Lie algebra

The concept of a Lie algebra is closely tied to a Lie group, but they are separate. Every Lie group has an associated Lie algebra, but the definition of a Lie algebra can be taken and used independently.

Definition

A Lie algebra is a vector space with a map called the Lie bracket. The Lie bracket is a map that satisfies the following properties :

• Bilinearity: (and also in the second component.)
• Antisymmetry:
• Jacobi Indentity:

Note that a Lie algebra is NOT(!!) an associative algebra, the Jacobi identity is a type of substitute for associativity.

Morphisms

A Lie algebra homomorphism is a linear map that preserves the Lie bracket, i.e.

Subalgebra

A Lie subalgebra is linear space that is closed under the Lie bracket.

Important constructions