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Notes on Lie Groups Paul Nelson February 19, 2019 Contents 1 Disclaimers 6 2 Summary of classes and homework assignments 6 2.1 9/20: The definition of a Lie group . .7 2.2 9/22: The connected component . .8 2.3 9/27: One-parameter subgroups and the exponential map . 10 2.4 9/29: The Lie algebra of a Lie group . 12 2.5 10/4 (half-lecture) and 10/6: Representations of SL(2) ...... 14 2.6 10/11 (half-lecture) and 10/13: The unitary trick . 17 2.7 10/18 (half-lecture): The adjoint representation . 19 2.8 10/20 (first half): Character theory for SL(2) (algebraic) . 21 2.9 10/20 (second half): Maurer-Cartan equations; lifting morphisms of Lie algebras . 23 2.10 10/25 (half-lecture): universal covering group . 24 2.11 10/27: Fundamental groups of Lie groups . 26 2.12 11/1: The Baker–Campbell–Hausdorff(–Dynkin) formula . 28 2.13 11/3: Closed subgroups are Lie; virtual subgroups vs. Lie subal- gebras . 30 2.14 11/8: Simplicity of Lie groups and Lie algebras . 32 2.15 11/10: Simplicity of the special linear Lie algebra . 33 2.16 11/22, 11/24: How to classify classical simple complex Lie algebras 35 2.17 11/29: 11/31: Why simple Lie algebras give rise to root systems 38 2.18 12/6: Complex reductive vs. compact real . 40 2.19 12/8: Compact Lie groups: center, fundamental group . 40 2.20 12/13 Maximal tori in compact Lie groups . 42 3 Selected homework solutions 43 4 Some notation 49 4.1 Local maps . 49 4.1.1 Motivation . 49 4.1.2 Definition . 49 4.1.3 Equivalence . 50 1 4.1.4 Images . 50 4.1.5 Composition . 50 4.1.6 Inverses . 50 5 Some review of calculus 50 5.1 One-variable derivatives . 50 5.2 Multi-variable total and partial derivatives . 51 5.3 The chain rule . 52 5.4 Inverse function theorem . 52 5.5 Implicit function theorem . 52 6 Some review of differential geometry 53 6.1 Charts . 54 6.2 Manifolds . 54 6.3 Smooth maps . 55 6.4 Coordinate systems . 55 6.5 Tangent spaces . 55 6.6 Derivatives . 57 6.7 Jacobians . 58 6.8 Inverse function theorem . 58 6.9 Local linearization of smooth maps . 59 6.9.1 Linear maps in terms of coordinates . 59 6.9.2 The constant rank theorem . 59 6.9.3 The case of maximal rank . 60 6.10 Submanifolds . 61 6.11 A criterion for being a submanifold . 62 6.12 Computing tangent spaces of submanifolds . 63 6.13 Summary of how to work with submanifolds . 63 7 Some review of differential equations 64 8 Some review of group theory 67 8.1 Basic definition . 67 8.2 Permutation groups . 68 8.3 Topological groups . 68 9 Some review of functional analysis 70 9.1 Definitions and elementary properties of operators on Hilbert spaces 70 9.2 Compact self-adjoint operators on nonzero Hilbert spaces have eigenvectors . 70 9.3 Spectral theorem for compact self-adjoint operators on a Hilbert space . 72 9.4 Basics on matrix coefficients . 72 9.5 Finite functions on a compact group are dense . 72 9.6 Schur orthogonality relations . 74 9.7 Peter–Weyl theorem . 75 2 10 Some facts concerning invariant measures 76 10.1 Definition of Haar measures . 76 10.2 Existence theorem . 77 10.3 Unimodularity of compact groups . 78 10.4 Direct construction for Lie groups . 78 10.5 Some exercises . 78 10.6 Construction of a Haar measure on a compact group via averaging 79 11 Definition and basic properties of Lie groups 80 11.1 Lie groups: definition . 80 11.2 Basic examples . 80 11.3 Lie subgroups: definition . 81 11.4 A handy criterion for being a Lie subgroup . 82 11.5 Lie subgroups are closed . 83 11.6 Translation of tangent spaces by group elements . 83 12 The connected component 84 12.1 Generalities . 84 12.2 Some examples . 85 12.3 Connectedness of SO(n) ....................... 86 13 Basics on the exponential map 88 13.1 Review of the matrix exponential . 88 13.2 One-parameter subgroups . 90 13.3 Definition and basic properties of exponential map . 92 13.4 Application to detecting invariance by a connected Lie group . 93 13.5 Connected Lie subgroups are determined by their Lie algebras . 95 13.6 The exponential map commutes with morphisms . 95 13.7 Morphisms out of a connected Lie group are determined by their differentials . 96 14 Putting the “algebra” in “Lie algebra” 96 14.1 The commutator of small group elements . 96 14.2 The Lie bracket as an infinitesimal commutator . 97 14.3 Lie algebras . 98 14.4 The Lie bracket commutes with differentials of morphisms . 99 15 How pretend that every Lie group is a matrix group and sur- vive 99 16 Something about representations, mostly SL2 100 16.1 Some preliminaries . 100 16.2 Definition . 100 16.3 Matrix multiplication . 102 16.4 Invariant subspaces and irreducibility . 102 16.5 Polynomial representations of SL2(C) ............... 103 16.6 Classifying finite-dimensional irreducible representations of SL2(C)105 3 16.7 Complete reducibility . 108 16.8 Linear reductivity of compact groups . 112 16.9 Constructing new representations from old ones . 115 16.9.1 Direct sum . 115 16.9.2 Tensor product . 115 16.9.3 Symmetric power representations . 115 16.10Characters . 116 17 The unitary trick 118 18 Ad and ad 121 18.1 Basic definitions . 121 18.2 Relationship between Ad and ad .................. 124 18.3 Interpretation of the Jacobi identity . 124 18.4 Ad; ad are intertwined by the exponential map . 124 18.5 Some low-rank exceptional isomorphisms induced by the adjoint representation . 125 19 Maurer–Cartan equations and applications 127 19.1 The equations . 127 19.2 Lifting morphisms of Lie algebras . 128 20 The universal covering group 130 21 Quotients of Lie groups 133 22 Homotopy exact sequence 134 23 Cartan decomposition 135 24 BCHD 138 25 Some more ways to produce and detect Lie groups 141 25.1 Summary . 141 25.2 Closed subgroups of real Lie groups . 142 25.2.1 Statement of the key result . 142 25.2.2 A toy example . 143 25.2.3 Proof of the key result . 143 25.3 Stabilizers . 145 25.3.1 The basic result . 145 25.3.2 Application to kernels . 145 25.3.3 Application to preimages . 145 25.3.4 Application to intersections . 145 25.3.5 Application to stabilizers of vectors or subspaces in rep- resentations . 146 25.4 Commutator subgroups . 146 4 26 Immersed Lie subgroups 146 27 Simple Lie groups 147 28 Simplicity of the special linear group 149 28.1 Some linear algebra . 149 28.2 Recap on sl2(C) ............................ 151 28.3 Reformulation in terms of representations of abelian Lie algebras 152 28.4 Roots of an abelian subalgebra . 153 28.5 The roots of the special linear Lie algebra . 154 28.6 The simplicity of sln(C) ....................... 157 28.7 The proof given in lecture . 159 29 Classification of the classical simple complex Lie algebras 160 29.1 Recap . 160 29.2 Classical simple complex Lie algebras . 161 29.3 How to classify them (without worrying about why it works) . 162 29.4 Dynkin diagrams of classical simple Lie algebras . 166 29.5 Classical algebras come with faithful representations and are cut out by anti-involutions . 168 29.6 Diagonalization in classical Lie algebras . 169 29.7 Semisimplicity of elements of classical Lie algebras . 170 29.8 Conjugacy of Cartan subalgebras . 171 29.9 Interpretation of Cartan matrix in terms of inner products . 172 29.10Independence with respect to the choice of simple system . 172 30 Why simple Lie algebras give rise to root systems 176 30.1 Overview . 176 30.2 The basic theorem on Cartan subalgebras . 177 30.3 Abstract root systems . 178 30.4 Illustration of the root system axioms.
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