Commutator 2-forms on Lie groups

Question

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?

Answer 1

Let $X$, $Y$, be left invariant vector fields, and let $f$ be a left invariant one-form. By Cartan's formula, $$(df)(X,Y)=X(\underbrace{f(Y)}_{\text{const}})-Y(\underbrace{f(X)}_{\text{const}})-f([X,Y])=-f([X,Y])\;,$$ so your constant commutator $2$-forms are exact. In the nonconstant case, Cartan's formula together with the Jacobi identity give a simple expression for $d(f([-,-]))$.

Answer 2

It seems to me more natural to dispose with the arbitrary choice of $f \in \mathfrak{g}^{\ast}$ and consider the corresponding $\mathfrak{g}$-valued $2$-form. If I'm not mistaken, this form is more or less the differential of the Maurer-Cartan $1$-form, and in particular is always exact.