1

Example .

G- = SU 12)

T -- { ( : E) I in - i }T is a maximal torus

.

Proofs.

AE G-,A commutes with

the torusT.

x f- I (8£) c-T - has different

eigenvalues for eigenvectors e , , ezto E)Ae , = A- ( to E) e , = a Ae ,⇒ Ae , = Xe , .

Also,in the same

way ,A- ez -- Nez AEG ⇒ A- ( II )

141=1 ⇒ A ET.

Tis maximal.

2

let N be the normalizer

of T . If A- e- N

Al: : ) at =L : E)has the same characteristic

polynomial ⇒ same eigenvalues .

Hence,either a =p or 2=5 .

Assume that at I. If a =P

as before we see that A ET .

If a=p , we take 13=(9-6)

CBA) (I E) CBA- '= BA (EF) A-

'

B-'

=

-

- B (EI ) B-'

= ( E-

g) fi !) --

= (I £) ⇒ BA ET ⇒ A c- Bi'T

N --Tutto)T

3

www.alcoueainggronpof#connected compact Lie group-

Lemuria .G connected Lie group

and C - discrete central Lie

subgroup of a- such that

GK is compact . Then thereexists a compact neighborhood D of l

im G such that

int(D) .C = G .

Proof : U open.

neighborhood of 1 in G-

such that T is compact . Since

p : a-→Gk is open , pw)

plv ) is a neighborhood of l E GK .

4{ 81466)) , keGk} is

an open cover of GK .

Since Gk is compactGK = Hkillpwl)

pcgi ) -ki can add g ,= I

Put D= gilt compact

intD o g,-U - open

pliantD) Spc giv) -- ate .

⇒ int LD) . C = G.

D8

Esg : a- connected Lie group,C

discrete central subgroup such

5

that Gk is compact .Equivalent

cil G is compactCii ) C is finite .

Pdf .C is closed aaid discrete

. . IfG- is compact , C must be finite .

If C is finite ,use lemma

.

Lenya.

Let G- be a connected

lie group andC a discrete

central subgroup such that

GK is compact . Then C is finitelygenerated .

Proof . We know that there euztg&

a compact set D such that

int (D) a C = Gi . mompact

Therefore, D'-- mlDx D) is

a compact net catered byint ne ice

-C . There is a

finite seta,. . .

, c.c-C such

that

DRC Dic.u

. . - uD.cnLet T be the subgroup of C generatedby as , -→ Ca .

Then DZCD .T.

⇒ Dna D. T for all we N

proof by induction ..

Dnt ' =D .Dna D? T a D. t'

= DF.

Since G- is connected 7

G- = int (D)E,

DnaD. T.

⇒ a- =D .P

ceC,c =D .y d ED

,yet

g-' =D c- C

d c-Dnc .Since C is

discrete and Dis compact⇒Dre is finite .

⇒ C is generated by Dncand T

. Since T is finitelygenerated , C if finitelygenerated .

Me

8

Let a- be a connected

compact lie group .Let E-

be the universal covering group ofG-

.Then Ker p is a central

~

G- discrete subgroup of CEte P and E-Iker

pE G -

G-Hence

, keep is finitelygenerated . Smee keep = IT, (G, i)we get the following result .

theorem.

Fundamental groupit,(Gi) of a connected compact Lie

group is a finitely generatedabelian group .