Lie algebra valued one forms

Question

Let $(P,\pi,M)$ be a principal $G$-bundle and $\omega$ be a Lie-algebra valued one-form. Then $\omega: \Gamma(TP) \rightarrow T_e G$ where $\Gamma(TP)$ are smooth sections of the principal bundle, is said to be a Lie algebra valued one form. It is also said to be linear. My question is in what sense this is linear? Linear with multiplication of reals of with Lie algebra elements in $T_e G$, and if so how is this multiplication defined?

Answer 1

Your interpretation of a Lie algebra valued one form is not quite correct. One interpretation is as a smooth map from $TP$ to the vector space $\mathfrak g:=T_eG$ which has the property that for each $u\in P$, the restriction of $\omega$ to $T_uP$ is a linear map (over the reals) $T_uP\to \mathfrak g$. If you want the form to act on sections of $TP$, then it has values in $C^\infty(P,\mathfrak g)$ but then you have to require that it is linear over $C^\infty(P,\mathbb R)$ (with respect to point-wise operations on both sides). Finally, you can also interpret $\omega$ as associating to each point $u\in P$ a linear map $\omega(u):T_uP\to\mathfrak g$ which depends smoothly on $u$ in the sense that plugging in vector fields, one obtains smooth smooth functions.