Lie group

Overview

Lie groups are smooth manifolds that are also groups and those two structures are compatible. They were first studied to help solve differential equations. They are useful, since we can use calculus techniques combined with group properties to study symmetries.

Definition

A Lie group $G$ is a group that is also a smooth manifold (or vice versa). The structures are compatible in that the multiplication map $m:G\times G⟶G$ and the inversion map $i:G⟶G$ are both smooth maps.

Morphisms

A Lie group homomorphism $\varphi :G\to H$ between Lie groups is a group homomorphism that is also a smooth map.

Lie subgroups

A Lie subgroup is a subset $H\subset G$ that is a subgroup ($H$ is closed under the group multiplication) such that the inclusion map $\iota :H\hookrightarrow G$ is a smooth immersion.

See also Closed subgroup theorem.

Examples

Most of the most common Lie groups used are matrix lie groups like those below:

• general linear group
• special linear group
• unitary group
• orthogonal group
• generalized orthogonal group
• symplectic group