Relationship Between Maurer-Cartan Forms on a Lie Group and its Central Extension Given a Lie group $G$ (or, if you like, reformulate everything below in terms of the Lie algebra $\mathfrak{g}$ of $G$), one can define a distinguished 1-form $\omega$ on $G$, the Maurer-Cartan form, as the pushforward of left multiplication. Furthermore, one may be able to find a non-trivial central extension $\tilde{G}$ of $G$, which would then have its own Maurer-Cartan form $\tilde{\omega}$.


Question: what is the explicit relationship between $\omega$ and $\tilde{\omega}$, if any?


I've looked around in the references I have, but have not come up with anything along these lines. I would be nice to have some sort of explicit discussion of this to help develop a bit of intuition for this topic... if anything, I'd imagine there must be some discussion somewhere of an important case (important for physicists, at least) like the Galilean group and its central extension.
 A: I also suppose this is a standard result in mathematics, but indeed, Google returns very little. The one reference where I know this to be done is section 3.3 of arXiv:1703.06142, though the specific example considered there is the Virasoro group.
It's actually easy to compute the Maurer-Cartan form for any centrally extended group. Here goes. Let $\hat G$ be a central extension of some Lie group $G$ (say $\hat G=G\times\mathbb{R}$ as a set). Let its elements be written as pairs $(f,a)$ where $f\in G$ and $a\in\mathbb{R}$, and let its group operation read
\begin{equation}
(f,a)\cdot(g,b)
=
\big(fg,a+b+C(f,g)\big)
\end{equation}
where $C$ is some real-velued 2-cocycle on $G$. Then, let $(f(t),a(t))$ be a path in $\hat G$, defining a tangent vector $(\dot f(t),\dot a(t))$ at time $t$. The left Maurer-Cartan form $\hat\Theta$ of $\hat G$ acting on that vector is, by definition,
\begin{equation}
\hat\Theta_{(f(t),a(t))}(\dot f,\dot a)
\equiv
\frac{\partial}{\partial\tau}\bigg|_{\tau=t}
(f(t),a(t))^{-1}(f(\tau),a(\tau))
=
\Big(
\frac{\partial}{\partial\tau}\bigg|_{\tau=t}\big(f(t)^{-1}f(\tau)\big),
\dot a(t)+\frac{\partial}{\partial\tau}\bigg|_{\tau=t}C(f(t)^{-1},f(\tau))
\Big)
\end{equation}
where we used the group operation written above. On the right-hand side of the last equation, the first entry of the parenthesis is the left Maurer-Cartan form $\Theta$ of the non-extended group $G$, acting on $\dot f$. The second entry involves the derivative of the cocycle with respect to its second argument. To rewrite that more `intrinsically', one may think of $C(f,\cdot):G\to\mathbb{R}:g\mapsto C(f,g)$ as a map depending parametrically on $f$; the differential of that map at $g\in G$ is a linear map
\begin{equation}
\big(\text{d}C(f,\cdot)\big)_g:
T_gG\to
T_{C(f,g)}\mathbb{R}=\mathbb{R}.
\end{equation}
Thus, with this notation, the Maurer-Cartan form of $\hat G$ is given by
\begin{equation}
\hat\Theta_{(f,a)}(V,v)
=
\Big(\Theta_f(V),v+\big(\text{d}C(f^{-1},\cdot)\big)_g(V)\Big)
\end{equation}
where we renamed $\dot f$ into $V\in T_gG$ and $\dot a$ into $v\in T_a\mathbb{R}=\mathbb{R}$.
Disclaimer: I'm a physicist, not a mathematician, so I was happy to formulate the notion of `central extension' in terms of an explicit cocycle function $C$. A proper mathematician would presumably prefer to formulate everything in more intrinsic terms, using short exact sequences and sections for the definition of central extensions and cocycles. I have avoided that formulation, for simplicity.
Also, I have written everything in terms of the left Maurer-Cartan form. The construction of the right Maurer-Cartan form is similar.
