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
- 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
Subalgebra
A Lie subalgebra is linear space