Lie algebra of a Lie group

Every Lie group has a Lie algebra associated with it that is compatible with the Lie group structure. This can be taken as a definition/ theorem.

Definition

For a Lie group , we define as the Lie algebra of . It is generally denoted .

Various constructions

The Lie algebra as a vector space is fairly easy to understand. It is simply the tangent space at the identity element of . However, this does not tell us how the Lie bracket acts on the Lie algebra. These are some constructions of the Lie bracket on .

Using vector fields

Let be a Lie group, and let denote the vector space of vector fields on . (For more info on this, look here).

However, is not the Lie algebra of the Lie group . It is too big, so instead we can use the subspace of left-invariant vector fields.

The evaluation map

is a vector space isomorphism.

Proof

todo

So thus, we can think of a left-invariant vector field as an element of the Lie algebra of .

Using representations

The adjoint representation of the Lie algebra gives an equivalent definition of the Lie bracket.

Using the Lie derivative

The Lie derivative gives yet another way to (equivalently) define the Lie bracket.