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Matrix Lie Groups and Lie Correspondences

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Matrix Lie Groups and Lie Correspondences MATRIX LIE GROUPS AND LIE CORRESPONDENCES T H E S I S Summited in partial fulfillment of the requirement for the master degree of science in basic mathematics A thesis by: Monyrattanak Seng Under direction of advisor: Professor. Raúl Quiroga Barranco Guanajuato, Gto., August, 02nd 2017 MATRIX LIE GROUPS AND LIE CORRESPONDENCES T H E S I S Summited in partial fulfillment of the requirement for the master degree of science in basic mathematics A thesis by: Monyrattanak Seng Under direction of advisor: Professor. Raúl Quiroga Barranco Authorization of the final version Guanajuato, Gto., August, 02nd 2017 Acknowledgement This thesis haven't been completed without the supports from my adviser pro- fessor.Raúl Quiroga Barranco. I would like to express my gratefulness to him for his patience, motivation, enthusiasm, and his valuable times. His guidance and ideas helped 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 thesis and acknowledge all professors and stus in CIMAT that gave me important lectures and supports. In addition, I would like to thanks all my friends who gave me useful resources. Finally, I would like to say "thank you" to all authors of the books that I used as 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 this area. i Abstract Lie group is a dierentiable manifold equipped with a group structure in which the group multiplication and inversion are smooth. The tangent space at the identity of a Lie group is called Lie algebra. Most Lie groups are in (or isomorphic to) the matrix forms that is topologically closed in the complex general linear group. We 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. The one-one correspondence between Lie group and it Lie algebra and 2. The one-one correspondence between Lie group homomorphism and Lie alge- bra homomorphism. However, the correspondences in the general case is not much dierent of those in the matrix case. To achieve these goals, we will present some matrix Lie groups and study their topological and algebraic properties. Then, we will construct their Lie algebras and develop some important properties that lead to the main result of the thesis. ii Contents Acknowledgement ......................................................... i Abstract.................................................................... ii Chapter 1. Differentiable manifolds ................................... 1 1. Dierentiable Manifolds................................................. 1 2. Tangent Spaces and Dierential Forms.................................. 5 3. Submanifolds........................................................... 12 4. Vector Fields, Brackets.................................................. 14 5. Connectedness of Manifolds............................................. 15 Chapter 2. Lie groups and matrix Lie groups .......................... 17 1. Lie Groups and Matrix Lie groups...................................... 17 2. Compactness............................................................ 23 3. Connectedness.......................................................... 25 4. Subgroups and Homomorphism......................................... 29 Chapter 3. Lie algebra .................................................... 30 1. The Exponential Map................................................... 30 2. Lie Algebra of Lie group................................................ 35 3. Properties of Lie algebra................................................ 39 4. The Closed Subgroup Theorem......................................... 46 5. Lie Group that is not a Matrix Lie Group............................... 48 Chapter 4. Lie correspondences ......................................... 51 1. Lie Group-Lie Algebra Correspondence.................................. 53 2. Lie Group-Lie Algebra Homomorphism Correspondence................. 59 Bibliography.................................................................. 65 iii CHAPTER 1 Differentiable manifolds The concept of dierentiable manifolds is useful because it allows us to locally describe and understand more complicated structure on those manifolds in term of relatively properties on Euclidean space. The goal of this chapter is to give a basic understanding of dierentiable manifolds. 1. Dierentiable Manifolds Denition 1.1. By a neighborhood of a point p in a topological space M, one means that an open set containing p. A topological space M is a n-dimensional locally Euclidean if every point p in M has an neighborhood U such that there is a homeo- morphism φ from U onto an open subset of Rn. We call the pair (U; φ : U ! Rn) a chart, U a coordinate neighborhood or a coordinate open set, and φ a coordinate map or a coordinate system on U. We say that a chart (U; φ) is center at p 2 U if φ(p) = 0. Denition 1.2. A Hausdor space is a topological space M such that whenever p and q are distinct points of M, there are disjoint open sets U and V in M with p 2 U and q 2 V .A dierentiable structure or smooth structure on a Hausdor, second countable (that is, its topology has a countable base), Locally Euclidean space M is a collection of chat F = f(Uα; φα)jα 2 Ig satisfying the following three properties: S P1: M = Uα. α2I P2: −1 is 1 for all with . φα ◦ φβ C α; β 2 I Uα \ Uβ 6= ? P3: The collection F is maximal with respct to P2; that is, if (U; φ) is a chart such that and are compatible for all , that is, −1 and −1 are 1 φ φα α 2 I φ ◦ φα φα ◦ φ C for all α 2 I, then (U; φ) 2 F . A pair (M; F ) is a dierentiable manifold or smooth manifold and is said to have dimension n if M is n-dimensional locally Euclidean. Remark 1.1. (1). A Hausdor, second countable, locally Euclidean space is called a topological manifold. (2). We think of the map φ as dening local coordinate functions x1; :::; xn where is the continuous function from into given by (the th compo- xk U R xk(p) = φ(p)k k nent of φ(p)). We call x1; :::; xn a local coordinate system. 1 1. DIFFERENTIABLE MANIFOLDS Figure 1. Dierentiable Manifold (3). In P2, the map −1 dene from to is called φα ◦ φβ φβ(Uα \ Uβ) φα(Uα \ Uβ) the change of coordinate. (4). To prove P3, it is suce to prove that φ ◦ −1 and ◦ φ−1 are smooth for a xed coordinate map since −1 −1 −1 and −1 φ ◦ φα = (φ ◦ ) ◦ ( ◦ φα ) φα ◦ φ = −1 −1 (φα ◦ ) ◦ ( ◦ φ ). (5). If Fo is a collection of chat (Uα; φα) that satises the properties P1 and P2, then we can extend Fo uniquely to F that in addition satises the condition P3. Namely, −1 and −1 are 1 for all F = f(U; φ)jφ ◦ φα φα ◦ φ C φα 2 Fog This remark tells us that to show that a Hausdor, second countable space M is a dierential manifold, it is suce to construct a collection of chat that satises the properties P1 and P2. Thus; without any doubt, we will also call F that satises the properties P1 and P2 a dierentiable structure. Example 1.1. (1). The cross 2 such that 2 2 Cp 2 R Cp = f(x; y) 2 R jx = p1g[f(x; y) 2 R jy = p2g where p = (p1; p2) is not a dierentiable manifold since it is not locally Eculidean at p. To see this, suppose that Cp is n-dimensional locally Euclidean at p and let φ be a homeomorphism from a neighborhood of to an open ball n that maps U p Br(0) 2 R p to 0. then φ induces homeomorphism from U n fpg ! Br(0) n f0g. This lead to a contradiction since Br(0) n f0g is connected if n ≥ 2 or has 2 connected components if n = 1 but U n fpg has 4 connected components. (2). The pendulum P that is a union of a sphere S2 ⊂ R3 with a semi vertical line L = f(xN ; yN ; z)jz ≥ zN g, where the north pole of sphere N = (xN ; yN ; zN ), is not a dierentiable manifold since it is not locally Euclidean at N. Suppose this is the case; as in example (1), U nfNg and Br(0)nf0g are homeomorphic. Now U nfNg has 2 connected components then the only case is when n = 1 that is Br(0) n f0g has 2 1. DIFFERENTIABLE MANIFOLDS 2 connected component which are open interval. However one connected component of is homeomorphic to the deleted open disk of ,(that is, 2). U nfNg 0 Dr(0)nf0g ⊂ R This is a contradiction. Figure 2. The cross and the pendulum (3). The Euclidean space Rn with the standard dierentiable structure F that is the maximal containing the single chat (Rn; id), where id : Rn ! Rn is the identity map satises the properties P1 and P2. Also, Rn n f0g is a dierentialble manifold. (4). An open set A of a dierentiable manifold (M; FM ) is itself a dierentiable manifold. Indeed, if (Uα; φα) are charts of dierentiable manifold M, we dene FA = f(A \ Uα; φαjA\Uα )j(Uα; φα) 2 FM g where is a restriction of in φαjA\Uα φα A \ Uα then Fα is a dierentiable structure on A. (5). The set M(n; R) which is isomorphic to Rn×n is a vector space of all n × n 2 2 real matrices. Since Rn×n isomophic to Rn , we give it a topology of Rn . Then M(n; R) is a dierentiable manifold. The real general linear group is a collection of invertible n × n real matrices that we can dene by GL(n; R) = fM 2 M(n; R)j det(M) 6= 0g since the matrix M (real or complex) is invertible if and only if det(M) 6= 0. Now, the determinant map det : M(n; R) ! R is continuous.
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