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Brian C. Hall Lie Groups, Lie Algebras, and Representations An Elementary Introduction With 31 Illustrations Springer Contents Part I General Theory Matrix Lie Groups 1.1 Definition of a Matrix Lie Group 1.1.1 Counterexamples 1.2 Examples of Matrix Lie Groups 1.2.1 The general linear groups GL(n; R) and GL(n; C) 1.2.2 The special linear groups SL(n; R) and SL(n; C) 1.2.3 The orthogonal and special orthogonal groups, 0(n) and SO(rc) 1.2.4 The unitary and special unitary groups, U(n) and SU(n) 1.2.5 The complex orthogonal groups, O(n;C) and SO(n;C) . 1.2.6 The generalized orthogonal and Lorentz groups 1.2.7 The symplectic groups Sp(n;R), Sp(n;C), and Sp(n) . . . 1.2.8 The Heisenberg group H 1.2.9 The groups R*, C*, S1, R, and R n 1.2.10 The Euclidean and Poincare groups E(ra) and P(n; 1)... . . 1.3 Compactness 1.3.1 Examples of compact groups 1.3.2 Examples of noncompact groups 1.4 Connectedness 1.5 Simple Connectedness 1.6 Homomorphisms and Isomorphisms 1.6.1 Example: SU(2) and SO(3) 1.7 The Polar Decomposition for SL(n; R) and SL(n; C) 1.8 Lie Groups 1.9 Exercises 3 3 4 4 4 5 5 6 6 7 7 8 9 9 11 11 11 12 15 17 18 19 20 23 Lie Algebras and the Exponential Mapping 2.1 The Matrix Exponential 2.2 Computing the Exponential of a Matrix 27 27 30 Contents 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11 XI 4.3.7 The unitary representations of the Heisenberg group . . . 100 The Irreducible Representations of su(2) 101 Direct Sums of Representations 106 Tensor Products of Representations 107 Dual Representations 112 Schur's Lemma 113 Group Versus Lie Algebra Representations 115* Complete Reducibility 118 Exercises 121 Part II Semisimple Theory 5 The Representations of SU(3) 5.1 Introduction 5.2 Weights and Roots 5.3 The Theorem of the Highest Weight 5.4 Proof of the Theorem 5.5 An Example: Highest Weight (1,1) 5.6 The Weyl Group 5.7 Weight Diagrams 5.8 Exercises 6 Semisimple Lie Algebras 6.1 Complete Reducibility and Semisimple Lie Algebras 6.2 Examples of Reductive and Semisimple Lie Algebras 6.3 Cartan Subalgebras 6.4 Roots and Root Spaces 6.5 Inner Products of Roots and Co-roots 6.6 The Weyl Group 6.7 Root Systems 6.8 Positive Roots 6.9 The sl(n; C) Case 6.9.1 The Cartan subalgebra 6.9.2 The roots 6.9.3 Inner products of roots 6.9.4 The Weyl group 6.9.5 Positive roots 6.10 Uniqueness Results 6.11 Exercises 7 Representations of Complex Semisimple Lie Algebras 7.1 Integral and Dominant Integral Elements 7.2 The Theorem of the Highest Weight 7.3 Constructing the Representations I: Verma Modules 127 127 129 132 135 140 142 149 152 155 156 161 162 164 170 ..173 . . 180 181 182 182 182 183 184 184 184 185 191 192 194 200 XII Contents 7.4 7.5 7.6 7.7 8 7.3.1 Verma modules 200 7.3.2 Irreducible quotient modules 202 7.3.3 Finite-dimensional quotient modules 204 7.3.4 The sl(2; C) case 208 Constructing the Representations II: The Peter-Weyl Theorem 209 7.4.1 The Peter-Weyl theorem 210 7.4.2 The Weyl character formula 211 7.4.3 Constructing the representations 213 7.4.4 Analytically integral versus algebraically integral elements 215 7.4.5 The SU(2) case 216 Constructing the Representations III: The Borel-Weil Construction 218 7.5.1 The complex-group approach 218 7.5.2 The setup 220 7.5.3 The strategy 222 7.5.4 The construction 225 7.5.5 The SL(2; C) case 229 Further Results 230 7.6.1 Duality 230 7.6.2 The weights and their multiplicities 232 7.6.3 The Weyl character formula and the Weyl dimension formula 234 7.6.4 The analytical proof of the Weyl character formula . . . . 236 Exercises 240 More on Roots and Weights 8.1 Abstract Root Systems 8.2 Duality 8.3 Bases and Weyl Chambers 8.4 Integral and Dominant Integral Elements 8.5 Examples in Rank Two 8.5.1 The root systems 8.5.2 Connection with Lie algebras 8.5.3 The Weyl groups 8.5.4 Duality .• 8.5.5 Positive roots and dominant integral elements 8.5.6 Weight diagrams 8.6 Examples in Rank Three 8.7 Additional Properties 8.8 The Root Systems of the Classical Lie Algebras 8.8.1 The orthogonal algebras so(2n; C) 8.8.2 The orthogonal algebras so(2n + 1; C) 8.8.3 The symplectic algebras sp(n; C) 8.9 Dynkin Diagrams and the Classification 243 243 248 249 254 255 255 257 257 258 258 259 262 263 265 265 266 268 269 Contents A XIII 8.10 The Root Lattice and the Weight Lattice 8.11 Exercises 273 276 A Quick Introduction to Groups A.I Definition of a Group and Basic Properties A.2 Examples of Groups A.2.1 The trivial group A.2.2 The integers A.2.3 The reals and Rn A.2.4 Nonzero real numbers under multiplication A.2.5 Nonzero complex numbers under multiplication A.2.6 Complex numbers of absolute value 1 under multiplication A.2.7 The general linear groups A.2.8 Permutation group (symmetric group) A.2.9 Integers mod n A.3 Subgroups, the Center, and Direct Products A.4 Homomorphisms and Isomorphisms A.5 Quotient Groups A.6 Exercises 279 279 281 282 282 282 282 282 B Linear Algebra Review B.I Eigenvectors, Eigenvalues, and the Characteristic Polynomial B.2 Diagonalization B.3 Generalized Eigenvectors and the SN Decomposition B.4 The Jordan Canonical Form B.5 The Trace B.6 Inner Products B.7 Dual Spaces B.8 Simultaneous Diagonalization C More on Lie Groups C.I Manifolds C.I.I Definition C.I.2 Tangent space C.1.3 Differentials of smooth mappings C.1.4 Vector fields C.I.5 The flow along a vector C.I.6 Submanifolds of vector spaces C.1.7 Complex manifolds C.2 Lie Groups C.2.1 Definition C.2.2 The Lie algebra C.2.3 The exponential mapping C.2.4 Homomorphisms field • 283 283 283 283 284 285 286 289 291 . 291 293 294 296 296 297 299 299 303 303 303 304 305 306 307 308 309 309 309 310 311 311 XIV Contents C.2.5 Quotient groups and covering groups C.2.6 Matrix Lie groups as Lie groups C.2.7 Complex Lie groups C.3 Examples of Nonmatrix Lie Groups C.4 Differential Forms and Haar Measure D E 312 313 313 314 318 Clebsch-Gordan Theory for SU(2) and the Wigner-Eckart Theorem 321 D.I Tensor Products of sl(2; C) Representations 321 D.2 The Wigner-Eckart Theorem 324 D.3 More on Vector Operators 328 C o m p u t i n g F u n d a m e n t a l G r o u p s o f M a t r i x L i e G r o u p s . . . . 331 E.I E.2 E.3 E.4 E.5 The Fundamental Group The Universal Cover Fundamental Groups of Compact Lie Groups I Fundamental Groups of Compact Lie Groups II Fundamental Groups of Noncompact Lie Groups 331 332 333 336 342 References 345 Index 347