I'm a physics student and I'm studying differential geometry, my main reference for the topics about manifolds and Lie groups is Nakahara textbook (chap.5). My problem is about Maurer-Cartan one-form, Nakahara defines it as a Lie algebra valued one form $\theta$: \begin{equation} \theta: X \rightarrow {(L_{g})}_{*}^{-1}X \end{equation} where $L_{g}$ is the left action and X a left invariant vector field on the Lie Group G. Nakahara shows that the Maurer-Cartan form satisfies two relations:
- $\theta= V_{\mu} \otimes \theta^{\mu}$ (it can we written as a tensor product of generators $V_{\mu}$ and left invariant one forms ${\theta}^{\mu}$)
- Satisfies the equation: \begin{equation} d\theta+ \frac{1}{2} [V_{\mu}, V_{\nu}] \otimes \theta^{\mu} \wedge \theta^{\nu}=0 \end{equation} where d is the exterior derivative.
My problem is that my professor always uses the fact that $\theta= g^{-1}dg$ ($*$) to perform different calculations, and I don't understand how this definition is connected to Nakahara. Besides, I find that the definition ($*$) is not a good one: in my understanding it doesn't make sense to consider an exterior derivative on a group element g, since it is defined only on one-forms. I think I'm missing something and I hope someone can help.
