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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. -
Introduction to Lie Theory Homework #6 1. Let G = G 1 ⊕ G2 Be a Direct
Introduction to Lie Theory Homework #6 1. Let g = g1 ⊕ g2 be a direct sum of two Lie algebras. Let V be a finite-dimensional g-module. Prove that V is completely reducible as a g-module if and only it is completely reducible both as a g1-module and as a g2-module. (Hint. Consider the finite-dimensional algebra A that is the image of U(g) under the induced algebra homomorphism ρ : U(g) ! EndC(V ), use Wedderburn theorem.) 2. Let g be a finite-dimensional semisimple Lie algebra with Killing form κ. We showed in L6-2 that g is the direct sum g1 ⊕· · ·⊕gn of its simple ideals, with gi ?κ gj for i 6= j. Let β be some other invariant bilinear form on g. Prove: (a) βjgi is a multiple of the Killing form κjgi for each i = 1; : : : ; n. (b) gi ?β gi for i 6= j. (c) β is a symmetric bilinear form. The remaining questions require the notion of a toral subalgebra t of a finite- dimensional semisimple Lie algebra g. Recall that this is a subalgebra con- sisting entirely of semisimple elements of g. As established in L7-1, such a subalgebra is necessarily Abelian. Hence, there is a root space decomposition M g = g0 ⊕ gα α2R ∗ where R = f0 6= α 2 t j gα 6= 0g is the set of roots of g with respect to t. Here, gα = fx 2 t j [t; x] = α(t)x for all t 2 tg for any α 2 t∗. -
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 ). -
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. -
Springer Undergraduate Mathematics Series Advisory Board
Springer Undergraduate Mathematics Series Advisory Board M.A.J. Chaplain University of Dundee K. Erdmann Oxford University A.MacIntyre Queen Mary, University of London L.C.G. Rogers University of Cambridge E. Süli Oxford University J.F. Toland University of Bath Other books in this series A First Course in Discrete Mathematics I. Anderson Analytic Methods for Partial Differential Equations G. Evans, J. Blackledge, P. Yardley Applied Geometry for Computer Graphics and CAD, Second Edition D. Marsh Basic Linear Algebra, Second Edition T.S. Blyth and E.F. Robertson Basic Stochastic Processes Z. Brze´zniak and T. Zastawniak Calculus of One Variable K.E. Hirst Complex Analysis J.M. Howie Elementary Differential Geometry A. Pressley Elementary Number Theory G.A. Jones and J.M. Jones Elements of Abstract Analysis M. Ó Searcóid Elements of Logic via Numbers and Sets D.L. Johnson Essential Mathematical Biology N.F. Britton Essential Topology M.D. Crossley Fields and Galois Theory J.M. Howie Fields, Flows and Waves: An Introduction to Continuum Models D.F. Parker Further Linear Algebra T.S. Blyth and E.F. Robertson Geometry R. Fenn Groups, Rings and Fields D.A.R. Wallace Hyperbolic Geometry, Second Edition J.W. Anderson Information and Coding Theory G.A. Jones and J.M. Jones Introduction to Laplace Transforms and Fourier Series P.P.G. Dyke Introduction to Lie Algebras K. Erdmann and M.J. Wildon Introduction to Ring Theory P.M. Cohn Introductory Mathematics: Algebra and Analysis G. Smith Linear Functional Analysis B.P. Rynne and M.A. Youngson Mathematics for Finance: An Introduction to Financial Engineering M. -
Contents 1 Root Systems
Stefan Dawydiak February 19, 2021 Marginalia about roots These notes are an attempt to maintain a overview collection of facts about and relationships between some situations in which root systems and root data appear. They also serve to track some common identifications and choices. The references include some helpful lecture notes with more examples. The author of these notes learned this material from courses taught by Zinovy Reichstein, Joel Kam- nitzer, James Arthur, and Florian Herzig, as well as many student talks, and lecture notes by Ivan Loseu. These notes are simply collected marginalia for those references. Any errors introduced, especially of viewpoint, are the author's own. The author of these notes would be grateful for their communication to [email protected]. Contents 1 Root systems 1 1.1 Root space decomposition . .2 1.2 Roots, coroots, and reflections . .3 1.2.1 Abstract root systems . .7 1.2.2 Coroots, fundamental weights and Cartan matrices . .7 1.2.3 Roots vs weights . .9 1.2.4 Roots at the group level . .9 1.3 The Weyl group . 10 1.3.1 Weyl Chambers . 11 1.3.2 The Weyl group as a subquotient for compact Lie groups . 13 1.3.3 The Weyl group as a subquotient for noncompact Lie groups . 13 2 Root data 16 2.1 Root data . 16 2.2 The Langlands dual group . 17 2.3 The flag variety . 18 2.3.1 Bruhat decomposition revisited . 18 2.3.2 Schubert cells . 19 3 Adelic groups 20 3.1 Weyl sets . 20 References 21 1 Root systems The following examples are taken mostly from [8] where they are stated without most of the calculations. -
Lie Group, Lie Algebra, Exponential Mapping, Linearization Killing Form, Cartan's Criteria
American Journal of Computational and Applied Mathematics 2016, 6(5): 182-186 DOI: 10.5923/j.ajcam.20160605.02 On Extracting Properties of Lie Groups from Their Lie Algebras M-Alamin A. H. Ahmed1,2 1Department of Mathematics, Faculty of Science and Arts – Khulais, University of Jeddah, Saudi Arabia 2Department of Mathematics, Faculty of Education, Alzaeim Alazhari University (AAU), Khartoum, Sudan Abstract The study of Lie groups and Lie algebras is very useful, for its wider applications in various scientific fields. In this paper, we introduce a thorough study of properties of Lie groups via their lie algebras, this is because by using linearization of a Lie group or other methods, we can obtain its Lie algebra, and using the exponential mapping again, we can convey properties and operations from the Lie algebra to its Lie group. These relations helpin extracting most of the properties of Lie groups by using their Lie algebras. We explain this extraction of properties, which helps in introducing more properties of Lie groups and leads to some important results and useful applications. Also we gave some properties due to Cartan’s first and second criteria. Keywords Lie group, Lie algebra, Exponential mapping, Linearization Killing form, Cartan’s criteria 1. Introduction 2. Lie Group The Properties and applications of Lie groups and their 2.1. Definition (Lie group) Lie algebras are too great to overview in a small paper. Lie groups were greatly studied by Marius Sophus Lie (1842 – A Lie group G is an abstract group and a smooth 1899), who intended to solve or at least to simplify ordinary n-dimensional manifold so that multiplication differential equations, and since then the research continues G×→ G G:,( a b) → ab and inverse G→→ Ga: a−1 to disclose their properties and applications in various are smooth. -
Introduction to Lie Algebras and Representation Theory (Following Humphreys)
INTRODUCTION TO LIE ALGEBRAS AND REPRESENTATION THEORY (FOLLOWING HUMPHREYS) Ulrich Thiel Abstract. These are my (personal and incomplete!) notes on the excellent book Introduction to Lie Algebras and Representation Theory by Jim Humphreys [1]. Almost everything is basically copied word for word from the book, so please do not share this document. What is the point of writing these notes then? Well, first of all, they are my personal notes and I can write whatever I want. On the more serious side, despite what I just said, I actually tried to spell out most things in my own words, and when I felt I need to give more details (sometimes I really felt it is necessary), I have added more details. My actual motivation is that eventually I want to see what I can shuffle around and add/remove to reflect my personal thoughts on the subject and teach it more effectively in the future. To be honest, I do not want you to read these notes at all. The book is excellent already, and when you come across something you do not understand, it is much better to struggle with it and try to figure it out by yourself, than coming here and see what I have to say about it (if I have to say anything additionally). Also, these notes are not complete yet, will contain mistakes, and I will not keep it up to date with the current course schedule. It is just my notebook for the future. Document compiled on June 6, 2020 Ulrich Thiel, University of Kaiserslautern. -
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. -
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. -
Killing Forms, W-Invariants, and the Tensor Product Map
Killing Forms, W-Invariants, and the Tensor Product Map Cameron Ruether Thesis submitted to the Faculty of Graduate and Postdoctoral Studies in partial fulfillment of the requirements for the degree of Master of Science in Mathematics1 Department of Mathematics and Statistics Faculty of Science University of Ottawa c Cameron Ruether, Ottawa, Canada, 2017 1The M.Sc. program is a joint program with Carleton University, administered by the Ottawa- Carleton Institute of Mathematics and Statistics Abstract Associated to a split, semisimple linear algebraic group G is a group of invariant quadratic forms, which we denote Q(G). Namely, Q(G) is the group of quadratic forms in characters of a maximal torus which are fixed with respect to the action of the Weyl group of G. We compute Q(G) for various examples of products of the special linear, special orthogonal, and symplectic groups as well as for quotients of those examples by central subgroups. Homomorphisms between these linear algebraic groups induce homomorphisms between their groups of invariant quadratic forms. Since the linear algebraic groups are semisimple, Q(G) is isomorphic to Zn for some n, and so the induced maps can be described by a set of integers called Rost multipliers. We consider various cases of the Kronecker tensor product map between copies of the special linear, special orthogonal, and symplectic groups. We compute the Rost multipliers of the induced map in these examples, ultimately concluding that the Rost multipliers depend only on the dimensions of the underlying vector spaces. ii Dedications To root systems - You made it all worth Weyl. -
Chapter 18 Metrics, Connections, and Curvature on Lie Groups
Chapter 18 Metrics, Connections, and Curvature on Lie Groups 18.1 Left (resp. Right) Invariant Metrics Since a Lie group G is a smooth manifold, we can endow G with a Riemannian metric. Among all the Riemannian metrics on a Lie groups, those for which the left translations (or the right translations) are isometries are of particular interest because they take the group structure of G into account. This chapter makes extensive use of results from a beau- tiful paper of Milnor [38]. 831 832 CHAPTER 18. METRICS, CONNECTIONS, AND CURVATURE ON LIE GROUPS Definition 18.1. Ametric , on a Lie group G is called left-invariant (resp. right-invarianth i )i↵ u, v = (dL ) u, (dL ) v h ib h a b a b iab (resp. u, v = (dR ) u, (dR ) v ), h ib h a b a b iba for all a, b G and all u, v T G. 2 2 b ARiemannianmetricthatisbothleftandright-invariant is called a bi-invariant metric. In the sequel, the identity element of the Lie group, G, will be denoted by e or 1. 18.1. LEFT (RESP. RIGHT) INVARIANT METRICS 833 Proposition 18.1. There is a bijective correspondence between left-invariant (resp. right invariant) metrics on a Lie group G, and inner products on the Lie al- gebra g of G. If , be an inner product on g,andset h i u, v g = (dL 1)gu, (dL 1)gv , h i h g− g− i for all u, v TgG and all g G.Itisfairlyeasytocheck that the above2 induces a left-invariant2 metric on G.