Yesterday
We detiue nation of a we
group ( G. µ ,
w
,
e )
.
Jg : E → E
Sgk ) =
µ lgih )
g8 : C- → G
Formlos :
TM ,
Tv
Recall
from Global Analysis .
f : M - M is a diffeoue .
of a smooth mfd .
M .
Pull back :
f-
*
: HIM ) → HIM )
( ff Kf
)
I
, of ) is linear and fis ) Effet
)
✓
sieht *(M) .
Det Se
pro se E is a hie
group
.
Then a Vector fiel d
4 E It (G) is called left (
resp .
right)
invariant , if
djs =
s (resp .
9 ) ; =
s ) Vage G .
Denon by HfG ) ( resp .
HRLG) ) the Sunset
of ACE)
of left ( resp .
right ) invariant vectorfield .
By lineority
of the pulbboar ,
Hd G) and HR (G) are subSpaces
of ALG ) .
They are even
subabgennes of the
infinite dimensionale
algebra (HLG ) ,
I , ] ) rede
,
by uauhpotihility e) E ,
] wih the pull back .
Nie
brocketoftiecq
Proton Suppen G is a hie
group
and set
of
: TEE
.
① For
my
XE
of ,
↳ Lg) : =
Teig X
ETGG (resp .
Rxlg) : -
Ifk
cTGG
)
is a left (
resp .
right) invariant vector fiele • G .
② The
mops Gxog → TG detined by (
GEIGY)
and Lg , x ) n
Rxlg) are di ffeomophism .
③ The
map X.→
↳ ( resp .
AHR ×
) detines
→ a linear isomorph in witn muerte 4ns le ) between
g
and H
,
(G) ( resp .
X-p ( G )
) .
Proof .
①
By ② ↳ and Rx are Smooth never Leeds .
Lehe )
neue theol ↳ is left - iuuaieut :
Telgte) X
ng
d
4)Lh) Tsg
)
!
↳ (gh ) -154534T-
-
=
IV
ofg.tn/X--4lh)V-heG
Id
19
'
Lx =
↳ Kg eG .
Simi Early ,
One shows
that Rx is
right
invariant
.
② Defilee the map
F : E-
xg
→
TG XTG ginen
by Flug ,
x ) : (0g
,
X) .
ETGG
It is Smooth and so is
TMOF :
Gxg → TG
vs
uaupositieu of Sunde
maps .
By Leuna 1.5
, Tarot : ( g. X ) → ↳ lg )
To
show Trio F is n
aeiffeauerp him we war> what a Kunde
inverse
of TM .
F .
Define E : TG - TEXTE
hy Ekg) : =
( 0g.
. , Sg ) ETg.li ×
Igf -
It is smooth ( since inversion is a made) and so is
1- µ .
F
,
which by Leuna 1.5 is
giveu by
Tm .
E .
Sgt Tg } i
} c- IG J
.
ETGG
=) The map sg → Lg ,
Tgslgig ) E Gx
J
* TG -
Gig
is Shoot and it is an inverse to (g)H ↳ (g)
Eeg → TG .
Sim italy , one proms
the Statement for Lg , + IN
Rxlg ) .
③ By ① ,
X n Lx defines a linear map
g
→ ALLE ) ↳ lg ) =
Te } X
(
m-
X
) .
ldglfsetzt E) ,
tun slg) -
kg.
.
)! Lg )
=
Teig sle ) =
4¥) kg c- G .
Similorly ,
fer XD Rx .
D
Def.tl G we group ,
TEE = :
g .
① The diffeaueopuisu Gx
] → TG
gina by
(g. x) ↳ Lg) kresp.lg.HN Rxlg)) of Pvp .
1.7
is called the natural lett)
tz→G :
TG
xg
\ Lp:/G②
For
KEY , Lx ( resp .
Rx ) is called the
left (
resp .
right )
, nveieuruectortiekdgeueretedhy XEJ .
Note that
any
↳ (
resp .
Rx ) is nowhere raus
hing
on E und
ausging
a basis Xs , . .
, Xu of the
Wow
Space g ,
txnlg) , . .
, 4dg ) ( resp .
Rxslg) , . .
.
Rxnlg) ) forma basis of TGE VGEG .
=
For
any ge
G
, bj Is ,
z
] =
tsj : ,
jz] =
Ihre]
By Pvp -
1.7
, t e¥2
is
+
sie C- Hd ) .
Lance a
finit-
dimensionalen
algebra of ( ACE ) , ),
Vice isauopnize GEILE) freue Prop .
1. 7. ③ ,
= =
we can Transport E ,
] to a brackel on
of .
Def.IN Suppose E is a hie
group .
Then the
tangent Space
of
: Te
G at the
identity e E E
togetue wite tue war
[ , ] :
g
+
g →
7
←
Exil] Thx
, Ly] (e)
is called the hie
algebra of G One
has
by Constructions La,
yj
[ ↳ 14] .
From the
pro parties of the lieb rocket of vector fields
it fellows :
Prop.1.cn The bucket T ,
] :
gxg →
g
in
Def . 1.9
is bi linear and the follow.mg yropoties hold :
His
Hew syuuuetrc
: [ X , Y] = -
EY,
×] YX ,
4cg .
LID It seti stieg tue Jacobi -
Identity
[X ,
EY,
Z]] +
EY ,
EZ,
x]] t [Z
,
[ 44]] =D
KX , Y ,
7 E
J -
Def.1.ec① A real ( resp -
Complex)
ggf algebra is
a real ( resp .
Complex) Vector Space equipped
with
a IR l
resp . E- ) bi linear map E , ] :
g xg
→
g
s .
t.IO and
of Pop .
1.10 hold .
② A hie
algebra homomorph: zu (resp .
isouuopuisue)
between hie
algebra s
of and
G is a linear
wo J
-
G si .
YLEX.BG/--E4khyHgD.
(
resp .
de linear
isomorphismen)
③ A e)
subalgebra of a hie
algebra g
is a sub Space
G of gs.tn.
IX. Y] EG ttx ,
YEG .
Examp①Gus: der a finite dimensionen
weder spie V .
• s a hie
greup.w.r.to t .
Then the left -
trividizatieu
of TV is the
und neue TV = Vx V and the left -
iuuorenl
-
-
42nd)
Carrey and to love > heul functions .
VTV .
In particulier ,
the hie brucker of two left iur
.
vechedf.ee,
Veurshes .
Heule ,
the hie
algenre of V is just V
eguipnad
with the Zero hnockel ( EYW] - O it v. w EV ) .
↳ her , we will see that hie
algebra of ony
45)
• www.taticlliegreuphosalwoyszeroliebrockef,
i. e .
is an abelian die
algebra los are
says) .
② G and It two hie
group , with hie
algebra
( g ,
E. Ig ) und ( G,
E.
Jg) .
Then the hie
group
Ex H ho, die
algebra :
Ex H ) =
IG ×
TEH =
g
⑦
G .
with the hie bucket
[ KM ) ,
lx
'
,
4
'
) ] = ( Exit
'
Ig ,
EY,
Y
'
}) .
K X , X
'
E
g ,
4,4
'
EG
Heule
,
the hie
algebra of Gx It is what one Calls
the direct sum of the hie
algebra g
and
f .
( Check this a on
exercise ) .
③ G- = ELIN ,
IR ) E Mu ( IR) .
g.
=
Minh TELlu.IR/--ELlu.lR)
„ +
Mulk).
igeln.IR)
For AE Gun .IR ) , ↳ : GLK.IR ) → Ellen IR)
is the restrichten of the linear map ) : MDR) → Mulk).
=)
Ist#
(B. × ) =
(AB ,
AX ) .
CTBGLlu.IR)
× E Mu ( pz ) and (A. AN
II .
= ) ↳ LA) =
T.at#X--AX/kIeutitylld.X)wihXan-d
Viewing ↳ as a function GLK.IR ) → Mal IR ) ,
we know that right
-
mit
× y
II „ „
[ ↳ , Ly] ( td ) =
Tdly ↳ Hd) Tdlx
Lylld)
=
XY YX
KX ,
Y
Eg
MDR
).
Lie
algebra of Ellie ,
IR ) is
g- gllu.IR) = MUIR)
equipned with the Liebrecht
giveu by the
↳ mmuhator
of matrices .