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 ~*