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 45 3. Haar measures on Lie groups 74 Chapter 3. Compact Lie groups 79 1. Compact Lie groups 79 Chapter 4. Basic Lie algebra theory 99 1. Solvable, nilpotent and semisimple Lie algebras 99 2. Lie algebras and field extensions 106 3. Cartan’s criterion 111 4. Semisimple Lie algebras 116 5. Cartan subalgebras 127 Chapter 5. Structure of semisimple Lie algebras 139 1. Root systems 139 2. Root system of a semisimple Lie algebra 148 3. Automorphisms of semisimple Lie algebras 155 4. Compact forms 156 iii CHAPTER 1 Basic differential geometry 1. Differentiable manifolds 1.1. Differentiablemanifoldsanddifferentiablemaps. LetM beatopo- 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∞- ∩ ∅ ∩ 6 ∅ ◦ ∩ −→ ∩ diffeomorphism. A family of charts on M is an atlas of M if the domains of charts form a A covering of M and any two charts in are compatible. A Atlases and of M are compatible if their union is an atlas on M. This is A B obviously an equivalence relation on the set of all atlases on M. Each equivalence classofatlasescontainsthelargestelementwhichisequaltotheunionofallatlases 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. ∈ ∈ Ifc=(U,ϕ,p)andc′ =(U′,ϕ′,p′)aretwochartsaroundm,thenp=p′. There- fore,allchartsaroundmhavethesamedimension. Therefore,wecallpthedimen- sion of M at the point m and denote it by dim M. The function m dim M m m 7−→ is locally constantonM. Therefore,itis constantonconnectedcomponents of M. If dim M =p for all m M, we say that M is an p-dimensional manifold. m ∈ LetM andN be twodifferentiablemanifolds. AcontinuousmapF :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 mapf :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 forma category. Isomorphismsin this categoryare called diffeomorphisms. 1 2 1. DIFFERENTIAL GEOMETRY 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 T (M) and call the tangent space to M at m. 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) ∂ (f)= ◦ (0). i ∂x i Clearly, ∂ are tangent vectors in T (M). i m 1.2.1.Lemma. The vectors ∂ ,∂ ,...,∂ for a basis of thelinear space T (M). 1 2 p m In particular, dimT (M)=dim M. m m Let F : M N be a morphism of differentiable manifolds. Let m M. −→ ∈ Then, for any ξ T (M), the linearform T (F)ξ :g ξ(g F) for g C∞(N), m m ∈ 7−→ ◦ ∈ isa tangentvectorinT (N). Clearly, T (F):T (M) T (N)is alinear F(m) m m F(m) −→ 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 T (F). m 1.2.2. Lemma. The function m rank F is lower semicontinuous on M. m 7−→ 1.3. Localdiffeomorphisms,immersions,submersionsandsubimmer- 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 −→ −→ T (N) is injective. A morphism F :M N is ansubmersion at m if T (F): F(m) m −→ T (M) T (N) is surjective. m F(m) −→ If F is an immersion at m, rank F = dim M, and by 1.2.2, this condition m m 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, rank F =dim N, and by 1.2.2, m F(m) this condition holds in an open neighborhoodof 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 −→ immersionifitisanimmersionateachpointofM. AdifferentiablemapF :M −→ N is an submersion if it is an submersion ant each point of M. A differentiable mapF :M N is ansubimmersion if it is ansubimmersion ateachpoint of M. −→ The rank of a subimmersion is constant on connected components of M. 1. DIFFERENTIABLE MANIFOLDS 3 1.3.2.Theorem. Let F :M N be a subimmersion at p M. Assume that −→ ∈ rank F = r. Then there exists charts c = (U,ϕ,m) and d = (V,ψ,n) centered at m 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. Letp:M N beasurjectivesubmersion. LetF :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- sumethatanypointn N hasanopenneighborhoodU inM andachart(U,ϕ,p) ∈ centered at n such that ϕ(N U) = ϕ(U) Rq 0 . If we equip N with the ∩ ∩ ×{ } induced topologyand define its atlas consistingof charts onopen 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 ofM ontof(M) N,f(M)is asubmanifoldin N andf :M f(M) ⊂ −→ is a diffeomorphism. Let f :M N is a differentiable map. Denote by Γ the graph off, i.e., the f −→ subset (m,f(m)) M N m M . Then, α:m (m,f(m)) is a continuous { ∈ × | ∈ } 7−→ bijectionofM ontoΓ . Theinverseofαistherestrictionofthecanonicalprojection f 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 T (α)(ξ) = (ξ,T (f)(ξ)) for m m any ξ T (M), hence α is an immersion. By 1.4.2, we get the following result. m ∈ 1.4.3.Lemma. Let f :M N be a differentiable map. Then the graph Γ of f −→ f is a closed submanifold of M N. × 1.4.4. Lemma. Let M and N be differentiable manifolds and F : M N −→ a differentiable map. Assume that F is a subimmersion. Then, for any n N, ∈ F−1(n) is a closed submanifold of M and T (F−1(n))=kerT (F). m m for any m F−1(n). ∈ In the case of submersions we have a stronger result. 1.4.5. Lemma. Let F : M N be a submersion and P a submanifold of N. −→ Then F−1(P) is a submanifold of M and the restriction f|F−1(P) : F−1(P)−→P is a submersion. For any m F−1(P) we also have ∈ T (F−1(P))=T (F)−1(T (P)). m m F(m) 4 1. DIFFERENTIAL GEOMETRY 1.5. Products and fibered products. Let M and N be two topological spaces and c = (U,ϕ,p) and d = (V,ψ,q) two charts on M, resp. N. Then (U V,ϕ ψ,p+q) is a chart on the product space M N. We denote this chart × × × by c d. × Let M and N be two differentiable manifolds with atlases and . Then A B c d c ,d is an atlas on M N. The corresponding saturated atlas { × | ∈ A ∈ B} × definesastructureofdifferentiablemanifoldonM N. Thismanifoldiscalledthe × product manifold M N of M and N. × Clearly dim (M N)=dim M +dim N for any m M and n N.. (m,n) m n × ∈ ∈ The canonical projections to pr :M N M and pr :M N N are 1 2 × −→ × −→ submersions. Moreover, (T (pr ),T (pr )):T (M N) T (M) T (N) (m,n) 1 (m,n) 2 (m,n) m n × −→ × is an isomorphism of linear spaces for any m M and n N. ∈ ∈ LetM,N andP bedifferentiablemanifoldsandF :M P andG:N P −→ −→ differentiable maps. Then we put M N = (m,n) M N f(m)=g(n) . P × { ∈ × | } This set is called the fibered product of M and N with respect to maps F and G. 1.5.1. Lemma. If F : M P and G : N P are submersions, the fibered −→ −→ product M N is a closed submanifold of M N. P × × The projections p:M N M and q :M N N are submersions. P P × −→ × −→ For any (m,n) M N, P ∈ × T (M N)= (X,Y) T (M N) T (f)(X)=T (G)(Y) . (m,n) P (m,n) m n × { ∈ × | } Proof. Since F andGaresubmersions,the productmapF G:M N × × −→ P P isalsoasubmersion. Since the diagonal∆is aclosedsubmanifoldinP P, × × from1.4.5we conclude that the fiber productM N =(F G)−1(∆) is a closed P × × submanifold of M N. Moreover,we have × T (M N)= (X,Y) T (M N) T (F)(X)=T (G)(Y) . (m,n) P (m,n) m n × { ∈ × | } Assume that (m,n) M N. Then p = f(m)= g(n). Let X T (M). Then, P m ∈ × ∈ sinceG isa submersion,thereexists Y T (N)suchthatT (G)(Y)=T (F)(X). n n m ∈ Therefore, (X,Y) T (M N). It follows that p : M N M is a (m,n) P P submersion. Analog∈ously, q :M× N N is also a submersi×on. −→ (cid:3) P × −→ 2. Quotients 2.1. Quotient manifolds. LetM be a differentiable manifold andR M ⊂ × M an equivalence relation on M. Let M/R be the set of equivalence classes of M with respect to R and p:M M/R the corresponding natural projection which −→ attaches to any m M its equivalence class p(m) in M/R. ∈ WedefineonM/Rthequotienttopology,i.e.,wedeclareU M/Ropenifand ⊂ only if p−1(U) is open in M. Then p : M M/R is a continuous map, and for −→ any continuous map F :M N, constant on the equivalence classes of R, there exists a unique continuous m−→ap F¯ : M/R N such that F = F¯ p. Therefore, −→ ◦ 2. QUOTIENTS 5 we have the commutative diagram F M //N . z<< z z p zz z (cid:15)(cid:15) zz F¯ M/R Ingeneral, M/R is not a manifold. For example,assume that M =(0,1) R, and ⊂ R the union of the diagonal in (0,1) (0,1) and (x,y),(y,x) for x,y (0,1), × { } ∈ x = y. Then M/R is obtained from M by identifying x and y. Clearly this 6 topological space doesn’t allow a manifold structure. x(cid:13) p(x)=p(y)(cid:13) p(cid:13) y(cid:13) M/R(cid:13) M(cid:13) Assume thatM/R hasa differentiable structuresuchthatp:M M/Ris a −→ submersion. Since p is continuous, for any open set U in M/R, p−1(U) is open in M. Moreover,pisanopenmapby1.3.5. Hence,foranysubsetU M/Rsuchthat ∈ p−1(U)isopeninM,thesetU =p(p−1(U))isopeninM/R. Therefore,asubsetU inM/Risopenifandonlyifp−1(U)isopeninM,i.e.,thetopologyonM/Risthe quotient topology. Moreover, by 1.3.4, if the map F from M into a differentiable manifold N is differentiable, the map F¯ :M/R N is also differentiable. −→ Weclaimthatsuchdifferentiablestructureisunique. Assumethecontraryand denote (M/R) and (M/R) two manifolds with these properties. Then, by the 1 2 above remark, the identity maps (M/R) (M/R) and (M/R) (M/R) 1 2 2 1 −→ −→ are differentiable. Therefore, the identity map is a diffeomorphism of (M/R) and 1 (M/R) , i.e., the differentiable structures on M/R are identical. 2 Therefore, we say that M/R is the quotient manifold of M with respect to R if it allows a differentiable structure such that p:M M/R is a submersion. In −→ this case, the equivalence relation is called regular. If the quotient manifold M/R exists, since p:M M/R is a submersion, it −→ is also an open map. 2.1.1. Theorem. Let M be a differentiable manifold and R an equivalence relation on M. Then the following conditions are equivalent: (i) the relation R is regular; (ii) R is closed submanifold of M M and the restrictions p ,p : R M 1 2 × −→ of the natural projections pr ,pr :M M M are submersions. 1 2 × −→ The proof of this theorem follows from a long sequence of reductions. First we remark that it is enough to check the submersion condition in (ii) on only one map p , i = 1,2. Let s : M M M M be given by s(m,n) = (n,m) for i × −→ × m,n M. Then, s(R)=R since R is symmetric. Since R is a closed submanifold ∈ 6 1. DIFFERENTIAL GEOMETRY ands:M M M M adiffeomorphism,s:R Risalsoadiffeomorphism. × −→ × −→ Moreover,pr =pr sandpr =pr s,immediatelyimpliesthatp isasubmersion 1 2 2 1 1 ◦ ◦ if and only if p is a submersion. 2 Wefirstestablishthat(i)implies(ii). ItisenoughtoremarkthatR=M M/R × M with respect to the projections p : M M/R. Then, by 1.5.1 we see that −→ R is regular, i.e., it is a closed submanifold of M M and p ,p : R M are 1 2 × −→ submersions. Now we want to prove the converse implication, i.e., that (ii) implies (i). This part is considerably harder. Assume that (ii) holds, i.e., R is a closed submanifold inM M andp ,p :R M aresubmersions. WeequipM/Rwiththequotient 1 2 × −→ topology. Hence, p : M M/R is a continuous map. We first observe the −→ following fact. 2.1.2. Lemma. The map p:M M/R is open. −→ Proof. Let U M be open. Then ⊂ p−1(p(U))= m M p(m) p(U) { ∈ | ∈ } = m M (m,n) R, n U =pr (R (M U))=p (R (M U)). 1 1 { ∈ | ∈ ∈ } ∩ × ∩ × Clearly, M U is open in M M, hence R (M U) is open in R. Since × × ∩ × p : R M is a submersion, it is an open map. Hence p (R (M U)) is an 1 1 −→ ∩ × open set in M. By the above formula it follows that p−1(p(U)) is an open set in M. Therefore, p(U) is open in M/R. (cid:3) Moreover,we have the following fact. 2.1.3. Lemma. The quotient topology on M/R is hausdorff. Proof. Let x = p(m) and y = p(n), x = y. Then, (m,n) / R. Since R 6 ∈ is closed in M M, there exist open neighborhoods U and V of m and n in M × respectively, such that U V is disjoint from R. Clearly, by 2.1.2, p(U) and p(V) × are open neighborhoods of x and y respectively. Assume that p(U) p(V) = . ∩ 6 ∅ Then there exists r M such that p(r) p(U) p(V). It follows that we can find ∈ ∈ ∩ u U and v V such that p(u) = p(r) = p(v). Therefore, (u,v) R, contrary ∈ ∈ ∈ to our assumption. Hence, p(U) and p(V) must be disjoint. Therefore, M/R is hausdorff. (cid:3) Now we are going to reduce the proof to a “local situation”. Let U be an open set in M. Since p is an open map, p(U) is open in M/R. Then we put R = R (U U). Clearly, R is an equivalence relation on U. U U ∩ × Let p : U U/R be the corresponding quotient map. Clearly, (u,v) R U U U −→ ∈ implies (u,v) R and p(u) = p(v). Hence, the restriction p : U M/R is U ∈ | −→ constant on equivalence classes. This implies that we have a natural continuous map i : U/R M/R such that p = i p . Moreover, i (U/R ) = p(U). U U U U U U U −→ | ◦ Weclaimthati isaninjection. Assumethati (x)=i (y)forsomex,y U/R . U U U U ∈ Then x=p (u) and y =p (v) for some u,v U. Therefore, U U ∈ p(u)=i (p (u))=i (x)=i (y)=i (p (v))=p(v) U U U U U U and (u,v) R. Hence, (u,v) R and x = p (u) = p (v) = y. This implies our U U U ∈ ∈ assertion. Therefore, i :U/R p(U) is a continuous bijection. We claim that U U −→