central extension

Obtain linear representation at the cost of homomorphism (w.r.t. original algebra) by lifting.
The lifting is classified by the second Cech cohomology group of the Lie group/algebra: $G\times G\to \mathbb{C}$, where $\mathbb{C}$ actually can be replaced by other centers of $G$.
eg1. In CFT, the Witt algebra has $\mathbb{C}$ kinds of different extensions, but among them $\mathbb{C}/\{0\}$ give algebras isomorphic to the Virasoro algebra. 如何寻找共形场论中的中心扩张和中心荷？ - 知乎
eg2. For orbifolds, the discrete torsion is very similar. But I don't know whether the lifting is also similar to the above. (Note on Orbifolds and Discrete Torsion - 知乎)