# Conditions for left-invariant one-forms to be closed.

Let $G$ be a connected (semisimple) Lie group with Lie algebra $\frak{g}$. For $\omega \in \frak{g}^*$, we may define a left invariant one-form on $G$ by $\left[ \omega (g)\right] (v)=\omega \left( L_{g^{-1}}'(v)\right)$. The question is: are there any nice conditions that will tell me when a form generated in this manner will be closed?

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As $d\omega (u,v)=\omega([u,v])$ ($u,v$ left-invariant vector fields), $\omega$ is closed iff $\omega([u,v])=0$ for all $u,v\in\mathfrak{g}$, i.e. iff $\omega:\mathfrak{g}\to\mathbb{R}$ is a Lie algebra homomorphism. Notice that this cannot happen (for $\omega\neq0$) if $\mathfrak{g}$ is semisimple.

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