Given a Lie group $G$, what is a $p$-form on the tangent space of the identity $T_e G$?

Question

I'm reading in Bredon at page $165$, where he writes that there is a one-to-one correspondence between $p$-forms on $T_e G$ and left-invariant $p$-forms on $G$. First of all, are forms generally not defined on manifolds? Secondly, I don't see the connection. He somehow argues that this correspondence comes from left-invariant forms being smooth, but don't we assume form to be differential in the first place?

Answer 1

A $p$-form on a vector space $V$ is an element of $\bigwedge V^*$. Equivalently, it is a skew-symmetric, multilinear map $V^k \to \mathbb{R}$. A differential $p$-form on a smooth manifold $M$ is a section of $\bigwedge^kT^*M$, i.e. a choice of $p$-form on each vector space $T_mM$ which varies smoothly in $m$.

On a Lie group $G$, a choice of $p$-form on $T_eG$, call it $w_e$, gives rise to a left-invariant differential $p$-form $w$ by defining $w_m = L_{m^{-1}}^*w_e$. Conversely, given a left-invariant differential $p$-form $w$, its value at $e$, namely $w_e$, is a $p$-form on $T_eG$.