Matrix Lie groups

A matrix Lie group is a subgroup $G\le\mathrm{GL}_n(\mathbb{C})$ that is closed in $\mathrm{GL}_n(\mathbb{C})$: if $A_m\in G$ and $A_m\to A$ in $M_n(\mathbb{C})$ with $A$ invertible, then $A\in G$. Basic examples: $\mathrm{GL}_n(\mathbb{C})$, $\mathrm{GL}_n(\mathbb{R})$, $\mathrm{SL}_n(\mathbb{C}) = \{\det A = 1\}$, $\mathrm{SL}_n(\mathbb{R})$, the invertible upper-triangular matrices, and the Heisenberg group of $3\times3$ upper unitriangular real matrices. Non-examples: $\mathrm{GL}_n(\mathbb{Q})$; the "irrational line" $\{\operatorname{diag}(e^{it}, e^{iat}) : t\in\mathbb{R}\}$ for $a\notin\mathbb{Q}$ (its closure is the whole torus $\{\operatorname{diag}(e^{i\alpha}, e^{i\beta})\}$).

Each classical group is the group preserving a bilinear or sesquilinear form:

• orthogonal $\mathrm{O}(n) = \{A\in M_n(\mathbb{R}) : A^TA = I\}$, special orthogonal $\mathrm{SO}(n) = \mathrm{O}(n)\cap\mathrm{SL}_n$;
• unitary $\mathrm{U}(n) = \{A\in M_n(\mathbb{C}) : A^*A = I\}$, special unitary $\mathrm{SU}(n) = \mathrm{U}(n)\cap\mathrm{SL}_n$;
• real symplectic $\mathrm{Sp}(2n, \mathbb{R}) = \{A : A^TJA = J\}$ with $J = \begin{pmatrix}0&I_n\\-I_n&0\end{pmatrix}$, and the compact symplectic group $\mathrm{Sp}(n) = \mathrm{Sp}(2n, \mathbb{C})\cap\mathrm{U}(2n)$;
• indefinite orthogonal $\mathrm{O}(p, q) = \{A : A^T\eta A = \eta\}$, $\eta = \operatorname{diag}(I_p, -I_q)$; the Lorentz group is $\mathrm{O}(1, 3)$. All are matrix Lie groups. $\mathrm{U}(1) = \{e^{i\theta}\}\cong\mathrm{SO}(2)$, the circle.

(i) $\mathrm{O}(n)$, $\mathrm{SO}(n)$, $\mathrm{U}(n)$, $\mathrm{SU}(n)$ and $\mathrm{Sp}(n)$ are compact; $\mathrm{GL}_n$, $\mathrm{SL}_n(\mathbb{R})$ ($n\ge2$), $\mathrm{Sp}(2n, \mathbb{R})$ and $\mathrm{O}(1, 3)$ are not. (ii) $\mathrm{SO}(n)$, $\mathrm{U}(n)$, $\mathrm{SU}(n)$ and $\mathrm{GL}_n(\mathbb{C})$ are path-connected. (iii) $\mathrm{O}(n)$ has two components ($\det = \pm1$) and $\mathrm{GL}_n(\mathbb{R})$ has two components ($\det > 0$, $\det < 0$).

Every element of $\mathrm{SU}(2)$ has the form

so $\mathrm{SU}(2)$ is homeomorphic to the unit sphere $S^3\subset\mathbb{C}^2 = \mathbb{R}^4$. Writing $\alpha = x_0 + ix_3$, $\beta = -x_2 + ix_1$ gives $U = x_0I + i(x_1\sigma_1 + x_2\sigma_2 + x_3\sigma_3)$ with $\sum x_j^2 = 1$: $\mathrm{SU}(2)$ is the group of unit quaternions ($i\sigma_1, i\sigma_2, i\sigma_3$ play the roles of quaternion units up to sign). In particular $\mathrm{SU}(2)$ is compact, connected and simply connected, with centre $\{\pm I\}$.