Linear Algebraic Groups I (Stanford, Winter 2010) notes typed by Sam Lichtenstein, lectures and editing by Brian Conrad February 8, 2020 Please send any errata (typos, math errors, etc.) to [email protected]. The sequel course \Linear Algebraic Groups II" treats the Borel-Tits structure theory of reductive groups over general fields (assuming knowledge of schemes, as in this course). Contents 1 January 4 4 1.1 Some Definitions . 4 1.2 Smoothness . 5 1.3 Connectedness . 5 1.4 Examples . 6 2 January 6 7 2.1 Translations . 7 2.2 Homomorphisms . 7 2.3 Normal subgroups . 8 2.4 Further discussion of examples . 8 2.5 How far is a general smooth connected algebraic group from being either affine (i.e. linear algebraic) or projective (i.e. an abelian variety)? . 9 3 January 8 10 3.1 How do linear algebraic groups arise in nature? . 10 3.2 Zariski closures of subgroups . 10 4 January 11 (Substitute lecture by A. Venkatesh) 12 4.1 Tori . 12 4.2 Maximal split tori . 12 4.3 Building up groups from tori . 13 4.3.1 Construction of Φ ...................................... 14 5 January 13 14 5.1 Mapping Ga and Gm into a reductive group . 14 5.2 Classification of 1-dimensional connected linear algebraic groups . 15 5.3 Examples of non-split 1-dimensional linear algebraic k-groups . 15 5.4 Start of proof of the classification theorem . 16 6 January 20 (Substitute lecture by A. Venkatesh) 16 6.1 End of proof of classification of 1-dimensional connected linear algebraic groups . 16 6.2 Smoothness criteria . 17 1 7 January 22 18 7.1 More on forms . 18 7.2 A loose end concerning smoothness . 19 7.3 How to apply Grothendieck's smoothness criterion . 20 7.4 How to show algebraic k-groups are smooth . 20 7.4.1 Smoothness of GLn; SLn in coordinates, via matrices . 20 7.4.2 Smoothness of GL(V); SL(V) without using coordinates (much) . 21 8 January 25 21 8.1 How to Show Connectedness . 21 8.2 Connectedness of SLn ........................................ 22 8.2.1 Method 1 - Flag varieties . 22 8.2.2 Method 2 - Projective Space . 22 8.3 Smoothness of Sp2n ......................................... 22 9 January 29 25 9.1 A little more about Sp2n ...................................... 25 9.2 Actions, Centralizers and Normalizers . 26 9.3 Closed orbit lemma . 26 9.4 Start of proof of closed orbit lemma (Theorem 9.3.5) . 27 10 February 1 28 10.1 Conclusion of proof of closed orbit lemma (Theorem 9.3.5) . 28 10.2 Criterion for a k-homomorphism of k-groups to be a closed immersion . 29 11 February 3 29 11.1 Proof of Proposition 10.2.1 . 29 11.2 Embedding linear algebraic groups in GLn ............................. 31 12 February 5 31 12.1 Actions of affine finite type k-groups on coordinate rings of affine k-schemes . 31 12.1.1 Application to embedding a smooth linear algebraic group into GLn . 31 12.2 Proof of Theorem 12.1.1 . 32 13 February 8 33 13.1 Jordan decomposition . 33 14 February 10 35 14.1 All subgroups of a smooth finite type k-group are stabilizers of a line . 35 15 February 12 36 15.1 Unipotent groups . 36 15.2 Proof of Theorem 15.1.8 on representations of unipotent groups . 37 15.3 Remaining ingredients necessary for structure theory . 38 16 February 17 38 16.1 Some motivation: a key calculation on SL2(k) ........................... 38 16.2 Subgroups generated by connected varieties . 39 16.3 Proof of Proposition 16.2.1 . 40 16.4 Improvements on Proposition 16.2.1 . 41 17 February 19 41 17.1 Conclusion of proof of Proposition 16.4.2 . 41 17.2 Solvable groups . 42 17.3 Structure of smooth connected commutative affine k-groups . 42 17.4 Coset spaces for closed subgroups (that is, quotients) . 43 2 18 February 22 45 18.1 Existence of quotients (of smooth affine groups) . 45 18.2 Lie algebras . 46 18.2.1 Functoriality of Lie(·) .................................... 47 19 February 24 47 19.1 Linear representations of split tori, over general rings . 47 19.2 Proof of Theorem 19.1.2 . 49 20 February 26 50 20.1 k-split solvable groups . 50 20.2 Borel Fixed Point Theorem . 52 21 March 1 52 21.1 Remark on quotients . 52 21.2 Lie-Kolchin Corollary 20.2.4 . 53 21.3 Structure theory of solvable groups . 53 21.4 Unipotent radical . 54 21.5 Reductive groups . 55 22 March 3 56 22.1 Borel subgroups . 56 23 March 5 58 23.1 Properties of parabolics . 58 23.2 Conjugacy of maximal tori . 59 24 March 8 60 24.1 Proof of Proposition 23.2.1 . 60 24.2 Cartan subgroups . 61 24.3 PG(λ);UG(λ);ZG(λ) for a 1-parameter subgroup λ ........................ 61 25 March 10 62 25.1 Proof of Lemma 24.3.3 on PG;UG;ZG ............................... 62 25.1.1 Proof that (i)+(ii) (iii) [except unipotence of UG(λ)].................. 62 25.1.2 Proof of (i) . 62 25.1.3 Proof of (ii) and unipotence) of UG(λ) ........................... 63 26 March 12 64 26.1 Classification of split reductive groups of rank 1 . 64 A Properties of Orthogonal Groups 67 A.1 Basic definitions . 67 A.2 Even rank . 69 A.3 Odd rank . 72 A.4 Center . 75 A.5 Accidental isomorphisms . 76 B Existence of Jordan decomposition 78 B.1 Variants in linear algebra over arbitrary fields . 78 C Affine quotients 79 C.1 Introduction . 79 C.2 Line stabilizers . 80 3 D Quotient formalism 80 D.1 Coset spaces and isomorphism theorems . 81 D.2 Classifying some extension structures . 84 D.3 Structure of split solvable groups . 86 E Grothendieck's theorem on tori 88 E.1 Introduction . 88 E.2 Start of proof of Theorem E.1.1 . 89 E.3 The case of infinite k ......................................... 91 E.4 Hypothesis (?) for G implies the existence of a nontrivial k-torus.