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Lie Groups, Condensed Lie Groups and Lie Algebras this document by Theo Johnson-Freyd based on: Mark Haiman, Math 261A: Lie Groups Fall 2008, UC-Berkeley Last updated December 23, 2009 ii Contents Contents v List of Theorems......................................... v Introduction vii 0.1 Notation........................................... vii 1 Motivation: Closed Linear Groups1 1.1 Definition of a Lie Group ................................. 1 1.1.1 Group objects.................................... 1 1.1.2 Analytic and Algebraic Groups.......................... 2 1.2 Definition of a Closed Linear Group ........................... 3 1.2.1 Lie algebra of a closed linear group........................ 3 1.2.2 Some analysis.................................... 4 1.3 Classical Lie groups .................................... 5 1.3.1 Classical Compact Lie groups........................... 5 1.3.2 Classical Complex Lie groups........................... 5 1.3.3 The Classical groups................................ 6 1.4 Homomorphisms of closed linear groups ......................... 6 Exercises ............................................. 7 2 Mini-course in Differential Geometry9 2.1 Manifolds.......................................... 9 2.1.1 Classical definition................................. 9 2.1.2 Sheafs........................................ 10 2.1.3 Manifold constructions............................... 10 2.1.4 Submanifolds.................................... 11 2.2 Vector Fields ........................................ 12 2.2.1 Definition...................................... 12 2.2.2 Integral Curves................................... 13 2.2.3 Group Actions ................................... 14 2.2.4 Lie algebra of a Lie group............................. 15 Exercises ............................................. 16 iii iv CONTENTS 3 General theory of Lie groups 19 3.1 From Lie algebra to Lie group............................... 19 3.1.1 The exponential map................................ 19 3.1.2 The Fundamental Theorem............................ 20 3.2 Universal Enveloping Algebras .............................. 22 3.2.1 The Definition ................................... 22 3.2.2 Poincar´e-Birkhoff-Witt Theorem ......................... 24 3.2.3 Ug is a bialgebra.................................. 25 3.2.4 Geometry of the Universal Enveloping Algebra................. 26 3.3 The Baker-Campbell-Hausdorff Formula......................... 26 3.4 Lie Subgroups........................................ 28 3.4.1 Relationship between Lie subgroups and Lie subalgebras............ 28 3.4.2 Review of Algebraic Topology........................... 30 3.5 A dictionary between algebras and groups........................ 32 3.5.1 Basic Examples: one- and two-dimensional Lie algebras ............ 33 Exercises ............................................. 34 4 General theory of Lie algebras 39 4.1 Ug is a Hopf algebra.................................... 39 4.2 Structure Theory of Lie Algebras............................. 41 4.2.1 Many Definitions.................................. 41 4.2.2 Nilpotency: Engel's Theorem and Corollaries.................. 42 4.2.3 Solvability: Lie's Theorem and Corollaries.................... 43 4.2.4 The Killing Form.................................. 44 4.2.5 Jordan Form .................................... 45 4.2.6 Cartan's Criteria.................................. 47 4.3 Examples: three-dimensional Lie algebras ........................ 48 4.4 Some Homological Algebra ................................ 48 4.4.1 The Casimir .................................... 49 4.4.2 Review of Ext ................................... 49 4.4.3 Complete Reducibility............................... 51 i 4.4.4 Computing Ext (K;M) .............................. 52 4.5 From Zassenhaus to Ado.................................. 55 Exercises ............................................. 59 5 Classification of Semisimple Lie Algebras 63 5.1 Classical Lie algebras over C ............................... 63 5.1.1 Reductive Lie algebras............................... 63 5.1.2 Guiding examples: sl(n) and sp(n) over C .................... 64 5.2 Representation Theory of sl(2) .............................. 70 5.3 Cartan subalgebras..................................... 72 5.3.1 Definition and Existence.............................. 72 5.3.2 More on the Jordan Decomposition and Schur's Lemma............ 75 5.3.3 Precise description of Cartan subalgebras .................... 77 CONTENTS v 5.4 Root systems........................................ 78 5.4.1 Motivation and a Quick Computation ...................... 78 5.4.2 The Definition ................................... 79 5.4.3 Classification of rank-two root systems...................... 80 5.4.4 Positive roots.................................... 83 5.5 Cartan Matrices and Dynkin Diagrams.......................... 84 5.5.1 Definitions ..................................... 84 5.5.2 Classification of finite-type Cartan matrices................... 85 5.6 From Cartan Matrix to Lie Algebra ........................... 90 Exercises ............................................. 94 6 Representation Theory of Semisimple Lie Groups 99 6.1 Irreducible Lie-algebra representations.......................... 99 6.2 Algebraic Lie Groups....................................108 6.2.1 Guiding example: SL(n) and PSL(n).......................108 6.2.2 Definition and General Properties of Algebraic Groups . 110 6.2.3 Constructing G from g ...............................113 6.3 Conclusion .........................................117 Exercises .............................................119 Index 121 Bibliography 125 List of Theorems Theorem 2.14 Inverse Mapping Theorem............................. 11 Theorem 3.5 Exponential Map.................................. 20 Theorem 3.12 Fundamental Theorem of Lie Groups and Algebras............... 20 Theorem 3.14 Baker-Campbell-Hausdorff Formula (second part only)............. 21 Theorem 3.24 Poincar´e-Birkhoff-Witt............................... 24 Theorem 3.31 Grothendieck Differential Operators........................ 26 Theorem 3.35 Baker-Campbell-Hausdorff Formula........................ 27 Theorem 3.37 Identification of Lie subalgebras and Lie subgroups............... 28 Theorem 4.25 Engel's Theorem................................... 42 Theorem 4.37 Lie's Theorem.................................... 43 Theorem 4.50 Jordan decomposition................................ 45 Theorem 4.53 Cartan's First Criterion............................... 47 Theorem 4.55 Cartan's Second Criterian............................. 47 Theorem 4.74 Schur's Lemma.................................... 51 Theorem 4.76 Ext1 vanishes over a semisimple Lie algebra................... 52 Theorem 4.78 Weyl's Complete Reducibility Theorem...................... 52 Theorem 4.79 Whitehead's Theorem................................ 52 vi CONTENTS Theorem 4.86 Levi's Theorem................................... 54 Theorem 4.88 Malcev-Harish-Chandra Theorem......................... 55 Theorem 4.89 Lie's Third Theorem................................ 55 Theorem 4.97 Zassenhaus's Extension Lemma.......................... 57 Theorem 4.99 Ado's Theorem................................... 58 Theorem 5.27 Existence of a Cartan Subalgebra......................... 74 Theorem 5.35 Schur's Lemma over an algebraically closed field................. 76 Theorem 5.79 Classification of indecomposable Dynkin diagrams................ 89 Theorem 5.89 Serre Relations.................................... 92 Theorem 5.94 Classification of finite-dimensional simple Lie algebras.............. 93 Theorem 6.15 Weyl Character Formula.............................. 103 Theorem 6.24 Weyl Dimension Formula.............................. 106 Theorem 6.69 Semisimple Lie algebras are algebraically integrable............... 116 Theorem 6.70 Classification of Semisimple Lie Groups over C .................. 117 Introduction In the Fall Semester, 2008, Prof. Mark Haiman taught Math 261A: Lie Groups, at the University of California Berkeley. The course covered the structure of Lie groups, Lie algebras, and their (complex) representations. The textbooks were [2] and [11]. I was one of many students in that class, and typed detailed notes [8], including all the motivation, discussion, questions, errors, and personal confusions and commentary. What you're reading right now is a first attempt to make those notes more presentable. It is also a study aid for my qualifying exam. As such, we present only the definitions, theorems, and proofs, with little motivation. I have made limited rearrangements of the material. Each subsection corresponds to one or two one-hour lectures. Needless to say, the pedagogy (and, since I was taking dictation, many of the words), are due to M. Haiman. In particular, I have quoted almost verbatim the problem sets M. Haiman assigned in the class (the reader may find my answers to some of the exercises in the appendices of [8]). Of course, any and all errors are mine. In addition to [2, 11], the reader might be interested in getting a sense of previous renditions of UC Berkeley's Math 261. In 2006 a three-professor tag-team taught a one-semester Lie Groups and Lie Algebras course; detailed notes are available [18], and occasionally I have referenced those notes, especially when I was absent or lost, or when my notes are otherwise lacking. They go quickly through the material | about twice as fast as we did | eschewing most proofs. For a very different version of the course, the reader may be
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