📂Representation Theory

Compact Lie Group

Definition¹

Lie group

Lie group $G$ is called compact if $G$, as a topological group, is a compact space.

Matrix Lie group

For a matrix Lie group $G \subset \operatorname{GL}(n, \mathbb{C})$, $G$ is called compact if $G$ is a compact subspace of $M_{n \times n}(\mathbb{C}) \cong \mathbb{R}^{2n \times 2n}$ when regarded as a topological space.

Description

By the Heine–Borel theorem, a matrix Lie group is compact if and only if it is bounded and closed as a subset of $M_{n \times n}(\mathbb{C})$. In other words, for a matrix Lie group $G$ to be compact the following two conditions must hold.

• Bounded: For every $A \in G$ there exists a constant $C > 0$ such that $|A_{ij}| \le C$ holds.
• Closed: If a sequence $\left\{ A_{m} \right\}$ in $G$ converges to $A$, then $A$ belongs to $G$.

Types

The following Lie groups are all compact Lie groups. They are closed subsets of the real or complex matrix space and are bounded. Closedness can be checked at each link, and boundedness follows because their elements are matrices whose columns are unit vectors, hence $|A_{ij}| \le 1$.

• Orthogonal group $\operatorname{O}(n)$
• Special orthogonal group $\operatorname{SO}(n)$
• Unitary group $\operatorname{U}(n)$
• Special unitary group $\operatorname{SU}(n)$
• Compact symplectic group $\operatorname{Sp}(n)$

Most other Lie groups are not compact. For example, the special linear group contains matrices of the following form for all $m \ne 0$, so it is not bounded.

$$ A_{m} = \begin{bmatrix} m & 0 & 0 \\ 0 & \frac{1}{m} & 0 \\ 0 & 0 & 1 \end{bmatrix}, \qquad \det A_{m} = 1 $$