DOCSLIB.ORG

Basic Lie Theory

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Basic Lie Theory Basic Lie Theory Hossein Abbaspour Martin Moskowitz ii To Gerhard Hochschild Contents Preface and Acknowledgments ix Notations xiii 0 Lie Groups and Lie Algebras; Introduction 1 0.1 Topological Groups . 1 0.2 LieGroups ......................... 6 0.3 CoveringMapsandGroups . 10 0.4 Group Actions and Homogeneous Spaces . 15 0.5 LieAlgebras......................... 25 1 Lie Groups 31 1.1 Elementary Properties of a Lie Group . 31 1.2 Taylor’s Theorem and the Coefficients of expX expY . 39 1.3 Correspondence between Lie Subgroups and Subalgebras 45 1.4 The Functorial Relationship . 48 1.5 The Topology of Compact Classical Groups . 60 1.6 The Iwasawa Decompositions for GL(n, R) and GL(n, C) 67 1.7 The Baker-Campbell-Hausdorff Formula . 69 2 Haar Measure and its Applications 89 2.1 Haar Measure on a Locally Compact Group . 89 2.2 Properties of the Modular Function . 100 2.3 Invariant Measures on Homogeneous Spaces . 101 2.4 Compact or Finite Volume Quotients . 106 v vi 2.5 Applications......................... 112 2.6 Compact linear groups and Hilbert’s 14th problem . 121 3 Elements of the Theory of Lie Algebras 127 3.1 Basics of Lie Algebras . 127 3.1.1 Ideals and Related Concepts . 127 3.1.2 Semisimple Lie algebras . 138 3.1.3 Complete Lie Algebras . 139 3.1.4 Lie Algebra Representations . 140 3.1.5 The irreducible representations of sl(2, k) . 142 3.1.6 Invariant Forms . 145 3.1.7 Complex and Real Lie Algebras . 147 3.1.8 Rational Forms . 148 3.2 EngelandLie’sTheorems . 150 3.2.1 Engel’s Theorem . 150 3.2.2 Lie’s Theorem . 153 3.3 Cartan’s Criterion and Semisimple Lie algebras . 158 3.3.1 SomeAlgebra.. .. .. .. .. .. 158 3.3.2 Cartan’s Solvability Criterion . 162 3.3.3 Explicit Computations of Killing form . 166 3.3.4 Further Results on Jordan Decomposition . 170 3.4 Weyl’s Theorem on Complete Reducibility . 173 3.5 Levi-Malcev Decomposition . 180 3.6 Reductive Lie Algebras . 188 3.7 The Jacobson-Morozov Theorem . 193 3.8 Low Dimensional Lie Algebrasover R and C ....... 198 3.9 Real Lie Algebras of Compact Type . 202 4 The Structure of Compact Connected Lie Groups 207 4.1 Introduction......................... 207 4.2 Maximal Tori in Compact Lie Groups . 208 4.3 Maximal Tori in Compact Connected Lie Groups . 210 4.4 TheWeylGroup ...................... 217 4.5 What goes wrong if G is not compact . 221 vii 5 Representations of Compact Lie Groups 223 5.1 Introduction......................... 224 5.2 The Schur Orthogonality Relations . 226 5.3 Compact Integral Operators on a Hilbert Space . 228 5.4 The Peter-Weyl Theorem and its Consequences . 234 5.5 Characters and Central Functions . 243 5.6 Induced Representations . 250 5.7 Some Consequences of Frobenius Reciprocity . 255 6 Symmetric Spaces of Non-compact type 261 6.1 Introduction......................... 261 6.2 The Polar Decomposition . 264 6.3 The Cartan Decomposition . 267 6.4 The Case of Hyperbolic Space and the Lorentz Group . 274 6.5 The G-invariant Metric Geometry of P .......... 278 6.6 The Conjugacy of Maximal Compact Subgroups . 289 6.7 The Rank and Two-Point Homogeneous Spaces . 294 6.8 The Disk Model for Spaces of Rank 1 . 299 6.9 Exponentiality of Certain Rank 1 Groups . 304 7 Semisimple Lie Algebras and Lie Groups 313 7.1 Root and Weight Space Decompositions . 313 7.2 CartanSubalgebras. 316 7.3 Roots of Complex Semisimple Lie Algebras . 323 7.4 Real Forms of Complex Semisimple Lie Algebras . 337 7.5 The Iwasawa Decomposition . 343 8 Lattices in Lie Groups 355 8.1 Lattices in Euclidean Space . 355 8.2 GL(n, R)/ GL(n, Z) and SL(n, R)/ SL(n, Z) ....... 360 8.3 Lattices in more general groups . 371 8.4 Fundamental Domains . 374 9 Density results for cofinite Volume Subgroups 377 9.1 Introduction......................... 377 9.2 A Density Theorem for cofinite Volume Subgroups . 379 viii 9.3 Consequences and Extensions of the Density Theorem . 389 A Vector Fields 397 B The Kronecker Approximation Theorem 403 C Properly discontinuous actions 407 D The Analyticity of Smooth Lie Groups 411 References 413 Index 421 Preface and Acknowledgments In the view of the authors, and as we hope to convince the reader, Lie theory, broadly understood, lies at the center of modern mathematics. It is linked to algebra, analysis, algebraic and differential geometry, topol- ogy and even number theory, and applications of some of these other subjects are crucial to many of the arguments we shall present here. This also holds true in the opposite direction: Lie theory can be used to clarify or derive results in these other areas. In a philosophical sense Lie groups are pervasive within much of mathematics, for whenever one has some system, the “automorphisms” of it will frequently be a Lie group. This even occurs in the oldest deductive system in mathematics, namely Euclidean geometry. Here the key issue is congruent figures, particularly triangles. Two such planar triangles are congruent if and only if they differ by an element of the Lie group E(2) = O(2, R) ⋉R2, the group of rigid motions of the Euclidean plane. The reader will find in these many interrelations a vast panorama well worth studying. This book is the result of courses taught by one of the authors over many years on various aspects of Lie theory at the City University of New York Graduate Center. The primary reader to which it is ad- dressed is a graduate student in mathematics, or perhaps physics, or a researcher in one of these subjects who wants a comprehensive ref- erence work in Lie theory. However, by a judicious selection of topics, some of this material could also be used to give an introduction to the subject to well-grounded advanced undergraduate mathematics majors. ix x Preface and Acknowledgments For example, Chapters 3 and most of 7 could form a semester’s course in Lie algebras. Similarly, Chapters 0, 2 and 5 (respectively 0, 2 and 8) could be a semester’s course in integration in topological groups and their homogeneous spaces (respectively lattices and their applications). For the reader’s convenience we have included a diagram of the inter- dependence of the chapters. We have also tried to make the text as self-contained as possible even at the cost of increased length. We shall assume the reader has some knowledge of basic group theory, topology, and linear algebra, and a general acquaintance with the grammar of mathematics. While reading this book one may wish to consult some of the other books on the subject for clarification, or to see another view- point or treatment; especially useful books are listed in the bibliography. We have not attempted to detail the historical development of our sub- ject, nor to systematically give credit to the individual researchers who discovered these results. The book’s organization is as follows: Chapter 0 introduces the players; topological and Lie groups, cover- ings, group actions, homogeneous spaces, and Lie algebras. Chapter 1 deals with the correspondence between Lie groups and their Lie algebras, subalgebras and ideals, the functorial relationship determined by the exponential map, the topology of the classical groups, the Iwasawa decomposition in certain key cases, and the Baker Campbell Hausdorff theorem, and local Lie groups. Chapter 2 concerns Haar measure both on a group and on cocom- pact and finite volume homogeneous spaces together with a number of applications. Chapter 3 gives the elements of Lie algebra theory in some consid- erable detail (except for the detailed structure of complex semisimple Lie algebras, which we defer until Chapter 7). Chapter 4 deals with the structure of a compact connected Lie group in terms of a maximal torus and the Weyl group. Chapter 5 contains the representation theory of compact groups. Chapter 6 concerns symmetric spaces of non-compact type. Chapter 7 presents the detailed structure of complex semisimple Lie algebras. Preface and Acknowledgments xi Chapter 8 gives an introduction to lattices in Lie groups. Chapter 9 presents a “density theorem” for cofinite volume sub- groups of certain Lie groups. Although we have included a rather detailed and extensive index, it might be helpful to inform the reader of what we are not doing here. We do not deal extensively with the theory of algebraic groups, nor with transformation groups, although each of these makes some appearance. Similarly, we do not prove that any connected Lie group is, as a man- ifold, the direct product of a Euclidean space and a maximal compact subgroup, K; nor that any two such Ks are conjugate. But we do this in important special cases. We do not deal, except by example, with the theory of faithful representations. We have also omitted the Weyl character formula, the universal enveloping algebra, the classification of complex simple algebras and, with the exception of one example, branching theorems. We would like to thank Frederick Greenleaf, Adam Kor´anyi, Keivan Mallahi and Grigory Margulis for reading various chapters of our book and making a number of valuable suggestions. Of course any errors or misstatements are the sole responsibility of the authors. The authors would like to thank Richard Mosak for his help in the final preparation of this manuscript. We also thank Isabelle and Anita for their extraordinary patience during the several years that it took to bring this project to completion. Hossein Abbaspour Martin Moskowitz Max-Planck Institut f¨ur Mathematik The Graduate School Bonn The City University of New York New York xii Interdependency Chart Chapter 0 Chapter 1 Chapter 3 Chapter 2 Chapter 6 Chapter 4 Chapter 8 Chapter 5 Chapter 7 Chapter 9 Notations xiii Notations For a complex number z, z is the real part and z is the imaginary ℜ ℑ part.
Load more

Recommended publications
  • 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]
  • A Taxonomy of Irreducible Harish-Chandra Modules of Regular Integral Infinitesimal Character
    A taxonomy of irreducible Harish-Chandra modules of regular integral infinitesimal character B. Binegar OSU Representation Theory Seminar September 5, 2007 1. Introduction Let G be a reductive Lie group, K a maximal compact subgroup of G. In fact, we shall assume that G is a set of real points of a linear algebraic group G defined over C. Let g be the complexified Lie algebra of G, and g = k + p a corresponding Cartan decomposition of g Definition 1.1. A (g,K)-module is a complex vector space V carrying both a Lie algebra representation of g and a group representation of K such that • The representation of K on V is locally finite and smooth. • The differential of the group representation of K coincides with the restriction of the Lie algebra representation to k. • The group representation and the Lie algebra representations are compatible in the sense that πK (k) πk (X) = πk (Ad (k) X) πK (k) . Definition 1.2. A (Hilbert space) representation π of a reductive group G with maximal compact subgroup K is called admissible if π|K is a direct sum of finite-dimensional representations of K such that each K-type (i.e each distinct equivalence class of irreducible representations of K) occurs with finite multiplicity. Definition 1.3 (Theorem). Suppose (π, V ) is a smooth representation of a reductive Lie group G, and K is a compact subgroup of G. Then the space of K-finite vectors can be endowed with the structure of a (g,K)-module. We call this (g,K)-module the Harish-Chandra module of (π, V ).
    [Show full text]
  • UTM Naive Lie Theory (John Stillwell) 0387782141.Pdf
    Undergraduate Texts in Mathematics Editors S. Axler K.A. Ribet Undergraduate Texts in Mathematics Abbott: Understanding Analysis. Daepp/Gorkin: Reading, Writing, and Proving: Anglin: Mathematics: A Concise History and A Closer Look at Mathematics. Philosophy. Devlin: The Joy of Sets: Fundamentals Readings in Mathematics. of-Contemporary Set Theory. Second edition. Anglin/Lambek: The Heritage of Thales. Dixmier: General Topology. Readings in Mathematics. Driver: Why Math? Apostol: Introduction to Analytic Number Theory. Ebbinghaus/Flum/Thomas: Mathematical Logic. Second edition. Second edition. Armstrong: Basic Topology. Edgar: Measure, Topology, and Fractal Geometry. Armstrong: Groups and Symmetry. Second edition. Axler: Linear Algebra Done Right. Second edition. Elaydi: An Introduction to Difference Equations. Beardon: Limits: A New Approach to Real Third edition. Analysis. Erdos/Sur˜ anyi:´ Topics in the Theory of Numbers. Bak/Newman: Complex Analysis. Second edition. Estep: Practical Analysis on One Variable. Banchoff/Wermer: Linear Algebra Through Exner: An Accompaniment to Higher Mathematics. Geometry. Second edition. Exner: Inside Calculus. Beck/Robins: Computing the Continuous Fine/Rosenberger: The Fundamental Theory Discretely of Algebra. Berberian: A First Course in Real Analysis. Fischer: Intermediate Real Analysis. Bix: Conics and Cubics: A Concrete Introduction to Flanigan/Kazdan: Calculus Two: Linear and Algebraic Curves. Second edition. Nonlinear Functions. Second edition. Bremaud:` An Introduction to Probabilistic Fleming: Functions of Several Variables. Second Modeling. edition. Bressoud: Factorization and Primality Testing. Foulds: Combinatorial Optimization for Bressoud: Second Year Calculus. Undergraduates. Readings in Mathematics. Foulds: Optimization Techniques: An Introduction. Brickman: Mathematical Introduction to Linear Franklin: Methods of Mathematical Programming and Game Theory. Economics. Browder: Mathematical Analysis: An Introduction. Frazier: An Introduction to Wavelets Through Buchmann: Introduction to Cryptography.
    [Show full text]
  • 132 Lie's Theory of Continuous Groups Plays Such an Important Part in The
    132 SHORTER NOTICES. [Dec, Lie's theory of continuous groups plays such an important part in the elementary problems of the theory of differential equations, and has proved to be such a powerful weapon in the hands of competent mathematicians, that a work on differential equations, of even the most modest scope, appears decidedly incomplete without some account of it. It is to be regretted that Mr. Forsyth has not introduced this theory into his treatise. The introduction of a brief account of Runge's method for numerical integration is a very valuable addition to the third edition. The treatment of the Riccati equation has been modi­ fied. The theorem that the cross ratio of any four solutions is constant is demonstrated but not explicitly enunciated, which is much to be regretted. From the point of view of the geo­ metric applications, this is the most important property of the equations of the Riccati type. The theory of total differential equations has been discussed more fully than before, and the treatment on the whole is lucid. The same may be said of the modified treatment adopted by the author for the linear partial differential equations of the first order. A very valuable feature of the book is the list of examples. E. J. WlLCZYNSKI. Introduction to Projective Geometry and Its Applications. By ARNOLD EMOH. New York, John Wiley & Sons, 1905. vii + 267 pp. To some persons the term projective geometry has come to stand only for that pure science of non-metric relations in space which was founded by von Staudt.
    [Show full text]
  • Representation Theory
    M392C NOTES: REPRESENTATION THEORY ARUN DEBRAY MAY 14, 2017 These notes were taken in UT Austin's M392C (Representation Theory) class in Spring 2017, taught by Sam Gunningham. I live-TEXed them using vim, so there may be typos; please send questions, comments, complaints, and corrections to [email protected]. Thanks to Kartik Chitturi, Adrian Clough, Tom Gannon, Nathan Guermond, Sam Gunningham, Jay Hathaway, and Surya Raghavendran for correcting a few errors. Contents 1. Lie groups and smooth actions: 1/18/172 2. Representation theory of compact groups: 1/20/174 3. Operations on representations: 1/23/176 4. Complete reducibility: 1/25/178 5. Some examples: 1/27/17 10 6. Matrix coefficients and characters: 1/30/17 12 7. The Peter-Weyl theorem: 2/1/17 13 8. Character tables: 2/3/17 15 9. The character theory of SU(2): 2/6/17 17 10. Representation theory of Lie groups: 2/8/17 19 11. Lie algebras: 2/10/17 20 12. The adjoint representations: 2/13/17 22 13. Representations of Lie algebras: 2/15/17 24 14. The representation theory of sl2(C): 2/17/17 25 15. Solvable and nilpotent Lie algebras: 2/20/17 27 16. Semisimple Lie algebras: 2/22/17 29 17. Invariant bilinear forms on Lie algebras: 2/24/17 31 18. Classical Lie groups and Lie algebras: 2/27/17 32 19. Roots and root spaces: 3/1/17 34 20. Properties of roots: 3/3/17 36 21. Root systems: 3/6/17 37 22. Dynkin diagrams: 3/8/17 39 23.
    [Show full text]
  • Perspectives on Geometric Analysis
    Surveys in Differential Geometry X Perspectives on geometric analysis Shing-Tung Yau This essay grew from a talk I gave on the occasion of the seventieth anniversary of the Chinese Mathematical Society. I dedicate the lecture to the memory of my teacher S.S. Chern who had passed away half a year before (December 2004). During my graduate studies, I was rather free in picking research topics. I[731] worked on fundamental groups of manifolds with non-positive curva- ture. But in the second year of my studies, I started to look into differential equations on manifolds. However, at that time, Chern was very much inter- ested in the work of Bott on holomorphic vector fields. Also he told me that I should work on Riemann hypothesis. (Weil had told him that it was time for the hypothesis to be settled.) While Chern did not express his opinions about my research on geometric analysis, he started to appreciate it a few years later. In fact, after Chern gave a course on Calabi’s works on affine geometry in 1972 at Berkeley, S.Y. Cheng told me about these inspiring lec- tures. By 1973, Cheng and I started to work on some problems mentioned in Chern’s lectures. We did not realize that the great geometers Pogorelov, Calabi and Nirenberg were also working on them. We were excited that we solved some of the conjectures of Calabi on improper affine spheres. But soon after we found out that Pogorelov [563] published his results right be- fore us by different arguments. Nevertheless our ideas are useful in handling other problems in affine geometry, and my knowledge about Monge-Amp`ere equations started to broaden in these years.
    [Show full text]
  • Orbits of Real Forms in Complex Flag Manifolds 71
    Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) Vol. IX (2010), 69-109 Orbits of real forms in complex flag manifolds ANDREA ALTOMANI,COSTANTINO MEDORI AND MAURO NACINOVICH Abstract. We investigate the CR geometry of the orbits M of a real form G0 of a complex semisimple Lie group G in a complex flag manifold X = G/Q.Weare mainly concerned with finite type and holomorphic nondegeneracy conditions, canonical G0-equivariant and Mostow fibrations, and topological properties of the orbits. Mathematics Subject Classification (2010): 53C30 (primary); 14M15, 17B20, 32V05, 32V35, 32V40, 57T20 (secondary). Introduction In this paper we study a large class of homogeneous CR manifolds, that come up as orbits of real forms in complex flag manifolds. These objects, that we call here parabolic CR manifolds, were first considered by J. A. Wolf (see [28] and also [12] for a comprehensive introduction to this topic). He studied the action of a real form G0 of a complex semisimple Lie group G on a flag manifold X of G.Heshowed that G0 has finitely many orbits in X,sothat the union of the open orbits is dense in X, and that there is just one that is closed. He systematically investigated their properties, especially for the open orbits and for the holomorphic arc components of general orbits, outlining a framework that includes, as special cases, the bounded symmetric domains. We aim here to give a contribution to the study of the general G0-orbits in X, by considering and utilizing their natural CR structure. In [2] we began this program by investigating the closed orbits.
    [Show full text]
  • Curvature-Free Margulis Lemma for Gromov-Hyperbolic Spaces
    Curvature-Free Margulis Lemma for Gromov-Hyperbolic Spaces G. Besson, G. Courtois, S. Gallot, A. Sambusetti December 2, 2020 Abstract We prove curvature-free versions of the celebrated Margulis Lemma. We are interested by both the algebraic aspects and the geometric ones, with however an emphasis on the second and we aim at giving quantitative (computable) estimates of some important invariants. Our goal is to get rid of the pointwise curvature assumptions in order to extend the results to more general spaces such as certain metric spaces. Essentially the upper bound on the curvature is replaced by the assumption that the space is hyperbolic in the sense of Gromov and the lower bound of the curvature by an upper bound on the entropy of which we recall the definition. Contents 1 Introduction 2 2 Basic definitions and notations8 3 Entropy, Doubling and Packing Properties 10 3.1 Entropies . 10 3.2 Doubling and Packing Properties . 13 3.3 Comparison between the various possible hypotheses : bound on the entropy, doubling and packing conditions, Ricci curvature . 14 3.4 Doubling property induced on subgroups . 20 4 Free Subgroups 21 4.1 Ping-pong Lemma . 21 arXiv:1712.08386v3 [math.DG] 1 Dec 2020 4.2 When the asymptotic displacement is bounded below . 24 4.3 When some Margulis constant is bounded below: . 27 4.3.1 When the Margulis constant L∗ is bounded below: . 28 4.3.2 When the Margulis constant L is bounded below: . 32 4.4 Free semi-groups for convex distances . 35 5 Co-compact actions on Gromov-hyperbolic spaces 36 5.1 A Bishop-Gromov inequality for Gromov-hyperbolic spaces .
    [Show full text]
  • Limit Sets of Discrete Groups of Isometries of Exotic Hyperbolic Spaces
    TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 351, Number 4, April 1999, Pages 1507{1530 S 0002-9947(99)02113-3 LIMIT SETS OF DISCRETE GROUPS OF ISOMETRIES OF EXOTIC HYPERBOLIC SPACES KEVIN CORLETTE AND ALESSANDRA IOZZI Abstract. Let Γ be a geometrically finite discrete group of isometries of hy- perbolic space n,where = ; ; or (in which case n =2).Weprove F F RCH O that the criticalH exponent of Γ equals the Hausdorff dimension of the limit sets Λ(Γ) and that the smallest eigenvalue of the Laplacian acting on square integrable functions is a quadratic function of either of them (when they are sufficiently large). A generalization of Hopf ergodicity theorem for the geodesic flow with respect to the Bowen-Margulis measure is also proven. 1. Introduction In a previous paper [C], the first author studied limit sets of discrete groups of isometries of rank one symmetric spaces of noncompact type. These symmet- ric spaces are the hyperbolic spaces associated with the reals, complexes, quater- nions, or Cayley numbers. The main point of interest in that paper was the Haus- dorff dimension (in an appropriate sense) of the limit set and its relationships with other invariants, such as the smallest eigenvalue of the Laplacian acting on square- integrable functions and the exponent of growth of the group. In the cases where the group of all isometries of the symmetric space has Kazhdan’s property, this led to conclusions about limitations on the possible exponents of growth, among other things. The most precise results were obtained for a class of groups which, in that paper, were called geometrically cocompact; these groups are also known as convex cocompact groups, or geometrically finite groups without cusps.
    [Show full text]
  • Math 222 Notes for Apr. 1
    Math 222 notes for Apr. 1 Alison Miller 1 Real lie algebras and compact Lie groups, continued Last time we showed that if G is a compact Lie group with Z(G) finite and g = Lie(G), then the Killing form of g is negative definite, hence g is semisimple. Now we’ll do a converse. Proposition 1.1. Let g be a real lie algebra such that Bg is negative definite. Then there is a compact Lie group G with Lie algebra Lie(G) =∼ g. Proof. Pick any connected Lie group G0 with Lie algebra g. Since g is semisimple, the center Z(g) = 0, and so the center Z(G0) of G0 must be discrete in G0. Let G = G0/Z(G)0; then Lie(G) =∼ Lie(G0) =∼ g. We’ll show that G is compact. Now G0 has an adjoint action Ad : G0 GL(g), with kernel ker Ad = Z(G). Hence 0 ∼ G = Im(Ad). However, the Killing form Bg on g is Ad-invariant, so Im(Ad) is contained in the subgroup of GL(g) preserving the Killing! form. Since Bg is negative definite, this subgroup is isomorphic to O(n) (where n = dim(g). Since O(n) is compact, its closed subgroup Im(Ad) is also compact, and so G is compact. Note: I found a gap in this argument while writing it up; it assumes that Im(Ad) is closed in GL(g). This is in fact true, because Im(Ad) = Aut(g) is the group of automorphisms of the Lie algebra g, which is closed in GL(g); but proving this fact takes a bit of work.
    [Show full text]
  • Lie Algebras from Lie Groups
    Preprint typeset in JHEP style - HYPER VERSION Lecture 8: Lie Algebras from Lie Groups Gregory W. Moore Abstract: Not updated since November 2009. March 27, 2018 -TOC- Contents 1. Introduction 2 2. Geometrical approach to the Lie algebra associated to a Lie group 2 2.1 Lie's approach 2 2.2 Left-invariant vector fields and the Lie algebra 4 2.2.1 Review of some definitions from differential geometry 4 2.2.2 The geometrical definition of a Lie algebra 5 3. The exponential map 8 4. Baker-Campbell-Hausdorff formula 11 4.1 Statement and derivation 11 4.2 Two Important Special Cases 17 4.2.1 The Heisenberg algebra 17 4.2.2 All orders in B, first order in A 18 4.3 Region of convergence 19 5. Abstract Lie Algebras 19 5.1 Basic Definitions 19 5.2 Examples: Lie algebras of dimensions 1; 2; 3 23 5.3 Structure constants 25 5.4 Representations of Lie algebras and Ado's Theorem 26 6. Lie's theorem 28 7. Lie Algebras for the Classical Groups 34 7.1 A useful identity 35 7.2 GL(n; k) and SL(n; k) 35 7.3 O(n; k) 38 7.4 More general orthogonal groups 38 7.4.1 Lie algebra of SO∗(2n) 39 7.5 U(n) 39 7.5.1 U(p; q) 42 7.5.2 Lie algebra of SU ∗(2n) 42 7.6 Sp(2n) 42 8. Central extensions of Lie algebras and Lie algebra cohomology 46 8.1 Example: The Heisenberg Lie algebra and the Lie group associated to a symplectic vector space 47 8.2 Lie algebra cohomology 48 { 1 { 9.
    [Show full text]
  • LIE GROUPOIDS and LIE ALGEBROIDS LECTURE NOTES, FALL 2017 Contents 1. Lie Groupoids 4 1.1. Definitions 4 1.2. Examples 6 1.3. Ex
    LIE GROUPOIDS AND LIE ALGEBROIDS LECTURE NOTES, FALL 2017 ECKHARD MEINRENKEN Abstract. These notes are under construction. They contain errors and omissions, and the references are very incomplete. Apologies! Contents 1. Lie groupoids 4 1.1. Definitions 4 1.2. Examples 6 1.3. Exercises 9 2. Foliation groupoids 10 2.1. Definition, examples 10 2.2. Monodromy and holonomy 12 2.3. The monodromy and holonomy groupoids 12 2.4. Appendix: Haefliger’s approach 14 3. Properties of Lie groupoids 14 3.1. Orbits and isotropy groups 14 3.2. Bisections 15 3.3. Local bisections 17 3.4. Transitive Lie groupoids 18 4. More constructions with groupoids 19 4.1. Vector bundles in terms of scalar multiplication 20 4.2. Relations 20 4.3. Groupoid structures as relations 22 4.4. Tangent groupoid, cotangent groupoid 22 4.5. Prolongations of groupoids 24 4.6. Pull-backs and restrictions of groupoids 24 4.7. A result on subgroupoids 25 4.8. Clean intersection of submanifolds and maps 26 4.9. Intersections of Lie subgroupoids, fiber products 28 4.10. The universal covering groupoid 29 5. Groupoid actions, groupoid representations 30 5.1. Actions of Lie groupoids 30 5.2. Principal actions 31 5.3. Representations of Lie groupoids 33 6. Lie algebroids 33 1 2 ECKHARD MEINRENKEN 6.1. Definitions 33 6.2. Examples 34 6.3. Lie subalgebroids 36 6.4. Intersections of Lie subalgebroids 37 6.5. Direct products of Lie algebroids 38 7. Morphisms of Lie algebroids 38 7.1. Definition of morphisms 38 7.2.
    [Show full text]