Grollarg1.21-letyn.tt → E be a Continuous
group homomorph.
between two hie
graeps Hand G .
Then
if is smooth .
Proof
First we show the statement for H (
R ,
t ) (o - e .
y is a Continuous 1- parameter sub
group ) .
Going If y
- x : LIR ,
t ) → E is a can hin . 1-
parameter
sub
gr .
,
then 2 is Smooth .
EG
By Thin - 1.19
, 7 r > O s .
t .
exp :
Bzrlo ) → explB.io))
with rodin, }" hallo sound 0
Eig
is a diffeom .
onlo on
open neighbled .
of e E E .
Since a is Continuous ( and a lo ) = e) ,
JE > 0
Set . < (t E.
c ] ) E exp
( Br (o )) = :
Br (e) .
Define
ß : TE.
E ] → Brlo)
ß =
exp
; od
Brlo)
For It) <
§ we love :
explplt))
vs
explßlzt) ) = akt ) = alt) alt)
=
exp 12pA)-
-
ßLZT) =
ZßH) =) ß (E) =
f- ß (s ) HSEEE.ae ].
By in ducken ,
one
gets
:
ß (G) =
# ßls) V-se-EE.ee]
- KKE IN \
For
YN one los :
+ E) -
AGE)
"
-
exr.ws#)T--exrLE.rskD
- Falt
) =
explttzßlc) ) ltte {II : KEIN
,
nzz
}⇐ =
since alt )
-
'
= al - t) and exptx) -
expk)
-1
Since DE IR is denn and botusides
of are
Continuous ,
we dednce that alt) =
expht f- ßcc) ) FTEIR.
So 2 is smooth ,
since
the
right
hand
siehe is .
Now Laesion tue
general Gte :
y : H → G- .
Toke a basis Xn , . . , Xu of ff -
Then
ü ltn . . .
.tn/--eapltsXu)..exrltnxn )
detinesadiffeau.frauaneigh-bld.at OEIR
"
to a
neighbhd .
of e E It and its inverse u is a mal .
Then
(you
-
1) Hr .
.
.
.tn/--ylexpltnx,)..expltnXnD=ylexpltsX)) . . .
ylexpltnxn))
i Continuous
1- parameter
subgreups,
heuce senden
by what we home
Shawn .
=)
you
-
^
is
Smooth and
-
keine
yis
Smooth ↳
colby
around e C- H .
=)
by, " , oy
=
yo In,
implies that y is smooth
↳ uol.ly around any
h E H -
D
Prop.1.LI For a
hie
group
G we elende by Go E G
the lonnected
Component of G Contra in
ing
e E G
,
when is
Called the home and Component of the
identity of G .
① Go is an
open and chered Sunset
of G .
Lu
per tiaeor ,
G. is a
submtd .
of G o
) tue seine dimension es G .
② E.
is o normal sub
group of G .
Heule ,
Go is a hie sub
group of E and
%
is a
diserettorologicd group ,
Called the
uauponeut group
of G .
Proof
① r
② geh E Go =) 7 Continuous Curve )
Cg , Cu : [ 0,17 → E
sie .
Cglo) Cnlo)
=
e E G-
and
Egli ) =
g , Cnl 1) = h .
=) tun
µ (GH) , Cult)) =
GH) -
Cult)
is a Continuous one connect
mg e E E
wir
gh .
=)
gh E Go since
. t → v Lcglt)) is continuous ,
also g- Z Go for
any GE Go .
It is a normal sub
group
:
her GEG>
,
k e G ,
+ →
kcglt) k
-
1
is Conti nous
Curve
Conway
e E G wie kgk
-
I
=)
kgk
It
Go .
①ad②
J
Go E G is a lieben
group and %.
is
group ,
while is a eliscret
homologgroup
.
( IT : G %
, quotient tordog.ae %.
.
UE %.
is open ⇐ IT
-
1/0 ) is open in G .
gGo
E % .
IT
-
1
( g.
Go ) =
G g.
h : h e Go }
=
Ig ( G. ) E G- .
p
=
4¥ G and H one we
group,
with hie
algebra ,
O .
J and
G .
Then one has :
① If y
: E → H is a hie
group homomorph isn
,
then
go
G
= ex pH .
y
"
←
where
y Tey :
g →
G .
② Es wincides with the subgraep generaked by exp EE .
③ If y .
y : E → It are die
group koueoueopzisu
s.ly'
=
y
'
.
then
9)
↳
=
41£.
'
Ie
petrenko , if G is conaeched ,
than
4=4 .
Prost.
① Recall that
Tgy ↳ Lg) =
Ly ,
Y ))
( see proof of ① of Pvp .
1.12 : Lx and
Lyck)are
y -
related
) .
www.impliesy.FI
"
=
FLIP"
!
y
.
-
Heuce ,
ylexrk )) =
y I FLY (e ) ) =
FI"
lyle))
-
-
e
=
exply
'
k ) )
V.
XE
J
.
② If Ä is the
Iubgreup generalrat by exp (g) E E
,
then I ± Ge ,
since the
xp Hx ) is a A-
arne
wmediwgetoexpkek.is to a
To see that Go E Ä ,
note that
,
since exp e
On
diffeom .
t.ro neighbhei .
.
f OE
7 ,
exp (g) Land thus also Ä ) contoins on
open
neighbhd.VEG of e E E .
Then for ge
Ä ,
Ig ( U ) is an
open neighbhd .
of
g containiend in Ä ,
since Ä is a sub
group .
Thies ,
Ä is on
g.eu
Kessel
of G .
But EEG is also aosed
,
equiualeutly GIÄ
is
open , since her ge GTG ,
Jg ( w (U )
) =
{ g-
hi
'
ihr}is
on
open neighbh .
of g. Containers in GIÄ .
9-
"
ZÄ
-
g.
ii.httThus ,
Ä , > also Closed .
=) Ä =
E- .
§
③ By ① , y
and
Y
Gina de on .
Since they one
group
how Thin
, 4) £
-
4)§
end
so
the uoim follow, freu ② .
D .
An application of ① of Then . 1.23 is for
example ,
txampledet: GLLn.IR) → (RVO }
,
•
) is a hie
group
det ( A B
) = detl A) der IB ) .
houuauophisu .
In Global analysis we saw :
det ! =
Tddet - Trace :
g RI .
By ① of Then .
1. 23 :
-
dethel-I~V-xegllu.IR
1. 2 Representation s of hie
group, and hie
algebra,
- Def.1.24-supp.ge
E is a hie
group .
Then a
representation of E on a finite -
dimension
real vector Space V is a hie
group homo pleiten
y : G- → GLIV) .
i.
Gxv → V s.
l .
Equivoleutly.it is a smooth nur 4 -
•
y Lg , _
) :
V - V is linear
trage G .
•
yle ,
v ) = v tue G
ylg , ylhiu )) =
ylgh ,
r ) kg, he G
,
keV .
Rennen Otten an nettes to Vos the representation of E
when
y
is under > hood .
Noten :
Ilg,
) = :
g. v-i.gr
Example.IOG- = G- LU )
Defiuiug ne >
presentation /standard representation of Gkv) :
y
: ELW ) ×
U → v
(A.) Av
Vie u uoice
of non;) ,
we can
identity
G-LU )- Elke ,
and VEIR
"
und
ye.GL/u.IR)xlR
"
→ IR
"
be comes luultpliuatieee of A with a weder v ERIK
EGLI u.IR
.
Similorly ,
any
matrix group H E GLIV) ho ) a
Standard representative •
V .
② Adjoiut representation of a hie
group G
Denohe by conjg : G → G ↳
njugatieu by GE G .
conjglh):-
ghg
"
the G .
It is a we
group homomorpheisn auch
Ad : E → ELI g) =
{
lin.isauopu.gg}
Adlg) : Icon5g
.
@ →
g .
is a
representation of G on
of ,
called the odjoint
representation of G .
Let us Check this is a .
representation :
conjg =
Dg .
19
-
I
f) Ig
Innig =
Tg-
nigel9-
'
=
Tage!
Teig .
(*)
conjgn
=
caisg.comin =)
t.to/u)
wenig.
,
=
@ujg )
-
^
=) Adlg"
) -
Adlg)
=) Ad : C- → CLIG)
is
group
hammer
puh .
To see that Ad is smooth
,
it sufhicieul to how
that Lg ,
x) n
Adlg) k) is smarte .
Set F : Gx
of-
TGTTGXTE
Flg ,
x ) n
Log ,
X
,
0g.
.
) .
It is Smooth and so is also
Tre .
lidax Tm ) ( Flg , x )) =
In} .
Tee
'
¥ Adlg) k) .
If E = E- Llu , IR) ,
then
Con B) = AB A- 1
is a linker
uop Wiz : Mut IR ) → Mal IR) .
Ad (A) K ) =
Tdcociz X =
Con = AXA
-
I
V-XEglhe.IR/V-AEGL/u.lR).