Lectures on Lie Groups - D. Milicic WW.pdf

Lectures on Lie Groups
Dragan Miliˇci´c
Contents
Chapter 1. Basic differential geometry 1
1. Differentiable 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 field 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
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CHAPTER 1
Basic differential geometry
1. Differentiable manifolds
. Differentiable manifolds and differentiable maps. Let M be a topo-
logical space. A chart on M is a triple c = (U, ϕ, p) consisting of an open subset
 
U M, an integer p + and a homeomorphism ϕ 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, ϕ, p) and c0 = (U 0, ϕ0, p0) on M patible if either
U U 0 = or U U 0 = and ϕ0 ϕ−1‘: ϕ(U U 0) ϕ0(U U 0) is a C∞-
diffeomorphism.∩∅∩ 6 ∅◦∩−→∩
A family of charts on M is an atlas of M if the domains of charts form a
covering of MAand all any two charts in patible.
Atlases and of M patibleA if their union is an atlas on M. This is
obviously anA equivalenceB 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 differentiable manifold M is a hausdorff topological space with a saturated
atlas.
Clearly, a differentiable manifold is a pact space. It is also locally
connected. Therefore, its ponents are open and closed subsets.
Let M be a differentiable manifold. A chart c = (U, ϕ, p) is