Download Lecture_2 - National University of Singapore

Mathematical Physics Seminar Notes
Lecture 2 Global Analysis and Lie Theory
Wayne M. Lawton
Department of Mathematics
National University of Singapore
2 Science Drive 2
Singapore 117543
Email [email protected]
Tel (65) 6874-2749
1
Lie Groups and Algebras
Definition. A Lie group is a manifold and a group
such that multiplication and inverses are smooth
Left and right multiplication induce bundle maps of
the tangent bundle, both left and right invariant vector
fields are in one-to-one correspondence with their
values at the identity, and the sets of left/right
invariant vector fields are closed under commutation
Definition. The Lie algebra of a Lie group G
is the Lie algebra of left invariant vector fields
identified with the tangent  space at the identity
2
One Parameter Subgroups
The integral trajectories of a vector field v G
are mapped into themselves by left translation
g , p  G, s  R
Fv (s)( gp)  gFv (s)( p)
Ev
R  G by E v ( t )  Fv ( t )(1)
hence for all
define
therefore
E v (s)E v (t )  E v (t  s)
3
Exponential Function
Definition.   G exp( v)  E v (1)
Then exp is a local analytic diffeomorphism and
exp
[u, v] [e
u s v s u s  v s
e
e
e
,0]
Campbell-Baker-Hausdorff For small u, v  
1
u v
t ad u ad v
log( e e )  v   g (e e )udt
0
logz
where g(z) 
(ad
u)(w)

[u,
w]
so
z -1
u v
1
log( e e )  u  v  2 [u, v] 
1


[
u
,
[
u
,
v
]]

[
v
,
[
v
,
u
]]
 4
12
Lie Groups from Lie Algebras
Lie
  span {u1 ,..., ud }
then
{ exp( t1u1   td ud : ti  R }
form a local Lie group, furthermore it is a linear
(matrix) group if the algebra is
Ado Every finite dimensional Lie algebra admits a
faithful representation and therefore is linear (this
result was discovered after Lie)
5