Abelian Lie Groups

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 23 November 2012

Abelian Lie groups

1. (a) If $G$ is a connected abelian Lie group then there exist $n,k\in {\mathbb{Z}}_{>0}$, with $0\le k\le n$ such that $$G\cong (S^1{)}^k\times {ℝ}^{n-k}.$$
2. (b) If $G$ is a compact abelian Lie group then there exist $k\in {\mathbb{Z}}_{\ge 0}$ and $m_1,\dots ,m_l\in {\mathbb{Z}}_{>0}$ such that $$G\cong (S^1{)}^k\times \mathbb{Z}/m_1\mathbb{Z}\times \mathbb{Z}/m_2\mathbb{Z}\times \cdots \times \mathbb{Z}/m_l\mathbb{Z}.$$

Proof (sketch)

1. $$0\to K\to 𝔤\stackrel{\exp }{\to }G\to 0,\text{where}K=\ker \left(\exp \right).$$ The map $\exp$ is surjective since the image contains the set of generators of $G$. The group $K$ is discrete since $\exp$ is a local bijection. So $K\cong {\mathbb{Z}}^k$ since it is a discrete subgroup of a vector space. So $G\cong 𝔤/K\cong {ℝ}^n/{\mathbb{Z}}^k\cong \left({ℝ}^k/{\mathbb{Z}}^k\right)\times {ℝ}^{n-k}$.
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), $T\cong (S^1{)}^k,$ and $G/T$ is finite. So $$G\cong (S^1{)}^k\times \mathbb{Z}/m_1\mathbb{Z}\times \mathbb{Z}/m_2\mathbb{Z}\times \cdots \times \mathbb{Z}/m_l\mathbb{Z},$$

1. The finite dimensional irreducible representations of $\mathbb{Z}/r\mathbb{Z}$ are $$\begin{array}{rcll}{X}^{\lambda }:& \mathbb{Z}/r\mathbb{Z}& \to & {ℂ}^{\times }\\ & e^{2\pi ik/r}& \mapsto & e^{2\pi ik\lambda /r}\end{array}\text{for}0\le \lambda \le r-1.$$
2. The finite dimensional irreducible representations of $S^1$ are $$\begin{array}{rcll}{X}^{\lambda }:& S^1& \to & {ℂ}^{\times }\\ & e^{2\pi i\beta }& \mapsto & e^{2\pi i\lambda \beta }\end{array}\text{for}\lambda \in \mathbb{Z}.$$
3. The finite dimensional irreducible representations of $\mathbb{Z}$ are $$\begin{array}{rcll}z:& \mathbb{Z}& \to & {ℂ}^{\times }\\ & r& \mapsto & z^r=e^{2\pi i\lambda r}\end{array}\text{for}z\in {ℂ}^{\times },\lambda \in ℂ.$$
4. The finite dimensional irreducible representations of $ℝ$ are $$\begin{array}{rcll}z:& ℝ& \to & {ℂ}^{\times }\\ & r& \mapsto & z^r=e^{2\pi i\lambda r}\end{array}\text{for}z\in {ℂ}^{\times },\lambda \in ℂ.$$

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.

page history