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 >0 , with 0kn such that G(S1 )k× n-k.
  2. (b) If G is a compact abelian Lie group then there exist k 0 and m1,, ml >0 such that G(S1 )k ×/m1 ×/ m2× × /ml.

Proof (sketch)

  1. 0K𝔤 exp G0, where K=ker(exp). 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 Kk since it is a discrete subgroup of a vector space. So G𝔤/K n/ k (k/ k) ×n-k .
  2. Let T=G0. Then 0T GG/T 0 and G/T is discrete and compact since T is open in G. Thus, by (a), T (S1)k , and G/T is finite. So G (S1)k × /m1 × /m2 ×× /ml,

  1. The finite dimensional irreducible representations of /r are Xλ: /r × e2πik /r e2πi kλ/r for 0λr-1.
  2. The finite dimensional irreducible representations of S1 are Xλ: S1 × e2πiβ e 2πiλβ for λ.
  3. The finite dimensional irreducible representations of are z: × r zr= e 2πiλr for z× , λ.
  4. The finite dimensional irreducible representations of are z: × r zr=e 2πiλr for z×, λ .

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.

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