In other words, why is the Maurer-Cartan form $\omega$ a Lie algebra homomorphism between $\mathfrak g=T_eG$ and the Lie algebra of left-invariant vector fields on $G$? Why does it preserve the Lie bracket?

  • $\begingroup$ See here under "Properties". By definition of the Lie bracket. $\endgroup$ – Dietrich Burde Sep 29 '16 at 21:09
  • $\begingroup$ @DietrichBurde The Wiki article just says If X and Y are both left-invariant, then $\omega([X,Y])=[\omega(X),\omega(Y)]$. I don't think that really explains it. If you take this equality as the definition of the Lie bracket on $\mathfrak g$ (as the Wiki article seems to suggest), then how do you show that this definition is equal to other definitions, for example the standard definition in matrix Lie algebras like $\mathfrak{gl}(n)$, where the definition is $[X,Y]=XY-YX$? $\endgroup$ – Andy Miles Oct 28 '16 at 20:37
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    $\begingroup$ Yes, you are right, Wiki suggests it. But then we have already proved why the Lie bracket for $\mathfrak{gl}(n)$ is given by $[X,Y]=XY-YX$, see here. $\endgroup$ – Dietrich Burde Oct 28 '16 at 20:41
  • $\begingroup$ @DietrichBurde Ah I see. The Maurer-Cartan form is just the inverse of the isomorphism that maps Lie algebra elements $X$ to left-invariant vector fields $X$. Because that's a Lie algebra isomorphism, the Maurer-Cartan form must preserve the Lie bracket, right? $\endgroup$ – Andy Miles Oct 28 '16 at 20:45

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