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Lectures on Lie Groups Lectures on Lie Groups Dragan Miliˇci´c Contents Chapter1. Basicdifferentialgeometry 1 1. Differentiable manifolds 1 2. Quotients 4 3. Foliations 11 4. Integration on manifolds 19 Chapter2.Liegroups 23 1.Liegroups 23 2. Lie algebra of a Lie group 49 3.HaarmeasuresonLiegroups 79 Chapter3.CompactLiegroups 83 1.CompactLiegroups 83 Chapter4. BasicLiealgebratheory 103 1. Solvable, nilpotent and semisimple Lie algebras 103 2. Lie algebras and field extensions 110 3. Cartan’s criterion 115 4. Semisimple Lie algebras 120 5.Cartansubalgebras 131 Chapter 5. Structure of semisimple Lie algebras 143 1.Rootsystems 143 2. Root system of a semisimple Lie algebra 152 3. BasesofrootsystemsandWeylgroups 158 4. Relations defining semisimple Lie algebra 165 5. Semisimple Lie algebras and their root systems 177 Chapter6. ClassificationofcompactLiegroups 181 1. Compact semisimple Lie groups 181 iii CHAPTER 1 Basic differential geometry 1. Differentiable manifolds 1.1. 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 Z+ and a homeomorphism ϕ of U onto an open set in Rp.⊂ 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 c′ = (U ′, ϕ′,p′) on M are compatible if either U U ′ = or U U ′ = and ϕ′ ϕ−1 : ϕ(U U ′) ϕ′(U U ′) 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 any two charts in are compatible. Atlases and of M are compatibleA 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 locally compact space. It is also locally connected. Therefore, its connected components are open and closed subsets. Let M be a differentiable manifold. A chart c = (U,ϕ,p) is a chart around m M if m U. We say that it is centered at m if ϕ(m) = 0. ∈If c = (U,ϕ,p∈ ) and c′ = (U ′, ϕ′,p′) are two charts around m, then p = p′. 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 dimm M. The function m dimm M is locally constant on M. Therefore, it is constant on connected components7−→ of M. If dimm M = p for all m M, we say that M is an p-dimensional manifold. Let M and N be two differentiable∈ manifolds. A continuous map F : M N is a differentiable map if for any two pairs of charts c = (U,ϕ,p) on M and−→d = (V,ψ,q) on N such that F (U) V , the mapping ⊂ ψ F ϕ−1 : ϕ(U) ϕ(V ) ◦ ◦ −→ is a C∞-differentiable map. We denote by Mor(M,N) the set of all differentiable maps from M into N. If N is the real line R with obvious manifold structure, we call a differentiable map f : M R a differentiable function on M. The set of all differentiable func- tions on M −→forms an algebra C∞(M) over R with respect to pointwise operations. Clearly, differentiable manifolds as objects and differentiable maps as mor- phisms form a category. Isomorphisms in this category are called diffeomorphisms. 1 2 1.DIFFERENTIALGEOMETRY 1.2. Tangent spaces. Let M be a differentiable manifold and m a point in M. A linear form ξ on C∞(M) is called a tangent vector at m if it satisfies ξ(fg)= ξ(f)g(m)+ f(m)ξ(g) for any f,g C∞(M). Clearly, all tangent vectors at m form a linear space which ∈ we denote by Tm(M) and call the tangent space to M at m. Let m M and c = (U,ϕ,p) a chart centered at m. Then, for any 1 i p, we can define∈ the linear form ≤ ≤ ∂(f ϕ−1) ∂i(f)= ◦ (0). ∂xi Clearly, ∂i are tangent vectors in Tm(M). 1.2.1. Lemma. The vectors ∂1, ∂2, . , ∂p for a basis of the linear space Tm(M). In particular, dim Tm(M) = dimm M. Let F : M N be a morphism of differentiable manifolds. Let m M. Then, for any ξ −→T (M), the linear form T (F )ξ : g ξ(g F ) for g C∞∈(N), ∈ m m 7−→ ◦ ∈ is a tangent vector in TF (m)(N). Clearly, Tm(F ): Tm(M) TF (m)(N) is a linear map. It is called the differential of F at m. −→ The rank rank F of a morphism F : M N at m is the rank of the linear m −→ map Tm(F ). 1.2.2. Lemma. The function m rank F is lower semicontinuous on M. 7−→ m 1.3. Local diffeomorphisms, immersions, submersions and subimmer- sions. Let F : M N be a morphism of differentiable manifolds. The map F is a local diffeomorphism−→ at m if there is an open neighborhood U of m such that F (U) is an open set in N and F : U F (U) is a diffeomorphism. −→ 1.3.1. Theorem. Let F : M N be a morphism of differentiable manifolds. Let m M. Then the following conditions−→ are equivalent: ∈ (i) F is a local diffeomorphism at m; (ii) T (F ): T (M) T (N) is an isomorphism. m m −→ F (m) A morphism F : M N is an immersion at m if T (F ): T (M) −→ m m −→ TF (m)(N) is injective. A morphism F : M N is an submersion at m if Tm(F ): T (M) T (N) is surjective. −→ m −→ F (m) If F is an immersion at m, rankm F = dimm M, and by 1.2.2, this condition holds in an open neighborhood of m. Therefore, F is an immersion in a neighbor- hood of m. Analogously, if F is an submersion at m, rankm F = dimF (m) N, and by 1.2.2, this condition holds in an open neighborhood of m. Therefore, F is an submersion in a neighborhood of m. A morphism F : M N is an subimmerson at m if there exists a neighbor- hood U of m such that the−→ rank of F is constant on U. By the above discussion, immersions and submersions at m are subimmersions at p. A differentiable map F : M N is an local diffeomorphism if it is a local diffeomorphism at each point of −→M. A differentiable map F : M N is an immersion if it is an immersion at each point of M. A differentiable map−→F : M N is an submersion if it is an submersion ant each point of M. A differentiable−→ map F : M N is an subimmersion if it is an subimmersion at each point of M. The rank of−→ a subimmersion is constant on connected components of M. 1.DIFFERENTIABLEMANIFOLDS 3 1.3.2. Theorem. Let F : M N be a subimmersion at p M. Assume that −→ ∈ rankm F = r. Then there exists charts c = (U,ϕ,m) and d = (V,ψ,n) centered at p and F (p) respectively, such that F (U) V and ⊂ (ψ F ϕ−1)(x ,...,x ) = (x ,...,x , 0,..., 0) ◦ ◦ 1 n 1 r for any (x ,...,x ) ϕ(U). 1 n ∈ 1.3.3. Corollary. Let i : M N be an immersion. Let F : P M be a continuous map. Then the following−→ conditions are equivalent: −→ (i) F is differentiable; (ii) i F is differentiable. ◦ 1.3.4. Corollary. Let p : M N be a surjective submersion. Let F : N P be a map. Then the following conditions−→ are equivalent: −→ (i) F is differentiable; (ii) F p is differentiable. ◦ 1.3.5. Corollary. A submersion F : M N is an open map. −→ 1.4. Submanifolds. Let N be a subset of a differentiable manifold M. As- sume that any point n N has an open neighborhood U in M and a chart (U,ϕ,p) centered at n such that∈ ϕ(N U) = ϕ(U) Rq 0 . If we equip N with the induced topology and define its∩ atlas consisting∩ of×{ charts} on open sets N U given by the maps ϕ : N U Rq, N becomes a differentiable manifold. With∩ this differentiable structure,∩ the−→ natural inclusion i : N M is an immersion. The manifold N is called a submanifold of M. −→ 1.4.1. Lemma. A submanifold N of a manifold M is locally closed. 1.4.2. Lemma. Let f : M N be an injective immersion. If f is a homeo- morphism of M onto f(M) N−→, f(M) is a submanifold in N and f : M f(M) is a diffeomorphism. ⊂ −→ Let f : M N is a differentiable map. Denote by Γf the graph of f, i.e., the subset (m,f(m−→)) M N m M . Then, α : m (m,f(m)) is a continuous { ∈ × | ∈ } 7−→ bijection of M onto Γf . The inverse of α is the restriction of the canonical projection p : M N M to the graph Γ . Therefore, α : M Γ is a homeomorphism. × −→ f −→ f On the other hand, the differential of α is given by Tm(α)(ξ) = (ξ,Tm(f)(ξ)) for any ξ T (M), hence α is an immersion.
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