DOCSLIB.ORG

Symmetric Spaces of the Non-Compact Type: Lie Groups Paul-Emile Paradan

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Symmetric Spaces of the Non-Compact Type: Lie Groups Paul-Emile Paradan Symmetric spaces of the non-compact type: Lie groups Paul-Emile Paradan To cite this version: Paul-Emile Paradan. Symmetric spaces of the non-compact type: Lie groups. Séminaires et congrès, Société mathématique de France, 2009, Géométries à courbure négative ou nulle, groupes discrets et rigidités (18), pp.39-76. hal-00773255 HAL Id: hal-00773255 https://hal.archives-ouvertes.fr/hal-00773255 Submitted on 12 Jan 2013 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. SYMMETRIC SPACES OF THE NON-COMPACT TYPE : LIE GROUPS by Paul-Emile PARADAN Abstract.— In these notes, we give first a brief account to the theory of Lie groups. Then we consider the case of a smooth manifold with a Lie group of symmetries. When the Lie group acts transitively (e.g. the manifold is homogeneous), we study the (affine) invariant connections on it. We end up with the particuler case of homogeneous spaces which are the symmetric spaces of the non-compact type. R´esum´e (Espaces sym´etriques de type non-compact : groupes de Lie) Dans ces notes, nous introduisons dans un premier les notions fon- damentales sur les groupes de Lie. Nous abordons ensuite le cas d’une vari´et´ediff´erentiable munie d’un groupe de Lie de sym´etries. Lorsque le groupe de Lie agit transitivement (i.e. la vari´et´eest homog`ene) nous ´etudions les connexions (affines) invariantes par ce groupe. Finalement, nous traitons le cas particulier des espaves sym´etriques de type non- compact. Contents 1. Introduction..................................... .... 2 2. Lie groups and Lie algebras: an overview............. 2 3. Semi-simple Lie groups............................. .. 23 4. Invariant connections............................. ... 33 5. Invariant connections on homogeneous spaces. 36 References......................................... ..... 46 2000 Mathematics Subject Classification.— 22E15, 43A85, 57S20. Key words and phrases.— Lie group, connection, curvature, symmetric space. 2 PAUL-EMILE PARADAN 1. Introduction This note is meant to give an introduction to the subjects of Lie groups and of equivariant connections on homogeneous spaces. The final goal is the study of the Levi-Civita connection on a symmetric space of the non-compact type. An introduction to the subject of “symmetric spaces” from the point of view of differential geometry is given in the course by J. Maubon [5]. 2. Lie groups and Lie algebras: an overview In this section, we review the basic notions concerning the Lie groups and the Lie algebras. For a more complete exposition, the reader is invited to consult standard textbooks, for example [1], [3] and [6]. Definition 2.1. — A Lie group G is a differentiable manifold(1)which is also endowed with a group structure such that the mappings G × G −→ G, (x, y) −→ xy multiplication G −→ G, x −→ x−1 inversion are smooth. We can define in the same way the notion of a topological group: it is a topological space(2) which is also endowed with a group structure such that the ‘multiplication’ and ‘inversion’ mappings are continuous. The most basic examples of Lie groups are (R, +), (C − {0}, ×), and the general linear group GL(V ) of a finite dimensional (real or complex) (1)All manifolds are assumed second countable in this text. (2)Here “topological space” means Hausdorff and locally compact. SYMMETRIC SPACES : LIE GROUPS 3 vector space V . The classical groups like SL(n, R) = {g ∈ GL(Rn), det(g)=1}, n t O(n, R) = {g ∈ GL(R ), gg = Idn}, n t U(n) = {g ∈ GL(C ), gg = Idn}, p+q t Idp 0 O(p, q) = {g ∈ GL(R ), gIp,qg = Ip,q}, where Ip,q = 0 −Idq 0 −Id Sp(R2n) = {g ∈ GL(R2n), tgJg = J}, where J = n Idn 0 are all Lie groups. It can be proved by hand, or one can use an old Theorem of E. Cartan. Theorem 2.2. — Let G be a closed subgroup of GL(V ). Then G is an embedded submanifold of GL(V ), and equipped with this differential structure it is a Lie group. The identity element of any group G will be denoted by e. We write the tangent spaces of the Lie groups G, H, K at the identity element e respectively as: g = TeG, h = TeH, k = TeK. Example : The tangent spaces at the identity element of the Lie groups GL(Rn), SL(n, R), O(n, R) are respectively gl(Rn) = {endomorphisms of Rn}, sl(n, R) = {X ∈ gl(Rn), Tr(X)=0}, o(n, R) = {X ∈ gl(Rn), tX + X =0}, n t o(p, q) = {X ∈ gl(R ), XIdp,q + Idp,qX =0}, where p + q = n. 2.1. Group action. — A morphism φ : G → H of groups is by defi- nition a map that preserves the product : φ(g1g2)= φ(g1)φ(g2). Exercise 2.3. — Show that φ(e)= e and φ(g−1)= φ(g)−1. Definition 2.4. — A(left)actionofagroup G on a set M is a mapping (2.1) α : G × M −→ M such that α(e, m) = m, ∀m ∈ M, and α(g,α(h, m)) = α(gh, m) for all m ∈ M and g,h ∈ G. 4 PAUL-EMILE PARADAN Let Bij(M) be the group of all bijective maps from M onto M. The conditions on α are equivalent to saying that the map G → Bij(M),g → αg defined by αg(m)= α(g, m) is a group morphism. If G is a Lie (resp. topological) group and M is a manifold (resp. topological space), the action of G on M is said to be smooth (resp. continuous) if the map (2.1) is smooth (resp. continuous). When the notations are understood we will write g · m, or simply gm, for α(g, m). A representation of a group G on a real (resp. complex) vector space V is a group morphism φ : G → GL(V ) : the group G acts on V through linear endomorphisms. Notation : If φ : M → N is a smooth map between differentiable manifolds, we denote by Tmφ : TmM → Tφ(m)N the differential of φ at m ∈ M. 2.2. Adjoint representation. — Let G be a Lie group and let g be the tangent space of G at e. We consider the conjugation action of G on itself, defined by −1 cg(h)= ghg , g,h ∈ G. The mappings cg : G → G are smooth and cg(e) = e for all g ∈ G, so one can consider the differential of cg at e Ad(g)= Tecg : g → g. Since cgh = cg ◦ch we have Ad(gh) = Ad(g)◦Ad(h). That is, the mapping (2.2) Ad : G −→ GL(g) is a smooth group morphism which is called the adjoint representation of G. The next step is to consider the differential of the map Ad at e: (2.3) ad= TeAd : g −→ gl(g). This is the adjoint representation of g. In (2.3), the vector space gl(g) denotes the vector space of all linear endomorphisms of g, and is equal to the tangent space of GL(g) at the identity. Lemma 2.5. — We have the fundamental relations • ad(Ad(g)X) = Ad(g) ◦ ad(X) ◦ Ad(g)−1 for g ∈ G,X ∈ g. SYMMETRIC SPACES : LIE GROUPS 5 • ad(ad(Y )X) = ad(Y ) ◦ ad(X) − ad(X) ◦ ad(Y ) for X, Y ∈ g. • ad(X)Y = −ad(Y )X for X, Y ∈ g. Proof. — Since Ad is a group morphism we have Ad(ghg−1) = Ad(g) ◦ Ad(h) ◦ Ad(g)−1. If we differentiate this relation at h = e we get the first point, and if we differentiate it at g = e we get the second one. For the last point consider two smooth curves a(t), b(s) on G d d with a(0) = b(0) = e, dt [a(t)]t=0 = X, and dt [b(t)]t=0 = Y . ∂2f We will now compute the second derivative ∂t∂s (0, 0) of the map f(t, s) = a(t)b(s)a(t)−1b(s)−1. Since f(t, 0) = f(0, s) = e, the term ∂2f ∂t∂s (0, 0) is defined in an intrinsic manner as an element of g. For ∂f the first partial derivatives we get ∂t (0, s) = X − Ad(b(s))X and ∂f ∂2f ∂s (t, 0) = Ad(a(t))Y − Y . So ∂t∂s (0, 0) = ad(X)Y = −ad(Y )X. Definition 2.6. — If G is a Lie group, one defines a bilinear map, [−, −]g : g × g → g by [X, Y ]g = ad(X)Y . It is the Lie bracket of g. The vector space g equipped with [−, −]g is called the Lie algebra of G. We have the fundamental relations • anti-symmetry : [X, Y ]g = −[Y,X]g • Jacobi identity : ad([Y,X]g) = ad(Y ) ◦ ad(X) − ad(X) ◦ ad(Y ). On gl(g), a direct computation shows that [X, Y ]gl(g) = XY − YX. So the Jacobi identity can be rewritten as ad([X, Y ]g) = [ad(X), ad(Y )]gl(g) or equivalently as (2.4) [X, [Y, Z]g]g + [Y, [Z,X]g]g + [Z, [X, Y ]g]g = 0 for all X,Y,Z ∈ g. Definition 2.7.
Load more

Recommended publications
  • Cohomology Theory of Lie Groups and Lie Algebras
    COHOMOLOGY THEORY OF LIE GROUPS AND LIE ALGEBRAS BY CLAUDE CHEVALLEY AND SAMUEL EILENBERG Introduction The present paper lays no claim to deep originality. Its main purpose is to give a systematic treatment of the methods by which topological questions concerning compact Lie groups may be reduced to algebraic questions con- cerning Lie algebras^). This reduction proceeds in three steps: (1) replacing questions on homology groups by questions on differential forms. This is accomplished by de Rham's theorems(2) (which, incidentally, seem to have been conjectured by Cartan for this very purpose); (2) replacing the con- sideration of arbitrary differential forms by that of invariant differential forms: this is accomplished by using invariant integration on the group manifold; (3) replacing the consideration of invariant differential forms by that of alternating multilinear forms on the Lie algebra of the group. We study here the question not only of the topological nature of the whole group, but also of the manifolds on which the group operates. Chapter I is concerned essentially with step 2 of the list above (step 1 depending here, as in the case of the whole group, on de Rham's theorems). Besides consider- ing invariant forms, we also introduce "equivariant" forms, defined in terms of a suitable linear representation of the group; Theorem 2.2 states that, when this representation does not contain the trivial representation, equi- variant forms are of no use for topology; however, it states this negative result in the form of a positive property of equivariant forms which is of interest by itself, since it is the key to Levi's theorem (cf.
    [Show full text]
  • Lecture Notes on Compact Lie Groups and Their Representations
    Lecture Notes on Compact Lie Groups and Their Representations CLAUDIO GORODSKI Prelimimary version: use with extreme caution! September , ii Contents Contents iii 1 Compact topological groups 1 1.1 Topological groups and continuous actions . 1 1.2 Representations .......................... 4 1.3 Adjointaction ........................... 7 1.4 Averaging method and Haar integral on compact groups . 9 1.5 The character theory of Frobenius-Schur . 13 1.6 Problems.............................. 20 2 Review of Lie groups 23 2.1 Basicdefinition .......................... 23 2.2 Liealgebras ............................ 25 2.3 Theexponentialmap ....................... 28 2.4 LiehomomorphismsandLiesubgroups . 29 2.5 Theadjointrepresentation . 32 2.6 QuotientsandcoveringsofLiegroups . 34 2.7 Problems.............................. 36 3 StructureofcompactLiegroups 39 3.1 InvariantinnerproductontheLiealgebra . 39 3.2 CompactLiealgebras. 40 3.3 ComplexsemisimpleLiealgebras. 48 3.4 Problems.............................. 51 3.A Existenceofcompactrealforms . 53 4 Roottheory 55 4.1 Maximaltori............................ 55 4.2 Cartansubalgebras ........................ 57 4.3 Case study: representations of SU(2) .............. 58 4.4 Rootspacedecomposition . 61 4.5 Rootsystems............................ 63 4.6 Classificationofrootsystems . 69 iii iv CONTENTS 4.7 Problems.............................. 73 CHAPTER 1 Compact topological groups In this introductory chapter, we essentially introduce our very basic objects of study, as well as some fundamental examples. We also establish some preliminary results that do not depend on the smooth structure, using as little as possible machinery. The idea is to paint a picture and plant the seeds for the later development of the heavier theory. 1.1 Topological groups and continuous actions A topological group is a group G endowed with a topology such that the group operations are continuous; namely, we require that the multiplica- tion map and the inversion map µ : G G G, ι : G G × → → be continuous maps.
    [Show full text]
  • The Pure State Space of Quantum Mechanics As Hermitian Symmetric
    The Pure State Space of Quantum Mechanics as Hermitian Symmetric Space. R. Cirelli, M. Gatti, A. Mani`a Dipartimento di Fisica, Universit`adegli Studi di Milano, Italy Istituto Nazionale di Fisica Nucleare, Sezione di Milano, Italy Abstract The pure state space of Quantum Mechanics is investigated as Hermitian Symmetric K¨ahler manifold. The classical principles of Quantum Mechan- ics (Quantum Superposition Principle, Heisenberg Uncertainty Principle, Quantum Probability Principle) and Spectral Theory of observables are dis- cussed in this non linear geometrical context. Subj. Class.: Quantum mechanics 1991 MSC : 58BXX; 58B20; 58FXX; 58HXX; 53C22 Keywords: Superposition principle, quantum states; Riemannian mani- folds; Poisson algebras; Geodesics. 1. Introduction Several models of delinearization of Quantum Mechanics have been pro- arXiv:quant-ph/0202076v1 14 Feb 2002 posed (see f.i. [20] [26], [11], [2] and [5] for a complete list of references). Frequentely these proposals are supported by different motivations, but it appears that a common feature is that, more or less, the delinearization must be paid essentially by the superposition principle. This attitude can be understood if the delinearization program is worked out in the setting of a Hilbert space as a ground mathematical structure. However, as is well known, the groundH mathematical structure of QM is the manifold of (pure) states P( ), the projective space of the Hilbert space . Since, obviously, P( ) is notH a linear object, the popular way of thinking thatH the superposition principleH compels the linearity of the space of states is untenable. The delinearization program, by itself, is not related in our opinion to attempts to construct a non linear extension of QM with operators which 1 2 act non linearly on the Hilbert space .
    [Show full text]
  • Cartan and Iwasawa Decompositions in Lie Theory
    CARTAN AND IWASAWA DECOMPOSITIONS IN LIE THEORY SUBHAJIT JANA Abstract. In this article we will be discussing about various decom- positions of semisimple Lie algebras, which are very important to un- derstand their structure theories. Throughout the article we will be assuming existence of a compact real form of complex semisimple Lie algebra. We will start with describing Cartan involution and Cartan de- compositions of semisimple Lie algebras. Iwasawa decompositions will also be constructed at Lie group and Lie algebra level. We will also be giving some brief description of Iwasawa of semisimple groups. 1. Introduction The idea for beginning an investigation of the structure of a general semisimple Lie group, not necessarily classical, is to look for same kind of structure in its Lie algebra. We start with a Lie algebra L of matrices and seek a decomposition into symmetric and skew-symmetric parts. To get this decomposition we often look for the occurrence of a compact Lie algebra as a real form of the complexification LC of L. If L is a real semisimple Lie algebra, then the use of a compact real form of LC leads to the construction of a 'Cartan Involution' θ of L. This involution has the property that if L = H ⊕ P is corresponding eigenspace decomposition or 'Cartan Decomposition' then, LC has a compact real form (like H ⊕iP ) which generalize decomposition of classical matrix algebra into Hermitian and skew-Hermitian parts. Similarly if G is a semisimple Lie group, then the 'Iwasawa decomposition' G = NAK exhibits closed subgroups A and N of G such that they are simply connected abelian and nilpotent respectively and A normalizes N and multiplication K ×A×N ! G is a diffeomorphism.
    [Show full text]
  • Symmetric Spaces with Conformal Symmetry
    Symmetric Spaces with Conformal Symmetry A. Wehner∗ Department of Physics, Utah State University, Logan, Utah 84322 October 30, 2018 Abstract We consider an involutive automorphism of the conformal algebra and the resulting symmetric space. We display a new action of the conformal group which gives rise to this space. The space has an in- trinsic symplectic structure, a group-invariant metric and connection, and serves as the model space for a new conformal gauge theory. ∗Electronic-mail: [email protected] arXiv:hep-th/0107008v1 1 Jul 2001 1 1 Introduction Symmetric spaces are the most widely studied class of homogeneous spaces. They form a subclass of the reductive homogeneous spaces, which can es- sentially be characterized by the fact that they admit a unique torsion-free group-invariant connection. For symmetric spaces the curvature is covari- antly constant with respect to this connection. Many essential results and extensive bibliographies on symmetric spaces can be found, for example, in the standard texts by Kobayashi and Nomizu [1] and Helgason [2]. The physicists’ definition of a (maximally) symmetric space as a metric space which admits a maximum number of Killing vectors is considerably more restrictive. A thorough treatment of symmetric spaces in general rel- ativity, where a pseudo-Riemannian metric and the metric-compatible con- nection are assumed, can be found in Weinberg [3]. The most important symmetric space in gravitational theories is Minkowski space, which is based on the symmetric pair (Poincar´egroup, Lorentz group). Since the Poincar´e group is not semi-simple, it does not admit an intrinsic group-invariant met- ric, which would project in the canonical fashion to the Minkowski metric ηab = diag(−1, 1 ..
    [Show full text]
  • Introduction to Representations Theory of Lie Groups
    Introduction to Representations Theory of Lie Groups Raul Gomez October 14, 2009 Introduction The purpose of this notes is twofold. The first goal is to give a quick answer to the question \What is representation theory about?" To answer this, we will show by examples what are the most important results of this theory, and the problems that it is trying to solve. To make the answer short we will not develop all the formal details of the theory and we will give preference to examples over proofs. Few results will be proved, and in the ones were a proof is given, we will skip the technical details. We hope that the examples and arguments presented here will be enough to give the reader and intuitive but concise idea of the covered material. The second goal of the notes is to be a guide to the reader interested in starting a serious study of representation theory. Sometimes, when starting the study of a new subject, it's hard to understand the underlying motivation of all the abstract definitions and technical lemmas. It's also hard to know what is the ultimate goal of the subject and to identify the important results in the sea of technical lemmas. We hope that after reading this notes, the interested reader could start a serious study of representation theory with a clear idea of its goals and philosophy. Lets talk now about the material covered on this notes. In the first section we will state the celebrated Peter-Weyl theorem, which can be considered as a generalization of the theory of Fourier analysis on the circle S1.
    [Show full text]
  • LIE GROUPS and ALGEBRAS NOTES Contents 1. Definitions 2
    LIE GROUPS AND ALGEBRAS NOTES STANISLAV ATANASOV Contents 1. Definitions 2 1.1. Root systems, Weyl groups and Weyl chambers3 1.2. Cartan matrices and Dynkin diagrams4 1.3. Weights 5 1.4. Lie group and Lie algebra correspondence5 2. Basic results about Lie algebras7 2.1. General 7 2.2. Root system 7 2.3. Classification of semisimple Lie algebras8 3. Highest weight modules9 3.1. Universal enveloping algebra9 3.2. Weights and maximal vectors9 4. Compact Lie groups 10 4.1. Peter-Weyl theorem 10 4.2. Maximal tori 11 4.3. Symmetric spaces 11 4.4. Compact Lie algebras 12 4.5. Weyl's theorem 12 5. Semisimple Lie groups 13 5.1. Semisimple Lie algebras 13 5.2. Parabolic subalgebras. 14 5.3. Semisimple Lie groups 14 6. Reductive Lie groups 16 6.1. Reductive Lie algebras 16 6.2. Definition of reductive Lie group 16 6.3. Decompositions 18 6.4. The structure of M = ZK (a0) 18 6.5. Parabolic Subgroups 19 7. Functional analysis on Lie groups 21 7.1. Decomposition of the Haar measure 21 7.2. Reductive groups and parabolic subgroups 21 7.3. Weyl integration formula 22 8. Linear algebraic groups and their representation theory 23 8.1. Linear algebraic groups 23 8.2. Reductive and semisimple groups 24 8.3. Parabolic and Borel subgroups 25 8.4. Decompositions 27 Date: October, 2018. These notes compile results from multiple sources, mostly [1,2]. All mistakes are mine. 1 2 STANISLAV ATANASOV 1. Definitions Let g be a Lie algebra over algebraically closed field F of characteristic 0.
    [Show full text]
  • Holonomy and the Lie Algebra of Infinitesimal Motions of a Riemannian Manifold
    HOLONOMY AND THE LIE ALGEBRA OF INFINITESIMAL MOTIONS OF A RIEMANNIAN MANIFOLD BY BERTRAM KOSTANT Introduction. .1. Let M be a differentiable manifold of class Cx. All tensor fields discussed below are assumed to be of class C°°. Let X be a vector field on M. If X vanishes at a point oCM then X induces, in a natural way, an endomorphism ax of the tangent space F0 at o. In fact if y£ V„ and F is any vector field whose value at o is y, then define axy = [X, Y]„. It is not hard to see that [X, Y]0 does not depend on F so long as the value of F at o is y. Now assume M has an affine connection or as we shall do in this paper, assume M is Riemannian and that it possesses the corresponding affine connection. One may now associate to X an endomorphism ax of F0 for any point oCM which agrees with the above definition and which heuristically indicates how X "winds around o" by defining for vC V0, axv = — VtX, where V. is the symbol of covariant differentiation with respect to v. X is called an infinitesmal motion if the Lie derivative of the metric tensor with respect to X is equal to zero. This is equivalent to the statement that ax is skew-symmetric (as an endomorphism of F0 where the latter is provided with the inner product generated by the metric tensor) for all oCM. §1 contains definitions and formulae which will be used in the remaining sections.
    [Show full text]
  • Symmetric Spaces of Hermitian Type
    Differential Geometry and its Applications 1 (1991) 195-233 195 North-Holland Symmetric spaces of Hermitian type G. ‘Olafsson Mathematisches Znstitut der Uniuersitit Giittingen, Bunsenstraj?e 3-5, D-3400 Giittingen Received 7 February 1991 ‘Olafsson, G., Symmetric spaces of Hermitian type, Diff. Geom. Appl. 1 (1991) 195-233. Abstract: Let M = G/H be a semisimple symmetric space, r the corresponding involution and D = G/K the Riemannian symmetric space. Then we show that the following are equivalent: M is of Hermitian type; r induces a conjugation on D; there exists an open regular H-invariant cone R in q = h’ such that k n 0 # 0. We relate the spaces of Hermitian type to the regular and parahermitian symmetric spaces, analyze the fine structure of D under r and construct an equivariant Cayley transform. We collect also some results on the classification of invariant cones in q. Finally we point out some applications in representations theory. Keywords: Symmetric spaces, semisimple Lie groups, invariant convex cones, causal orientation, ordering, convexity theorem. MS classification: 53635, 57S25, 22E15, 06AlO. Introduction Bounded symmetric domains and their unbounded counterparts, the Siegel domains, have long been an important part of different fields of mathematics, e.g., number theory, algebraic geometry, harmonic analysis and representations theory. So the holomorphic discrete series and other interesting representations of a group live in spaces of holo- morphic functions on such domains. In the last years some interplays with harmonic analysis on affine symmetric spaces have also become apparent, e.g., a construction of non-zero harmonic forms related to the discrete series of such spaces (see [44] and the literature there).
    [Show full text]
  • Symmetric Spaces
    CHAPTER 2 Symmetric Spaces Introduction We remark that a prerequisite for this course is a basic understanding of Rie- mannian geometry, see for instance [Boo86]or[Hel79], which at the same time is a standard reference for the theory of symmetric spaces. See also [Bor98]fora much more condensed version of the latter. In 1926, É. Cartan began to study those Riemannian manifolds,forwhich central symmetries are distance preserving. The term central symmetry is going to be made precise later but you can think of it as a reflection about a point, as in Rn for instance. These spaces are now known as symmetric spaces;henoticedthatthe classification of these is essentially equivalent to the classification of real semisimple Lie algebras. The local definition of symmetric spaces is a generalisation of constant sectional curvature. Just as a differentiable function from R to R is constant if its derivative vanishes identically, one may ask for the Riemanncurvaturetensorto have covariant derivative zero. As incomprehensible as thisstatementmaybeatthe moment, it yields an important equivalent characterizationofsymmetricspaces. Example. The following list of examples of symmetric spaces contains the constant curvature cases. n n (i) Euclidean n-space E =(R ,geucl) has constant sectional curvature zero and isometry group Iso(En) = O(n) !Rn. n n ∼ (ii) The n-sphere S =(S ,geucl) has constant sectional curvature one. Its Riemannian metric arises through restriction of the Riemannian metric of the ambient space Rn+1 Sn to the tangent bundle of Sn.Theisometry group of Sn is O(n). ⊃ (iii) Hyperbolic n-space Hn has constant sectional curvature minus one.
    [Show full text]
  • Finite Order Automorphisms on Real Simple Lie Algebras
    TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 364, Number 7, July 2012, Pages 3715–3749 S 0002-9947(2012)05604-2 Article electronically published on February 15, 2012 FINITE ORDER AUTOMORPHISMS ON REAL SIMPLE LIE ALGEBRAS MENG-KIAT CHUAH Abstract. We add extra data to the affine Dynkin diagrams to classify all the finite order automorphisms on real simple Lie algebras. As applications, we study the extensions of automorphisms on the maximal compact subalgebras and also study the fixed point sets of automorphisms. 1. Introduction Let g be a complex simple Lie algebra with Dynkin diagram D. A diagram automorphism on D of order r leads to the affine Dynkin diagram Dr. By assigning integers to the vertices, Kac [6] uses Dr to represent finite order g-automorphisms. This paper extends Kac’s result. It uses Dr to represent finite order automorphisms on the real forms g0 ⊂ g. As applications, it studies the extensions of finite order automorphisms on the maximal compact subalgebras of g0 and studies their fixed point sets. The special case of g0-involutions (i.e. automorphisms of order 2) is discussed in [2]. In general, Gothic letters denote complex Lie algebras or spaces, and adding the subscript 0 denotes real forms. The finite order automorphisms on compact or complex simple Lie algebras have been classified by Kac, so we ignore these cases. Thus we always assume that g0 is a noncompact real form of the complex simple Lie algebra g (this automatically rules out complex g0, since in this case g has two simple factors). 1 2 2 2 3 The affine Dynkin diagrams are D for all D, as well as An,Dn, E6 and D4.By 3 r Definition 1.1 below, we ignore D4.
    [Show full text]
  • Strongly Exponential Symmetric Spaces
    STRONGLY EXPONENTIAL SYMMETRIC SPACES YANNICK VOGLAIRE Abstract. We study the exponential map of connected symmetric spaces and characterize, in terms of midpoints and of infinitesimal conditions, when it is a diffeomorphism, generalizing the Dixmier–Saito theorem for solvable Lie groups. We then give a geometric characterization of the (strongly) exponential solvable symmetric spaces as those spaces for which every triangle admits a unique double triangle. This work is motivated by Weinstein’s quantization by groupoids program applied to symmetric spaces. 1. Introduction In the late 1950’s, Dixmier [5] and Saito [16] characterized the strongly exponential Lie groups: the simply connected solvable Lie groups for which the exponential map is a global diffeomorphism. They are characterized by the fact that the exponential map is either injective or surjective, or equivalently the fact that the adjoint action of their Lie algebra has no purely imaginary eigenvalue. In contrast, the exponential map of semisimple Lie groups is never injective, but may be surjective or have dense image. As a result, the latter two weaker notions of exponentiality have been studied for general Lie groups by a number of authors (see [6, 21] for two surveys and for references). Recently, a number of their results were generalized to symmetric spaces in [15]. The first main result of the present paper intends to complete the picture by giving a version of the Dixmier–Saito theorem for symmetric spaces (see Theorems 1.1, 1.2 and 1.3). The second main result gives a geometric characterization of the strongly ex- arXiv:1303.5925v2 [math.DG] 6 Apr 2014 ponential symmetric spaces which are homogeneous spaces of solvable Lie groups, in terms of existence and uniqueness of double triangles (see Theorem 1.4).
    [Show full text]