TITLE: Gauge connections and Lie algebras?
QUESTION [6 upvotes]: I'm probably missing something obvious, but I've been wondering what the motivation is for requiring the components $A_\mu$ in a local trivialization of a gauge connection on a smooth principal $G$-bundle to lie in $\mathfrak{g}$, the Lie algebra of $G$. I can see that this gives a couple of nice properties; for example, in a local trivialization it ensures that under a gauge transformation $A'_\mu=gA_\mu g^{-1}+g\partial_\mu g$ lies in $\mathfrak{g}$, and that the  curvature form $F=dA+A\wedge A$ lies in $\mathfrak{g}$ (since $\mathfrak{g}$ is closed under the Lie bracket). But is there a more intrinsic or geometric reason that $A_\mu$ must be in $\mathfrak{g}$? Thanks.

REPLY [2 votes]: I always find it helpful to think about Cartan geometries first - they are less "abstract" than principal bundles and shed new light on things like Riemannian geometry.
For a nice introduction see
http://www.emis.de/journals/SIGMA/2009/080/sigma09-080.pdf
(look for the hamster on page 4!)
or the following nice book
http://www.amazon.com/Differential-Geometry-Generalization-Erlangen-Mathematics/dp/0387947329