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 .

Lie group/algebra homomorphisms

Let and be Lie groups with Lie algebra and . Consider a Lie group homomorphism . Then the differential is a Lie algebra homomorphism.

Conversely, let and be finite dimensional Lie algebras. By Lie’s third theorem, there exists a connected and simply connected Lie group such that . Let be a Lie group such that (again, guaranteed to exist, but does not necessarily need to be connected/ simply connected).

Let be a Lie algebra homomorphism. Then there exists a unique Lie group homomorphism that lifts .

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.