00313 - Lectures on Lie Groups.pdf

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lie.dvi
Lectures on Lie Groups
Dragan Milicic
Contents
Chapter 1. Basic dierential geometry
1
1. Dierentiable manifolds
1
2. Quotients
4
3. Foliations
11
4. Integration on manifolds
19
Chapter 2. Lie groups
23
1. Lie groups
23
2. Lie algebra of a Lie group
43
3. Haar measures on Lie groups
72
Chapter 3. Compact Lie groups
77
1. Compact Lie groups
77
Chapter 4. Basic Lie algebra theory
97
1. Solvable, nilpotent and semisimple Lie algebras
97
2. Lie algebras and eld extensions
104
3. Cartan’s criterion
109
4. Semisimple Lie algebras
113
5. Cartan subalgebras
125
Chapter 5. Structure of semisimple Lie algebras
137
1. Root systems
137
2. Root system of a semisimple Lie algebra
145
iii
CHAPTER 1
Basic dierential geometry
1. Dierentiable manifolds
1.1. Dierentiable manifolds and dierentiable maps. Let M be a topo-
logical space. A chart on M is a triple c = ( U,',p ) consisting of an open subset
U M , an integer p 2
and a homeomorphism ' of U onto an open set in
p . The open set U is called the domain of the chart c , and the integer p is the
dimension of the chart c .
The charts c = ( U,',p ) and c 0 = ( U 0 ,' 0 ,p 0 ) on M are compatible if either
U \ U 0 =;or U \ U 0 6=;and ' 0 ' − 1 ‘ : ' ( U \ U 0 )−! ' 0 ( U \ U 0 ) is a C 1 -
dieomorphism.
A familyAof charts on M is an atlas of M if the domains of charts form a
covering of M and all any two charts inAare compatible.
AtlasesAandBof M are compatible if their union is an atlas on M . This is
obviously an equivalence relation on the set of all atlases on M . Each equivalence
class of atlases contains the largest element which is equal to the union of all atlases
in this class. Such atlas is called saturated.
A dierentiable manifold M is a hausdor topological space with a saturated
atlas.
Clearly, a dierentiable manifold is a locally compact space. It is also locally
connected. Therefore, its connected components are open and closed subsets.
Let M be a dierentiable manifold. A chart c = ( U,',p ) is a chart around
m 2 M if m 2 U . We say that it is centered at m if ' ( m ) = 0.
If c = ( U,',p ) and c 0 = ( U 0 ,' 0 ,p 0 ) are two charts around m , then p = p 0 . There-
fore, all charts around m have the same dimension. Therefore, we call p the dimen-
sion of M at the point m and denote it by dim m M . The function m 7−!dim m M
is locally constant on M . Therefore, it is constant on connected components of M .
If dim m M = p for all m 2 M , we say that M is an p -dimensional manifold.
Let M and N be two dierentiable manifolds. A continuous map F : M −! N
is a dierentiable map if for any two pairs of charts c = ( U,',p ) on M and d =
( V, ,q ) on N such that F ( U ) V , the mapping
F ' − 1
: ' ( U )−! ' ( V )
is a C 1 -dierentiable map. We denote by Mor( M,N ) the set of all dierentiable
maps from M into N .
If N is the real line with obvious manifold structure, we call a dierentiable
map f : M −! a dierentiable function on M . The set of all dierentiable func-
tions on M forms an algebra C 1 ( M ) over with respect to pointwise operations.
Clearly, dierentiable manifolds as objects and dierentiable maps as mor-
phisms form a category. Isomorphisms in this category are called dieomorphisms.
1
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