Infinitesimally Central Extensions of Chevalley Groups
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356 II
W. L. J. van der Kallen
Mathematisch Instituut der Rijksuniversiteit Utrecht De Uithof, Utrecht/Netherlands
Infinitesimally Central Extensions of Chevalley Groups
Springer-Verlag Berlin .Heidelberg-New York 1973
AMS Subject Classifications (1970): Primary: 20G 10, 20G 15, 17B45, t7B55 Secondary: 20G05, 20H15, 50B30, 17B20, 20F25
ISBN 3-540-06559-8 Springer-Verlag Berlin • Heidelberg • New York ISBN 0-387-06559-8 Springer-Verlag New York • Heidelberg . Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. @ by Springer-Verlag Berlin - Heidelberg t973. Library of Congress Catalog Card Number 73-19034. Printed in Germany. Offsetdruck: Julius Beltz, Hemsbach/Bergstr.
Contents 1
Conventions Section
1
Universal
central
Section
2
Degenerate
sums.
Section
3
The
Ad.
Section
4
Admissible
action
extensions. The
E-connected 5
The
Section
6
G-invariant
Section
7
The
Section
8
Extensions
Section
9
The
Hochschild
Section
10
The
existence
Section
11
Relations
Section
12
Representatives
Section
13
The
Theorem
and
its c o n s e q u e n c e s
The
group
14
References List Index
of N o t a t i o n s
and
* of gz'
: g ~T
r
trick
2
-~
5
gz
22
gR*
the c a t e g o r y
£V
39
components
Section
Section
extension
Structure
lattices
Central
G-module
ker
~
46
[ p]-structures
extension
~
51
: G* -~ G
54
of G by a G - m o d u l e
59
groups
61
of % : G* -~ G
in the
open
cell
97
in G* of the Weyl
of g e n e r a t o r s
functor
G*
65
group
119
and r e l a t i o n s
124
138
141
144 147
Introduction In these notes we study the c o n n e c t i o n between i n f i n i t e s i m a l l y central extensions of Chevalley groups and u n i v e r s a l central extensions of their Lie algebras.
Here an i n f i n i t e s i m a l l y central
extension is a m o r p h i s m of algebraic groups % : H ~ G such that, if ~, ~ denote the Lie algebra of G, H respectively, (i)
~ is surjective and separable,
(ii) the kernel of the derivative d~ of ~ is contained in the centre of the Lie algebra h. will restrict ourselves to the case that ~ = [~,h] .
We
Assume that ~ = [~,~] . Then a universal central extension : ~* ~ ~ exists. :
It may be c h a r a c t e r i z e d as a h o m o m o r p h i s m
~* ~ ~ such that
(i)
~ is surjeetive,
(ii)
~* = [~*,~*] ,
(iii) the kernel of ~ is contained in the centre of ~*, (iv)
~* is u n i v e r s a l with respect to (i),
Condition
(ii),
(iii).
(iv) is equivalent to
(iv)' If T : ~' ~ ~* is a h o m o m o r p h i s m satisfying
(i), (ii),
(iii) with ~ r e p l a c e d by ~ and ~*
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