Representation theories between Lie algebra and its 1-dim central extension

Question

Let the ground field is $\mathbb{C}$ and $\mathfrak{g}$ be a Lie algebra. Assume that we have the following central extension $$ 0\rightarrow L \rightarrow E \rightarrow \mathfrak{g} \rightarrow 0.$$ Namely, $L$ is a one-dimensional Lie algebra, $E$ is a Lie algebra, and the image of $L$ lies in the center of $E$. What is the relation between category of $E$-modules and that $\mathfrak{g}$-modules ? Can we obtain one's information from another? Thanks!

Answer 1

Let $p:L\rightarrow \mathfrak{g}$ be the morphism of the exact sequence and $V$ a $\mathfrak{g}$-module, it is defined by a morphism of Lie algebras $f:\mathfrak{g}\rightarrow gl(V)$, $f\circ p$ endows $V$ with a structure of an $E$-module.

Conversely, consider $U$ an $E$-module defined by $h:E\rightarrow gl(U)$. Let $x\in\mathfrak{g}$, $y,y'\in E, y\neq y'$ such that $p(y)=p(y')=x$, $h(y)=h(y')$ i.e $h(y-y')=0$, this implies that $h(L)=0$ since $y-y'\in L$ and $L$ is $1$-dimensional. If $h(L)=0$, we can define $\bar h:\mathfrak{g}\rightarrow gl(U)$ by $\bar h(x)=h(y)$ where $p(y)=x$.