Matrix Lie Groups And Lie Correspondences
MATRIX LIE GROUPSANDLIE CORRESPONDENCEST H E S ISSummited in partial fulfillment of therequirement for the master degreeof science in basic mathematicsA thesis by:Monyrattanak SengUnder direction of advisor:Professor. Raúl Quiroga BarrancoGuanajuato, Gto., August, 02nd 2017
MATRIX LIE GROUPSANDLIE CORRESPONDENCEST H E S ISSummited in partial fulfillment of therequirement for the master degreeof science in basic mathematicsA thesis by:Monyrattanak SengUnder direction of advisor:Professor. Raúl Quiroga BarrancoAuthorization of the final versionGuanajuato, Gto., August, 02nd 2017
AcknowledgementThis thesis haven't been completed without the supports from my adviser professor.Raúl Quiroga Barranco.I would like to express my gratefulness to him forhis patience, motivation, enthusiasm, and his valuable times. His guidance and ideashelped me in all the time for research and writing this thesis.At the same time, I would like to thanks myself for working hard on this thesisand acknowledge all professors and stu s in CIMAT that gave me important lecturesand supports. In addition, I would like to thanks all my friends who gave me usefulresources.Finally, I would like to say "thank you" to all authors of the books that I usedas the references of this thesis, especially, Brian C. Hall with his book Lie groups,Lie algebra, and representations . His book inspires me to extend my Ph.D on thisarea.i
AbstractLie group is a di erentiable manifold equipped with a group structure in whichthe group multiplication and inversion are smooth. The tangent space at the identityof a Lie group is called Lie algebra. Most Lie groups are in (or isomorphic to) thematrix forms that is topologically closed in the complex general linear group. Wecall them matrix Lie groups.The Lie correspondences between Lie group and itsLie algebra allow us to study Lie group which is an algebraic object in term of Liealgebra which is a linear object.In this work, we concern about the two correspondences in the case of matrix Liegroups; namely,1. The one-one correspondence between Lie group and it Lie algebra and2. The one-one correspondence between Lie group homomorphism and Lie algebra homomorphism.However, the correspondences in the general case is not much di erent of thosein the matrix case. To achieve these goals, we will present some matrix Lie groupsand study their topological and algebraic properties. Then, we will construct theirLie algebras and develop some important properties that lead to the main result ofthe thesis.ii
ContentsAcknowledgement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .iAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .iiChapter 1.1Differentiable manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1.Di erentiable Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .12.Tangent Spaces and Di erential Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .53.Submanifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .124.Vector Fields, Brackets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .145.Connectedness of Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .15Chapter 2.Lie groups and matrix Lie groups . . . . . . . . . . . . . . . . . . . . . . . . . .171.Lie Groups and Matrix Lie groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .172.Compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .233.Connectedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .254.Subgroups and Homomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .29Chapter 3.Lie algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .301.The Exponential Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .302.Lie Algebra of Lie group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .353.Properties of Lie algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .394.The Closed Subgroup Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .465.Lie Group that is not a Matrix Lie Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .48Chapter 4.Lie correspondences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .511.Lie Group-Lie Algebra Correspondence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .532.Lie Group-Lie Algebra Homomorphism Correspondence . . . . . . . . . . . . . . . . .59Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .65iii
CHAPTER 1Differentiable manifoldsThe concept of di erentiable manifolds is useful because it allows us to locallydescribe and understand more complicated structure on those manifolds in term ofrelatively properties on Euclidean space. The goal of this chapter is to give a basicunderstanding of di erentiable manifolds.1. Di erentiable ManifoldsDe nition 1.1.By aneighborhoodmeans that an open set containingEuclideanif every pointmorphismφfromUpp.of a pointpin a topological spaceA topological spaceMisM,onea n-dimensional locallyhas an neighborhood U such that there is a homeonnonto an open subset of R . We call the pair (U, φ : U R ) ainMchart, U a coordinate neighborhood or a coordinate open set, and φ a coordinate mapor a coordinate system on U . We say that a chart (U, φ) is center at p U if φ(p) 0.De nition 1.2. A Hausdor space is a topological spaceMsuch that wheneverp and q are distinct points of M , there are disjoint open sets U and V in M with p Uand q V . A di erentiable structure or smooth structure on a Hausdor , secondcountable (that is, its topology has a countable base), Locally Euclidean space Mis a collection of chat F {(Uα , φα ) α I} satisfying the following three properties:M SUα .α I 1 P2: φα φβ is Cfor all α, β I with Uα Uβ 6 .P3: The collection F is maximal with respct to P2; that is, if 1such that φ and φα arefor all α I , that is, φ φα andfor all α I , then (U, φ) F .P1:compatibleA pair(M, F ) is a di erentiable manifold or smooth manifoldis n-dimensional locally Euclidean.dimension n if M(U, φ) is a chartφα φ 1 are C and is said to haveRemark 1.1.(1). A Hausdor , second countable, locally Euclidean space is called amanifold.topologicalφ as de ning local coordinate functions x1 , ., xn wherexk is the continuous function from U into R given by xk (p) φ(p)k (the k th component of φ(p)). We call x1 , ., xn a local coordinate system.(2). We think of the map1
1.DIFFERENTIABLE MANIFOLDSFigure 1. Di erentiable Manifold(3). In P2, the maptheφα φ 1βchange of coordinate.de ne fromφβ (Uα Uβ )toφα (Uα Uβ )is called 1 1To prove P3, it is su ce to prove that φ ψand ψ φare smooth 1 1 1for a xed coordinate map ψ since φ φα (φ ψ) (ψ φα ) and φα φ 1 (φα ψ 1 ) (ψ φ 1 ).(4).(5). IfFois a collection of chatthen we can extendFo(Uα , φα ) that satis es the properties P1 andF that in addition satis es the conditionuniquely toP2,P3.Namely,F {(U, φ) φ φ 1αandφα φ 1areC for allφα Fo }This remark tells us that to show that a Hausdor , second countable spaceMisa di erential manifold, it is su ce to construct a collection of chat that satis es theproperties P1 and P2. Thus; without any doubt, we will also callFthat satis esthe properties P1 and P2 a di erentiable structure.Example 1.1.(1). The crossCp R2 such that Cp {(x, y) R2 x p1 } {(x, y) R2 y p2 }p (p1 , p2 )is not a di erentiable manifold since it is not locally Eculidean atwherep.n-dimensional locally Euclidean at p and let φ be aU of p to an open ball Br (0) Rn that mapsp to 0. then φ induces homeomorphism from U \ {p} Br (0) \ {0}. This lead to acontradiction since Br (0) \ {0} is connected if n 2 or has 2 connected componentsif n 1 but U \ {p} has 4 connected components.To see this, suppose thatCpishomeomorphism from a neighborhoodP that is a union of a sphere S 2 R3 with a semi verticalline L {(xN , yN , z) z zN }, where the north pole of sphere N (xN , yN , zN ), isnot a di erentiable manifold since it is not locally Euclidean at N . Suppose this isthe case; as in example (1), U \{N } and Br (0)\{0} are homeomorphic. Now U \{N }has 2 connected components then the only case is when n 1 that is Br (0) \ {0} has(2). The pendulum2
1.DIFFERENTIABLE MANIFOLDS2 connected component which are open interval. However one connected component2of U \ {N } is homeomorphic to the deleted open disk of 0,(that is, Dr (0) \ {0} R ).This is a contradiction.Figure 2. The cross and the pendulumwith the standard di erentiable structure F that isnnnthe maximal containing the single chat (R , id), where id : R R is the identitynmap satis es the properties P1 and P2. Also, R \ {0} is a di erentialble manifold.(3). The Euclidean space(4). An open setmanifold. Indeed, ifRnA of a di erentiable manifold (M, FM ) is itself a di erentiable(Uα , φα ) are charts of di erentiable manifold M , we de neFA {(A Uα , φα A Uα ) (Uα , φα ) FM }wherethenFαφα A Uαis a restriction ofis a di erentiable structure onφαinA UαA.M (n, R) which is isomorphic to Rn n is a vector space of all n nn nn2n2Since Risomophic to R , we give it a topology of R . Then(5). The setreal matrices.M (n, R)is a di erentiable manifold. Theinvertiblen nreal general linear groupis a collection ofreal matrices that we can de ne byGL(n, R) {M M (n, R) det(M ) 6 0}det(M ) 6 0.Now, the determinant map det : M (n, R) R is continuous. Since GL(n, R) det 1 (R \ {0}) where R \ {0} is an open set in R, then GL(n, R) is an open set inM (n, R) which itself is a di erentiable manifold.since the matrixM(real or complex) is invertible if and only ifcomplex general linear group GL(n, C) is a subset of a complex vector(6). Then n space C 2R2nn n complex maGL(n, C) is a di erentiableis de ned to be a collection of all invertibletrices. The similar argument in the real case tell us thatmanifold.(7). Then sphere S n Rn 1(Sn is de ned by(x1 , x2 , ., xn 1 ) Rn 1n 1Xi 13)x2i 1
1.DIFFERENTIABLE MANIFOLDSis a di erentiable manifold. To see this, Let N (0, ., 0, 1) be the north pole andS (0, ., 0, 1) be the south pole of S n and consider thestereographic projectionfrom the north pole and south poleφN : S n \ {N } Rn , (x1 , x2 , ., xn 1 ) 7 (x1xn, .,)1 xn 11 xn 1xnx1, .,)1 xn 11 xn 1nnthat take p (x1 , ., xn 1 ) S \ {N } (or S \ {S}) into the intersection of thehyperplane xn 1 0 with the line that pass through p and N (or S )These maps φN and φS are di erentiable, injective and map onto the hyperplanexn 1 0 1 1It is easy to check that the inverse maps φN and φS are also di erentiable. Thisnnnnimplies that S is locally Euclidean. Moreover, S (S \ {N }) (S \ {S}) and 1nthe change of coordinates (by a direct calculation) φN φS on R is given byyj, j 1, ., nyj0 Pn2yiφS : S n \ {S} Rn , (x1 , x2 , ., xn 1 ) 7 (i 1 1 1φS φ 1is smooth.N (φN φS )nnF {(S \ {N }, φN ), (S \ {S}, φS )} is di erentiableis smooth and alsoTherefore,structure onFigure 3. The stereotype projections from north and south poles of(8). For a subset ofA Rnand a functionf : A Rm , the graphoffSnis de nedto be the subsetIfUis an open subsetΓ(f ) {(x, f (x)) A Rm }nm of R and f : U Ris C , thenφ : Γ(f ) U, (x, f (x)) 7 x4S n.the two maps
2.TANGENT SPACES AND DIFFERENTIAL FORMSandψ : U Γ(f ), x 7 (x, f (x))are continuous and inverse to each other, and so are homeomorphisms. The set Γ(f ) mof a Cfunction f : U Rhas a single chat (Γ(f ), U ) that satis es the propertyP1 and P2.Figure 4. The graph of a smooth functionProduct manifolds ).f : Rn U Rm(M k , F ) and (N l , G ) be di erentiable manifolds ofkldimensions k and l , respectively. Then M N becomes a di erentiable manifold ofdimension k l , with di erentiable structure H the maximal collection containing:(9)(Let{(Uα Vβ , φα ψβ : Uα Vβ Rk Rl ) (Uα , φα ) F , (Vβ , ψβ ) G }is,(10).nThe1n-torus T n12can be consider as a n-product of circle S R that11S is a di erentiable manifold since S is a di erentiable1 S . } S {zTn-timesmanifolds.Remark 1.2.One may ask if there exists a topological manifold that is not adi erentiable manifold.The answer is yes.A triangulable closed manifolddimension 10 is a topological manifold (in fact, it is athat does not have any smooth structures.M0 ,M0ofpiecewise linear manifold )For the construction of this manifold16], A manifold which does not admit any Di erentiable Structure byconsult [Michel A. Kerviare.Note that in examples above, we do not concern about Hausdor and secondcountibility since the subspace with relative topology of a Hausdor and secondcountable space is Hausdor and second countable space.From now on, "manifold" is refered to "di erentiable manifold".2. Tangent Spaces and Di erential FormsDe nition 1.3. Let M k be k-dimensional manifold and N lbe l -dimensionalf : M k N l is said to be C or smooth at p M k if given a(V, ψ) about f (p) N l , there exists a chat (U, φ) about p M k such thatmanifold. A mapchart5
2.TANGENT SPACES AND DIFFERENTIAL FORMSf (U ) V and the compositionmap f is said to be smooth if itψ f φ 1 : φ(U ) Rl is C ksmooth at every point of M .mapisatφ(p).TheRemark 1.3.(1). It follows from P2 of de nition 1.2 that the above de nition of the smoothklkness of a map f : M N at a point p M is independent of the choice of charts. 1Indeed, if (U , ϕ) is any chart about p then ψ f ϕ (ψ f φ 1 ) (φ ϕ 1 ) isC since it is the composition of C maps.(U, φ) is a chart at p M n with φ (x1 , . . . , xn ). It follows from the above 1de nition and the condition P2 that φ and φare smooth and then the coordinatefunctions xi D(M ) where D(M ) denotes the set of real value smooth function onM.(2). IfFigure 5. A smooth map between two manifoldsDe nition 1.4.Thetangent vectoratp Mu : D(M ) RM (D(M ) or simply D )is the linear mapsfrom the set of real value smooth function on a manifoldto a set of real number that satis es the production rule of derivation. That is, forallf, g Dandλ R,1.u(f λg) u(f ) λu(g)2.u(f g) u(f )g(p) f (p)u(g)(linearity),(product rule of derivation).6
2.TANGENT SPACES AND DIFFERENTIAL FORMSThe collection of all tangent vectors atdenoted bypis said to be atangent spacepatandTp M .Remark 1.4. The linear map 0 Tp M and if we de ne (u v)(f ) : u(f ) v(f )(λu)(f ) : λu(f ) for all u, v Tp M and λ R,(u v)(f ) and (λu(f )) again are tangent vectors at p.andthen it is easy to see thatThus,Tp Mis a real vectorspace.De nition 1.5.atp.TheLetφ (x1 , . . . , xn )partial di erentialbe a coordinate system in a manifoldof a smooth functionfon f (f φ 1 )(p) (φ(p)) xi uiwhereu1 , . . . , unMatpMnis de ned by:1 i n,are the natural coordinate functions ofRn .A straightforward computation then shows that the function i p sending eachpicture i pf D(M )as an arrow atLemma 1.1.(1). If(2). Iftop xi: D(M ) Rp( f / xi )(p)is a tangent vector totangent to thexi coordinatecurveMp. Wethrough p.atcanv Tp M .f, g D(M ) are equal on a neighborhood U of p, then v(f ) v(g).h D(M ) is constant on a neighborhood of p, then v(h) 0.LetProof.To prove (1), we make use of the result that for any neighborhoodUofp M,f D , called a bump function at p, such that0 f 1 on M .b. f 1 on some neighborhood of p.c. suppf {x f (x) 6 0} U .Let h be a bump function at p such that supph U then (f g)h 0But v(0) v(0 0) v(0) v(0) implies v(0) 0. Thus,there exists a functiona.onM.0 v((f g)h) v(f g)h(p) (f g)(p)v(h) v(f g) v(f ) v(g)Therefore,v(f ) v(g)For (2), observe thatv(1) v(1.1) v(1)1 1v(1) 2v(1)which impliesv(1) 0Thus, ifh conMthenv(h) v(c.1) cv(1) 0.This completes the proof. Proposition 1.1. Let M be an n-dimensional manifold and p M .Let (U, φ) be a chart about p with a coordinate system x1 , ., xn . Then Tp M has basis, . . . , x n x1and any tangent vector u Tp M can be expressed (uniquely) as7
2.TANGENT SPACES AND DIFFERENTIAL FORMSu(f ) nXaii 1whereai u(xi ) RIn this wayofTp Mand f xif D.is a real vector space of dimensionn(as the same as dimensionM ).Remark 1.5.: φ(U ) R, (φ(p)) in ui nR for i 1, · · · , n eventually yield a base xi (p) for i 1, · · · , n of Tp M . Thus, letf D where f is de ned on a neighborhood V of p then f is smooth on U V Uso that we can write f in term of φ (x1 , · · · , xn ) where xi ui φ. Therefore, forφ(U ) which is an open set in Rn , the function g f φ 1 : φ(U ) R is smooth onφ(U ) and f g φ g(x1 , · · · , xn ).If we taking a chart (U, φ) of M at p such that for uinthe natural coordinate function on R , the partial derivative operatorProof. Without loss of generality, we can assume thatxi yi t yields / xi / yi .R kqk } for some .Let g be a smooth function on φ(U )lationnZ1gi (q) ShrinkingUand for each g(tq)dt uiφ(p) 0since a trans-if necessary give1 i n,φ(U ) {q we de neq φ(U )for all0It follows from the fundamental theorem of calculus that:Z1g(q) g(0) Z10g (tq)dt 0since g (tq) g g(tq)) ( u(tq), · · · , un1g(q) g(0) Z1 Xni 10sinceui (q) qi .Now, letf D0andq (q1 , · · · , qn )nX g(tq)qi g(0) uii 1g g(0) Thus,and set 1f φg f gqdt (tq)Z10then:nX g(tq)qi g(0) gi (q)ui (q) uii 1nPgi uii 1 φ 1 then: 1 f φ (0) nXgi ui f (p) i 1nXgi uii 1So we obtain:f f (p) nX(gi ui ) φ f (p) i 0sincef (p)is constant,fi gi φnXfi (ui φ) f (p) i 0andxi u i φ8nXi 0f i xi
2.TANGENT SPACES AND DIFFERENTIAL FORMSThus, from lemma 1.1, applyu(f ) 0 nXu(fi xi ) i 1sincexi (p) 0Sincefandu Tp MnXwe obtain:u(fi )xi (p) i 1i 1ai u(xi ) Rthen0 i 1 1 g(0) uiIt remains to prove that the coordinate vectornPpose thatαi i 0 then apply to xj yieldsi 0nXnX f(p)u(xi )fi (p)u(xi ) xii 1 φ ) (f u(φ(p)) inP fu(f ) (p)u(xi ) xifi (p) gi φ(p) gi (0) is arbitrary, letnX f(p) xii 1are linearly independent. Sup- in xj Xαi αi σij αjxii 0 j De nition 1.6.f : M m R be a smooth function. We de ne the di erential of f at p M to be the map dfp : Tp M Tf (p) R R by (dfp )(v) v(f ).mnMore general, if f : M N be a smooth function and let p M . The di erential of f at p is the map dfp : Tp M Tf (p) M such that for any u Tp M , dfp (u)is to be a tangent vector at f (p). On the other hand, if g is a smooth function onneighborhood of f (p), we de ne dfp (u)(g) u(g f ).LetProposition 1.2.(V, ψ)is a chart atdfp is a linear map. In addition, if (U, φ) is a chart at p andf (p), then dfp has a matrix which is the Jacobian matrix of frepresented in these coordinates.Proof. Letu, v Tp Mthen forλ Randg D(N ),we have:dfp (u λv)(g) (u λv)(g f ) u(g f ) λv(g f ) dfp (u)(g) λdfp (v)(g)This proves the linearity ofdfp .φ (x1 , . . . , xm ) and ψ (y1 , . . . , yn ) be the given coordinate functionsf can be expressed in term of coordinates in the neighborhood V as:Now, letso thatfi y i fLeti 1, . . . , n / xjand / yi be a basis for Tp M and Tf (p) N , respectively and let [aij ] be adfp . We will provePthat aij fi / xjhave dfp ( / xj ) aij / yi Tf (p) N . Using the fact that yk D(N ), wematrix ofWeiobtain:X yk fk (yk f ) dfp ()(yk ) aij akj xj xj xj yii9
2.TANGENT SPACES AND DIFFERENTIAL FORMSThe last equality is followed by usingThus,aij fi / xj yk / yi δkiis the desired Jacobian matrix. Proposition 1.3.Then for eachLetf : Mm Nnandg : N n Kkbe smooth functions.p M,d(g f )p dgf (p) dfpProof. First, note that ifh to u(h f )λ R then:u Tp M then the map uf : D(N ) R sending eachN at f (p). To see this, let h1 , h2 D(N ) andis a tangent vector touf (h1 λh2 ) u((h1 λh2 ) f ) u((h1 f ) (λh2 f )) u(h1 f ) λu(h2 f ) uf (h1 ) λuf (h2 )uf (h1 h2 ) u(h1 h2 f ) u((h1 f )(h2 f )) u(h1 f )(h2 (f (p)) h1 (f (p))u(h2 f ) uf (h1 )(h2 (f (p)) h1 (f (p))uf (h2 )Now, letu Tp Mandh D(P ),then:d(g f )p (u)(h) u(h g f ) dfp (u)(h g) [dgf (p) dfp (u)](h)Thus,d(g f )p dgf (p) dfp De nition 1.7.(1).Thecurvesubmanifold ofR, Ia smooth mapα:I Mbe a curve. Thevelocity vector By the de nition ofdα,dduαuof Iofαatt Iis Tα(t) Mthe tangent vectorα0 (t)applied to a functiongives: d(f α)dα (t)f dαtf (t)dudu0with say α (0) v , then: 0Thus, ifopen Remark 1.6.f D(M )is an(d/du)(t) Tt (R).α0 (t) dαt(1).α : I M where IR). As anbe half in nity or all ofhas a coordinate system consisting of the identity mapthen the coordinate vector(2). LetM isR( I canon a manifoldopen interval in the real lineis any curvev(f ) d(f α)(0)dt.(2). Iff :M Nα is a curve on M then f α is a(f α)0 (t) dfα(t) (α0 (t)) T(f α)(t) N .is smooth andand from the chain rule, we have10curve onN
2.TANGENT SPACES AND DIFFERENTIAL FORMSLemma 1.2. consider the product manifold M m N n and the canonical projec-π1 : M N M and π2 : M N N .(1). The map α : u (dπ1 (u), dπ2 (u)) is an isomorphism of T(m,n) (M N ) Tm M Tn N .(2). If (m0 , n0 ) M N , and de ne injections in0 : M M N and im0 : N M N by:tionsin0 (m) (m, n0 )im0 (n) (m0 , n)Letu T(m0 ,n0 ) (M N ), u1 dπ1 (u), u2 dπ2 (u)andf D(M N ).Thenu(f ) u1 (f in0 ) u2 (f im0 )Proof.(1). First, it is easy to see that the projectionsare smooth. Thedπi (i 1, 2) are linear map. Now, observe thatdπ2 are [Im 0]m,m n and [0 In ]n,m n respectively.0Thus, dπi are surjective and so is α. To see that α is injective, let α(u) α(u ) for00000some u (x1 , . . . , xm , y1 , . . . , yn ) and u (x1 , . . . , xm , y1 , . . . , yn ) in T(m,n) (M N )0000then dπ1 (u) dπ1 (u ) and dπ2 (u) dπ2 (u ) so that (x1 , . . . , xm ) (x1 , . . . , xm ) and000(y1 , . . . , yn ) (y1 , . . . , yn ) which imply u u .mapαπi (i 1, 2)is clearly a linear map sincethe Jacobian matrices ofdπ1and(2). It is not di cult to see that injections im0 and in0 are smooth. We have fromde nition ofu,linearity ofdf(m0 ,n0 )and (1) that:u(f ) df(m0 ,n0 ) (u) df(m0 ,n0 ) (u1 , u2 ) df(m0 ,n0 ) [(u1 , 0) (0, u2 )] df(m0 ,n0 ) (u1 , 0) df(m0 ,n0 ) (0, u2 )Now, choose the curveα(t) (α1 (t), n0 ) ino (α1 (t))withα10 (0) u1then:df(m0 ,n0 ) (u1 , 0) df(m0 ,n0 ) (α0 (0)) α0 (0)(f )d(f α)d((f in0 ) α1 )(0) (0)dtdt α10 (0)(f in0 ) u1 (f in0 ) Similarly, if we chose the curveβ(t) (m0 , β2 (t)) im0 (β2 (t))withβ20 (0) u2 ,then:df(m0 ,n0 ) (0, u2 ) u2 (f im0 ) De nition 1.8. Let M k and N l be manifolds of dimension k and l, respectively.A mapff : Mk Nldi eomorphismis said to beif it is bijective and bi-smooth i.e.f 1 are smooth. On the other hand, f is said to be M k if there exist a neighborhood U of p and V of f (p) suchand its inverse mapdi eomorphismatthatis di eomorphism.f :U Vp11locally
3.SUBMANIFOLDSRemark 1.7.klIt is immediate that if f : M N is a di eomorphism, thenlkTf (p) N is an isomorphism for all p M ;in particular, the dimensionskdfp : Tp M klof M and N are equal. The converse is not true; however, the local converse is truekkkkki.e. if f : M N be smooth map and let p M such that dfp : Tp M Tf (p) Nis an isomorphism, then f is locally di eomorphism at p. This result is followedimmediately from the Inverse Function Theorem.Proposition 1.4.di eomorphsim. IfLet(U, φ)(M n , F )be a manifold and a functionis a chart inF,then(U, f φ) F .f : Rn Rnbe(V, ψ) be a chart in F such that U V 6 then:ψ (f φ) (ψ φ 1 ) f 1 and (f φ) ψ 1 f (φ ψ 1 ) are smooth 1since f, f, ψ φ 1 , φ ψ 1 are smooth. From the maximality of F , we obtain(U, f φ) F . Proof. Let 13. SubmanifoldsDe nition 1.9. Let (M, F ) be a n k-dimensional manifolds.n-dimensionalembedded submanifold in M is a subset N M such that for each p N , there is annkchart (U, φ : U V ) of F with p U such that φ(U N ) V (R {0}) R R .On the other hand, An n-dimensional immersed submanifold in M is a topological space N M such that for each p N , there is a chart (U, φ : U V )of F with p U such that for a neiborhood W of p in N with W U , we haveφ(W ) V (Rn {0}) Rn Rk .Ann knknnkIn this de nition, we identify Rwith R R and often write R R Rnnkinstead of R {0} R R to signify the subset of all points with the k lastcoordinates equal to zero.Remark 1.8.(1). Ifcan giveN(M, F )is a manifold andN Mis a embedded submanifold, then wea di erentiable structureFN { (N Uα , φα N Uα ) (Uα , φα ) F }i : N , M is smooth. Also notein N is a induced topology from M .Note that the inclusion mapsmooth structure, the topology(2). The topological of a immersed submanifoldN Mthat from thisneed not be the inducedtopology of the containing manifold. Also note that the dimension of a submanifold(embedded or immersed) is less than or equal to the dimension of its containingmanifold and in the case of equality we just obtain open submanifolds.Example 1.2.n22be a natural number. Then Kn {(x, x ) R x R} R is2a submanifold. Indeed, if we give R a di erentiable structure with a single chart(1).Letn12
3.SUBMANIFOLDS(R2 , φ)where(2).φ : R2 R2given byConsider the unit spherestructureF(x, y) 7 (x, y xn )S 1 R2 .InR2 ,thenφ(Kn ) R {0}.we can construct a smoothto be the maximal collection containingF {(R2 , id)}φi : Ui R2 as φ1 (x, y) (x, y 1 x2 ),pφ3 (x, y) (x 1 y 2 , y),Now consider the mapsandUibelow: φ2 (x, y) (x, y 1 x2 ),pφ4 (x, y) (x 1 y 2 , y)are open sets indicating in the following picture:φ 1 φ2 , φ 1 φ4 and all φi are smooth on13It is easy to see that(Ui , φi ) FsinceφiandidUi .Thenare compatible.p S 1 , p is contained in one of Ui .case p is contained in the upper or theLetThelower half of the circle, we have:φ1 (U1 S 1 ) φ1 (U1 ) {(x, 0) 1 x 1} φ1 (U1 ) (R {0})φ2 (U2 S 1 ) φ2 (U2 ) {(x, 0) 1 x 1} φ2 (U2 ) (R {0})p is in the left or the right half of the circle, we have:(U3 , f φ3 ), (U4 , f φ4 ) F , where f : (x, y) (y, x) is the interchange of coordinateFor the casefunction (Proposition 1.4) and:f φ3 (U3 S 1 ) f φ3 (U3 ) {(y, 0) 1 y 1} f φ3 (U3 ) (R {0})f φ4 (U4 S 1 ) f φ4 (U4 ) {(y, 0) 1 y 1} f φ4 (U4 ) (R {0})This proves thatS1is a submanifold ofR2 .13Also,S1is a submanifold ofR2 \ {0}.
4.VECTOR FIELDS, BRACKETS4. Vector Fields, BracketsDe nition 1.10.Avector led Xon a manifoldassociates to each point p M a vectorφ : M U Rn , we can writeX(p) X(p) Tp M .nXai (p)i 1where eachai : U Ri 1, ., n.It is said to be smooth ifis a function onaiUandMis a correspondence thatIn term of a coordinate map xi{ x i }φ,is the basis associated toare smooth.Occasionally, it is convenient to use this idea and think of a vector eld as a mapX : D F from the set D of smoothM , de ned in the following wayMfunction onnXXp (f ) (Xf )(p) ai (p)i 1It is easy to check that the functionXfφ. In this context, it is immediate that Xf D.Proposition 1.5. Let X and Ythere exists a unique vector eldZFof function on f(p) xidoes not depend on the choice of coordi-nate mapfor allto the setis smooth if and only ifbe smooth vector elds on a manifoldsuch that for allf D,Xf DM n.ThenZf (XY Y X)f X(Y f ) Y (Xf ).p Mx1 , ., xn be a local coordinate system about p.X and Y uniquely asnnXX ,Y (p) bj (p)X(p) ai (p) xi xjj 1i 1Proof. LetandThen,we can express the vector eldThen for allf D,X(Y f )(p) XXjY (Xf )(p) YXi fbj (p) xj! f xi!ai (p)(p) Xi,j(p) XX 2f bj fai (p) ai (p)bj (p) xi xj xi xji,jbj (p)i,j ai f xj xiX2ai (p)bj (p)i,j f xi xj!(p)!(p)Thus, X bj ai f bj(p)(Zf )(p) X(Y f )(p) Y (Xf )(p) ai xi xj xji,jis a vector eld in coordinate neighborhood ofwe can de ne{(Uα , φα )}onZpMpand is unique. Sincein each coordinate neighborhoodby the above expression.14Uppis arbitrary,of di erentiable structureThe uniqueness impliesZp Zqon
5.CONNECTEDNESS OF MANIFOLDSUp Uq 6 ,Z is unique.which allows us to de neZover the entire manifoldMand also this De nition 1.11.and is de ned byThe vector eld[X, Y ] XY Y XProposition 1.6.numbers and(1).(2).(3).(4).f, gX, YIfandZZde ned in proposition 1.5 is calledofXandY . [X, Y ]bracketis obviously smooth.are smooth vector elds onM , a, bare realare smooth functions, then:[X, Y ] [Y, X] (anticommutativity),[aX bY, Z] a[X, Z] b[Y, Z] (linearity),[[X, Y ], Z] [[Z, X], Y ] [[Y, Z], X] 0 (Jacobi[f X, gX] f g[X, Y ] f (Xg)Y g(Y f )X .identity),Proof.(1) is obvious and (2) is immediate from the linearity of derivation. For (3), wehave:[[X, Y ], Z] [X, Y ]Z Z[X, Y ] (XY Y X)Z Z(XY Y X) XY Z Y XZ ZXY ZY Xand by interchangingX, YandZ,we obtain:[[Z, X], Y ] ZXY XZY Y ZX Y XZ[[Y, Z], X] Y ZX ZY X XY Z XZYThus,[[X, Y ], Z] [[Z, X], Y ] [[Y, Z], X] 0(4). Leth D,then:[f X, gY ]h [(f X)(gY ) (gY )(f X)]h f X(gY h) gY (f Xh) f (Xg)(Y h) f gX(Y h) g(Y f )(Xh) f gY (Xh) f (Xg)(Y h) f g(X(Y h) Y (Xh)) g
call them matrix Lie groups. The Lie correspondences between Lie group and its Lie algebra allow us to study Lie group which is an algebraic object in term of Lie algebra which is a linear object. In this work, we concern about the two correspondences in the case of matrix Lie groups; namely, 1.
Chapter II. Lie groups and their Lie algebras33 1. Matrix Lie groups34 1.1. Continuous symmetries34 1.2. Matrix Lie groups: de nition and examples34 1.3. Topological considerations38 2. Lie algebras of matrix Lie groups43 2.1. Commutators43 2.2. Matrix exponentiald and Lie's formulas43 2.3. The Lie algebra of a matrix Lie group45 2.4.
Chapter 1. Introduction 7 Chapter 2. Lie Groups: Basic Definitions 9 §2.1. Lie groups, subgroups, and cosets 9 §2.2. Action of Lie groups on manifolds and representations 12 §2.3. Orbits and homogeneous spaces 13 §2.4. Left, right, and adjoint action 14 §2.5. Classical groups 15 Exercises 18 Chapter 3. Lie Groups and Lie algebras 21 §3.1 .
Chapter 1. Introduction 7 Chapter 2. Lie Groups: Basic Definitions 9 §2.1. Lie groups, subgroups, and cosets 9 §2.2. Action of Lie groups on manifolds and representations 12 §2.3. Orbits and homogeneous spaces 13 §2.4. Left, right, and adjoint action 14 §2.5. Classical groups 15 Exercises 18 Chapter 3. Lie Groups and Lie algebras 21 §3.1 .
CONTENTS CONTENTS Notation and Nomenclature A Matrix A ij Matrix indexed for some purpose A i Matrix indexed for some purpose Aij Matrix indexed for some purpose An Matrix indexed for some purpose or The n.th power of a square matrix A 1 The inverse matrix of the matrix A A The pseudo inverse matrix of the matrix A (see Sec. 3.6) A1 2 The square root of a matrix (if unique), not elementwise
CONTENTS CONTENTS Notation and Nomenclature A Matrix Aij Matrix indexed for some purpose Ai Matrix indexed for some purpose Aij Matrix indexed for some purpose An Matrix indexed for some purpose or The n.th power of a square matrix A 1 The inverse matrix of the matrix A A The pseudo inverse matrix of the matrix A (see Sec. 3.6) A1/2 The square root of a matrix (if unique), not elementwise
A Matrix A ij Matrix indexed for some purpose A i Matrix indexed for some purpose Aij Matrix indexed for some purpose An Matrix indexed for some purpose or The n.th power of a square matrix A 1 The inverse matrix of the matrix A A The pseudo inverse matrix of the matrix A (see Sec. 3.6) A1/2 The square root of a matrix (if unique), not .
(1) R and C are evidently Lie groups under addition. More generally, any nite dimensional real or complex vector space is a Lie group under addition. (2) Rnf0g, R 0, and Cnf0gare all Lie groups under multiplication. Also U(1) : fz2C : jzj 1gis a Lie group under multiplication. (3) If Gand H are Lie groups then the product G H is a Lie group .
Fiction Excerpt 1: The Adventures of Tom Sawyer (retold with excerpts from the novel by Mark Twain) Saturday morning was come, and all the summer world was bright and fresh, and brimming with life. There was a song in every heart; and if the heart was young the music issued at the lips. There was cheer in every face and a spring in every step. The locust trees were in bloom and the fragrance .
