Abelian Lie groups

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

and

Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA
ram@math.wisc.edu

Last updates: 1 April 2010

Abelian Lie groups

1. If $G$ is a connected abelian Lie group then $$G\cong {\left(S^1\right)}^k\times {ℝ}^{n-k},$$ for some $n\in {\mathbb{Z}}_{>0},0\le k\le n.$
2. If $G$ is a compact abelian Lie group then $$G\cong {\left(S^1\right)}^k\times \mathbb{Z}/m_1\mathbb{Z}\times \mathbb{Z}/m_2\mathbb{Z}\times \dots \dots \mathbb{Z}/m_l\mathbb{Z},$$ for some $k\in {\mathbb{Z}}_{\ge 0},m_1,\dots ,m_l\in {\mathbb{Z}}_{>0}.$

Proof (sketch)

1. $$0\to H\to 𝔤\stackrel{\exp }{\to }G\to 0,\text{where}K=\ker \left(\exp \right).$$ 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 os a discrete subgroup of a vector space. So $G\cong 𝔤/K\cong {ℝ}^n/{\mathbb{Z}}^k\cong \left({ℝ}^n/{\mathbb{Z}}^k\right)\times {ℝ}^{n-k}.$
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), $T\cong {\left(S^1\right)}^k,$ and $G/T$ is finite. So $$G\cong {\left(S^1\right)}^k\times \mathbb{Z}/m_1\mathbb{Z}\times \mathbb{Z}/m_2\mathbb{Z}\times \dots \dots \mathbb{Z}/m_l\mathbb{Z},$$

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

References [PLACEHOLDER]

[BG] A. Braverman and D. Gaitsgory, Crystals via the affine Grassmanian, Duke Math. J. 107 no. 3, (2001), 561-575; arXiv:math/9909077v2, MR1828302 (2002e:20083)

page history