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Let $G$ be a connected Lie Group of dimesion $m<\infty$ and let $g\in G$. The Maurer-Cartan form allows us to define a map from $G$ to the space of $\mathfrak{g}$-valued forms, via $$g\rightarrow g^{-1}dg$$

Is this map surjective, i.e. can every $\mathfrak{g}$-valued form be written as $g^{-1}dg$ for some $g\in G$?

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These forms are at different points $g$ of $G$ for different values of $g$, so these are not in the same cotangent space, and the question is not meaningul. If you pick only one point $g$ of $G$, you only get one $\mathfrak{g}$-valued 1-form.

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    $\begingroup$ If you left translate them to identify different tangent spaces of $G$, you always get the same 1-form, by left invariance, so the answer is no. If you right translate, you get 1-forms varying in the adjoint representation, but you never get the zero 1-form, so then the answer is no. $\endgroup$
    – Ben McKay
    Commented Jan 14, 2017 at 15:32
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    $\begingroup$ Moreover, under right translation, the Maurer--Cartan form always remains a linear isomorphism of each tangent space to the Lie algebra, so many $\mathfrak{g}$-valued 1-forms do not appear in this way. $\endgroup$
    – Ben McKay
    Commented Jan 14, 2017 at 15:36

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