Let $G$ be a compact Lie group and $\mathfrak g$ its Lie algebra. For any $f$ in the dual space $\mathfrak g^*$, we can define a skew-symmetric bi-linear form on $\mathfrak g$ by $(A,B)\mapsto f([A,B])$. Let us say that a differential 2-form on $G$ is a commutator 2-form if it has this structure at every point. We can identify the tangent space by left translations, so commutator 2-forms correspond to functions in $C^\infty(G,\mathfrak g^*)$. (The function corresponding to a given 2-form is not generally unique although every function corresponds to some form. For a silly example, consider any abelian group: all commutator 2-forms vanish identically.)
I have some questions about this concept:
- Have commutator 2-forms been studied, perhaps under another name? If yes, where can I read more? I have not managed to find anything in this direction, but I may be missing a crucial keyword in my searches.
- What kinds of commutator 2-forms are closed? In particular, can a non-constant commutator 2-form be closed? If exactness is better understood than closedness, answers in that case are welcome as well.
- If the answer to the previous question is complicated in general, are there examples of non-abelian Lie groups where every non-constant commutator 2-form is closed?