Notes on Symmetric Spaces
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Hyperpolar Homogeneous Foliations on Symmetric Spaces of Noncompact Type
HYPERPOLAR HOMOGENEOUS FOLIATIONS ON SYMMETRIC SPACES OF NONCOMPACT TYPE JURGEN¨ BERNDT Abstract. A foliation F on a Riemannian manifold M is homogeneous if its leaves coincide with the orbits of an isometric action on M. A foliation F is polar if it admits a section, that is, a connected closed totally geodesic submanifold of M which intersects each leaf of F, and intersects orthogonally at each point of intersection. A foliation F is hyperpolar if it admits a flat section. These notes are related to joint work with Jos´eCarlos D´ıaz-Ramos and Hiroshi Tamaru about hyperpolar homogeneous foliations on Riemannian symmetric spaces of noncompact type. Apart from the classification result which we proved in [1], we present here in more detail some relevant material about symmetric spaces of noncompact type, and discuss the classification in more detail for the special case M = SLr+1(R)/SOr+1. 1. Isometric actions on the real hyperbolic plane The special linear group a b G = SL2(R) = a, b, c, d ∈ R, ad − bc = 1 c d acts on the upper half plane H = {z ∈ C | =(z) > 0} by linear fractional transformations of the form a b az + b · z = . c d cz + d By equipping H with the Riemannian metric dx2 + dy2 dzdz¯ ds2 = = (z = x + iy) y2 =(z)2 we obtain the Poincar´ehalf-plane model of the real hyperbolic plane RH2. The above action of SL2(R) is isometric with respect to this metric. The action is transitive and the stabilizer at i is cos(s) sin(s) K = s ∈ R = SO2. -
Arxiv:Math/0206135V1
PROJECTIVE PLANES, SEVERI VARIETIES AND SPHERES MICHAEL ATIYAH AND JURGEN¨ BERNDT Abstract. A classical result asserts that the complex projective plane modulo complex conjugation is the 4-dimensional sphere. We generalize this result in two directions by considering the projective planes over the normed real division algebras and by considering the complexifications of these four projective planes. 1. Introduction There is an elementary but very striking result which asserts that the quotient of the complex projective plane CP 2 by complex conjugation is the 4-dimensional sphere. This result has attracted the attention of many geometers over the years, rediscovered afresh each time and with a variety of proofs. For some historical remarks about the origins of this theorem we refer to Arnold [3], where it is put in a more general context. The purpose of the present paper is to give two parallel generalizations of this theorem. In the first we view CP 2 as the second member of the family of projective planes over the four normed real division algebras. This is close to Arnold’s treatment and completes it by dealing with the octonions. The second generalization views CP 2 as the complex algebraic variety obtained by complexifying the real projective plane RP 2 and hence as the first member of the four algebraic varieties got by complexifying the four projective planes. This family of algebraic varieties has appeared in several contexts. First, in relation to Lie groups and the “magic square” of Freudenthal [13], somewhat independently in the characterization by Zak [26] of “Severi varieties”, and also in the classification by Nakagawa and Takagi [19] of K¨ahler submanifolds with parallel second fundamental form in complex projective spaces. -
Invariant Differential Operators for Quantum Symmetric Spaces, I
Invariant Differential Operators for Quantum Symmetric Spaces, I Gail Letzter∗ Mathematics Department Virginia Tech Blacksburg, VA 24061 [email protected] January 1, 2018 Abstract This is the first paper in a series of two which proves a version of a theorem of Harish-Chandra for quantum symmetric spaces in the maximally split case: There is a Harish-Chandra map which induces an isomorphism between the ring of quantum invariant differential op- erators and the ring of invariants of a certain Laurent polynomial ring under an action of the restricted Weyl group. Here, we establish this result for all quantum symmetric spaces defined using irreducible sym- metric pairs not of type EIII, EIV, EVII, or EIX. A quantum version of a related theorem due to Helgason is also obtained: The image of the center under this Harish-Chandra map is the entire invariant ring arXiv:math/0406193v1 [math.QA] 9 Jun 2004 if and only if the underlying irreducible symmetric pair is not one of these four types. Introduction In [HC2, Section 4], Harish-Chandra proved the following fundamental result for semisimple Lie groups: the Harish-Chandra map induces an isomorphism ∗supported by NSA grant no. MDA904-03-1-0033. AMS subject classification 17B37 1 between the ring of invariant differential operators on a symmetric space and invariants of an appropriate polynomial ring under the restricted Weyl group. When the symmetric space is simply a complex semisimple Lie group, this result is Harish-Chandra’s famous realization of the center of the enveloping algebra of a semisimple Lie algebra as Weyl group invariants of the Cartan subalgebra ([HC1]). -
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. -
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. -
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 . -
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. -
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 .. -
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. -
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. -
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. -
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).