Tensor product used in this notation

Question

A connection form is a Lie-algebra-valued one form $\omega\in\Omega^1(P,\mathfrak{g})$ on a principal bundle P. I have also seen this written this as $\Omega^1(P)\otimes\mathfrak{g}$, but I do not see the reasoning behind this notation.

Answer 1

If $M$ is a manifold and $V$ a vector space, you can think of a "$V$-valued one-form" on $M$ as element of $T^*M \otimes V$ in the following way. If $(x_1,\dots,x_n)$ are local coordinates on $M$ and $e_1,\dots,e_m$ is a basis of $V$, then the one-form which sends the basis vector $\frac{\partial}{\partial x_i}$ (of $TM$) to the basis vector $e_j$ (of V) can be thought of as the element $dx_i \otimes e_j$.

In other words, if $\omega \in T^*M$ is an ordinary one-form, and $v \in V$, and $X \in TM$ a tangent vector, then $\omega \otimes v$ is the $V$-valued one-form which acts on $X$ via

$$ (\omega \otimes v)(X) = \omega(X) \, v $$