Lectures on Lie Groups and Lie Algebras (London Mathematical Society Student Texts, #32)

In This Excellent Introduction To The Theory Of Lie Groups And Lie Algebras, Three Of The Leading Figures In This Area Have Written Up Their Lectures From An Lms/serc Sponsored Short Course In 1993. Together These Lectures Provide An Elementary Account Of The Theory That Is Unsurpassed. In The First Part Roger Carter Concentrates On Lie Algebras And Root Systems. In The Second Graeme Segal Discusses Lie Groups. And In The Final Part, Ian Macdonald Gives An Introduction To Special Linear Groups. Anybody Requiring An Introduction To The Theory Of Lie Groups And Their Applications Should Look No Further Than This Book. Roger Carter, Graeme Segal, Ian Macdonald. Based On The Three Introductory Lecture Courses Given At The Lms-serc Instructional Conference On Lie Theory And Algebraic Groups Held At Lancaster University In September 1993--fwd. Includes Bibliographical References (p.187 - 188) And Index. Cover......Page 1 List of Series Publications......Page 3 Title: Lectures on Lie Groups and Lie Algebras......Page 4 ISBN 0 521 49922 4......Page 5 Contents......Page 6 Foreword......Page 8 Lie Algebras and Root Systems by R.W. Carter......Page 10 Contents: Lie Algebras and Root Systems......Page 11 Preface......Page 12 1.1 Basic concepts......Page 14 1.2 Representations and modules......Page 16 1.3 Special kinds of Lie algebra......Page 17 1.4 The Lie algebras sln(C)......Page 19 2.1 Cartan subalgebras......Page 21 2.2 The Cartan decomposition......Page 22 2.3 The Killing fom......Page 24 2.4 The Weyl group......Page 25 2.5 The Dynkin diagram......Page 27 3.1 The universal enveloping algebra......Page 34 3.2 Verma modules......Page 35 3.3 Finite dimensional irreducible modules......Page 36 3.4 Weyl's character and dimension formulae......Page 38 3.5 Fundamental representations......Page 41 4.1 A Chevalley basis of g......Page 45 4.2 Chevalley groups over an arbitrary field......Page 47 4.3 Finite Chevalley groups......Page 48 4.4 Twisted groups......Page 50 4.5 Suzuki and Ree groups......Page 52 4.6 Classification of finite simple groups......Page 53 Lie Groups by Graeme Segal......Page 54 Contents: Lie Groups......Page 55 Introduction......Page 56 1 Examples......Page 58 Matrix groups......Page 59 Low dimensional examples......Page 60 Local isomorphism......Page 61 2 SU2, S03, and SL2R......Page 62 A picture of SL2R.......Page 65 3 Homogeneous spaces......Page 68 Symmetric spaces......Page 69 Complex structures on R^2n......Page 70 B The Gram-Schmidt process......Page 72 C Reduced echelon form: the Bruhat decomposition......Page 73 D Diagonalization and maximal tori......Page 76 Smooth manifolds......Page 78 Tangent spaces......Page 81 One-parameter subgroups and the exponential map......Page 82 Lie's theorems......Page 84 6 Fourier series and representation theory......Page 91 General remarks about representations......Page 93 7 Compact groups and integration......Page 94 A formula for integration on U,.......Page 95 8 Maximal compact subgroups......Page 98 9 The Peter-Weyl theorem......Page 100 The structure of Calg( G)......Page 104 10 Functions on R^n and S^(n-1)......Page 109 The Radon transform......Page 112 11 Induced representations......Page 113 12 The complexification of a compact group......Page 117 Weyl's correspondence......Page 119 Quantum groups......Page 122 14 The Borel-Weil theorem......Page 124 15 Representations of non-compact groups......Page 129 16 Representations of S L2R......Page 133 17 The Heisenberg group, the metaplectic representation, and the spin representation......Page 137 The spin representation......Page 141 Linear Algebraic Groups by I. G. Macdonald......Page 142 Contents: Linear Algebraic Groups......Page 143 Preface......Page 144 Introduction......Page 146 1 Affine algebraic varieties......Page 148 Products......Page 152 The image of a morphism......Page 153 Dimension......Page 154 Examples......Page 155 Jordan decomposition......Page 160 Interlude......Page 163 3 Projective algebraic varieties......Page 166 Prevarieties and varieties......Page 167 Projective Varieties......Page 168 Complete varieties......Page 169 4 Tangent spaces. Separability......Page 171 Separability......Page 173 5 The Lie algebra of a linear algebraic group......Page 175 The adjoint representation......Page 179 6 Homogeneous spaces and quotients......Page 181 7 Borel subgroups and maximal tori......Page 186 Borel subgroups......Page 187 Maximal tori......Page 188 Characters and one-parameter subgroups of tori......Page 191 The root datum B(G, T)......Page 192 Notes and references......Page 195 Bibliography......Page 196 Index......Page 198 Cover; Series Page; Title; Copyright; Contents; Foreword; Lie Algebras and Root Systems R.W. Carter; Preface; 1 Introduction to Lie algebras; 1.1 Basic concepts; 1.2 Representations and modules; 1.3 Special kinds of Lie algebra; 1.4 The Lie algebras sln(C); 2 Simple Lie algebras over C; 2.1 Cartan subalgebras; 2.2 The Cartan decomposition; 2.3 The Killing form; 2.4 The Weyl group; 2.5 The Dynkin diagram; 3 Representations of simple Lie algebras; 3.1 The universal enveloping algebra; 3.2 Verma modules; 3.3 Finite dimensional irreducible modules; 3.4 Weyl's character and dimension formulae 3.5 Fundamental representations4 Simple groups of Lie type; 4.1 A Chevalley basis of g; 4.2 Chevalley groups over an arbitrary field; 4.3 Finite Chevalley groups; 4.4 Twisted groups; 4.5 Suzuki and Ree groups; 4.6 Classification of finite simple groups; Lie Groups Graeme Segal; Introduction; 1 Examples; Matrix groups; Low dimensional examples; Local isomorphism; 2 SU2, S03, and SL2R; 3 Homogeneous spaces; Symmetric spaces; Complex structures on R2n; 4 Some theorems about matrices; A The polar decomposition; B The Gram-Schmidt process; C Reduced echelon form: the Bruhat decomposition 14 The Borel-Weil theorem15 Representations of non-compact groups; 16 Representations of S L2R; 17 The Heisenberg group the metaplectic representation, and the spin representation; The spin representation; Linear Algebraic Groups I.G. Macdonald; Preface; Introduction; 1 Affine algebraic varieties; Morphisms; Products; The image of a morphism; Dimension; 2 Linear algebraic groups: definition and elementary properties; Jordan decomposition; Interlude; 3 Projective algebraic varieties; Prevarieties and varieties; Projective Varieties; Complete varieties; 4 Tangent spaces. Separability D Diagonalization and maximal tori5 Lie theory; Smooth manifolds; Tangent spaces; One-parameter subgroups and the exponential map; Lie's theorems; 6 Fourier series and representation theory; General remarks about representations; 7 Compact groups and integration; A formula for integration on Un; 8 Maximal compact subgroups; 9 The Peter-Weyl theorem; The structure of Calg(G); 10 Functions on Rn and sn-l; The Radon transform; 11 Induced representations; 12 The complexification of a compact group; 13 The unitary groups and the symmetric groups; Weyl's correspondence; Quantum groups Three of the leading figures in the field have composed this excellent introduction to the theory of Lie groups and Lie algebras. Together these lectures provide an elementary account of the theory that is unsurpassed. In the first part, Roger Carter concentrates on Lie algebras and root systems. In the second Graeme Segal discusses Lie groups. And in the final part, Ian Macdonald gives an introduction to special linear groups. Graduate students requiring an introduction to the theory of Lie groups and their applications should look no further than this book. Separability5 The Lie algebra of a linear algebraic group; The adjoint representation; 6 Homogeneous spaces and quotients; 7 Borel subgroups and maximal tori; Borel subgroups; Maximal tori; 8 The root structure of a linear algebraic group; Characters and one-parameter subgroups of tori; The root system R(G, T); The root datum R(G, T); Notes and references; Bibliography; Index