## Lie Groups, Lie Algebras, and Representations: An Elementary IntroductionThis book provides an introduction to Lie groups, Lie algebras, and repre sentation theory, aimed at graduate students in mathematics and physics. Although there are already several excellent books that cover many of the same topics, this book has two distinctive features that I hope will make it a useful addition to the literature. First, it treats Lie groups (not just Lie alge bras) in a way that minimizes the amount of manifold theory needed. Thus, I neither assume a prior course on differentiable manifolds nor provide a con densed such course in the beginning chapters. Second, this book provides a gentle introduction to the machinery of semi simple groups and Lie algebras by treating the representation theory of SU(2) and SU(3) in detail before going to the general case. This allows the reader to see roots, weights, and the Weyl group "in action" in simple cases before confronting the general theory. The standard books on Lie theory begin immediately with the general case: a smooth manifold that is also a group. The Lie algebra is then defined as the space of left-invariant vector fields and the exponential mapping is defined in terms of the flow along such vector fields. This approach is undoubtedly the right one in the long run, but it is rather abstract for a reader encountering such things for the first time. |

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### Contents

Matrix Lie Groups | 5 |

111 Counterexamples | 6 |

122 The special linear groups SLnℝ and SLn ℂ | 7 |

124 The unitary and special unitary groups Un and SUn | 8 |

126 The generalized orthogonal and Lorentz groups | 9 |

128 The Heisenberg group H | 10 |

129 The groups ℝ ℂ Są ℝ and ℝn | 11 |

13 Compactness | 13 |

Representations of Complex Semisimple Lie Algebras | 193 |

71 Integral and Dominant Integral Elements | 194 |

72 The Theorem of the Highest Weight | 196 |

Verma Modules | 202 |

733 Finitedimensional quotient modules | 206 |

734 The sl2 ℂ case | 210 |

The PeterWeyl Theorem | 211 |

741 The PeterWeyl theorem | 212 |

14 Connectedness | 14 |

15 Simple Connectedness | 17 |

16 Homomorphisms and Isomorphisms | 19 |

SU2 and SO3 | 20 |

17 The Polar Decomposition for SLnℝ and SLnℂ | 21 |

18 Lie Groups | 22 |

19 Exercises | 25 |

Lie Algebras and the Exponential Mapping | 29 |

22 Computing the Exponential of a Matrix | 32 |

X is nilpotent | 33 |

X arbitrary | 34 |

24 Further Properties of the Matrix Exponential | 37 |

25 The Lie Algebra of a Matrix Lie Group | 40 |

251 Physicists Convention | 41 |

253 The special linear groups | 42 |

256 The generalized orthogonal groups | 43 |

259 The Euclidean and Poincare groups | 44 |

26 Properties of the Lie Algebra | 45 |

27 The Exponential Mapping | 50 |

28 Lie Algebras | 55 |

281 Structure constants | 58 |

210 Exercises | 60 |

The BakerCampbellHausdorff Formula | 65 |

32 The General BakerCampbellHausdorff Formula | 69 |

33 The Derivative of the Exponential Mapping | 72 |

34 Proof of the BakerCampbellHausdorff Formula | 75 |

35 The Series Form of the BakerCampbellHausdorff Formula | 76 |

36 Group Versus Lie Algebra Homomorphisms | 78 |

37 Covering Groups | 82 |

38 Subgroups and Subalgebras | 84 |

39 Exercises | 90 |

Basic Representation Theory | 93 |

42 Why Study Representations? | 96 |

43 Examples of Representations | 97 |

432 The trivial representation | 98 |

434 Some representations of SU2 | 99 |

435 Two unitary representations of SO3 | 101 |

436 A unitary representation of the reals | 102 |

44 The Irreducible Representations of su2 | 103 |

45 Direct Sums of Representations | 108 |

46 Tensor Products of Representations | 109 |

47 Dual Representations | 114 |

48 Schurs Lemma | 115 |

49 Group Versus Lie Algebra Representations | 117 |

410 Complete Reducibility | 120 |

411 Exercises | 123 |

The Representations of SU3 | 129 |

52 Weights and Roots | 131 |

53 The Theorem of the Highest Weight | 134 |

54 Proof of the Theorem | 137 |

Highest Weight 11 | 142 |

56 The Weyl Group | 144 |

57 Weight Diagrams | 151 |

Semisimple Lie Algebras | 157 |

61 Complete Reducibility and Semisimple Lie Algebras | 158 |

62 Examples of Reductive and Semisimple Lie Algebras | 162 |

63 Cartan Subalgebras | 162 |

64 Roots and Root Spaces | 166 |

65 Inner Products of Roots and Coroots | 172 |

66 The Weyl Group | 175 |

67 Root Systems | 182 |

68 Positive Roots | 183 |

69 The slnℂ Case | 184 |

693 Inner products of roots | 185 |

694 The Weyl group | 186 |

611 Exercises | 187 |

742 The Weyl character formula | 213 |

743 Constructing the representations | 215 |

744 Analytically integral versus algebraically integral elements | 217 |

745 The SU2 case | 218 |

The BorelWeil Construction | 220 |

752 The setup | 222 |

753 The strategy | 224 |

754 The construction | 227 |

755 The SL2 ℂ case | 231 |

76 Further Results | 232 |

762 The weights and their multiplicities | 234 |

763 The Weyl character formula and the Weyl dimension formula | 236 |

764 The analytical proof of the Weyl character formula | 238 |

More on Roots and Weights | 245 |

82 Duality | 250 |

83 Bases and Weyl Chambers | 251 |

84 Integral and Dominant Integral Elements | 256 |

85 Examples in Rank Two | 258 |

852 Connection with Lie algebras | 259 |

854 Duality | 260 |

856 Weight diagrams | 261 |

86 Examples in Rank Three | 264 |

87 Additional Properties | 265 |

88 The Root Systems of the Classical Lie Algebras | 267 |

882 The orthogonal algebras so2n + 1 ℂ | 268 |

883 The symplectic algebras spn ℂ | 270 |

89 Dynkin Diagrams and the Classification | 271 |

810 The Root Lattice and the Weight Lattice | 275 |

811 Exercises | 278 |

Quick Introduction to Groups | 281 |

A2 Examples of Groups | 283 |

A21 The trivial group | 284 |

A26 Complex numbers of absolute value 1 under multiplication | 285 |

A3 Subgroups the Center and Direct Products | 286 |

A4 Homomorphisms and Isomorphisms | 287 |

A5 Quotient Groups | 288 |

A6 Exercises | 291 |

Linear Algebra Review | 293 |

B2 Diagonalization | 295 |

B3 Generalized Eigenvectors and the SN Decomposition | 296 |

B4 The Jordan Canonical Form | 298 |

B6 Inner Products | 299 |

B7 Dual Spaces | 301 |

More on Lie Groups | 305 |

C12 Tangent space | 306 |

C13 Differentials of smooth mappings | 307 |

C14 Vector fields | 308 |

C15 The flow along a vector field | 309 |

C16 Submanifolds of vector spaces | 310 |

C17 Complex manifolds | 311 |

C22 The Lie algebra | 312 |

C23 The exponential mapping | 313 |

C25 Quotient groups and covering groups | 314 |

C26 Matrix Lie groups as Lie groups | 315 |

C3 Examples of Nonmatrix Lie Groups | 316 |

ClebschGordan Theory for SU2 and the WignerEckart Theorem | 323 |

D2 The WignerEckart Theorem | 326 |

D3 More on Vector Operators | 330 |

Computing Fundamental Groups of Matrix Lie Group | 333 |

E2 The Universal Cover | 334 |

E3 Fundamental Groups of Compact Lie Groups I | 335 |

E4 Fundamental Groups of Compact Lie Groups II | 338 |

E5 Fundamental Groups of Noncompact Lie Groups | 344 |

347 | |

349 | |

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Lie Groups, Lie Algebras, and Representations: An Elementary Introduction Brian Hall No preview available - 2010 |

### Common terms and phrases

acting action adjoint algebra g angle apply associated assume base basis called Chapter character coefficients commutative compact complex condition connected consider constant construction contained continuous corresponding cover define Definition denote determinant diagonal dimension dimensional direct sum dominant integral element eigenvalue eigenvectors entries equal equivalent example Exercise exists expressed finite finite-dimensional formula function fundamental given gives GL(n group G highest weight identity inner product invariant subspace inverse irreducible representation isomorphic lattice Lemma Let G Lie algebra linear manifold matrix Lie group means Mn(C multiple namely nonzero Note numbers operator orthogonal positive precisely Proof Proposition prove Recall representation of g result root system satisfies semisimple Lie algebra shown simply simply connected SL(n smooth SO(n span standard subalgebra subgroup Suppose Theorem theory unique unitary vector space Weyl group zero