In an online lecture given by a famous physicist Youshi Wu, he says:
Consider a map $g:\quad M\rightarrow G:x\rightarrow g(x)\in G$ ($G$ is a Lie Group, $M$ can be physical space or spacetime), we have the pull-back $$\text{Tr}\left\{[g^{-1}(x)dg(x)]^{k}\right\}$$
I am not clear why the pull-back appear like this.
Three questions:
$g(x)$, which I think, is zero form on $G$, right?
What is $k$? By searching previous lectures given by Wu, I didn't find what it is here.
Why trace here? How to write that formulas down from the definition of pull-back?