📂Representation Theory

Matrix Lie Groups

Definition¹

• The field of real numbers $\mathbb{R}$ may be replaced by the field of complex numbers $\mathbb{C}$ without loss of generality.

A subgroup $G$ of $\operatorname{GL}(n, \mathbb{R})$ that satisfies the following property is called a matrix Lie group. For a sequence $\left\{ A_{n} \in G \right\}$ of elements of $G$,

$$ \lim\limits_{n \to \infty} A_{n} = A \implies A \in G \text{ or } A \notin \operatorname{GL}(n, \mathbb{R}) $$

The convergence above means convergence of matrices. $\operatorname{GL}(n, \mathbb{R})$ is the general linear group.

Explanation

The above condition means that $G$ must be a closed subset of $\operatorname{GL}(n, \mathbb{R})$. In other words, a closed subgroup of $\operatorname{GL}(n, \mathbb{R})$ is called a matrix Lie group. Since the whole set is closed, $\operatorname{GL}(n, \mathbb{R})$ itself is a matrix Lie group.

Types

• General linear group $\operatorname{GL}(n, \mathbb{R})$
  The general linear group is itself a matrix Lie group.

• Special linear group $\operatorname{SL}(n, \mathbb{R})$

• Orthogonal group $\operatorname{O}(n)$

• Special orthogonal group $\operatorname{SO}(n)$

• Unitary group $\operatorname{U}(n)$

• Special unitary group $\operatorname{SU}(n)$