Is conjugation by gauge transformation of $G$-bundle contained in $\mathfrak{g}$?

Question

Before painstakingly defining all these terms, let me ask my question in plain english: given a $G$-bundle, is every conjugate of a vector field by a gauge transformation an element of the Lie algebra?

For a little more context, the question arises from a calculation in Exercise 86 of Part II (Gauge Fields) of Gauge fields, knots, and gravity by Baez and Muniain. Spurred on by the comments, I have decided to spell out the terminology in full detail as follows.

Let $M$ be a smooth manifold, let $G$ be a Lie group, and let $ E\to M$ be a $G$-bundle, viz. a vector bundle whose model fiber $V$ is a (finite dimensional) vector space equipped with a representation $\rho\colon G\to \mathrm{GL}(V)$ such that all bundle transition maps belong to $\rho(G)$. Denote the endormorphism bundle of $E$ by $\mathrm{End}(E)$ and let $G(E)\subset \Gamma\bigl(\mathrm{End}(E)\bigr)$ denote the gauge group of the bundle. Elements $g\in G(E)$ are sections of $\mathrm{End}(E)$ such that for every local trivialization $\phi\colon \mathrm{End}(E)\restriction U\to U\times \mathrm{GL}(V)$ and for every $p\in U$, it holds that $\phi(g_p)\in \{p\}\times \rho(G)$. The derivative $d\rho\colon\mathfrak{g}\to \mathrm{End}(V)$ furnishes an embedding of the Lie algebra of $G$ into the set of linear transformations of $V$. Paralleling the definition of $G(E)$, we define the infinitesimal gauge group $\mathfrak{g}(E)$ to be the set of sections $\mathcal{G}\in\mathrm{End}(E)$ such that for $\phi(\mathcal{G}_p)\in \{p\}\times d\rho(\mathfrak{g})$ for every $p$ and for every local trivialization $\phi$ around $p$. Finally, on any trivialization $\phi$ of $\mathrm{End}(E)\restriction U$ let $D^{\phi}$ denote the standard flat connection on the trivialization.

Question. For all vector fields $X$ and $\rho,\phi, U$ as above is $\rho(g)^{-1}D^\phi_X\rho(g)\in \mathfrak{g}(E\restriction U)$?

Answer 1

I suggest an alternative definition of ''$G$-bundle'', not using trivializations, that may help you sort out the confusion. It appears in many places, e.g. Kobayashi Nomizu, Fundation of Dfferential Goemetry, vol 1, chap.2. Here is a quick summary.

Let $E\to M$ be a rank $n$ vector bundle and $F\to M$ the associated frame bundle. The fiber $F_x$ over a point $x\in M$ consists of all linear isomorphisms $\varphi:\mathbb{R}^n\to E_x$. The group $\mathrm{GL}_n(\mathbb{R})$ acts freely on $F$ on the right by $\varphi\mapsto \varphi\circ g$. Let $G\subset \mathrm{GL}_n(\mathbb{R})$ be a subgroup (instead of $\rho(G)$ of your question). A reduction of the structure group of $F$ to $G$ consists of a $G$-subbundle of $F$. That is, a submanifold $F^G\subset F$ such that each $F^G\cap F_x$ consists of a single $G$-orbit. The vector bundle $E$ is said to be a $G$-bundle, or equipped with a $G$-structure, if a reduction of the structure group of its frame bundle to $G$ has been chosen. Now you can show that $E$ admits a $G$-structure if and only if there exists a cover of $M$ by open subsets, over each of which $E$ can be trivialized, i.e. is isomorphic to the trivial vector bundle with fiber $\mathbb{R}^n$, and such that the transition functions between the trivializations take value in $G$.

To make all the above rigorous you need to add smoothness conditions on maps and local triviality on all bundles. I will skip it here.

You can make a rigorous definition of a $G$-structure on a vector bundle using trivializations alone, avoiding the frame bundle, but it's a bit tedious. The definition you gave in your question is not of a $G$-bundle but the condition of existence of a $G$-structure on a vector bundle. It is not the same. $G(E)$ does not make sense unless a $G$-structure has been chosen, it is not enough to know that there is one.

Anyway, using the above definition of a $G$-bundle, it is rather easy to sort out the rest. $\mathrm{Aut}(E)$ is the bundle whose fiber at a point $x\in M$ consists of all linear automorphisms of $E_x$. $G(E)\subset\mathrm{ Aut}(E)$, for a vector bundle $E$ with a given $G$-structure $F^G\subset F$, is defined as the bundle whose fiber at a point $x\in M$ consists of all linear automorphisms $f:E_x\to E_x$ such that $\varphi\circ f\circ \varphi^{-1}\in G$ for all $\varphi\in F^G_x.$ Similarly, $\mathrm{End}(E)$ is the vector bundle whose fiber at a point $x\in M$ consists of the linear transformations $T:E_x\to E_x$. Finally, $\mathfrak{g}(E)\subset\mathrm{End}(E)$ is the vector bundle whose fiber at a point $x\in M$ are the linear transformation $T:E_x\to E_x$ such that $\varphi\circ T\circ \varphi^{-1}\in\mathfrak{g}$ for all $\varphi\in F^G_x,$ where $\mathfrak{g}\subset \mathrm{End}(\mathbb{R}^n)$ is the Lie algebra of $G$.