Basic theory of algebraic groups and lie algebras, gerhard p hochschild

Graduate Texts in Mathematics 75 Editorial Board F W Gehring P R Halmos (Managing Editor) C C Moore Gerhard P Hochschild Basic Theory of Algebraic Groups and Lie Algebras Springer-Verlag New Yark Heidelberg Berlin Gerhard P Hochschild Department of Mathematics University of California Berkeley, CA 94720 USA Editorial Board P R Halmos F W Gehring Managing Editor Department of Mathematics Indiana University Bloomington, IN 47401 USA Department of Mathematics University of Michigan Ann Arbor, MI48109 USA c C Moore Department of Mathematics University of California Berkeley, CA 94720 USA AMS Subject Classification (1981): 14~0\, 20~01, 20GXX Library of Congress Cataloging in Publication Data Hochschild, Gerhard Paul, 1915~ Basic theory of algebraic groups and Lie algebras (Graduate texts in mathematics: 75) Bibliography: p Includes index I Lie algebras Linear algebraic groups I Title II Series QA252.3.H62 512'.55 80~27983 All rights reserved No part of this book may be translated or reproduced in any form without written permission from Springer-Verlag © 1981 by Springer-Verlag New York Inc Softcover reprint of the hardcover 1st edition 1981 543 ISBN-13: 978-1-4613-8116-7 DOl: 10.1007/978-1-4613-8114-3 e-ISBN-13: 978-1-4613-8114-3 Preface The theory of algebraic groups results from the interaction of various basic techniques from field theory, multilinear algebra, commutative ring theory, algebraic geometry and general algebraic representation theory of groups and Lie algebras It is thus an ideally suitable framework for exhibiting basic algebra in action To that is the principal concern of this text Accordingly, its emphasis is on developing the major general mathematical tools used for gaining control over algebraic groups, rather than on securing the final definitive results, such as the classification of the simple groups and their irreducible representations In the same spirit, this exposition has been made entirely self-contained; no detailed knowledge beyond the usual standard material of the first one or two years of graduate study in algebra is presupposed The chapter headings should be sufficient indication of the content and organisation of this book Each chapter begins with a brief announcement of its results and ends with a few notes ranging from supplementary results, amplifications of proofs, examples and counter-examples through exercises to references The references are intended to be merely suggestions for supplementary reading or indications of original sources, especially in cases where these might not be the expected ones Algebraic group theory has reached a state of maturity and perfection where it may no longer be necessary to re-iterate an account of its genesis Of the material to be presented here, including much of the basic support, the major portion is due to Claude Chevalley Although Chevalley's decisive classification results, contained in [6J, have not been included here, a glimpse of their main ingredients can be had from Chapters XVII and XIII The subject of Chapter XIII is Armand Borel's fundamental theory of maximal solvable subgroups and maximal toroids, which has made it v vi Preface possible tc recreate the combinatorial features of the Cartan-Weyl theory of semisimple Lie algebras, dealt with in Chapter XVII, in terms of subgroups of semisimple algebraic groups In particular, this has freed the theory from the classical restriction to base fields of characteristic O I was encouraged to write this exposition chiefly by the appearance of James Humphreys's Linear Algebraic Groups, where the required algebraic geometry has been cut down to a manageable size In fact, the algebraicgeometric developments given here have resulted from Humphreys's treatment simply by adding proofs of the underlying facts from commutative algebra Moreover, much of the general structure theory in arbitrary characteristic has been adapted from Borel's lecture notes [1] and Humphreys's book I have made use of valuable advice from my friends, given in the course of several years on various occasions and in various forms, including print It is a pleasure to express my thanks for their help to Walter Ferrer-Santos, Oscar Goldman, Bertram Kostant, Andy Magid, Calvin Moore, Brian Peterson, Alex Rosenberg, Maxwell Rosenlicht, John B Sullivan, Moss Sweedler and David Wigner However, it must be emphasized that no one but me has had an opportunity to remedy any of the defects of my actual manuscript Gerhard P Hochschild Contents Chapter I Representative Functions and Hopf Algebras Chapter II Affine Algebraic Sets and Groups Chapter III Derivations and Lie Algebras Chapter IV Lie Algebras and Algebraic Subgroups 15 28 44 Chapter V Semisimplicity and Unipotency 59 Chapter VI Solvable Groups 78 Chapter VII Elementary Lie Algebra Theory 93 Chapter VIII Structure Theory in Characteristic 106 Chapter IX Algebraic Varieties 122 vii viii Contents Chapter X Morphisms of Varieties and Dimension Chapter XI Local Theory 137 155 Chapter XII Coset Varieties 173 Chapter XIII Borel Subgroups 188 Chapter XIV Applications of Galois Cohomology 200 Chapter XV Algebraic Automorphism Groups Chapter XVI The Universal Enveloping Algebra Chapter XVII Semisimple Lie Algebras 210 221 233 Chapter XVIII From Lie Algebras to Groups 249 References 263 Index 265 Chapter I Representative Functions and Hopf Algebras This chapter introduces the basic algebraic machinery arising in the study of group representations The principal notion of a Hopf algebra is developed here as an abstraction from the systems of functions associated with the representations of a group by automorphisms of finite-dimensional vector spaces This leads to an initializing discussion of our main objects of study, affine algebraic groups Given a non-empty set S and a field F, we denote the F-algebra of all F-valued functions on S by F S • In the statement of the following lemma, and frequently in the sequel, we use the symbol 6ij , which stands for if i = j, and for U if i :f= j Lemma 1.1 Let V be a non-zero finite-dimensional sub F-space of F S There is a basis (vt> , Vn) of V and a corresponding subset (st> ,Sn) of S such that v;(s) = 6ij for an indices i and j PROOF Suppose that we have already found elements S1' , Sk of S and a basis (v1.k>' , • k) of V such that the Vi,k'S and the s/s satisfy the requirements of the lemma for each i from (1, , n) and each j from (1, , k) If k < n, there is an element sk+ in S such that Vk+ 1,k(Sk+ 1) :f= O We set Vk+1.k+1 For the indices i other than k = Vk+1,k(Sk+l)-1 Vk +1.k' + 1, we set Now the sets (S1"'" Sk+ 1) and (V1.k+ 1, , Vn.k+ 1) satisfy our requirements at level k + The lemma is obtained by induction, starting with an arbitrary basis of V at level k = O I.l For non-empty sets Sand T, we examine the canonical morphism of F-algebras, n, from the tensor product F S ® FT to F SX T, where n(L f ® g)(s, t) = L f(s)g(t) Proposition 1.2 The canonical morphism n: F S ® FT ~ F S x T is injective, and its image consists of all functions h with the property that the F-space spanned by the partial functions ht, where t ranges over T and hls) = h(s, t), is finite-dimensional PROOF Let L,J= fi ® gj be an element of the kernel of n, and let V be the sub F-space of F S spanned by f1' ,fm If V = (0) then our element is O Otherwise choose (Vl> , vn) and (SI' ' sn) as in Lemma 1.1, and write our element in the form L7=1 Vi ® hi Applying n and evaluating at (Sk' t) yields hk(t) = O This shows that each hi is 0, and we conclude that n is injective It is clear that if h is an element of the image of n then it has the property stated in the proposition Conversely, suppose that h is an element of FSX T having this property This means that there are elementsfl, ,j" in F S such that each ht is an F-linear combination of the fi's Choosing coefficients from F for each t in T, we obtain elements g 1, • ,gn of FT such that, for each t, n ht This means that h = = Lgi(t)!; i= n(.i,= h ® g;) D Let us consider the above in the case where both Sand T coincide with the underlying set of a monoid G, with composition m: G x G ~ G This composition transposes in the natural fashion to a morphism of F -algebras m*: FG ~ FGXG , where m*(f) is the compositef m We abbreviate m(x, y) by xy, so that m*(f)(x, y) = f(xy) By transposing the right and left translation actions of G on itself, we obtain a two-sided G-module structure on F G, which we indicate as follows (x f)(y) = f(yx), (f x)(y) = f(xy) Now we see from Proposition 1.2 that m*(f) belongs to the image of n: FG ® FG + FG x G if and only if the F -space spanned by the functions x·J, with x ranging over G, is finite-dimensional If this is so, we say thatfis a representative function We denote the F-algebra of all F -valued representative functions on G by ~F(G), but we shall permit ourselves to suppress the subscript F when there is no danger of confusion Clearly, ~F(G) is a twosided sub G-module of F G , as well as a sub F-algebra 254 XVIII.2 This reduces the theorem to the case where p(2(P» is the Lie algebra of a linearly reductive subgroup Q of H such that H is the semidirect product H u > £'(G) The inverse of this isomorphism yields an injective Lie algebra homomorphism p: £'(G) -> £,(H) Both G and H have trivial center, and £'(G) is semisimple Here again, p cannot be the differential of a homomorphism of algebraic groups (J": G -> H, because then (J" would yield an inverse of the group covering Q -> G, which is impossible Theorem 1.1 is due to M Goto References [I] A Borel, Linear Algebraic Groups (lecture notes taken by H Bass), Benjamin, New York, 1969 [2] A Borel and J-P Serre, Theoremes de Finitude en Cohomologie Galoisienne, Comment Math Helv 39, 111-163 (1964) [3] N Bourbaki, Groupes et Algebres de Lie, Ch I, Act Sc et Ind 1285, Hermann, Paris, 1960 [4] C Chevalley, Theorie des Groupes de Lie, Tome II, Hermann, Paris, 1951 [5] C Chevalley, Theorie des Groupes de Lie, Tome III, Hermann, Paris, 1955 [61 C Chevalley, Classification des Groupes de Lie Algebriques (mimeographed lecture notes), Ecole Norm Sup., Paris, 1956-1958 [7] J E Humphreys, Linear Algebraic Groups, GTM 21, Springer-Verlag, New York, 1975 [8] N Jacobson, Lie A~qebras, Interscience, New York, 1962 [9] J W Milnor and J C Moore, On the Structure of Hopf Algebras, Ann Math 81, 211-264 (1965) [10] G D Mostow, Fully Reducible Subgroups of Algebraic Groups, Am J Math 78,200-221 (1956) [11] D Mumford, Introduction to Algebraic Geometry (preliminary version of the first three chapters) Math Dept., Harvard Univ., Cambridge, MA [12] M Rosenlicht, Some Basic Theorems on Algebraic Groups, Am J Math 78, 401-443 (1956) [13] M Rosenlicht, Extensions of Vector Groups by Abelian Varieties, Am J Math 80, 685-714 (\ 958) [14] J-P Serre, Groupes Algebriques et Corps de Classes, Act Sc et Ind 1264, Hermann, Paris, 1959 [15] J-P Serre, Algebres de Lie Semi-simples Complexes, Benjamin, New York, 1966 [16] R Steinberg, Lectures on Chevalley Groups (notes by J Faulkner and R Wilson), Math Dept., Yale Univ., New Haven, CT, 1967 [17] M E Sweedler, HopfAlgebras, Benjamin, New York, 1969 [18] H Weyl, The Classical Groups, Princeton Univ Press, Princeton, NJ, 1946 263 Index Abelian Lie algebra 53 Adjoint representation 51, 98 Affine algebraic group 10 Affine algebraic set 10 Affine patch 123 Affine variety 127 Algebraic automorphism group 211 Algebraic hull 112 Algebraic subgroup 12 Algebraic sub Lie algebra 112 Algebraic vector group 89 Antimorphism Antipode Artin - Rees lemma 156 Artin ring 157 Artin-Tate proposition 22 Associated representative function 11 Bialgebra Biideal 12 Borel's fixed point theorem Borel subgroup 190 189 Campbell- Hausdorff formula 228 Cartan's solvability criterion 96 Cartan subalgebra 233 Casimir element 99 Casimir operator 99 Center of a Lie algebra 94 Characteristic degree 159 Coalgebra Coboundary 80, 101 Cocyc1e 80, 100 Coideal 12 Commutator ideal 95 Comodule Complete variety 133 Comultiplication 3, Conductor 147 Constructible 124 Counit Defined at a point 123 Defined over a subfield 200 Derivation 28, 35 Differential 36, 170 Differentiation 36 Dimension 38, 139 Dominant morphism 142 Dominant weight 242 Dual comodule Elementary open set 125 Engel's theorem 96 Exceptional automorphism Extended differential 41 Exterior algebra 17 Finite morphism Flag 190 252 143 265 266 Index Fonn 200 Full linear group 13 Fundamental system of roots Parametric dimension 159 Perfect 31 Poincare-Birkhoff-Witt theorem 222 Polynomial character 74 Polynomial function 10 Polynomial group module 13 Polynomial map 10 Polynomial representation 13 Prevariety 123 Primitive Hopf algebra element 87 Principal ideal theorem 161 Principal open set 124 Projective variety 130 Properly nonnal 70 241 Galois cocycle 202 Grassmann variety 131 Hilbert Nullstellensatz Hopf algebra Hopf ideal 12 22 Ideal of a Lie algebra 53 Irreducible affme variety 123 Irreducible component 16 Irreducible space 15 Quasiaffine 181 Quasiprojective 176 Jordan components 66 Jordan decomposition 63 Radical 103, 255 Rational function 123, 124, 139 Regular function 123 Regular local ring 165 Representation of a Lie algebra 42 Representative function Root 236 Krull dimension 159 Krull topology 202 Length of a module 157 Lie algebra 35, 36 Lie homomorphism 221 Lie's theorem 95 Linearly reductive 67, 115 Locally closed 124 Locally finite Locally unipotent 64 Local ring 147 Maximum nonnal unipotent subgroup 65 Morphism of affine algebraic groups 13 Morphism of affine algebraic sets 10 Morphism of prevarieties 125 Morphism of sheaves 125 Multiplication Nilpotent action 95 Nilpotent component 63 Nilpotent Lie algebra 104 Noetherian space 16 Noether's nonnalization theorem Nonnal integral domain 147 Nonnal point 148 Nonnal variety 148 Observable 184 138 Semidirect product 70 Semi invariant 18 Semisimple component 63 Semisimple Lie algebra 94 Semisimple module 59 Separable 28 Separable morphism 170 Sheaf 122, 123 Singular point 167 Solvable Lie algebra 95 Stalk 147 Strict variety 182 Tangent 167 Tensor product of comodules Trace fonn 98 Triangular group 84 Type Unipotent component 63 Unipotent group 64 Unit Universal enveloping algebra 221 Index Valuation subring Variety 126 Wall 241 Weight 235 20 267 Weight of a semiinvariant Weyl chamber 241 Weyl group 199, 241 Zariski topology 10 18 Graduate Texts in Mathematics Soft and hard cover editions are available for each volume up to Vol 14, hard cover only from Vol 15 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 TAKEUTI/ ZARING Introduction to Axiomatic Set Theory OXTOBY Measure and Category 2nd ed SCHAEFFER Topological Vector Spaces HILTON/STAMMBACH A Course in Homological Algebra MACLANE Categories for the Working Mathematician HUGHES/PIPER Projective Planes SERRE A course in Arithmetic TAKEUTI/ZARING Axiomatic Set Theory HUMPHREYS Introduction to Lie Algebras and Representation Theory 2nd printing, 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Functions and Hopf Algebras Chapter II Affine Algebraic Sets and Groups Chapter III Derivations and Lie Algebras Chapter IV Lie Algebras and Algebraic Subgroups 15 28 44 Chapter V Semisimplicity and. .. Gehring P R Halmos (Managing Editor) C C Moore Gerhard P Hochschild Basic Theory of Algebraic Groups and Lie Algebras Springer-Verlag New Yark Heidelberg Berlin Gerhard P Hochschild Department of. .. Section applies this to yield an important tool theorem about polynomial maps between algebraic groups, and then establishes the principal general result concerning factor groups of algebraic groups