Lie algebra ideal

Definition

Let be a Lie algebra and be a linear subspace. is called a Lie algebra ideal if .

Properties

If and are ideals in a Lie algebra , the following are ideals:

Lie algebra ideals allow for the construction of

Examples

Let be a Lie algebra. The following are ideals in :

1. The center of defined as is an ideal.
2. is an idea called the commutator ideal.
3. For a Lie algebra homomorphism , is an ideal.

Note, in some way, the 3rd example is all ideals. Any ideal can be expressed as the kernel of a Lie algebra homomorphism.