Simple Lie algebra

A finite-dimensional Lie algebra $\mathfrak g$ (over a field, typically of characteristic $0$) is simple if:

1. $\mathfrak g$ is not abelian (see abelian Lie algebra ), and
2. the only ideals in $\mathfrak g$ are $0$ and $\mathfrak g$ itself (see ideal ).

Simplicity is the Lie-algebra analogue of “no nontrivial normal subgroups” in group theory, but formulated in terms of invariant subspaces under the adjoint action (see adjoint representation ). A basic immediate consequence is that any nonzero homomorphism out of a simple Lie algebra is injective (see Lie algebra homomorphism ).

Simple Lie algebras are the building blocks of semisimple ones: every semisimple Lie algebra decomposes as a direct sum of simple ideals (see semisimple direct sum simple ). Over $\mathbb C$, the complete list of finite-dimensional simple Lie algebras is given by the Cartan–Killing classification .