Lie Groups and Matrix Groups - Key Properties

Lie groups exhibit several fundamental properties that make them particularly tractable objects of study. Understanding these properties is essential for developing the deeper theory of Lie groups and their applications.

Definition

Let $G$ be a Lie group and $g \in G$ . The left translation by $g$ is the map $L_g: G \to G$ defined by $L_g(h) = gh$ . Similarly, the right translation by $g$ is $R_g: G \to G$ defined by $R_g(h) = hg$ .

Both left and right translations are diffeomorphisms of $G$ . This means that the geometry of $G$ is "homogeneous" - the manifold structure looks the same at every point. This property is crucial for defining geometric structures on Lie groups.

Definition

A Lie subgroup $H$ of a Lie group $G$ is a subgroup of $G$ that is also a submanifold (not necessarily closed). If $H$ is also a closed subset of $G$ , it is called a closed Lie subgroup.

Example

$SL_n(\mathbb{R}) \subset GL_n(\mathbb{R})$ is a closed Lie subgroup (defined by $\det = 1$ )
$SO(n) \subset O(n)$ is a closed Lie subgroup (the connected component of the identity)
The group of rotations by angles $\alpha \in \mathbb{R}$ on the circle is isomorphic to $\mathbb{R}$ , while the subgroup of rotations by rational multiples of $\pi$ is dense but not closed

A fundamental theorem in Lie theory, known as Cartan's theorem or the closed subgroup theorem, states that every closed subgroup of a Lie group is automatically a Lie subgroup. This is remarkable because it requires no smoothness assumption on the subgroup - the closure condition alone suffices.

Remark

The tangent space at the identity element $T_e G$ of a Lie group $G$ inherits a special algebraic structure called a Lie bracket, making it into a Lie algebra. This Lie algebra encodes much of the local structure of the Lie group.

Connectedness is an important topological property for Lie groups. A Lie group $G$ may have multiple connected components. The connected component containing the identity $e$ forms a normal subgroup denoted $G^0$ , called the identity component. For example, $O(n)$ has two components (determinant $\pm 1$ ), while $SO(n)$ is connected for $n \geq 2$ .

Another key property is compactness. Compact Lie groups (such as $SO(n)$ , $U(n)$ , $SU(n)$ ) have especially nice properties: they admit a bi-invariant Haar measure, their representation theory is completely reducible (every representation decomposes into irreducibles), and they can be completely classified.

The center of a Lie group $G$ is defined as $Z(G) = \{g \in G : gh = hg \text{ for all } h \in G\}$ . For example, the center of $SU(n)$ consists of scalar matrices $e^{2\pi i k/n}I$ for $k = 0, 1, \ldots, n-1$ , forming a cyclic group of order $n$ . Understanding the center is crucial for studying covering spaces and representations.

Matrix Lie groups inherit additional structure from the ambient matrix space, including natural metrics and norms, which facilitate both theoretical analysis and computational approaches.