RepresentationLiegroupandLie
alesie
A representation of G on a finite-dim. real vector space
V is a lie
group homomorphism y
: G + GL(V).
Equivalently ,
it is a smooth map 4 : GxV-V s .
t
.
·
y(g ,
-) : V-V linear
FgeE
·
y(e , v) =
v ver
· 4(g , y(h , v)) =
y(gh , v) FgineG ,
veV .
RemarkQueofleuutretetorepresentate
mop
y : G + GL(V).
Examples
①GL(V) = G dim (V) = n
Defining Istended repres on
V :
4 :
Gh(V) + V +
(A , v) +- Av
=
y(t , v)
via choice of basis , one con
identity
Gh(v) = GL(u , IR)
and
4
with GL(uIR) +IRh >
IR"
(A ,
v) + Av
(Matrix meuhtp- of AEGLn, IR)
with a
vector in
IRK)
Similarly ,
any matrix
group
HIGL(V) has a
standard representation , namely
V.
② Adjoint representation of a lie
group
'G au
its lie
algebra o
Denoe by Long : G >
G Conjugation in G :
cong(h)= gugt The G.
It is a
lie
group homomorphism
Ad : G +
GL(g)
Ad(g) : =
T
- cong :
g +
g
is called the adjoint representation of G on its lie
alg .
g.
Let us Check this is
really a
representation :
cong
=
↓go 19 =19
glatt
cougy =
con Coun = Ad(gh) =
Ad(g)oAd(h)
conge
=
(ong)= > Ad (gz) = Ad(g)
1
=> Ad : G +
GL(g) is a
group homonarpeisen.
To see
that Ad is smooth , we can
equir . Show that
(g .
x) + Ad(g)(X) is smooth .
Setting F:
Gxg+ TEXTEXTG
,
F(g ,
x) =
(og ,
X ,
Og) ,
we have
Im(ida + Tr) · F)(g,
x) =
Tyng 0
Tep9X =
Ad(g)x
which is smooth as a
composition of smooth maps.
If G = GLu ,
IR) ,
then
cu(B) = ABA-1
is lineer as a
mop cout : MuxulIR) >
Mux)
=> Ad(A)(X) =
Tau(X) =
cona(X) =
AXA
-
1
VX
gl(u, (R)
VAEGLLu , IR)
.
P1 .
25
Suppose g is a real (or complex) lie
algebra
over (K =
1R(r = () .
A representation of
gou
a finite-aim .
vectospace
V over IK is a Lie
algebra homomarpuises
4 :
q
-+
g)(v)
i .
e .
a lineor mop s .
r .
U(tX ,Y]) =
[4() , U(Y)]
=
y(x)04(y) -
y(y)-
4(x)
VX ,
Yeg
Equivalently y a
bilinear map 4 :
gxV V s .
t .
P([x ,43 , v) = 4(x , P(Y, v)) -
P(Y,
PIX, r))
EU
By Prop . 1 . 12
, any representation y : G + GL(V)
of a lie
group
G induces a representation
4 =
Tel :
g
g((v)
of its Lie algubrag .
For G =
GL(u,
IR)
,
the standard representation 4 of
GL(uR)
gives
rise to standard representation of
geln.
IR)
y :
gen,
IR) x IR" - IRm
(x, v) + Xv
Similarly , for ary motrx
group
and its standard
representation .
For the adjoint representation of ahie
group
G,
Ad :
G-Gh(g) ,
the induced representation
of g ,
called the adjoint representation of
g , is
given by
ad :
g
+ gl(y)
ad(x)(y) = [X , 4] EXYeg .
os
the
following proposition shows .
Prop.
1 . 26 G Lie
group
with lie
alg .
(g .
[ , ]) .
① For XegaudgeG , (x(g) =
Ra
② For X ,
Yeg ,
ad(x)(y) = [X , 4]
③ For
Xeg , geG one has
④ For X ,
Ye
one has [X ,
[x ,
..
[x , Y]. .
]
dieh+ [x,
[x, 43+ - -
ca
=>
TelgX =
TepTcongX =
Ra eg
1
--
(y(g)
Ad(g)(x)
=
② Choose a basis X..... In of
g ,
then
Ad(g) :
g-g corresponds to an neu wohi
(Mij(g) for any get and Rij : G + IR ane smooth
Matrix representation of ad(x) :
g
+
& equals
X .
nij =
TajX =
(j) (e) Exeg
.
Any Yeg can be written as
YZyX
-
=>
Ly(g)
=
MYg
R,a
iis -
=>
[]=
R
Evoluting at e
yields :
[4] =
[3) Xi
=
ad(x)(Y)
③ Since
Ad(g) =
Tecong ,
the result follow from
① of Them .
1 .
23 ,
log(exp(+x) = expg)(+x)) = exp(+ Ad(g)(x)
VgeG,
Xeg
.
⑪ Ad : G +
Gh(g) hie group houn.
Tete = ad :
g+
ge(g)
ByD of Thus . 1. 23 :
Ad (exp(xs)(y) =
exp(ad(x))(y)
= gadk)(y)
=
W
Prop.
1 . 27 G hie
group with hie algebra o
Let 4 : G + GLIV) be a
represent - of G with
induced represent .
4' :
g+
gl(v) ofg .
① y(exp(+x(()= exp(tyk)(G) fte(R , Yxeg , vel
② y'(x)v =dexp
Pof
① follows from D of Thu .
1 .
23 and & is
an immedide lausegu .
D
hiesubgroups and Virtual Lie
subgreups
P1 . 28 Suppose It is a hie
subgroup of a hie
group G .
Then It is closed as subset of the
hopolog .
space G.
of
Any submite N of a mild M is
locally coed
,
i .
e.
open in its closure # (ED every poinc x + N Los
a
neighbled W in M s. t. Un N
is closed in
U
For
any subgraup It of a
hopolog group
G ,
#T is also
Elt
a
subgroup of G (*th 9 highn +
1 n +
0
If It is a lie
subgreup of G
.
It is
open and deuse
in # .
Heuce ,
for get ,
(g(H) # is
open in #
Since It is dcuse in # ,
(g(H) n H + 0 ,
which implies
geH.
*
Conversely ,
one has :
Thre .
1 .
29
Suppose It is a
subgroup of a
lie
group
G
that
is closed as a subet of the topolog ,
space G.
Then It is a
Lie
subgroup .
Pof We write
o for the
liealg. of G and set
2 : =
[c'(0) : c : IR-G is smooth ,
clo) -e
3andc has values in It
g
-Heins1
G is a lincer
subspace of
g.
If Ge :
IR-HEG Co-curves ,
Geld-clo = e.
Then ((E) : =
G(t)G(at) neIR
is Co-curve with values in It clo) =
e.
= c'(0) +
211
Tem (c(0) . ncilo) =
clo) + a
=(0)
Aum2 Suppose (Xninew is
sequence in S. 2.
lim In =
X Eq and let (tune be e
sequence o
IRyo With
limtn
=0
Then
,
if expLtutu) eH UneIN,
the explteH FteIR.
Fix
telR .
For ne let on be the
largest
integer
=> inth It and t-outh
/to ,
so
limanti
=>
line(n) =
liuexplanten) =
expl
continuity CH
of exp Since It is
Gold .
Pu3
G =
Sxeg :
exp(+ x) - H Ft -
IR]
RAS 2G by Def. of G.
To show
y[RIS ,
let c : IR-H-G be usmooth
curve with clo) =
e . Then c'(0) -g .
Then 7930 and a co-cive v : (E .
c) +
g
St .
e(t) =
explu(t)) f +
etqa)(v(d =
0q)
=>
c(d)) =
Texplo
Sollt
= liven u (m)
n +0
ser
te= and Xn =
n for sulf .
Lage n
=> expltutn) = exp(r()) =
c(z) - H
fer suff.
Legen
By claim ,
exp(tc(o) -H FtelR
imWunvecheine
Then 7 an
open neighbld Wot Oth in & s .
L .
exp(W)nH =
23.
conversely
, assume that's not the love .
Then I a
sequence of elements YntR s .
r . Live Yn = 0 and
= O h+0
exp(Yn)eH .
For a
norm 11 .
11 onk , put tu = Yu
.
By passing11Yull
to a
subsequence if
necessary , we can assume that
limtn =: Xek .
Then 11X11 = 1
,
in
particular,
X = 0.
h-y
Set tu = /Mall .
Than expltnYn) =
exp(Y) =H
and Claim Q and ⑤ show that XEG , which is
a controduction to X10 and XER .
we define the
following smooth wap
F:
G +
k + G
F(x , y) =
exp(x) exp(y)
Since Tof is a linedisemaphism ,
7 open neighbleds
V and W of OEG and Dek S . L.
F w
: VW - F(Yw) ==
is a
differaphism onto on
open neighbled Wote in G.
By shrinking
W
,
we
way assume that
exp(W)nH =
Ses
by Claim 4
.
↑ restricted to Ve30] is a
bijection autoHt.
Indeed
, exp(u) [Unl, since VEG .
Moreover ,
ony -Utt
cas be written
reiguely as x =
exp(x) exp(y) for Kev.
YeW.
=> exp(y) =
exp(-) x
eH
-H
By construction
,
this implies exply) = e
,
Lance Y = 0.
Therefore , (W ,
n : =
Flam) is a submeld .
Chart for
defined around eG and (in(U) , noin) for helt
is a
submitel. Chor around halt-
*