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Let $omega:Grightarrow Lambda^1 T^*G$ be a left invarinat $1$-form on $G$ i.e., $(L_g)^*omega=omega$ for each $gin G$.

Is it true that $omega(Y):Grightarrow mathbb{R}$ is constant for any vector field $Y:Grightarrow TG$ of $G$?

I see that if $Y$ is of the form $A^*$ for some $Ain mathfrak{g}$ then $omega(Y)$ is the constant function $A$.

Given $Ain mathfrak{g}$, we have $A^*:Grightarrow TG$ is defined as $A^*(g)=(L_g)_{*,e}(A)$.

We have $omega(Y):Grightarrowmathbb{R}$ given by $gmapsto omega(g)(Y(g))$.

As $omega$ is left invariant, we have $omega(g)(v)=omega(e)((L_{g^{-1}})_{*,g}(v))$.

For $Y=A^*$, we have $$omega(g)(Y(g))=omega(g)(A^*(g))
=omega(e)((L_{g^{-1}})_{*,g}((L_g)_{*,e}(A)))=omega(e)(A).$$
Thus, $omega(Y):Grightarrow mathbb{R}$ is the constant function $omega(e)(A)$ when $Y=A^*$.

Is it true for an arbitrary vector field $Y$ on $G$ that $omega(Y)$ is constant for a left invariant $1$-form $omega$ on $G$?

We have $domega(X,Y)=frac{1}{2}(X(omega(Y))-Y(omega(X))-omega[X,Y])$.

As mentioned above, $omega(A^*)$ is constant, so, $B^*(omega(A^*))$ is zero map and for similar reason $A^*(omega(B^*))$ is zero map for any $A,Bin mathfrak{g}$. So, $domega(A^*,B^*)=-frac{1}{2}omega[A^*,B^*]$.

In case $omega(Y)$ is constant for any vector field $Y$ on $G$ we will then have $domega(X,Y)=-frac{1}{2}omega[X,Y]$.

I do not think this is true but wikipedia article says it is true for any vector fields $X,Y$ on $G$ not necessarily left invariant vector fields i.e., of the form $A^*,B^*$ for some $A,Bin mathfrak{g}$.

EDIT : Wikipedia section says the following :

In particular, if $X$ and $Y$ are left-invariant, then $X(omega (Y))=Y(omega (X))=0,$ so $$domega (X,Y)+frac{1}{2}[omega (X),omega (Y)]=0$$
but the left-invariant fields span the tangent space at any point (the push-forward of a basis in $T_eG$ under a diffeomorphism is still a basis), so the equation is true for any pair of vector fields $X$ and $Y$. This is known as the Maurer–Cartan equation.

No. Let $G=mathbb{R}$ with coordinate $t$. The 1-form $omega=mathrm{d}t$ is left invariant. Now choose any non-constant vector field, e.g. $Y=tfrac{partial}{partial t}$. Then $omega(Y)=t$.

Of course, as you say, evaluating a left-invariant 1-form on a left-invariant vector field does result in a constant function.

EDIT: To conclude the Maurer-Cartan equation just take arbitrary vector fields $X,Y$ and express them as $X=f_iX^i,Y=g_iX^i$ for some left-invariant vector fields $X^i$. Then notice that the equation is (bi)linear over functions so the case of left-invariant vector fields implies the general case.