Let G be a Lie group .

x e- An TFCG) n

-- dive G-

Wg = Mtg lol g-'DI

w is left invariant in - formson Ct

⇒ corresponding positivemeasure

µ= Iwl is a left

Haar measure on G-.

This completes the proof

ofeaistence.lt- Lie group , µ

- leftHaar measure

X : f te { fight drug) a leftinvariantmeasure

2

oehifefcghldmeg = Sflgjdprlg)

an >o o : G → IRIO - continuous (exercise)

OCgigs) Safl hog,gfdµCh) == aged Sg flhgz)dµCh) =

= 0cg ,)OCg,) Saf Ch) dm(h)

⇒gig ,ga) =D Ig) .Olga)

O is a group homomorphismD is a Lie group homomorphismModular OG : a-→ IREfunction a- malt . group of positivef-G-

reals

G- is unimodalar3

-if

OG = I n

theorem .Let G- be a

compact Lie group . ThenG- is unimodalar .

Proof-

.

0cg) {11hg) d h = 01g) -MCG)"

{Hh) dh = aCat Dg

Exercise

connected

-

.G-atwo dimensional

momabelian Lie group

⇒ G is not unimodalar !

4if a Lie group A- is anine o dulled

andµ a left Haar measure onASfig) dm Cg) = f Hgh) dm Cg)

= Sf (hg) du Ig)

⇒ µ is right invariant,i.e.

, µ is bi invariant-

Haar measure on G-

Assume that a- is a compactLie group , µ a Haar measure

on G. ThemMLG)>0 .

By replacing µ with its

multiple , we can assume that

5

µ CG) = I.

Such Haar

measure is unique- normalized Haar measure-

on G-.

theorem .

Let a- be a Lie

group .Then the following

conditions are equivalent :

Cil G is compact ,Cii) play is finite .Proof . We proved Ci)⇒ Cii)

.

Assume that µCGI is finite .

Let V be a compact neighborhoodof 1 .

Thenµ (V) >O .

6Denote by 9 the familyof finite sets Igi , gas . gm }

such that g .-V n g ;V

= 0

for all , it j , it is j Em .

Then

miff giv ) = m.mu) salat .Hence

meAGIMCV)

i.e . m is bounded.

Let m be maximal possible .Then for {g . . - - n, gnn} in 'T ,

and

g c- G- we have

GV n giV t & for some i .

7-

This implies

g e giv V-'

⇒ a- = IF givv"

since V is compact , V-'

is

also compact ⇒ VV-'

is

compact ⇒ g ;VV- '

is compact⇒ G is compact . Ed

8

www.aartinuer-produetout (G) , a- compactLet a- be a compact Lie

group . Take an arbitraryinner product C. , . ) on

4G) -

g'→ ( Ad (g) 3 , Ad (g) z)

is a continuous functionou G .

Put

<3 ,z>= Sq ( Ad (g) 3, Ad (GD dyCg)

where ris the normalized

Haar measure on G .

c.,

. > : 4G) x LCf) → IR9

is a bilinear form .

Since C; . ) is symmetric ,

C;

. > is also symmetric .

<3,97 = Salad (g) S , Adly)§) fulg) == Sq HAD (g) ELFIN Cg) 70

Meant innous

positive

Cg is > = o implies that

{ HAHg)3112dm(g) =0Assume that 3 to .

Then

10

1181170 . Therefore ,there exists an open weigh .

U of 1 such that

HANg)3117 I 11511for g EV .

⇒

{ Hard lad 'sRda Cg) >7 f 4Ad Ig) 315dm Cg) 3¥ So 11515dm (g) == f. pled 1151T .

Hence < 3,3.

> so and

c.,.> is an inner product

on L (G) .

'I

A-dig) 3 , Adcgly > =

= S (Adh) Adlgl } , Adlh) Adlglz) fun)a-

= Sq ( Adlhg) 8 , Ad (ng) # dah) == SaladHas , Adhdg) drink= 28,27 .

Hence L .

,. > satisfies

( Adige , Adcg) n> =LSinsfor any geG , 3 , ZELIG) .

Hence it is an G - insouciantimmerge .

12

By differentiationwe getsadB) 3 ,n> t

( 3,ad (3)z> = o

for all 3 ,z , } c- LCG) .

⇒ ad (y) , y c- Kat, is

antisymmetric linear

map with respect to theinvariant inner product .

The existence of iuoaoiaut

inner product ou L (G) allowsto say a lot on its structure .

13

Let or be an ideal in L (A) .Let of be the orthogonalcomplement of oeSecret

, z eor ⇒ 2g , y> =o

Let S E hCG) .Then I 8 ,y] E OL

⇒

0=55,Cad 9)Cnb = - scad 9)Cg ),z>

= - C E 's is ], 27

⇒ as is ] c- oh .

at is an ideal .

((G) = or⑤ out as linear

spaces . § c- oh, ye of ⇒

[ 8 ,y] c- anof= { o } .

UG) is a direct sum of

14two ideals .

By induction

((G) is a direct sum

of minimal ideals .

M - minimal ideal in LlG).

le cM ideal in M

LlG) = M to Mt ⇒b is an ideal in L (G)

.

⇒ b --fo} or le = Me .

There are two options① dimM = I

,M is abelian

{ EM ,ad 5 Im = o , ad { Inf =0

ad } = o ⇒ § E z - cetera of L l G)

m c Z .

15

② M is not abelian.

Then dim M > I and Me

has no nontrivial ideals

- simple Lie algebra .

-

4G) is a direct sum ofits center Z and simpleideals

.

-

Example ① of two dimensional

Lie algebra with basise, , ez and Tee

, ,ez ] = e ,

Then Rie,is an ideal rn

og - of is most simple .

16

This implies that

the dimension of a simpleLie algebra =3 .

② of=L (subs)

og-- ft; Ei:) ; x.meet}

of is simple .