Lie Groups and Matrix Groups - Core Definitions

A Lie group is a smooth manifold $G$ equipped with a group structure such that the group operations are smooth maps. Specifically, the multiplication map $\mu: G \times G \to G$ given by $\mu(g, h) = gh$ and the inversion map $\iota: G \to G$ given by $\iota(g) = g^{-1}$ must both be smooth (infinitely differentiable).

Definition

A Lie group is a pair $(G, \cdot)$ where $G$ is a smooth manifold and $\cdot$ is a group operation on $G$ such that:

The multiplication map $\mu: G \times G \to G$ , $(g, h) \mapsto g \cdot h$ is smooth
The inversion map $\iota: G \to G$ , $g \mapsto g^{-1}$ is smooth

The dimension of a Lie group $G$ is defined as its dimension as a manifold. Lie groups provide the natural framework for studying continuous symmetries in mathematics and physics.

Definition

A matrix Lie group is a closed subgroup of $GL_n(\mathbb{C})$ , the general linear group of invertible $n \times n$ complex matrices. By a theorem of Cartan, every closed subgroup of $GL_n(\mathbb{C})$ is a smooth submanifold, hence a Lie group.

Example

Classical matrix Lie groups:

$GL_n(\mathbb{R})$ = invertible real $n \times n$ matrices (dimension $n^2$ )
$SL_n(\mathbb{R})$ = matrices with determinant 1 (dimension $n^2 - 1$ )
$O(n)$ = orthogonal matrices: $A^T A = I$ (dimension $\frac{n(n-1)}{2}$ )
$SO(n)$ = special orthogonal matrices: $A^T A = I$ , $\det(A) = 1$
$U(n)$ = unitary matrices: $A^* A = I$ (dimension $n^2$ )
$SU(n)$ = special unitary matrices: $A^* A = I$ , $\det(A) = 1$ (dimension $n^2 - 1$ )

The general linear group $GL_n(\mathbb{R})$ consists of all invertible real $n \times n$ matrices. It is an open subset of $\mathbb{R}^{n^2}$ (the space of all $n \times n$ matrices) defined by the condition $\det(A) \neq 0$ . Since the determinant is a continuous function, $GL_n(\mathbb{R})$ is an open submanifold of $\mathbb{R}^{n^2}$ .

Remark

The topology on a matrix Lie group is inherited from the ambient space $\mathbb{C}^{n^2}$ (or $\mathbb{R}^{n^2}$ ). The group operations are smooth because matrix multiplication and inversion involve only polynomial operations (and division by the determinant for inversion).

Matrix Lie groups are particularly important because Ado's theorem states that every finite-dimensional Lie algebra has a faithful representation, which means every finite-dimensional Lie group can be realized as a matrix group (at least locally). This justifies the study of matrix Lie groups as representatives of the general theory.

The exponential map, which we will study later, provides a bridge between Lie groups and their associated Lie algebras, allowing us to linearize the nonlinear group structure near the identity element.