Lie algebra extension

Summary

An extension of a Lie algebra $\mathfrak g$ by a Lie algebra $\mathfrak a$ is a Lie algebra $\mathfrak e$ containing $\mathfrak a$ as an ideal with quotient $\mathfrak g$. It is written as a short exact sequence $0 \to \mathfrak a \to \mathfrak e \to \mathfrak g \to 0$. Extensions are the Lie algebra analogue of group extensions, and those with abelian kernel are classified by Lie algebra cohomology. Central extensions, where $\mathfrak a$ lies in the center, give the Heisenberg algebra, affine Kac–Moody algebras and the Virasoro algebra.