The torus $T^n = \underbrace{S^1 \times \cdots \times S^1}_{n \text{ times}}$

The $n$-torus is the fundamental abelian compact Lie group. It appears as:

• Maximal torus in any rank-$n$ compact group
• Classifying space for $\mathbb{Z}^n$ lattices
• Phase space in classical mechanics

Lie algebra: $\mathfrak{t}^n = \mathbb{R}^n$ (tangent space at identity)

Special unitary group $SU(n)$:

Matrices $A \in M_n(\mathbb{C})$ satisfying $A^* A = I$ and $\det(A) = 1$. This is:

• Simply connected for all $n$ (universal cover of $PSU(n)$)
• Dimension $n^2 - 1$
• Rank $n - 1$
• Compact (unitary matrices have norm-bounded entries)

Fundamental group: $\pi_1(SU(n)) = 0$ (simply connected)

Special orthogonal group $SO(n)$:

Real orthogonal matrices with determinant 1. Properties:

• Connected for $n \geq 2$
• Simply connected for $n \geq 3$: $\pi_1(SO(n)) = 0$
• For $SO(3)$: $\pi_1(SO(3)) = \mathbb{Z}/2\mathbb{Z}$ (double covered by $SU(2)$)
• Dimension $\frac{n(n-1)}{2}$

Physical interpretation: rotations in $n$-dimensional space.

Symplectic group $Sp(n)$ (compact form):

Quaternionic unitary matrices. Can be realized as: $Sp(n) = U(2n) \cap Sp_{2n}(\mathbb{C})$ where $Sp_{2n}(\mathbb{C})$ preserves a complex symplectic form.

Properties:

• Dimension $n(2n+1)$
• Rank $n$
• Simply connected

The exceptional compact Lie groups corresponding to $G_2, F_4, E_6, E_7, E_8$:

• $G_2$: Compact form of 14-dimensional exceptional algebra (automorphisms of octonions)
• $F_4$: 52-dimensional, related to exceptional Jordan algebra
• $E_6, E_7, E_8$: Simply connected compact groups of dimensions 78, 133, 248

These appear in:

• String theory (heterotic $E_8 \times E_8$)
• Exceptional holonomy manifolds
• Moonshine phenomena