• Lie
group
G
•
its hie
algebra :
g- IG E ,
] :
Gag g
.
=
GEILE )
P
Ta-
Prop.1.cn Suppose G and It are hie
group,
und
q
: G → A is a hie
group homomorph isn .
① Then
y
!Tey
: IG =
g- IH =L ( de) -
e).
Is a homomorphisn
of hie
algebra .
② If E is wmmutetivela.be Lion
,
then the hie brocket on
J is Zero
( i. e . ( g ,
[ , ]) is what one calls
on abelian hie
algebra ) .
Proof
① Recall from Global
Analysis , If f : M → N G-
ueop
¥between mfds .
and
si C- KLM ) und
R ; e- HIN) for
i - 1.2 are f- related li e
.
Ifs ; k ) =
R,
Ifk) ) THEM)then Iss , sz] is f- related to [
Es .kz] .
By assumption ,
ylgk) =
y (g) ylh ) ,
which
Says
90dg =
tag; y Vg e- G .
-
Different
iating gtye} =
Thug; 4
"
=)
Tgy ↳ Lg) Te
!]
'
k)
erwürgten,
_
-
=
4¥19))
oh XE
of
So , Lx and
Ly,
#
are
y
related
KX
Eeg .
By H ) , for any
X
Meg ,
[ Lx , Ly] is
y
-
related
to
The") ,
Lyyy)
] I
that is
'
qq.jp/JTyELx.LyJ=ILyiq,LyyyIoYEveheheateEG:-yEXyJ)--[ Lyn) .LY/fy)JLeI
② If E is abelian
,
then
U : G → E is a
group homomorph her .
✓ lgn) -
_
hi !
g-
1
=
g- ? h
-
'
=
vlg) Kh ) .
µ
G- oihelian
By ① ,
U
'
:
of
→
g. is a hie
algebra
komm .
"
→
LIFE -
1dg
=) KAM cg.
,
EX
,
y]
= EX, Y]
=) It it] =D V-x.VE
g. .
Gr.1.13-supp.de E is a We
group
and A E G a hie
Sub
group
Then
the hie
algebra f of It is
notwally
a sub
algebra of J .
In particulier ,
the hie
algebra of my
matrix
group
HE Elle, IR)
is a sub
algebra of (geln.IR) ,
E , ] ) , when E , ]
is the comme hat er
a) matrices .
Prof. Ap ply ① of Pro
p . 1.12 to the in Austen i : Hans G .
| Lance ,
Ii = ;
)
:
G ↳
g
is the natural Mansion
Tettau TEE .
)
Prop.1.14-supg.ae E is a hie
group wih hie
dg .
1g ,
E , ]) .
① Rx = so
*
L
-
×
F X E
J
⑦ [ Rx ,
Ry] = -
R
µ
]
KAYE
of .
③ Thx , Ry] = 0
V-x.YEg.pro#.see home work .
Now lotus Andy the flow of left (resp .
right
)
invariant
weder
fields .
Prop.tl E a we
group ,
s a lett.
( resp .
right)
inuoiaet
weder Jiehd on E .
① F)(g) =
g. File) trage G ( resp .
F)G) =
FILE) ]
tgt
G) .
② s is
Complete .
Prof. ① If s is left -
invariant
,
then s is
Ig
-
related
to its elf kg e- E . Heule ,
FLI.bg =
Dj FI Kg e- G .
Eva make de :
FLI Lg) =
g. FLI (e )
② Follow, from a criteria we psroved in Global
Analysis
und ① : If ZE > o si .
any inhgrd one
through any
point of a ne das find } is definiert an C-E. E
)
,
tun 4 is landete •
Def.1.IE we
group .
A one parameter sub
group of G is a hie
group
homomorphismen
< :
( R ,
t ) → G (i. e .
a : IR → E i)
u Co Curve
S L
.
< Ist f) = als) -
alt)
In putin ,
x ( o) =
e .
Vs,
TEIR)
Lennart E hie
group ,
a : R → E -
ueop
with a lo) -
e
,
Xzg .
Then the folkwang are
eguiv. :
① × is a one
parameter sub
group
with a
'
lo ) =
X
② alt) =
File)
③ alt) =
FIX le )
Proof .
①
②
Juke
(s ))
<
'
H)
=L ,
lstt ) =
7)*
µ
!als ) =
Te! #
=
↳ KHD
=) d it on
integral • me
of ↳ and since a lo ) -
e
,
we neuste home alt) =
FEY(e ) by uniqueness .
② ① alt) = F
(e) is smooth Curve in G with
* lo) e
and a
'
lo ) =
↳ Lab ) ) = X .
Since it is a flow ( of .
Global
Analysis ) we how :
altes =
FLI le ) =
FLIEGE!) ) griffin
By excuouging
rdes
of s and t one
power
timing hat
① ⇐ ③ .
.
D
Det.IE a we
group with hie
dg .
(
J .
I ,
] ) .
Then the
exponentiellen p of G is
gruen by
exp :
g → G
expk ) : =
FLY ( e )
By definition , explo) = e
Thani E hie
group
wih hie
algebra g , exp
:
of
→ G
tue exponentiellemp . Then the
folkwang hotels :
① The
map oxp is Smooth and Toe xp
:
Ig → IG
=]IS
g
-
aquds
ldg
:
ign g .
Heine
, exprestn.cn to a
diffeouuerpuism from
ten
open neighborhood of O C-
J
in
of
to an
open
neigend .
of e e E in G .
② For XEJ , ge
E one has :
F
Lg ) =
gexpltx )
FLEX (g) =
expltx ) -
g
-
Proof
We know that IX. g) ↳ ↳ '
g) is a Smooth
way
g.
× G- → TG by Pvp. 1.7 .
Heuce ,
IX. g) m ( O
,
Lxlg) ) is a smooth vectofidd
on
g.
× E . Hs integral curues are tns ( X ,
FL# Ig))
and
they are smooth .
In
particulier ,
Kit ) n LX , FLI
"
Lg )) is ssuooth
bund
Lance exp is Smooth .
Now ,
F (e) =
FLY"
le ) =
expltx)
µ a
If c : I - G is an
integre Curve of
↳ ,
then the Cat) ts an
Integral one
of a
↳ =
↳×
Hat IR .
und so FL
Lg ) =
g. exp HX ) by P-rop.1.AT .
For
FLEX (g) =
Flöhe )
g
=
expltx ) -
g
µ
P
Bop .
1.15
kennen 1.17 .
Toexp µ,
rltx )
d) Le ) ↳
le)
HXEJ .
=)
Iexp =
1dg .
es warnend .
Evm
① 6ns der the
www.hotiveliegreep
( R
> • ,
-
)
Its hie
algebra is IR with trivial hie bnecket .
The lettiuueieuh
weder bind geuedad by × ER is
↳ la) =
ax
„
F! ×
Integral Curve
of Lx :
↳ KH) ) = <
'
H
)
„
through Me R > •
c)× clo ) =
1
Solution is alt) = eHenie
, exp : IR → IR
> •
is the und
exponentieller .
② E- = GLlu.IR ) XE
g- gllu.IR )
↳( A
) = AX fer AE Gllu , R ) .
↳ ICH) ) = eilt )
"
and
< 6) = Id
c
Unique solution is the matrix exponierend
FEI
( If II .
It is the
operator norm an
M
.
LIR) ,
then
DX
"
H E DX II
"
and so this
power sei es
uaevrges
absolutely and mit windig an
uaepoce Sets ) .
exp LXTY) =/ e
xp
k) .
exp ( y ) un less X and
Y www.he .
Def.1.LI ( Exponent ich werdindes )
G we group
with hie
algebra J ,
V
Ey is on
open
neigend .
of OEG Set .
expyv
:
V → explv)= :
U
is a
diffeau.euho on
open
neighbwal
U
of e C- G ①
Then LU ,
KPI ) is a bad aarr her G- wih
e EU and (
Sgv) , Ig.
expy; ) a bad warte
around
GE G .
„
Canonical Coordinator of the trist Kind
"
.
② Chose u basis { Xs .
. . , Xu } of the weder
Space
@ ,
then v : IR
"
→ G
giveu by
P
✓ (t! . .
.tn/:=explt1Xn)..expLtuXn)resIrrcrstoadiffeau.
freu on
open neighbwd .
V
of
OER"
onho an
open neighbwd . eof e E G- .
tudeed ,
¥76) = X ; auch so
Fr ( a
! .
.
. - a
"
)
= a
"
X
,
+ . .
t a
"
Xu
( U ,
Tj = : n ) und
1dg LU ) ,
dgon ) are Local
Charts around EEG und
ge
G .
„
Coordinator of the secaud Kind .
"
.
€-1.1 Levy : H → G- a Continuous
group
homomorphie between hie
group Hand G .
Then
if is smooth .