Compact Lie Groups - Core Definitions

Compact Lie groups form a particularly well-behaved class with exceptional properties. Their representation theory is completely reducible, they admit bi-invariant metrics, and they can be completely classified.

Definition

A compact Lie group is a Lie group $G$ that is compact as a topological space. Equivalently, $G$ is a closed and bounded subset of some $\mathbb{R}^N$ when realized as a matrix group.

Compactness has profound implications for the structure and representations of the group.

Example

Classical compact Lie groups:

$U(n)$ = unitary $n \times n$ matrices: $A^* A = I$
$SU(n)$ = special unitary matrices: $A^* A = I$ , $\det(A) = 1$
$SO(n)$ = special orthogonal matrices: $A^T A = I$ , $\det(A) = 1$
$Sp(n)$ = compact symplectic group (quaternionic unitary matrices)

All are closed and bounded in matrix space, hence compact.

Theorem

Haar Measure for Compact Groups

Every compact Lie group $G$ admits a unique (up to scaling) bi-invariant Haar measure $\mu$ : $\mu(gE) = \mu(Eg) = \mu(E)$ for all $g \in G$ and measurable sets $E$ . This measure can be normalized so $\mu(G) = 1$ (probability measure).

The bi-invariant measure allows integration on compact Lie groups, enabling harmonic analysis and averaging arguments.

Definition

A Lie algebra $\mathfrak{g}$ over $\mathbb{R}$ is called compact if it is the Lie algebra of a compact Lie group. Equivalently, the Killing form of the complexification $\mathfrak{g}_\mathbb{C}$ is negative semidefinite on $\mathfrak{g}$ .

Remark

For compact connected Lie groups, the exponential map $\exp: \mathfrak{g} \to G$ is surjective. Every element of $G$ can be written as $\exp(X)$ for some $X \in \mathfrak{g}$ . This is a special property of compact groups.

Theorem

Structure of Compact Lie Groups

Every compact connected Lie group $G$ has a finite cover that decomposes as: $\tilde{G} = (G_1 \times \cdots \times G_k) \times T^m$ where $G_i$ are compact simple Lie groups and $T^m$ is an $m$ -dimensional torus. The original $G$ is obtained as a quotient by a finite central subgroup.

This reduces the classification of compact Lie groups to:

Classification of compact simple Lie groups (related to simple Lie algebras)
Understanding toroidal factors
Computing fundamental groups and central quotients

Example

The group $SO(3) = SU(2)/\{\pm I\}$ is a quotient of the simply connected compact group $SU(2)$ by its center $\mathbb{Z}/2\mathbb{Z}$ .

Definition

A maximal torus $T$ in a compact Lie group $G$ is a maximal connected abelian subgroup. All maximal tori in $G$ are conjugate and have the same dimension, called the rank of $G$ .

Maximal tori play the role in compact Lie groups that Cartan subalgebras play in semisimple Lie algebras—they provide coordinates for classification and representation theory.