Algebraic Groups and Arithmetic Groups J.S. Milne Version 1.01 June 4, 2006 These notes provide an introductory overview of the theory of algebraic groups, Lie algebras, Lie groups, and arithmetic groups. They are a revision of those posted during the teaching of a course at CMS, Zhejiang University, Hangzhou in Spring, 2005. v0.00 (February 28 – May 7, 2005). As posted during the course. v1.00 May 22, 2005. Minor corrections and revisions; added table of contents and index. v1.01 June 4, 2006. Fixed problem with the diagrams. Please send comments and corrections to me at [email protected] Available at http://www.jmilne.org/math/ The photo is of the famous laughing Buddha on The Peak That Flew Here, Hangzhou. Copyright c 2005, 2006 J.S. Milne. This electronic version of this work is licensed under a Creative Commons Licence (Attribution- NonCommercial-NoDerivs 2.5). 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For more details, see http://creativecommons.org/licenses/by-nc-nd/2.5/ Contents Introduction 1 Notations 2; Prerequisites 2; References 3 1 Overview and examples 4 The building blocks 4; Semisimple groups 5; Extensions 6; Summary 7; Exercises 8 2 Definition of an affine algebraic group 10 Principle of permanence of identities 10; Affine algebraic groups 10; Homomorphisms of al- gebraic groups 13; The Yoneda lemma 13; The coordinate ring of an algebraic group 14; Very brief review of tensor products. 14; Products of algebraic groups 15; Fibred products of al- gebraic groups 15; Extension of the base field (extension of scalars) 15; Algebraic groups and bi-algebras 16; Homogeneity 18; Reduced algebras and their tensor products 19; Reduced alge- braic groups and smooth algebraic groups 20; Smooth algebraic groups and group varieties 20; Algebraic groups in characteristic zero are smooth 22; Cartier duality 23; Exercises 24 3 Linear representations 25 Linear representations and comodules 25; Stabilizers of subspaces 30 4 Matrix Groups 32 An elementary result 32; How to get bialgebras from groups 32; A little algebraic geometry 33; Variant 34; Closed subgroups of GLn and algebraic subgroups 35 5 Example: the spin group 36 Quadratic spaces 36; The orthogonal group 40; Super algebras 40; Brief review of the tensor algebra 41; The Clifford algebra 42; The Spin group 46; The Clifford group 47; Action of O.q/ on Spin.q/ 48; Restatement in terms of algebraic groups 48 6 Group Theory 49 Review of group theory 49; Review of flatness 49; The faithful flatness of bialgebras 51; Definitions; factorization theorem 51; Embeddings; subgroups. 52; Kernels 52; Quotient maps 54; Existence of quotients 55; The isomorphism theorem 56 7 Finite (etale)´ algebraic groups 58 Separable k-algebras 58; Classification of separable k-algebras 59; Etale algebraic groups 60; Examples 60 8 The connected components of an algebraic group 62 Some algebraic geometry 62; Separable subalgebras 64; The group of connected components of an algebraic group 65; Connected algebraic groups 66; Exact sequences and connectedness 68; Where we are 69 9 Diagonalizable groups; tori 70 A remark about homomorphisms 70; Group-like elements in a bialgebra 70; The characters of an algebraic group 70; The algebraic group D.M / 71; Characterizing the groups D.M / 72; Diagonalizable groups 73; Diagonalizable groups are diagonalizable 74; Split tori and their rep- resentations 75; Rigidity 76; Groups of multiplicative type 76 10 Jordan decompositions 78 Jordan normal forms 78; Jordan decomposition in GLn.V / (k k) 79; Jordan decomposition in GL.V /, k perfect 80; Infinite-dimensional vector spaces 81; TheD regular representation contains all 81; The Jordan decomposition in the regular representation 82 11 Solvable algebraic groups 85 Brief review of solvable groups (in the usual sense) 85; Remarks on algebraic subgroups 85; Commutative groups are triangulizable 86; Decomposition of a commutative algebraic group 87; The derived group of algebraic group 88; Definition of a solvable algebraic group 89; Independence of characters 90; The Lie-Kolchin theorem 91; Unipotent groups 92; Structure of solvable groups 93; Tori in solvable groups 93; The radical of an algebraic group 94; Structure of a general (affine) algebraic group 94; Exercises 95 12 The Lie algebra of an algebraic group: basics 96 Lie algebras: basic definitions 96; The Lie algebra of an algebraic group 97; The functor Lie 98; Examples 98; Extension of the base field 101; Definition of the bracket 101; Alternative construction of the bracket. 102; The unitary group 103; Lie preserves fibred products 104 13 The Lie algebra of an algebraic group 106 Some algebraic geometry 106; Applications 107; Stabilizers 108; Isotropy groups 109; Normalizers and centralizers 110; A nasty example 111 14 Semisimple algebraic groups and Lie algebras 112 Semisimple Lie algebras 112; Semisimple Lie algebras and algebraic groups 112; The map ad 113; The Lie algebra of Autk.C / 113; The map Ad 114; Interlude on semisimple Lie algebras 115; Semisimple algebraic groups 119 15 Reductive algebraic groups 121 Structure of reductive groups 121; Representations of reductive groups 122; A criterion to be reductive 124 16 Split reductive groups: the program 126 Split tori 126; Split reductive groups 127; Program 129 17 The root datum of a split reductive group 130 Roots 130; Example: GL2 130; Example: SL2 130; Example: PGL2 131; Example: GLn 131; Definition of a root datum 132; First examples of root data 132; Semisimple groups of rank 0 or 1 134; Centralizers and normalizers 134; Definition of the coroots 135; Computing the centre 137; Semisimple and toral root data 137; The main theorems. 138; Examples 138 18 Generalities on root data 142 Definition 142 19 Classification of semisimple root data 146 Generalities on symmetries 146; Generalities on lattices 147; Root systems 147; Root systems and semisimple root data 148; The big picture 149; Classification of the reduced root system 149; The Coxeter graph 153 20 The construction of all split reductive groups 155 Preliminaries on root data/systems 155; Brief review of diagonalizable groups 156; Construction of all almost-simple split semisimple groups 157; Split semisimple groups. 157; Split reductive groups 157; Exercise 157 21 Borel fixed point theorem and applications 158 Brief review of algebraic geometry 158; The Borel fixed point theorem 159; Quotients 159; Borel subgroups 160; Parabolic subgroups 162; Examples of Borel and parabolic subgroups 162 22 Parabolic subgroups and roots 164 Lie algebras 165; Algebraic groups 166 23 Representations of split reductive groups 167 The dominant weights of a root datum 167; The dominant weights of a semisimple root datum 167; The classification of representations 167; Example: 168; Example: GLn 168; Example: SLn 169 24 Tannaka duality 170 Recovering a group from its representations 170; Properties of G versus those of Repk.G/ 170; (Neutralized) Tannakian categories 171; Applications 172 25 Algebraic groups over R and C; relation to Lie groups 174 The Lie group attached to an algebraic group 174; Negative results 174; Complex groups 175; Real groups 176 26 The cohomology of algebraic groups; applications 177 Introduction 177; Non-commutative cohomology. 177; Applications 180; Classifying the forms of an algebraic group 181; Infinite Galois groups 182; Exact sequences 183; Examples 183; (Weil) restriction of the base field 184; Reductive algebraic groups 184; Simply connected semisimple groups 184; Absolutely almost-simple simply-connected semisimple groups 185; The main theo- rems on the cohomology of groups 186 27 Classical groups and algebras with involution 188 The forms of Mn.k/ 188; The inner forms of SLn 189; Involutions of k-algebras 190; All the forms of SLn 190; Forms of Sp2n 191; The forms of Spin./ 192; Algebras admitting an in- volution 192; The involutions on an algebra 193; Hermitian and skew-hermitian forms 194; The groups attached to algebras with involution 194; Conclusion. 195 28 Arithmetic subgroups 196 Commensurable groups 196; Definitions and examples 196; Questions 197; Independence of and L. 197; Behaviour with respect to homomorphisms 198; Adelic` description of congruence subgroups 199; Applications to manifolds 200; Torsion-free arithmetic groups 200; A fundamen- tal domain for SL2 201; Application to quadratric forms 202; “Large” discrete subgroups 203; Reduction theory 204; Presentations 206; The congruence subgroup problem 207; The theorem of Margulis 208; Shimura varieties 209 Index of definitions 211 1 Introduction For one who attempts to unravel the story, the problems are as perplexing as a mass of hemp with a thousand loose ends. Dream of the Red Chamber, Tsao Hsueh-Chin. Algebraic groups are groups of matrices determined by polynomial conditions. For example, the group of matrices of determinant 1 and the orthogonal group of a symmetric bilinear form are both algebraic groups. The elucidation of the structure of algebraic groups and the classification of them were among the great achievements of twentieth century mathematics (Borel, Chevalley, Tits and others, building on the work of the pioneers on Lie groups). Algebraic groups are used in most branches of mathematics, and since the famous work of Hermann Weyl in the 1920s they have also played a vital role in quantum mechanics and other branches of physics (usually as Lie groups). Arithmetic groups are groups of matrices with integer entries. They are an important source of discrete groups acting on manifolds, and recently they have appeared as the sym- metry groups of several string theories in physics. These are the notes for a 40 hour course that I gave at CMS, Zhejiang University, Hangzhou, in the spring of 2005.