mleft[\mu, \nu\mright]_{\mathfrak{g}}
%\ea
%for all constant sections $\mu, \nu \in \Gamma(E)$. We call this structure \textbf{action Lie algebroid}.
%\end{propositions}
\section{Lie group bundles (LGBs)}
\subsection{Definition}
%\pagebreak
\begin{definitions}{Lie group bundle, \cite[\S 1.1, Def.\ 1.1.19; p. 11]{mackenzieGeneralTheory}}{LieGroupBundle}
Let $G, \mathcal{G}, M$ be smooth manifolds. A fibre bundle
\begin{center}
\begin{tikzcd}
G \arrow{r} & \mathcal{G} \arrow{d}{\pi} \\
& M
\end{tikzcd}
\end{center}
is called a \textbf{Lie group bundle} if:
\begin{enumerate}
\item $G$ and each fibre $\mathcal{G}_x \coloneqq \pi^{-1}\mleft( \{x\} \mright)$, $x\in M$, are Lie groups;
\item there exists a bundle atlas $\mleft\{ \mleft( U_i, \phi_i \mright) \mright\}_{i \in I}$ such that the induced maps
\bas
\phi_{ix}
&\coloneqq
\mathrm{pr}_2 \circ \mleft. \phi_i\mright|_{\mathcal{G}_x}: \mathcal{G}_x \to G
\eas
are Lie group isomorphisms, where $I$ is an (index) set, $U_i$ are open sets covering $M$, $\phi_i: \mathcal{G}|_U \to U \times G$ subordinate trivializations, and $\mathrm{pr}_2$ the projection onto the second factor. This atlas will be called \textbf{Lie group bundle atlas} or \textbf{LGB atlas}.
\end{enumerate}
We also often say that \textbf{$\mathcal{G}$ is an LGB (over $M$)}, whose structural Lie group is either clear by context or not explicitly needed; and we may also denote LGBs by $G \to \mathcal{G} \stackrel{\pi}{\to} M$.
\end{definitions}
\begin{remarks}{Principal and Lie group bundles}{LiegroupbundlesNotPrincipalBundles}
Beware, a Lie group bundle is \textbf{not} the same as a principal bundle $P \to M$ with the same fibre type $G$. First of all, the fibres of $P$ are just diffeomorphic to a Lie group, a priori they carry no Lie group structure, while the fibres of $\mathcal{G}$ carry a Lie group structure.
\newline
Second, on $P$ we have a multiplication given as an action of $G$ on $P$
\bas
P \times G \to P,
\eas
preserving the fibres $P_x$ ($x\in M$) and simply transitive on them. Restricted on $P_x$ we have
\bas
P_x \times G \to P_x.
\eas
For $\mathcal{G}$ we have canonically a multiplication over $x$ given by
\bas
\mathcal{G}_x \times \mathcal{G}_x \to \mathcal{G}_x,
\eas
also clearly simply transitive. Observe, the second factor is not "constant", \textit{i.e.}\ we do not have $\mathcal{G}_x \times G \to \mathcal{G}_x$ in general. Hence, there is in general no well-defined product $\mathcal{G} \times \mathcal{G} \to \mathcal{G}$ or $\mathcal{G} \times G \to \mathcal{G}$.
\newline
All of that is also resembled in the existence of sections. The existence of a section of $P$ has a 1:1 correspondence to trivializations of $P$, which is why $P$ in general only admits sections locally; see \textit{e.g.}\ \cite[\S 4.2, Thm.\ 4.2.19; page 219f.]{Hamilton}. $\mathcal{G}$ clearly admits always a global section, even if $\mathcal{G}$ is non-trivial; just take the section which assigns each base point the neutral element of its fibre.
\end{remarks}
If $M$ is a point we recover the notion of Lie groups, and, as usual, we have the notion of trivial LGBs:
\begin{examples}{Trivial LGB}{TrivialLGBundle}
The \textbf{trivial LGB} is given as the product manifold $M \times G \to M$ with canonical multiplication $(x, g) \cdot (x, q) \coloneqq (x, gq)$.
\end{examples}
We are also interested into LGB bundle morphisms:
\begin{definitions}{LGB morphism, \newline \cite[\S 1.2, special situation of Def.\ 1.2.1 \& 1.2.3, page 12]{mackenzieGeneralTheory}}{LGB morphism}
Let $\mathcal{H} \stackrel{\pi_{\mathcal{H}}}{\to} N$ and $\mathcal{G} \stackrel{\pi_{\mathcal{G}}}{\to} M$ be two LGBs over two smooth manifolds $N$ and $M$. An \textbf{LGB morphism $F$ over $f$} is a pair of smooth maps $F: \mathcal{H} \to \mathcal{G}$ and $f: N \to M$ such that
\ba\label{FibreRelationOverf}
\pi_{\mathcal{G}} \circ F &= f \circ \pi_{\mathcal{H}},\\
F(gq) &= F(g) ~ F(q)\label{LGBHomomorph}
\ea
for all $g, q \in \mathcal{H}$ with $\pi_{\mathcal{H}}(g) = \pi_{\mathcal{H}}(q)$. We then also say that \textbf{$F$ is an LGB morphism over $f$}. If $N = M$ and $f = \mathrm{id}_M$, then we often omit mentioning $f$ explicitly and just write that \textbf{$F$ is a (base-preserving) LGB morphism}.
We speak of an \textbf{LGB isomorphism (over $f$)} if $F$ is a diffeomorphism.
\end{definitions}
\begin{remark}\label{LGBMOrphismRemark}
\leavevmode\newline
\indent $\bullet$ The right hand side of Eq.\ \eqref{LGBHomomorph} is well-defined because of Eq.\ \eqref{FibreRelationOverf}.
$\bullet$ It is clear that condition 2 in Def.\ \ref{def:LieGroupBundle} is equivalent to say that $\mathcal{G}$ is locally isomorphic to a trivial LGB; as one may have expected already.
$\bullet$ If $F$ is a diffeomorphism, then also $f$: By Eq.\ \eqref{FibreRelationOverf} surjectivity of $f$ is clear; for $y \in M$ just take any $g \in \mathcal{G}_y$, and since $F$ is a bijective, we have a $q \in \mathcal{H}_x$ for some $x\in N$ with $F(q)=g$. By Eq.\ \eqref{FibreRelationOverf} we have $y = \pi_{\mathcal{G}}(F(q)) \stackrel{\eqref{FibreRelationOverf}}{=} f(x)$, thence, surjectivity follows. For injectivity we know by Eq.\ \eqref{LGBHomomorph} and \eqref{FibreRelationOverf} that $F\mleft(e^{\mathcal{H}}_x\mright) = e^{\mathcal{G}}_{f(x)}$, where $e^{\mathcal{H}}_x$ and $e^{\mathcal{G}}_{f(x)}$ denote the unique neutral elements of $\mathcal{H}_x$ and $\mathcal{G}_{f(x)}$, respectively. Assume that there are $x, x^\prime \in N$ with $f(x) = f(x^\prime)$, then we can derive
\bas
F\mleft(e^{\mathcal{H}}_x\mright)
&=
e^{\mathcal{G}}_{f(x)}
=
e^{\mathcal{G}}_{f(x^\prime)}
=
F\mleft(e^{\mathcal{H}}_{x^\prime}\mright).
\eas
Then we have $e^{\mathcal{H}}_x = e^{\mathcal{H}}_{x^\prime}$ due to that $F$ is bijective, and hence $x = x^\prime$. Therefore $f$ is bijective. Finally, $F^{-1}$ is by assumption also a diffeomorphism, Eq.\ \eqref{LGBHomomorph} clearly carries over, and Eq.\ \eqref{FibreRelationOverf} is w.r.t.\ $f^{-1}$, that is
\bas
\pi_{\mathcal{H}} \circ F^{-1} &= f^{-1} \circ \pi_{\mathcal{G}}.
\eas
Since $\pi_{\mathcal{H}} \circ F^{-1}$ is smooth and $\pi_{\mathcal{G}}$ is a smooth surjective submersion, it follows that $f^{-1}$ is smooth; this is a well-known fact for right-compositions with surjective submersions, see \textit{e.g.}\ \cite[\S 3.7.2, Lemma 3.7.5, page 153]{Hamilton}. We can conclude that $f$ is a diffeomorphism. Observe that we also concluded that $F^{-1}$ is an LGB isomorphism, too.
\end{remark}
Similar to the case of Lie groups, \textit{the} example of an LGB are the automorphisms of a vector bundle.
\begin{examples}{Automorphisms of a vector bundle,\newline \cite[\S 1.1, special situation of Ex.\ 1.1.12, page 8]{mackenzieGeneralTheory}}{AutOfVectorBundleAnLGB}
Let $V\to M$ be a vector bundle and $\mathrm{Aut}(V) \to M$ its bundle of fibre-wise automorphisms (not to be confused with the sections of $\mathrm{Aut}(V)$ which are the base-preserving automorphisms of $V$). Denote with $W$ the structural vector space of $V$, then $\mathrm{Aut}(V)$ is an LGB with structural Lie group $\mathrm{Aut}(W)$. It is clear that each fibre of $\mathrm{Aut}(V)$ is a Lie group, and the LGB atlas is directly inherited by a vector bundle atlas $\mleft\{ \mleft( U_i, L_i \mright) \mright\}_{i \in I}$ of $V$, where we use a similar notation as for LGB atlases, especially we have vector bundle trivializations $L_{i}: \mleft.V\mright|_{U_i} \to U_i \times W$. Then define an LGB atlas over the same open covering $\mleft( U_i \mright)_i$ by
\bas
\mleft. \mathrm{Aut}(V) \mright|_{U_i} &\to U_i \times \mathrm{Aut}(W),\\
T &\mapsto \mleft.L_{i} \circ T \circ L_{i}^{-1}\mright|_{\{x\} \times W},
\eas
where $T \in \mleft.\mathrm{Aut}(V)\mright|_x = \mathrm{Aut}(V_x)$,
and $U_i \times \mathrm{Aut}(W)$ acts canonically on $U_i \times W$ in a fibre-wise sense. Then it is trivial to check that these give local trivializations such that $\mathrm{Aut}(V)$ carries the structure as an LGB.
\end{examples}
\subsection{Associated Lie group bundles}
For another important example recall that there is the notion of associated fibre bundles; following and stating the results of \cite[\S1, Construction 1.3.8, page 20]{mackenzieGeneralTheory} and \cite[\S 4.7, page 237ff.; see also Rem.\ 4.7.8, page 242f.]{Hamilton}: Let $P \stackrel{\pi_P}{\to} M$ be a principal bundle with structural Lie group $G$ over a smooth manifold $M$, $N$ another smooth manifold, and $\Psi$ a smooth left $G$-action denoted by
\bas
G \times N &\to N,\\
(g, v) &\mapsto \Psi(g, v) \coloneqq g \cdot v.
\eas
Then we have a right $G$-action on $P \times N$ given by
\bas
(P \times N) \times G &\to P \times N,\\
(p,v,g) &\mapsto \mleft( p \cdot g, g^{-1} \cdot v \mright),
\eas
and one can show that the quotient under this action, $P\times_\Psi N \coloneqq ( P \times N) \Big/ G$, yields the structure of a fibre bundle
\begin{center}
\begin{tikzcd}
N \arrow{r} & P\times_\Psi N \arrow{d}{\pi_{P\times_\Psi N}} \\
& M
\end{tikzcd}
\end{center}
such that the map to the equivalence classes $P \times N \to P \times_\Psi N$, $(p, v) \mapsto [p, v]$, is a smooth surjective submersion,
where the projection $\pi_{P\times_\Psi N}: P\times_\Psi N \to M$ is given by
\bas
\pi_{P\times_\Psi N}\mleft( [p, v] \mright)
&\coloneqq
\pi_P(p)
\eas
for all $[p, v] \in P\times_\Psi N$. For $x \in M$, the fibre $\mleft(P\times_\Psi N\mright)_x$ is given by $\mleft( P_x \times N \mright) \Big/ G = P_x \times_\Psi N$, and the fibre is diffeomorphic to $N$ by $N \ni v \mapsto [p, v] \in \mleft(P\times_\Psi N\mright)_x$ for a fixed $p \in P_x$. We will frequently use this diffeomorphism in the following without further notice.
A very important example are of course associated vector bundles, related to $N$ being a vetor space. We need a similar concept for Lie groups.
\begin{definitions}{Lie group representation on Lie groups, \newline \cite[special situation of the comment after Ex.\ 1.7.14, page 47]{mackenzieGeneralTheory}}{LieGroupActingOnLieGroup}
Let $G, H$ be Lie groups. Then a \textbf{Lie group representation of $G$ on $H$} is a smooth left action $\psi$ of $G$ on $H$
\bas
G \times H
&\to H,\\
(g,h)
&\mapsto
\psi_g(h)
\coloneqq
\psi(g, h)
\eas
such that
\ba
\psi_g(hq)
&=
\psi_g(h)
~ \psi_g(q)
\ea
for all $g \in G$ and $h,q \in H$.
\end{definitions}
\begin{remarks}{Note about labeling}{WhyRepresentation}
Observe that we have by the definition of group actions
\bas
\psi_{gg^\prime}
&=
\psi_g \circ \psi_{g^\prime}
\eas
for all $g, g^\prime \in G$, viewing $\psi_g$ as a map $H \to H$. Therefore we can view the action $\psi$ as a group homomorphism
\bas
G &\to \mathrm{Aut}(H),
\eas
where $\mathrm{Aut}(H)$ is the set of Lie group automorphisms. The similarity to Lie group representations on vector spaces is obvious, thence the name.
\newline
This definition is of course also motivated by various references pointing out that Lie group representations define Lie group actions with extra properties; see for example \cite[\S 3, Ex.\ 3.4.2, page 143f.]{Hamilton}. In \cite[comments after Ex.\ 1.7.14, page 47]{mackenzieGeneralTheory} this definition is also called \textit{action by Lie group isomorphisms}.
\end{remarks}
With this we can discuss and define associated Lie group bundles; the following definition is clearly motivated by the definition of associated vector bundles as \textit{e.g.}\ provided in \cite[\S 4, Thm.\ 4.7.2, page 239f.]{Hamilton}.
\begin{theorems}{Associated Lie group bundle as quotient}{AssociatedGroupBundlesHaveGroupStructure}
Let $G, H$ be Lie groups, $P \stackrel{\pi_P}{\to} M$ a principal $G$-bundle over a smooth manifold $M$, and $\psi$ a $G$-representation on $H$. Then $\mathcal{H} \coloneqq P \times_\psi H$ is an LGB
\begin{center}
\begin{tikzcd}
H \arrow{r} & \mathcal{H} \arrow{d}{\pi} \\
& M
\end{tikzcd}
\end{center}
with projection $\pi$ given by
\ba
\mathcal{H} &\to M,\nonumber\\
[p, h] &\mapsto \pi_P(p),
\ea
and fibres
\ba
\mathcal{H}_x
&=
P_x \times_\psi H
\ea
for all $x \in M$, which are isomorphic to $H$ as Lie groups. The Lie group structure on each fibre $\mathcal{H}_x$ is defined by
\ba\label{LiegroupStructureOnFibresofAssociated}
[p, h] \cdot \mleft[p, q\mright]
&\coloneqq
\mleft[ p, hq \mright]
\ea
for all $h, q \in H$ and $p \in P_x$ (that is, $\pi_P(p) = x$).
\end{theorems}
\begin{remarks}{Neutral and inverse elements}{NeutralAndInverseInAssocLGB}
The neutral element for $\mathcal{H}_x$ ($x \in M$) is given by
\bas
e_x
&=
[p, e],
\eas
where $p \in P_x$ is arbitrary and $e$ is the neutral element of $H$. This is clearly independent of the choice of $p$ due to
\bas
\mleft[ p, e \mright]
&=
\mleft[ p \cdot g, \psi_{g^{-1}}(e) \mright]
=
\mleft[ p \cdot g, e \mright]
\eas
for all $g \in G$. Thence, the fact that $e_x$ is the neutral element follows immediately.
By Def.\ \eqref{LiegroupStructureOnFibresofAssociated} the inverse of $[p, h] \in \mathcal{H}_x$ is clearly given by
\bas
\mleft( [p, h] \mright)^{-1}
&=
\mleft[ p, h^{-1} \mright].
\eas
\end{remarks}
\begin{proof}
\leavevmode\newline
\indent $\bullet$ That $\pi$ is the well-defined projection and that the fibres are precisely $P_x \times_\psi H$ for all $x \in M$ is well-known, see our discussion before Def.\ \ref{def:LieGroupActingOnLieGroup} and the references therein; it is also very straightforward to check. We also discussed that $\mathcal{H}$ is a fibre bundle with structural fibre $H$. Hence, if one knows that the proposed group structure in Def.\ \eqref{LiegroupStructureOnFibresofAssociated} is well-defined, then the smoothness of the group structure is implied by the smoothness structures of $H$ and $\mathcal{H}$. Thence, let us check whether Def.\ \eqref{LiegroupStructureOnFibresofAssociated} is well-defined. Let $x \in M$, $p \in P_x$ and $p^\prime \coloneqq p \cdot g^\prime$ be another element of $P_x$, where $g^\prime \in G$. Also let $[p_1,h_1], [p_2, h_2] \in P_x \times_\psi H$; then we have unique elements $q_i, q_i^\prime$ of $G$ such that ($i \in \{1,2\}$)
\bas
p_i &= p \cdot q_i,&
p_i &= p^\prime \cdot q_i^\prime,
\eas
especially, it follows $q_i = g^\prime q_i^\prime$.
On the one hand, if we use $p$ as fixed element of $P_x$ to calculate the multiplication, we get
\ba\label{MultiPlicationInAssocGroup}
[p_1,h_1] \cdot [p_2,h_2]
&=
\mleft[ p, \psi_{q_1}(h_1) \mright]
\cdot \mleft[ p, \psi_{q_2}(h_2) \mright]
=
\mleft[ p, \psi_{q_1}(h_1) ~ \psi_{q_2}(h_2) \mright],
\ea
on the other hand, using Def.\ \ref{def:LieGroupActingOnLieGroup} and $p^\prime = p \cdot g^\prime$ instead of $p$,
\bas
[p_1,h_1] \cdot [p_2,h_2]
&=
\mleft[ p \cdot g^\prime, \psi_{q_1^\prime}(h_1) ~ \psi_{q_2^\prime}(h_2) \mright]
\\
&=
\Bigl[ p, \underbrace{\psi_{g^\prime} \mleft( \psi_{q_1^\prime}(h_1) ~ \psi_{q_2^\prime}(h_2) \mright)}_{= \psi_{g^\prime} \mleft( \psi_{q_1^\prime}(h_1) \mright) ~ \psi_{g^\prime} \mleft( \psi_{q_2^\prime}(h_2) \mright)} \Bigr]
\\
&=
\mleft[ p, \psi_{g^\prime q_1^\prime}(h_1) ~ \psi_{g^\prime q_2^\prime}(h_2) \mright]
\\
&=
\mleft[ p, \psi_{q_1}(h_1) ~ \psi_{q_2}(h_2) \mright],
\eas
which implies that Def.\ \eqref{LiegroupStructureOnFibresofAssociated} is well-defined, and thus defines a Lie group structure on each fibre of $\mathcal{H}$.
$\bullet$ That the fibres $\mathcal{H}_x$ are isomorphic to $H$ as Lie groups for all $x \in M$ also quickly follows. Recall by our discussion before Def.\ \ref{def:LieGroupActingOnLieGroup} that the fibres are diffeormorphic to $H$ by $H \ni h \mapsto [p, h] \in \mathcal{H}_x$ for a fixed $p \in P_x$. By Def.\ \eqref{LiegroupStructureOnFibresofAssociated} it is clear that this map is a Lie group homomorphism and hence a Lie group isomorphism.
$\bullet$ Let us now construct an LGB atlas for $\mathcal{H}$ by using a principal bundle atlas for $P$. That is, for some $U \subset M$ open and a local trivialization $\varphi_U: P_U \to U \times G$ of $P$ we write
\bas
\varphi_U(p)
&=
\bigl( \pi_P(p), \beta_U(p) \bigr)
\eas
for all $p \in P$, where $\beta_U: P_U \to G$ is an equivariant map, \textit{i.e.}\ $\beta_U(p \cdot g) = \beta_U(p) ~ g$ for all $g \in G$. Then define $\phi_U$ as a map by
\bas
\mathcal{H}_U
&\to
U \times H,\\
[p, h]
&\mapsto
\mleft(
\pi_P(p), \psi_{\beta_U(p)} (h)
\mright).
\eas
$\phi_U$ is well-defined: Let $\mleft[p^\prime, h^\prime\mright] \in \mathcal{H}_U$ with $\mleft[p^\prime, h^\prime\mright] = \mleft[p, h\mright]$. Then there is a $g \in G$ such that
\bas
\mleft(p^\prime, h^\prime\mright)
&=
\mleft( p \cdot g, \psi_{g^{-1}}(h) \mright),
\eas
hence, using the equivariance of $\beta_U$ and Def.\ \ref{def:LieGroupActingOnLieGroup},
\bas
\phi_U\mleft( \mleft[p^\prime, h^\prime\mright] \mright)
&=
\Bigl(
\underbrace{\pi_P\mleft(p \cdot g\mright)}_{= \pi_P(p)}, \underbrace{\mleft(\psi_{\beta_U\mleft(p \cdot g\mright)} \circ \psi_{g^{-1}} \mright)}_{= \psi_{\beta_U(p)} \circ \psi_g \circ \psi_{g^{-1}} } (h)
\Bigr)
=
\mleft(
\pi_P(p), \psi_{\beta_U(p)} (h)
\mright)
=
\phi_U\bigl( [p, h] \bigr),
\eas
which proves that $\phi_U$ is well-defined. Denote the projection onto equivalence classes $P \times H \to \mathcal{H}$ by $\varpi$, then observe
\bas
\phi_U \circ \varpi
&=
L,
\eas
where $L_U: P_U \times H \to U \times H$ is given by $L_U(p,h) \coloneqq \mleft( \pi_P(p), \psi_{\beta_U(p)} (h) \mright)$ for all $(p, h) \in P_U \times H$. $L_U$ is clearly smooth and recall that $\varpi$ is a smooth surjective submersion, therefore $\phi_U$ is smooth; this is a well-known fact for right-compositions with surjective submersions, see \textit{e.g.}\ \cite[\S 3.7.2, Lemma 3.7.5, page 153]{Hamilton}. We define a candidate of the inverse $\phi_U^{-1}: U \times H \to \mathcal{H}_U$ by
\bas
\phi_U^{-1}(x, h)
&=
\mleft[ \varphi_U^{-1}\mleft(x, e\mright), h \mright]
\eas
for all $(x, h) \in U \times H$, where $e$ is the neutral element of $G$.
By the definition of $\varphi_U$ we immediately get
\bas
\mleft( \varphi_U \circ \varphi^{-1}_U \mright)(x, e)
&=
\Bigl(
\pi_P\mleft( \varphi_U^{-1}(x, e) \mright), \beta_U\mleft( \varphi_U^{-1}(x, e) \mright)
\Bigr)
=
(x, e),
\eas
for all $x \in U$, and, also using again the equivariance of $\beta_U$,
\bas
\varphi^{-1}_U\mleft(\pi_P(p), e\mright)
&=
\varphi^{-1}_U\Bigl(\pi_P\mleft(p \cdot \beta_U^{-1}(p) \mright), \beta_U(p)~\beta_U^{-1}(p)\Bigr)
\\
&=
\varphi^{-1}_U\Bigl(\pi_P\mleft(p \cdot \beta_U^{-1}(p) \mright), \beta_U\mleft(p \cdot \beta_U^{-1}(p)\mright)\Bigr)
\\
&=
\mleft(\varphi^{-1}_U \circ \varphi_U\mright)\mleft( p \cdot \beta_U^{-1}(p) \mright)
\\
&=
p \cdot \beta_U^{-1}(p)
\eas
for all $p \in P_U$.
Then altogether
\bas
\mleft(\phi_U \circ \phi_U^{-1}\mright)(x, h)
&=
\mleft(
\pi_P\mleft( \varphi^{-1}_U(x, e) \mright), \psi_{\beta_U\mleft( \varphi^{-1}_U(x, e) \mright)} (h)
\mright)
=
\bigl(
x, \psi_e(h)
\bigr)
=
(x, h),
\eas
for all $(x, h) \in U \times H$, and
\bas
\mleft(\phi_U^{-1} \circ \phi_U\mright)([p, h])
&=
\bigl[
\underbrace{\varphi_U^{-1}\mleft( \pi_P(p), e \mright)}_{= p \cdot \beta_U^{-1}(p) },
\psi_{\beta_U(p)}(h)
\bigr]