Trop1
.
7 Suppose G is a Lie
group and es es
① For
any
Keg
,
4(9) : =
TelgX e TgG (resp .
Ry(g)=
[93X
-
>
TgG)
is a left-(resp.
right) invariant vector
field
on
G.
② The maps Gxg >
TG ,
defined be
(9 .
X)-
>
Lolg) resp .
(g .
X) + Rx(g) ,
are differmorphisus .
⑤ The map XreLx Cresp . XRx) define
linear icomorphism with inverse +ste)
between
g
and (ty(G) (resp .
(tip(G) .
Iroof
② (g ,
x) + Lx(g) is a differu.
Define the mop
F :
Gxg
+
TExTG given by
F(g , x) =
(0g , x) (OgfTgG zero taugent vecteaty)
It is smooth and so is
TroF :
Gxg
>
TG ,
which
equals (g
. x) + L(g) by Leana 1 .
5
.
To show Trot is a differe .
We construct a smooth
inverse .
Define F :
TG + TGxTG by
↑ (g) =
(Og , 4g). It is smooth (by Bofhemnes
1 .
5)
*
gG
and so is TroF ,
which is
given by Egte Ti
=
Egt (g , Tgig(g) -Gxg is smooth was
TG Gxg
,
which is inverse (g , x)h(g)=
Tig
Similarly ,
one proves
the statement for (g ,
x) +
Ry(g).
⑦ By ② ↳ and Ry ane smooth vector fields on G.
(ig
*
(x)(h) =
[ig) ((x(ig(4))) =
=
TutgeTigaX =
Telgin)x =
4(4)
Vgin - G. 20 in
Similarly , one checks
right-inv .
of Rx
③ By D , Xi define a linear mop
g
+ (t
,
(E) and L() = X
If seCtuLG) , theg =
(igs) (g) =
Tigg(e)
=
Lget
-
#
P1. 8 G Lie
group . TeG =
ig
① The differepuism Erg -+ TG
given by
(g ,
x) +
2x (g) (resp .
(g ,
X)+
Rx(g) of Prop.
1 . 7
is called the left-(resp.
right) trivialization of
tangent bundle TG + G :
TG-Gxy vector bundle
isamorphisa.
&↓P - natural proj.to first factor
② For
Xeg , La Cresp. Rx) is called the
left-Creep .
right) invorient rector field on G
generated by X.
Note that
any L tresp . Rx) is nowhere
vanishing
on G and
choosing a basis X
1, .... An of the
Vector space f , Lxe(g) ,
.... h(g) (resp.
Rx(g). ... Rx3)
forms a basis
of TgG
For
any geG ,
Ig
*
[sin] =
[Sg ,
ijn] =
[5n]
Vs ,
y
+ (ty(t).
The subspace (ty(G) e (t(G) is a
subelgebra of(Ct(G)
·
[ . J) . Vir isomorphim ,
g
=
(ty(G) of3
of Prop . 1.
7 ,
we can transport I ,
I to a brocket on
g
Be.
1 . 4
Suppose G is a
lie
group.
Then the taugent
space g
:
TeG at e@G together with the mop
[ ,
J :
gxg
+
g
[x , 47 : =
[(x , Ly](e)
is called the lie
abgebre of G.
Onehas
Lexys =
[hehy] by construction.
From the properties of the lie bracket of rector fields
it follows :
Prop.
1 .
10 The mop [ ,
J :
gxg
+
gas
in Def .
1.
9
hos the following properties :
(i) it is bilinear (overIR)
(ii) Skew-symmetric :
[xiT] = -[4 , x]
ExYeg
(iii) it
satisfies the Jacobi-identity :
[x ,
[4 , z]] +
[4 ,
[z , x]] + [z ,
[X , 4]] = 0
VX ,
Y ,
zeg
Def .
1 . 11
①
A real (resp . couplex) Lie
algebra is real (resp .
complex
rector space
o equipped with a
IR-(resp (-)
bilinear map [ , 7 .
gxg
+
& satisfying
(ii) and (iii) of Prop . 1. 10.
② A Lie
algebra homomorphism (resp.
is
amarphise)
between Lie
algebras gand G is a linear map
Cresp . Linear
isamorphism) 4 :
g
+
G s.
t
.
P([x ,
43) =
[4(x) , 4(3778x,
78
③ A subalgebra of a lie algebra g
is a
subspace
zofg .. 2 .
[X]eg Exeg
Examples①
Consider a Vector
space V (dim(V) <b) as
Lie
group wir to addition + .
Then the
left-trivialization is
just the usual identification
of TV =
VXV and the left-ivv.
Vector fields
corresp .
to constant functions V+ V.
In particular , the Lie bracket of two left-inv. Vf.
vanishes and hance the lie
algenre is TV = V
equipped with the zero brocker [v, w] = 0 Fuwer .
② Let G and It be lie
groups with hie
algenies
(g.
[ .
Ig) and (G .
[ ,
Ig)
Then GxH is a lie group with lie
algebra
TeeGxH) =
TeGxTelt =
g]
with Lie brocket [(X, 4) ,
(x !
41)] =
([Xx]g .
[Y,
41]VX ,
x g ,
4, 4tg .
Heace ,
the lie
algebra of Ex I is what one calls the
direct sor of the lie algebras of g
and
G
Check this as an
exercise)
③ G = GL(n ,
IR) Maxn(IR)
g
=
Maxu(R)
Tonau
For AEGL(n .
IR)
, Ja : GLIn ,
IR) +
GLIn,
IR)
is the restriction of the linear mop Sp :
MuxnUR) -
MLR)
=>
Ta(B ,
X) = (AB , Ax) - TABGL(n,
(2)
ETizGL(u, IR)
=
k(t) =
Tax =
*
~
Viewing
Lx as a function GR) +
Man(1)
we know that
[x4] : =
[Lx , Ly7 ((d) =
TraLyLxe)
-
Tax
Lyll=
xy -
YX
VX ,
4 g=
Muxu((R) .
Lie
algebra of GL(u,
IR)
is
=
Maxn(R) =:
ge (n, 1)
with the commutator
of matrices as hie brocket.
Prop.
1 .
12
Suppose G and H are lie
groups and y
: G+ H
is
a lie
group homomorphism .
① Then 4 :=
TeY :
TeG =
-Tel =
G
(q(e) =
e) is a homomorphism of Lie
algebras ·
② If G is commutative /abelian ,
then the lie bracket
ou
g is zero (i. e .
(g ,
[ , 1) is an abelian liedly.
as one
says)
Prof Exercise
ClarySuppeGinliegend
subalgebra of g.
In particular,
the lie
algebra of
any matrix
group H1GL(u ,
IR) is a
subalgebra
of (geIniR) , [ ,
2) , where [ ,
J is the commutete
of motrices.
Pof Apply ① of Prop .
1. 12 to the incusion i :
H4G.
Proph
Suppose G is a lie
group
with lie e
⑦Ry = r
+
Lyfxeq
② [Rx , Ry] = Rex,
y]
Fileg
.
③ [Ly , Ry] = 0
Yeg
Rot Exercise
Prop. 1. 15 Suppose G is a hie group
and 5 is a
Left-(resp .
right)- invariant vector field on G.
⑦ FE(g) =
gF((e) VgeG (resp .
F((g) =
Fl(e)g
Fg =
G)
② is complete
Prof
⑦ If h is left-invenient ,
theu I is
lg-related to
itself
(1g
*
(h) = >(n) ()
Tig
=
S(gn)
↑
Sy- - ((gu) =
S(u)
which implies
that the flow of s commute with
bg :
Folg =
IgOFL VgeG.
Evaluating at eThows :
F((g) =
gF(e),
② Follows from Criteria we
proved in Global
anylisis :
If EG)0 s .
t .
any integral curve
through only
point of a vector fieldis defined on 79.
a) ,
then
S is complete
&1 . 16 Suppose G is a lie
group .
A one paramete subgroup of G is a lie
group
homomorphism a :
(IR +) -> G (i. e . a smooth
Curve 2 : 1R + G s.
r .
a(s+ +) =
a(t)a(s) Vs. te(R) ·
In particular ,
a (0) =
e.
Lemma 1. 17 G lie
group ,
a : IR- > G smoothwith
a loze and X Eg
.
Then the
following are
equivalent :
⑦ a is a bue parameter subgroup with(0) = X.
② a(t) =
F(e) V + EIR
③
a(t) =
FLX(e) FtEIR
Prof
① ② Jaslals)
a (t)
=dstt)ata
=
Tec
=
k(x(+)
= < is an
integral wive of Lx and since alo) = e,
we must hore alt) =
Fut(e) by uniqueness.
② a(t) =
Fl(e) is a smooth curve in G
with
< (0) =
e and a'(0) =
Lx(a(0) = X .
Since
It is a flow we have
alt + s) =
F((e) = (()) = a(s)a(t)
↑
=
als) @ of Prop.
1. 15
By exchanging roles
ofs andt
, one
proves
similarly that D E.
V