I've been trying to understand the following 2 expressions for the Maurer-Cartan form:

$$\omega = g^{-1}dg \quad (1)$$


$$\omega = T_a\otimes\eta^a \quad (2)$$

where $T_a \in T_eG$ and $\eta^a$ is a left-invariant 1-form.

I know about a similar post in this forum (Expression for the Maurer-Cartan form of a matrix group) but I don't get the answer, so that's why I create a new one explaining my difficulties. Sorry.

I know that Maurer-Cartan form is defined as a map $\omega: T_gG \rightarrow T_eG$, so one can think about it as $\omega = L_{g^{-1}*}$. From this I understand the neccesity of $g^{-1}$ in Eq. (1) but I don't see why $dg$ belongs to $T_gG$. Wouldn't it be a 1-form?

On the other hand, I don't know how to connect the Eq. (2) with the idea of the mapping between tangent spaces.

Any help will be great! I'm Physicist not Mathematician, so if you could explain it in detail, you'll help me a lot.

Thank you in advance! ;)


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