Chapter 2 Lie Groups

Definition 2.1 (Lie Group) A Lie group is a smooth manifold $G$ which is also a group, such that the group multiplication \[\mu: G \times G \to G, \quad (g,h) \mapsto gh\] and the inverse map \[\iota: G \to G, \quad g \mapsto g^{-1}\] are both smooth.

Example 2.1 Let $G = \mathbb{R} \times \mathbb{R} \times S$, equipped with the group operation \[(x_1, y_1, \lambda_1) \cdot (x_2, y_2, \lambda_2) = (x_1 + x_2,\, y_1 + y_2,\, e^{ix_1y_2}\lambda_1 \lambda_2). \text{ and } \] \[(x,y,\lambda)^{-1}=\left(-x,\; -y,\; e^{ixy}\lambda^{-1}\right)\]$$ Then $G$ is a Lie group.

Definition 2.2 Let $A_m$ be a sequence of complex matrices in $\mathbb{M}_n(\mathbb{C})$. We say that $A_m \to A$ (i.e., $A_m$ converges to a matrix $A$) if each entry of $A_m$ converges to the corresponding entry of $A$ as $m \to \infty$; In other words, \[(A_m)_{jk} \to A_{jk} \quad \text{for all } 1 \leq j, k \leq n.\]

Definition 2.3 (Matrix Lie Group) A matrix Lie group is a subgroup $G \subseteq \mathrm{GL}(n, \mathbb{C})$ with the following property: If $\{A_m\}$ is any sequence of matrices in $G$ such that $A_m \to A$, then either $A \in G$ or $A$ is not invertible.

Example 2.2 (Example of Matrix Linear Groups)

General Linear Groups over $\mathbb{C}$ and $\mathbb{R}$.

$GL_n(\mathbb{C})$ is sub group itsself.
if $A_m$ is a sequence of matrices in $GL_n(\mathbb{C}$ and $A_m$ converges to $A$, then by the definition of $GL_n(\mathbb{C}$, either $A$ is in $GL_n(\mathbb{C}$,or $A$ is not invertible.

Special linear Groups over $\mathbb{C}$ and $\mathbb{R}$.

Example 2.3 (Non example) Let $\alpha \in \mathbb{R} \setminus \mathbb{Q}$, and define the subgroup $G \subseteq \mathrm{GL}(2, \mathbb{C})$ consisting of matrices of the form

\[G = \left\{ \begin{pmatrix} e^{it} & 0 \\ 0 & e^{i\alpha t} \end{pmatrix} : t \in \mathbb{R} \right\}.\]

Clearly, $G$ is a subgroup of $\mathrm{GL}(2, \mathbb{C})$. The closure of $G$ in the standard topology is the torus subgroup

\[\overline{G} = \left\{ \begin{pmatrix} e^{i\theta} & 0 \\ 0 & e^{i\phi} \end{pmatrix} : \theta, \phi \in \mathbb{R} \right\} = \left\{ \begin{pmatrix} z_1 & 0 \\ 0 & z_2 \end{pmatrix} : |z_1| = |z_2| = 1 \right\},\]

which is a compact Lie group. Since $G$ is not closed, it fails the closure condition required for matrix Lie groups. Thus, $G$ is not a matrix Lie group.

This subgroup $G$ is sometimes referred to as an irrational line in a torus.

I had problem synchronized with yellow dots. Sorry about that.

– $A small portion of the group G inside \bar{G} (left) and a larger portion (right)$

Definition 2.4 (Lie Group homomorphism and Isomorphisms) Let $G$ and $H$ be matrix Lie groups. A map $\varphi : G \to H$ is called a Lie group homomorphism if

$\varphi$ is a group homomorphism, and
$\varphi$ is continuous.

If, in addition, $\varphi$ is one-to-one and onto, and the inverse map $\varphi^{-1}$ is continuous, then $\varphi$ is called a Lie group isomorphism.

Example 2.4 (Examples of Lie Group Homomorphisms)

The determinant is a homomorphism $\det : \mathrm{GL}_n(\mathbb{C}) \to \mathbb{C}\setminus \{0\}$
The map $\varphi : \mathbb{R} \to \mathrm{SO}(2)$, defined by \[\varphi(t) = \begin{pmatrix} \cos t & -\sin t \\ \sin t & \cos t \end{pmatrix}\] is also a group homomorphism.