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May 4, 2006

Introduction to Compact Groups

A character on a compact (or, more generally, on a topological) group G is a continuous morphism χ:G → T. The set Hom(G,T) of all characters is an abelian group Gˆ , the character group of the dual of G, de- noted Gb. We calculated Tb ∼= Z by attaching to each endomorphism χ:T → T a “winding number n which we identified with the endomorphism r 7→ n·r : R → R in such a fashion that χ(r + Z) = n·r + Z which amounts to the commuting of the diagram

R −−−→

r7→n·r R

quot

 y

 yquot T

−−→χ T.

This required the proof of the following

Extension Lemma. Let U be a nondegenerate in- terval on R containing O and f:U → G a func- tion into a group such that r, s, r + s ∈ U implies f(r + s) = f(r)f(s). Then there is a unique mor- phism F:R → G such that F|U = f .

A continuous morphism F:R → G into a topological group is called a one parameter subgroup of G. This is an example where a morphism is called a “subgroup”.

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We discussed the fact that passing to the dual is an operation that also acts on morphisms, reversing ar- rows: If f:A → B is a morphism of abelian groups, then fb:Bb → Ab, and f gc = gb ◦ fb. In other words, b is a contravariant functor, mapping the category AB of abelian groups into the category CAB of compact abelian groups—and vice versa.

Program for today.

Biduals. Projective Limits. Character groups of abelian groups as projective limits.

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