¥1.29 Suppen H is a Closed sub
group of a hie
group E .
Then A is a hie sub
group of G .
Prof
G : =
{ c
'
(O ) : c : IR → HEG is suwotu ,
e lo ) = e }
←
g .
is a linear
subjroce at J .
G={XEgiR3 .
Clair Write
g
=
G ⑦ k as a vector
space ( k is a
linear lomplimeur of Ging ) .
Then 7 an
open neighbhd .
W of OER in k s .
h .
exp (W) n H =
{ e } .
Converse
} ,
ass neue tweet
'
s not the con .
Then I a
Sequence
of dtueemls Yn E k s .
t.li#.Yu=OaudexpLYu)EH KEIN .
¥0
For
u norm II . H au k
, puh Xu : _
1Yu
.
" Yul )
By possiwgho a Seed
Sequence if necessery ,
we con assad
that
li:X,
= :X tk .
Tun 11×4=1 ( in
pwtiaew ,
X # O).
Set tu: =
11411 .
Then expltu Xu ) =
exp ( Yu ) e H KEIN .
[ heim 2 and 3 show that XEG ,
which is n coutrodidiae
#
4¥! H
) 4g to O # X and * k .
De f-in the
folkwang cuop
:
F :
Gxlz → G
FIX , y ) : =
expk) exp
( Y ) =
µ leer exply))
IF is oliver isomorphen ,
haue 7 open neighbds .
V
and W of Off and O c- k respect . s .
L .
FI "
Vxw → Flyw ) = : U
is a diffeoun ouho
an
open
neigubwd .
U of e in G .
By possible shrinkiug
W , we
way ossuue exrlw) n It -
les
by deine 4 .
=
Ehestand to €04 is a
hijectieuonko Us H .
Iudeed , exp W) E Us H ,
since WEG .
Moreau
,
any
× E Us H war he unique Ly Witten vs +
=
exp 4)exp (D
for
X EV and b YEW . =)
exp 4) =
explx
) - x e H
- EH
C- H
=)
ein IY ) e
4=0 EH
=) ( U ,
u : = F
w
) is a
submfd.am for H
de find eraud e e G and ( In LU ) , u.tn.
. ) is one
for any
h EH =)
H E G is a
suwotusubmfd.DE/amples①
y : E- → H hie
group homomorphismen
Then Kerl g) =
4-
'
(e) is a normal sub
group of E
,
which is closed - Heere ,
Kerl
y ) is a hie
subgrag of G .
⑤ Center of a
group
G :
2- (G ) : =
{ ge
C- :
gh
=
hg the G } .
This subgraep of G .
Now that to any
he G :
fn :
G → G- is snuoote .
9A g-
'
high
(in parkuhr Continuous ) .
=
file ) =
lege G :
gh
=
hg } EE is Closed .
→ 2- ( G ) =
h fiele) ⇐ G is aozad .
h EG
=3 ZLG ) is a
liesusgnaepef G .
③ Any chered
subgraep of
G-Lluc IR ) is a
hie •
ubgrags
of ELIn.IR) .
For some purposes the nation of a
hie
subgrenp of a
we
group
is too restriktive :
Def.1.30-supp.ae E is a hie
group .
Then a virtual hie sub
group of G is the
image
of an iüective Lie
group hauouuupunei :
H - G .
Prop.1.31-letC.be a we
group
and i : H - G a
Virtual hie Sub
group .
① i is an immersion ( i. e .
Tui : TNH →
Info is iujechn
the H )
In particulier , i
'
=
Tei :
G →
J is on in
jede
hie
algebra hauaeorpn.sn .
( Henie , G can he ideutitied with the sub .
legen.
i
'
(G)
Eg).
② Assur G and It one conueched .
Then
i LH) is normal
subgrap ←
G ( = i
'
( G )) Eg
is an ideal ( i.e. EYYIEG
+ +
EG , YE g) .
Prof ①
i =
! iosn.
.
implies
i
'
Ii
:
G -
of
" interne ⇐ Ii : IH -9nF
- is ins . KHEH .
By Thin .
1.2310 ,
ilexrltx ) ) =
expci (ti
'
k ) )
So ,
i
'
k ) =D
impliesexp.tt/tx)=e-VtE1R
*
EG
- Htc IR .
by ins ectivity of i .
=, X - O ②
By definiten ,
ILH ) EG is anormal
hebgrap
⇐
wniglilh) ) EILH ) Kg EG,
the H -
„
=3
'
For
my XEG and TEIR
,
expltx) EH
and
conjg (i (expltx))) =
conjglexplti
'
k ))) =
Tun
.
Ei ( H ) 1.23
=
explt Adlg) ( i
'
k ))
Prop .
1.26
Theiß:
ttoeukeuydt.co yiehds , Adlg) (i
'
k)) Ei
'
(G)
For D= explt Y) for
Key ,
t-VJEG.tw>
ging
Adkexplhe)) ( i
'
(G)) ⇐
IYG) Htt IR
,
tyeg.
Different . otto yidds :
µ adv) Li 'S)) Ei
'
(G) Khg
"
EY,
i
'
LG)] E i
'
(G ) klag .
i.e .
i
'
(G) Eig is on ideal .
⇐
'
iodid KYE
J .
=) A = äd
")
( ing))
Prop.
1.26 c-
i '
(G)
Since G is umrechnet und Ad a
group homomorph .
,
② of Them .
1.23 Shows
teud
¥4) Ei
'
(G) ttge G .
=) Ez
exp (Adl g) i
'
k)) =
coujglilexpk)) EILH )Prep
.
1.26
EIYG) .
KXEG .
Since It is lonnecheed and
loujg a
group Kanonieren.su
,
this shows that con
}
( i ( H) ) Eil A) tg e- G .
D .
Def.1.32-supp.se J
is a hie
algebra and
G =
of
on ideal
( IX. 4) CG
Ka eg.
Yt G) .
Then the quotient ueao
spon
71g
=
{ X.
G :
AEG}
has a natural hie
algebra Strecke
TXTG ,
Yt G)⇐ IX. Y] +
G
Check this is well debuet
if G is an ideal .
Tor ( 71g ,
[ , ] ) is called the
quotient of of by
the ideal
G .
QUESTION :
For a hie
group
G
,
is
any Sub
algen re
G
of of
a hie
algebra ofeliesubgraep.GG ?
Tweets Seppose E is a we
group
with hie
algebra J
and let
GE of be a
susalgebre - Then there ex;D s
"
Unique connected Virtual Lie sun
group
i : H → G
s - t .
i
'
LIH) =L ( i
'
'
IH -7J isaeerguisn .
) .
More over
,
i ( H) ⇐ G is an initial sabmfd .
Proof Left trivielizotieee of TG :
←
TEIG -
g.
P
↳Lpr,
} ETGG
↳ Als ,
Tgrg.is )
↳Lg) ← Lg ,
×
,
„
Tebgt .
(et ECTE he the smooth distribution wrresj.to
C- xlg under the left trivial >
atien .
E : =
{ 3g ETGE .
Tg }, } EG }g
ETGG
Chase u hosis IX. , .
. .
# } of G CJ
,
then
↳fg) , . .
, 4g ) fern o bonus of Eg .
.
Clair
ECTG is integroble .
By the Frobenius Then .
,
it is sufti Lied to know that
E is invehtive .
g. ye
TLE) ( i e
-
sgihg eEg
Kg cG)
.
slg ) =
Ä )
Lxilg )
ygef.si ; ni
: GTR
KG ) =
Äh ↳ : Ig )
are smooth Ki .
[ 5 ,
q] =
Isis : ↳ ; ,
§ Li La ;]
=
=
.
[ " ↳ i i Ri , ↳
j ]
s
.
-44 )
.si/Lx;--SiRjELxi.LxjIt3ilLxi2j)4y.tKiL-y-)-si)4;- =
L
[Xi
, Xj ]
-
CG.
, since
G is a
subelgesre .
=) Es ,
y
]
(g) CEg
Kg eG .
CIaim-2.tt : =
FEE leef of the fdietieee FE 6.
resp .
i : Hans G ho E
through e e- G .
is a wnn.ec ted virtual hie sub
group of G s .
t i
i ( H) is an initial submtd and i
'
LTEH ) -
G
.
Frau G- A class
,
we know that i :
Fe ↳ G
is connected in: tielsubmtd of
Goal Ii (Tete ) =L .
H remains to show that It is a smooth hie
group :
Note that
,
Egn
Tn}
Eu Kg,
he G .
→
Farge G
,
Ig IEE) =
TEE .
It
ge FE ,
then
Fj FE
and Lance
g.
ne FEE ,
gh
=
GEE) Fj
=
FF
-
-
-
g.
e- FE .
g-
'
EFF since
rg LFF ) FF
.
=) HE FEE - G is a
group homomorph:L .
µ
#
f
/ ↳ G is a Smooth os anstreichen
of a smarten uvp ( µ :
Ex E →
G) and
here µ
"
: Hr H - H i) Smooth Iueootu
by the universal
properhy of initial subuufol .
It remains to show uuigueuess :
Ferrari :
H → G Is
u nennend virtual hie
subgr .
winke
aeg
IH
-
GE J .
ß
Then
TGH
=
Telgte # =
Eg .
=) H is an
integre subuefd of
E .
Also,
e EH
,
impu.es HEFE .
Recall exp (G) EHEFEE.
-
-
=
Since up (G) generales FEE by Them . 1.23 ,
we laechele that H =
FEE .
D .
Yesterday : G hie
group
with t.ie
alg .
g .
If
GEG is a sabo
Genre ,
then 7 ! wnnecheed virtual
wie sub
group
i : HTG s .
h .
i
'
LTE H ) =
G .
Tweets
( Ado
'
s Theorem )
Suppose of
is a finite dein
.
hie
algebra .
Then
g.
adeuits an injective representation y :
g- ge µ )
ouho seine finite - dich .
weder space V .
In particulier , g. is isomorphe to a sub
algebra of
gew) .
Proof see literature .
Thm.1. ( Lie
'
s 3rd Fundamental Theorem
)
Let
of
be a fiuik dim
.
hie
algebra .
Then I
a hie
group
G with hie
algebra g .
Prof By Them .
1.34
,
we can
identity of
with
a sub
algebra of some
egllv) . Now
at Theorem 1.33
im
plies 7 a virtual hie
subgraep G- → GLLV)
with hie
algebra g. Egllv) .
REI But it is not true that
any
come cheat we
group
is isomorphe
to a virtual hie Sun
group of some GLCV) .
-
1. 4 .
Homogeneaus Spaces and Klein
geometry-
we
group, arise es transformation graeps of a
space
(Vector space or mfd .
) preservicug
some additional Structure
on that space - seeIntroduction
-
e :
G- →
Bij k)
Def.1.ec G is a
group ,
X a Set .
^ P
A left action of C- on X is a
ueop l : Ex X → X s.
t .
lle , x ) = x the X and llg ,
llh ,
x ) ) =
llgh ,
x ) KLEI .
For tixed GEG we wrihe
lg
: =
elg ,
_
) :X → ×
and for fixed xe X we w.ie Ü : = Cl-
,
×
) : G- → X .
Simihrly , au has the metten of a
right action off an X
It is
giveu by a
ueop
r : Xx C- → X s - h .
rk , e) = × and
rlrk.gl ,
h ) =
rlx ,
gh ) KXEX
itg , HEG .
We seh r9 :
= rl ,
g) : X → X and
rx . =
rk ,
_
) : G → ×
hier fixed GEG resp . XEX .
Notation : We anwende llg , x ) : g.x
=
gx resp .
rk ,
g) : = x -
g
=
xg
Note that
gm eg and
gr,
rot de line mops Gz→B)
( Lgs
.
r
9 "
one the inverses
of Lg resp .
r 9)
It is a
group homomorphen for
grieg ( gut lg,
=
§ .
G)
and a anti group
homomorphen her
gtt rot ( r
9 "
= r ! rot) .
Rennen Einen a
right
actien
r
,
then
lg : = r 9
"
is
a Left adieu und
uaeuusdy .
Det.1.37-G.eu a
leftachten
l : Gx X → X
,
the or bit of × EX is
giveu by
⇐ × = im LE ) = { gx
:
ge
G } EX ⇐
Simi Carly ,
die olefine, tue arhil
of a
right- actien .
Prop.1.38-Giveumsupp.ae l : Ex X → X is a leftocker
.
Then for point , in X „ being in
the san orbit
"
dehne,
an
eeguivdeuce relation on
X .
( ×
ey , if 7g e- Es.
) .
+ =
gy ).
The set of egnivdence Classe GV is called
the orbit space Kf den actien) .
Simi
lorly ,
to a
night adieu and then we elende the
arnie space
by XYG .
Proof If x. y EX s .
t .
Ex n
Ey # ¢ ,
then
7 g.
h E G- s.tn .
g-
=
kg - =
) x -
5) y
= : × eGy
and Gx E
Gy ( 5×-5 g-1h
y e-
Ey
Bysyuuidry ,
also Gy E Ex ( y Vg
×
eG)
t.IE
G) .
Haue ,
Gx =
Ey .
D .
Eta : G
group ,
HEG
seibgreup
[ : H x E → G de line a left (resp .
night)4. g)
i-mlh.gr/--hgactieuafH an C- .
r : Ex H → G(
Gh ) 1- 9h .
HIE right
but Space
EAT left cosel spence .
Def.1.IE : Ex X - X left adrien .
① e is called transitive ,
if Ex = X
for a (heuceyny ) × EX .
② For
onyx e- X ,
the stahilizer or
isotropes group Gx
of x is
giveu by
G-i.
=
{ ge
G :
gx
- × } E G .
Sim :
harley , die defilees tue Corre
:p .
objeas for
right
octieees
.
Note
yz
Gx
,
i e
.
y
=
gx for some
ge
G
,
Ey
=
Gg; 9
"
C+
J . |
g #
gx
Mnahe
& : G → G- × iuduce, a hijeotien
% Ex .
I
-
*
In deed
, geh EG sie.
lxlg) = =
? ( h
) ,
then
of × =
g-thx ,
i. e .
g- the Ex .
=
) h E
g Ex and so
g Ex =
h Ex .
Def.1.40-sunr.ae G is a
group
.
Then a
G- homogene
aus space is a set X
eguipred with
a
transitiv Heft) actien l : Gx X - X of G .
In this hase
,
fer a
y point XEX
,
we
get
abi jeden
%!
X .
Under this identification
,
the left adieu
of G- an X
be comes left multiplikation by elements
of Gen %×
:
C : G- × %+
→
%-
×
( g , JE× ) -195 Ex -
Now ,
if E is a
torolog .
group
( resp .
a hie
group )
and X a
topolog .
space ( resp .
a smooth cnfd .
) /
We von
nequ.ve an actien to be Continuous ( resp .
smooth ) ,
i. e . we l : Gxx → X ( resp .
r :X × G - X ) to
be continuous (
resp.
.
tuootu ) .
EI.
C- hie
group .
A representation of G is a smooth left action on a never space
X - V
, l y
: E × V >
V s ! . ylg , _ ) =
lg
: VTV
is linear
Vg e- G- .
Gruen a continuous lett actien
of a
hopdog .
Joop an
a
hopdg . Space ,
then GTX is
uotwdly egnipred weh
a.
hopdogy :
IT : X → GH
Eeuipp
GIX with the
quotient torology f- find
tordog .
W . r . to IT) ,
i.e . the finest to
podogys.TT is
Continuous .
Due los : U E GLX is
open ⇐ IT
-
1
LU ) EX is
open .
For
any honolg .
space Y
,
widme map f :
GUN Y is
but innen ) ← fort , X - Y is Continuous -
Topologie an
GIX might he
"
had
"
,
even if E and X
are
"
nice
"
topolog . Spaces .
Prop.1.41-Etor.bg .
group ,
H E E a
horolog .
subgrarp .
and IT : E -
GIH,
the natural Continuous
project ieee .
-
①
lg :
GIH →
% lglg
'
H ) =
gg
'
H tg E E .
is continuous
( Ex
% →
G) Continuous left
-
action
) .
② % is
Hausdorff =-D His aosed
lkrdog )
Proof
① U E % open suhzel ,
then we weed ho show that
4g)
"
Lu) is open Age C- .
Set V : =
{ lg.gl/EGxG:g-g'EITYu)}
=
µ
"
( IT
Yu))
E Ex E -
is
open in Ex E
, since IT and µ are continuous .
IT
'
: Ex E → Ex HH is
continuous and
open"
(
dax IT
Since IT : C- →
% is
gar .
=) IT
'
LV) -
{
lg.gl/t):gg'HeU3--flgTLU)is open .
② ,
=)
/
GIH is Hausdorff point' are closed .
Since IT : G → GIH is Continuous
,
H - IT
-
1
↳ H )
is Closed .
F-
"
Assur H E E it Closed .
4 : Ex C- → G
ylg.ge/--g-ng~ is
continuous .
4
"
LH) {
(g. j ) :
GH JH}
e- Ex G
is Closed -
For
any par ( g. g) e- G- + G)4-
'
LH
) .
7 open neighbh .
-
"
Wand Üofg resp-5in G .
Sir . U ×
Ü is on
open neigen .
of lgig ) in Gx E und
internen
wg 4-
'
LH) .
=\ IT LU) and TLÜ) are open neighbu .
of GH # JH
rette die
leg heut don't in Used ( by lorswucti
+ hop.
group O Note
that , if E is a
ching on a
torologicol Hausdorffspace X
,
then
§ E E is a Closed
subgreup cand Lance
% is Hausdorff .
( %.
-7 Ex laut innen, hrjedia
,
but not u
homeocaopldze in
general ) .
Tkm.1.42-supp.se G is a hie
group
and H E G- an
Closed sub
group (Lance a hie sub
group by Then .
1.29 ) .
Then the
homogene
aus Space % odmits
a
Unique structure of a Smooth uefd .
s.LT : G- → GIH
is a Smooth sub messen ( i e
.
TGT :
TGG →
GIGA is
swj .
+
g e- G) .
In particulier
,
dein (G) = dim (G) - dimltt) .
Moreau ,
l : Ex % →
4¥ , llg.g.lt/--gg'H,
is a smooth left -
action
of G an GIH .