Basic Theory of Algebraic Groups and Lie Algebras

275 Pages·1981·12.33 MB·English

by  Gerhard P. Hochschild (auth.)

75
Graduate Texts in Mathematics
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
Department of Mathematics University of Michigan
Indiana University Ann Arbor, MI48109
Bloomington, IN 47401 USA
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. 2. 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 1s t edition 1981
9 8 7 6 543 2 1
ISBN-13: 978-1-4613-8116-7 e-ISBN-13: 978-1-4613-8114-3
DOl: 10.1007/978-1-4613-8114-3
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 do 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 pre
supposed.
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 algebraic
geometric 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 charac
teristic 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 15
Chapter III
Derivations and Lie Algebras 28
Chapter IV
Lie Algebras and Algebraic Subgroups 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 0 106
Chapter IX
Algebraic Varieties 122
vii
viii Contents
Chapter X
Morphisms of Varieties and Dimension 137
Chapter XI
Local Theory 155
Chapter XII
Coset Varieties 173
Chapter XIII
Borel Subgroups 188
Chapter XIV
Applications of Galois Cohomology 200
Chapter XV
Algebraic Automorphism Groups 210
Chapter XVI
The Universal Enveloping Algebra 221
Chapter XVII
Semisimple. Lie Algebras 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.
1. Given a non-empty set S and a field F, we denote the F-algebra of all
F -valued functions on S by FS• In the statement of the following lemma,
and frequently in the sequel, we use the symbol 6 which stands for 1 if
ij,
i = j, and for U if i :f= j.
Lemma 1.1. Let V be a non-zero finite-dimensional sub F-space of FS. 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>' .. , vn•k) of V such that the Vi,k'S and the s/s satisfy the require
ments of the lemma for each i from (1, ... , n) and each j from (1, ... , k). If
k < n, there is an element sk+ 1 in S such that Vk+ 1,k(Sk+ 1) :f= O. We set
Vk+1.k+1 = Vk+1,k(Sk+l)-1Vk+1.k'
For the indices i other than k + 1, we set
Now the sets (S1"'" Sk+ 1) and (V1.k+ 1, ... , Vn.k+ 1) satisfy our requirements
at level k + 1. The lemma is obtained by induction, starting with an arbitrary
basis of V at level k = O. 0
2 I.l
For non-empty sets Sand T, we examine the canonical morphism of
F-algebras, n, from the tensor product FS ® FT to FSX T, where
n(L L
f ® g)(s, t) = f(s)g(t).
Proposition 1.2. The canonical morphism n: FS ® FT ~ FS 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= 1 fi ® gj be an element of the kernel of n, and let V be
the sub F-space of FS 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 FS such
that each h is an F -linear combination of the fi's. Choosing coefficients
t
from F for each t in T, we obtain elements g 1, ..• ,gn of FT such that, for
each t,
n
h = Lgi(t)!;.
t
i= 1
This means that
n(.i
g;).
h = h ® D
,=
1
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)
0
by xy, so that m*(f)(x, y) = f(xy). By transposing the right and left trans
lation actions of G on itself, we obtain a two-sided G-module structure on
FG, 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 if and only if the F -space spanned by the functions
--+ x G
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 representa
tive 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 two
sided sub G-module of FG, as well as a sub F-algebra.
1.1 3
Proposition 1.3. The image of the morphism of F -algebras
n-I 0 m*: &tF(G) --+ FG ® FG
actually lies in &fF(G) ® &tF(G).
PROOF. Letfbe an element of &tF(G). Proceeding as in the proof of Proposi
tion 1.2, we find elements S10 ••• 'Sn in G and elements VI' ... , Vn in FG as
in Lemma 1.1 such that we may write
n
n-1( m*(f)) = L Vi ® hi.
i= 1
Evaluating this at (Sj' t), we find h/t)= f(sjt), whence hj = f· Sj. This
shows that hj belongs to &f p( G). The conclusion is that the image of n - 1 0 m*
lies in FG ® &tF(G). Changing sides throughout, we find that this image
also lies in &tp(G) ® FG. Clearly, the last two conclusions imply the assertion
of Proposition 1.3. 0
The morphism of F-algebras &t(G) --+ &t(G) ® &t(G) defined by Proposi
tion 1.3 is called the comuitiplication of &t(G), and we shall denote it by b.
For an element S of G, let s*: &t(G) --+ F denote the evaluation at s, so that
s*(f) = f(s). Then 15 is characterized by the formula
(x* ® y*)(b(f)) = f(xy).
More explicity, if
b(!) = L f; ®f;',
i
then
L
f(xy) = f;(x)f;'(Y)·
i
We adopt some general terminology, as follows. The structure of an
F-algebra A is understood to consist of
(1) the structure of A as an F -space;
(2) the multiplication of A, viewed as an F-linear map /1: A ® A --+ A;
(3) the unit of A, viewed as an F -linear map u:F --+ A sending each element
a of F onto the a-multiple a1 A of the identity element of A.
In writing the axioms, it is convenient to name the canonical identification
maps coming from the F -space structure of A. These are
PI: F ® A --+ A and P2: A ® F --+ A.
Generally, we use is to denote the identity map on a set S. The axioms
of an F-algebra structure may now be written as follows