Theory of Hopf Algebras Attached to Group Schemes
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614 Hiroshi Yanagihara
Theory of Hopf Algebras Attached to Group Schemes
Springer-v'erlag Berlin Heidelberg New York 1977
Author
Hiroshi Yanagihara Department of Mathematics Faculty of Science Hiroshima University Hiroshima/Japan
Library of Congress Cataloging in Publication Data
Yanagihara, 1934Theory of Hopf' algebras attached to group schemes. (Lecture notes in mathematics ; 614) Includes bibliographical references and index. 1. Hopf algebras. 2. Group schemes (Mathematics) 1. Title. II. Series: Lecture notes in mathematics (Berlin) ; 614. no. 614 [QA251.5] 510'.88 [5l2'.55] 77-14028
AMS Subject Classifications (1970): 14 L05, 14 L 10,14 L 15,16A24, 17 B50
ISBN 3-540-08444-4 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-08444-4 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 1977 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3140-543210
TABLE OF CONTENTS
Preface •••••......•••....•.•.....................•...•........ §
1
v
Group schemes over a field ...................•............ 1
§2
Group subschemes of a group scheme ••••••••••••••••••••••• 15
3
§4
Coalgebras over a field .....................•............ 29 Ropf algebras over a field ••••••••••••••••••••••••••••••• 45
§5 §6
Structures of colocal bigebras ••••••••••••••••••••••••••• 57 Formal groups over a field ••••••••••••••••••••••••••••••• 68
§7 §8
Structures of formal groups as local rings ••••••••••••••• 82 Algebraic Ropf suba Lgebr-as ••••••••••••••••••••••••••••••• 97
§9
Rational representations of group schemes ••••••••••••••• 113
§10
Joins and intersections of group subschemes ••••••••••••• 126
§ll
Adjoint representations of group schemes •••••••••••••••• 142
§
§12
Normalizers of Ropf subalgebras and group subschemes •••• 154
§13 §14 §15
Centralizers of Ropf subalgebras and group subschemes ••• 166 Commutators of Ropf subalgebras and group subschemes •••• 175 Lie algebras attached to group schemes •••••••••••••••••• 191
§16
Left invariant semi-derivations of group varieties
§17 §18
Algebraic schemes on which a group scheme operates 216 Isogenies of group varieties •••••••••••••••••••••••••••• 231
§19
Kernels of purely inseparable isogenies ••••••••••••••••• 243
203
Appendix I ••••.•••••.••..••••••••......•••••..••••••••••••..• 258 Appendix II •••••.•.••.•.••.•.••••..•.•.•.•..•...••••.•..••... 279
Appendix III •••.•..••..••.•....•••.....•....•.•.•...••..•..•. 294
References ••.•..•..........••••....•...........••....•.....•. 299 Index •••••••••••••••.•••..
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