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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 $\rho(g)^{-1}D^\phi_X\rho(g)\in \mathfrak{g}(E\restriction U)$?

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  • $\begingroup$ Can you please change "$g$ be a smooth section of $\text{End}(E)$" to "$s$ be a smooth section of $\text{End}(E)$".. What does it mean to say $g(p)\in G$ for all $p\in M$? $\endgroup$ – Praphulla Koushik Jan 13 at 7:50
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    $\begingroup$ A $G$-bundle (as defined in II.2 of the aforementioned book) consists of a bundle together with a representation of $G$ on its model fiber such that the bundle transition maps belong to (the image of) $G$. Thus it is meaningful to stipulate that a section of $\textrm{End}(E)$ take values in (the image of) $G$ (see page 214 in the aforementioned book for a discussion of this point). $\endgroup$ – pre-kidney Jan 13 at 8:08
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    $\begingroup$ What do you mean by $g^{-1} X g$? As you said, $g(p)$ is a linear (?) map between the fibers $E_p \to E_p$. How does such a map act on a tangent vector to $M$ at $p$? Do you assume that $E$ is a natural bundle so that you have a lift of vector fields on $M$ to vector fields on $E$? $\endgroup$ – Tobias Diez Jan 13 at 11:16
  • $\begingroup$ In local coordinates around $p$ write $g=g^i_je_i\otimes e^j$ and let $Xg:= X(g^i_j)e_i\otimes e^j$. An implicit part of the question is to understand whether the condition $(g^{-1}Xg)_p\in\mathfrak{g}$ using the above definition is independent of the choice of chart. $\endgroup$ – pre-kidney Jan 13 at 11:48
  • $\begingroup$ I think if you add all you said in comments to question it will become easier for other to read.. you might get more attention.. $\endgroup$ – Praphulla Koushik Jan 16 at 9:47
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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 $G$.

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