In a paper I'm working through they define the Maurer-Cartan 1-form in coordinate-basis as $$ j_{i}=-(\partial_{i}g)g^{-1}$$ where g is some arbitrary group element and the 1-form takes its values in the corresponding Lie algebra. The authors then say that by writing the 1-from in differential form notation, so that $d*j=0$, that $$dj +j\wedge j=0.$$ I'm fine with the above vanishing of the curvature expression, as I can calculate it explicitly, but I don't understand the comment that $d*j=0$. I understand that the Hodge-Star simply maps p-forms to (n-p)-forms, but here I don't see what the Hodge-Star is doing.

My question is a simple one and likely stems from not having a great working knowledge of exterior calculus. Additionally, if anyone has a recommendation for getting quickly up-to-speed on exterior calculus I would love to hear it.



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