Abelian Lie Groups

Last update: 23 November 2012

Abelian Lie groups

1. (a) If $G$ is a connected abelian Lie group then there exist $n,k\in {ℤ}_{>0}$, with $0\le k\le n$ such that $G≅(S1 )k× ℝn-k.$
2. (b) If $G$ is a compact abelian Lie group then there exist $k\in {ℤ}_{\ge 0}$ and ${m}_{1},\dots ,{m}_{l}\in {ℤ}_{>0}$ such that $G≅(S1 )k ×ℤ/m1 ℤ×ℤ/ m2ℤ× ⋯× ℤ/mlℤ.$

Proof (sketch)

1. $0→K→𝔤 →exp G→0, where K=ker(exp).$ The map $\mathrm{exp}$ is surjective since the image contains the set of generators of $G$. The group $K$ is discrete since $\mathrm{exp}$ is a local bijection. So $K\cong {ℤ}^{k}$ since it is a discrete subgroup of a vector space. So $G\cong 𝔤/K\cong {ℝ}^{n}/{ℤ}^{k}\cong \left({ℝ}^{k}/{ℤ}^{k}\right)×{ℝ}^{n-k}$.
2. Let $T={G}^{0}$. Then $0\to T\to G\to G/T\to 0$ and $G/T$ is discrete and compact since $T$ is open in $G$. Thus, by (a), $T\cong \left({S}^{1}{\right)}^{k},$ and $G/T$ is finite. So $G≅ (S1)k × ℤ/m1ℤ × ℤ/m2ℤ ×⋯× ℤ/mlℤ,$

1. The finite dimensional irreducible representations of $ℤ/rℤ$ are $Xλ: ℤ/rℤ → ℂ× e2πik /r ↦ e2πi kλ/r for 0≤λ≤r-1.$
2. The finite dimensional irreducible representations of ${S}^{1}$ are $Xλ: S1 → ℂ× e2πiβ ↦ e 2πiλβ for λ∈ℤ.$
3. The finite dimensional irreducible representations of $ℤ$ are $z: ℤ → ℂ× r ↦ zr= e 2πiλr for z∈ℂ× , λ∈ℂ.$
4. The finite dimensional irreducible representations of $ℝ$ are $z: ℝ → ℂ× r ↦ zr=e 2πiλr for z∈ℂ×, λ∈ℂ .$

Notes and References

These notes are from a small section on Abelian Lie groups from Chapter 4 of the Representation Theory notes Book2003/chap41.17.03.