Lie algebra of a Lie group-

G- a lie groupITCG) - tangent space at l

-

"

linearized version"

of G- additional structure

q : a-→ H Lie group

homomorphismTibet :T, (G)→T, CH ) .

ge IG InHg) , a- → G-

InHg)Ch) = ghgi'

Lie group automorphism ofG .

Ad Cg) =T, CINITGD i.That→Ift-

T,( Int Cgh)) --T,

(InHgDot, Ent th))Ad Cgh) = Ad (g) Ad th )Ad(g g-

') = Ad Igt Ad ( g-' l

H

Adl l ) = It, CG )

Ad : G→ GL ft,CG))

lie group homomorphism(TCG) ,

Ad ) says more about G

q : G → H

yftntatg ) fu)) = y l ghg- Y =ylglylhlycg

') = IntakeCgl)

THI . Adalgl = Ad* 41 g)lofty)

3

Flat -43 T.CH)G UAdalg) Ad

*( yl g)) .

Let k be a field . A LiealgeaA is a vector space over R

with a bilinear operationCay)-s Ix ,y] such that

cis Ex ,x] = 0 for all x E AgGil Haeobi identity )Ex,Ty ,ED tEy , Ez ,xD t Iz, Tex ,JD = 0

for all x , y ,te A .

4Ad : a-→ GL (Tila))T,( Ad) : T, IG) → LCT,CGD

linearmaps on

Tia)

lean) -741Ad) 157112)bilinear operation on T, IG) , i.e .

TIGHT,lat→ T, IG)

GmJ=GCAdXgDfy)TG ,H

Lie groups

y'

-G→ H morphism of Lie

groups

Inutile Cgd le Ch)) = eelg) elhlelg-D=

= .YCghg-D= @ otndacg)) Ch)t

i. e .

a- a-is

et tecommutative

.H→ H'ntlycg)

Taking the differential at 1

we get the coven .utaNoe diagramthat ICG)

Helf TIKI

THI→ tht)T,Kutty GD

⇒ The)o A-dig) = Adlylgdlotfy)

6

Tie) (Adalg) 2) = Ad# Icecap) (Tie) 2)Differentiating again and

applying to § we getTH ) (TifAdal les ) Ind) =

= TIAdid IT, hexed them, Smet, I a-7 .⇒Ice , teens) = Geeks ) ,TieMD

for E.yet, CG) .

. Therefore , Titel :T, IG)→ ITCH)is a morphine of CT , IGI, I;])into CT

,CH),I;D) . '

we want to prove that

I, fat , I;D) is a Lie algebra ,

7Let gEG . Then Int(g) : a-→ G-is aLie group automorphism .

T,(InHg)) = Ad Cg)

Hence

Ad (g) (en ,33) = [Ad (g) in , Ad Ig)'s)for any 8 ,he T, (G) .

By differentiating at LEG and

applying to } we getus, INSD = TIAd) lesleyisD=

= ETTA d)H1n1 ,'s ] + Er , T.CAd)G) list] =

= Hasid ,I ] t- Er, 23,533 .

8

Is , Iris's] - Ifs .5h33 thisBD

3,2is c- TCG) .E)

If we prove that I ;D is

anticommutative E)⇒

Jacobi identity .

We have to prove 25 in] = -Enid .

G- x G F G £2 ITR

f :G→ IR is co- function

} ET, (G)Hom)

's

(g) = es fffmlg, . )))Hom)n(g) =p If (m f. ,gD)

9

q( (fomfs) =} ( ( form) q) ERfor any 3 ,yet, (G) .

We define

⇐ *g) (f) = 54 f.m)") =

=p ( ( f -m) g ) .

- convolution of §and2

Lemmy : E3,y] =3 *y

- z*§ .

This implies immediatelythat Eg .n7= -This] . 4.3 is

anti commutative and

(T , (G) , [ .

,I) is a Lie algebra

Proof . Take feoff) .

9

Wcg)=( Ad g) (f) ==

ntlgbqllfl-zff.int/gD--yCfCg.g-Y) .

cd.co#l--fTilAdlkdCzDlfl---Eqn3lflgi-m(g,4gD--l4g)--g '⇒ Ten,Cm) -6,54151--3+7415=0⇒ T.ec)= - It.cat

cdahkst-eskf.my/-sClfom)n/-4*nKfl-Cz*SXf1Ging -- 3*2-2*3 ,

too

10

(TCG) , E, .T) is a Lie algebra -Litt) - Lie algebra of G-- .

q : G→ H morphism ofLie groups

T, let it , IG) → T, fit)

deftly .

.at IH )

We provedTitel les

, is) -- Ethel Csl,tidy]⇒ welts

,D) = Tied Kd , .ua/lzD .

He) is a morphism of Lie

algebras .

L -Lieaegeerafuuct?

If

A- Lie group , It a Lie subgroupi : H → G inclusion - Lie

group morphism Lci) :UH)→LlG)linear injection , can identifyVH1 with its image in L(G) .

Defy ' of Lie algebra , ya linear subspace of of ,x,y e b⇒ Tx ,y] c-b

y is aLiaof of .

LCH) is a Lie subalgebra ofL (G) .

Example : G- a Lie subgroupof a-LN) (V finite dimensional

12

real nectar space) .UG) is a subalgebra ofLW) (and commutator is

just [ABT = A oB -B .A ) .Examples : soCn) , SOCn)spent .

. .

I

a- Lie group , A- o - identitycomponent 4G.) =L(G).=

p fat covering of Lie groupa- by C't ⇐Kpl : LIG't→UG)is a Lie algebra isomorphism

13

A linear subspace y oflie algebra of is an idealif xeg , ye b ⇒ Gyp c-b .

-

Let G- be a lie group , H a

normal Lie subgroup of G .

Then Lf H) is an ideal in

KG) .

Brood : g c- G- ⇒ Int (g) CH) --Ha-EEE

'

a-

it Tiit→ ItInHg)ht

⇒ by differentiation

4G) Ed L(G)14

Ui) T T KilLCHl→ LCH)

Adlgl hat)⇒ LCH ) is a subspace invariant

for Ad (g) . Differentiatingwith respect to g at

1

Lades) (n) c- UH), re HH) .

§f 'LlG)

⇒HH) is an ideal . Dk

15

of Lie algebrab ideal

oyly - linear quotient×,x'c- of x - X

'e ly

y ,y'c- of y- y

'

Eb

Ex,y]- Ex

'

,y'] = Ex - x

'

, y]t

[x'

, y-y

' ] c- b

Gtb, yty) '→ Ex ,y] tb

Is a binary operation one

ofly - quotientalgebraic

I

16

G- a Lie group , H a normal

lie subgroup , GIH quotientgroup

a- Has →uaIiiiiagain+ P up)! FionnorphisM9TH UGH)

p is submersion , up) is surjectiveH = her p , LCH) = her up)

LCatH)=LCG)/LCT

17

G- Lie group ,H a Lie subgroup

A- x G -9 a- is an action ofG ou G

G x a- → G-ida. xp I

'

-

-

→

IG-x Ght →GIH

p in a submersion,ida-xp is

a submersion, GXGIH

is

+heguotieut of Gx G-the action of G ou G induces

the action of G- on G-ht.

The orbit map of identity cosetin GH is p .

her T, (p) = LCH) .

GH lie groups , y.. G- → H

'8

lie group morphism .

K a lie subgroup of It

G- acts on Hfk by(g ,h K) t y (g) h K .

the stabilizer of identity cosetin HK is e-

'

(K) .

⇒ y- ' fk) is a Lie subgroup of

G-

ily- 'CKD = but, f pay)-495'Chest

,l pH =L he)

- ' (HKD .

G- Lie group ,H

, ,Ha too Lie

subgroups

Let O = {Cg ,g) I g c- a-3 be the19

diagonal in G-xG .

Then O is a

Lie subgroup of G-xG- .

x : a-→ a- xG a Ig) -- Ig , g)is a lie group morphism

Hix Ha is a lie subgroupa-

'

f Hix Ha ) = H ,nHais a lie subgroup of G-with Lie algebra LCH ,) nLlHa) .

Equalizer : G- Lie groupH Lie group , y , if

i G- → It

Liegroup morphisms

K = { g c-G ly , I g) = yzlgd}ZO

is a subgroup of G -

claim .

K is a lie subgroupof G .

UK) = { SELIG) )Lte . ) (g) =Lleaks) } .

Proofs . p : G → H x H

play) = (y ,cgbyzlgl)B is a Lie group morphism

p-'fo) = { g EG l pig? c- 03

deignedGD

= K.

UK) =Llet'

( LHD =-

- { SELIG) / Lte .lk/--LlyIfs)}

21

Let G,It be Lie groups

y , if : G-→ H two Lie group

morphisms .

Assume that up. ) =L(q) .

Then the equalizer of y , and kis a Lie subgroup K of G-

suck that LIK) =L (G).

Hence,K > Go,lie

. 4 , I Go =44Go .

Prod .

Let G- be a connected

Lie group .Then the map

HourggfG , H) ay→ Lte) EHoyaKlas,UHDis injective .

G- - connected Lie group

22

Z center of G-

claim : Z is a Lie subgroup of Ct .

7-Z ⇒ Intfz) -_ ida ⇒

Ad G) = t.ca, ⇒ 2- c- Ker Ad

If 2- c- Kei Ad,Ad G) = It

, say

⇒ HINTED =L Cida) ⇒IntG) = idq ⇒ 2- c-Z

⇒ Z --bar Ad - Z is a Lie

subgroup .

Hz) = ko LCAD ) - her ad .

=

of lie algebra2=13 c- of ITS ,nT=o , mtg}

is the# of og .

23

Z = Karad = { § c- of lad E = o }ad : g → Leg)

=L(Z) = center of L (G) -

theorem .

G- connected Lie group .

G- is abelian ⇒ L (GI is abelian -

Proof .G- is abelian ⇒ G- =Z

UG) =3 =3 L (G) is abelian .

If ECG) is abelian,L (Ad) = ad = o

-

Ad : a- → GL (UG))Ad (G) = III ⇒ Ad Cg) = I

, gc- G-

Heung)) = Adlg) = they24

for any g EG .

1- (InHg)) =L ( idq)for g c- G-

Since G- is connectedwe see that

InHg) --idaghg'= h for any h c- It

g c-Z ⇒ 2- = G ⇒

G- is abelian.

KB

25

We are going to prove the

following theorem .

theorem .Let G- be a lie group .

Let y be a Lie subalgebra of L(G) .Then there exists a connected Lie

groupH and an injective Lie

group morphism i : H→Gsuch that Lci ) : UH) → Ll G)is an isomorphism of LCH )onto b .

-_

Weassume this result fornow and deduce some cornynerves .

theory .

Let ofbe a finite -

26

dimensional real tie algebra .

Then there exists a Lie group

G- such that L (G) ⇐ of .

Proofs . By Ado's theorem

there exists a faithful finitedimensional representation of of,i. e.there exists a finite- dimensional

real vector space V such that

of is isomorphic to a Lie subalgebra

of LW) .

Since LIGLCV)) =L(V) ,by preceding theorem there

evists a Lie group Gand

4

an injects we Lie group morphismis G → GUV) such that

Ui) : Lf G)→ LIV) is

an isomorphism of LlG) onto

a Lie subalgebra of LIV) isomorphicto g .

D8

theorem : Let G- be a simplyconnected Lie group .

Let It

be a lie group .

Then the map

Hornsey(GH)→ Hornet that ,4AMy

E

Y→ Lte )

is a bijection .

28

This result implies that the

functor L from simply convectedLie groups to Lie algebras is

fully faithful .

Bythe first theorem it is

essentially onto .

I::÷÷i÷÷÷.

29Pvoof_ofthetheoreG- x H is a Lie group4G x H) = L (G) x L ht) isits Lie algebraLet y : L lat → L l H) be

a Lie algebra morphism .

We have to construct a

Lie group morphism y : G → It

such that L ly) = y .

Let

Ty = { ( es , y Iss )) c- L (G) x L IH) )

§ E L lG) }be the graph of y .

30

Then Ty is a Lie subalgebra

of L (G) x L (H) .

⇒ There exists a connectedinjective

lie group K and an,tie group

morphism i : K→ Gx It

such that Lci) i LIK)→ LIE) xUH)

is an isomorphism out o

Ty

r

par, a Gx H → G projectionI = pro , . i i k→ G-

L (E) =L (pm) . L ( i) inUK)→ LIG)1 is:

.on Ty

is o . of Tyonto Ll G)

⇒ LCE) is au isomorphism !

OI : K → G-31

is a covering projection !

Since G- is simply connectedK is simply connected and

IT is a lie group isomorphism .

E-'

: a-→ K the inverse .

G-A a- x H

E tra- HI

Let I = przoi in K→ It

Theny = I . E

- '

is

a lie group morphism of a- into It .

44) =L ( yet . L (E-

Y32

Lci ) .HE'

) is the isomorphism

of hCG) onto My dgiveu by9.→ ( 3,4k )) .Hence

,UH (g) = 4131,3 ELCA)

and LH) =Y . KB

consequences-G- connected abelian Lie group .

E- - universal covering groupLCE) =L(G) ⇐ IRM a- dim G-

I ;] = 0Iberian Lie algebrapin- abelian Lie group( ita simply connected

33

L Chan) = IRM[g -T = O

⇒ L CE) E Ll R'n )

⇒ E ⇐ pin.

Discrete subgroups of IR'"

are Zk (up to a choice ofbasis ) k s n .

G±(sykxRn#this gives the classification ofconnected abelian Lie groups .

Compact connected abelian Lie

groups - tori Tn = (S'T --LT'T .

Low dimensional examples⇒

-

dim G connected Lie group

UG) -- IR' as a vector space

3 ,ne hCG) - proportional[ { in] = o ⇒ Lff) is abelian

⇒ a- is abelian ⇒

G- = IR or G E T'

,

dim=2 .

G- connected Lie group

a- abelian : G- ERR,Rat's T2

G- isnotaheliay.nl (G) is not

abelian.F 's

,ne UG) such that

⑨ in] to , 3 ,z must be linearlyindependent .

KS tf z , yes + Say ] =35

•SEE ,y] t Prez is] =

= Cad -er) 28.23Hence all commentators in LCG)

span I - dim. subspace .

Let e,be a nonzero vector in

this subspace . ez lin . independent[e , , ez] = X e ,

X t O

by sealing ez can assume that

[e , , ea] = e , !

There exists a unique Cup to iso .)2- dimensional nonabelian Lie

algebra !

H = { ( : at ) / a > o , be its }36

H is a lie group , as a manifoldit is isomorphic to Rix RI R2

- simply connected

HHI -- Ho -f) la , per}

a.⇒ e: :D =L: :] - e: :] =L: :]e.-

- f: : ) ez-

- E: :]

UH) is isomorphic to LLG) !It is the universal coveringgroup af G- .

EFF'

H :

37

2- in the center

2- = ( : ah)sf: :Dgz-g-i-f://a.fi/t:g-=i:::..x::t=c::.:T=zfor all c>o ⇒ 6=00⇒ z -- ( a. Ei)g-- ki) gag

- i-

- f : :X: :X:-D=

-

- f: t.it.ca.- a

-act a-'c = o for all c

(a-t- a) c = o ⇒ a-

I= a ⇒ a' = I ⇒ a -- I

.

center of It is trivial .38

Therefore G E H,i.e

.

, up

to an isomorphism 9- is

unique .

This a the group of affinetransf . of real line

X te ax t le,a > 0

.