Jordan Triple Systems by the Grid Approach
Grids are special families of tripotents in Jordan triple systems. This research monograph presents a theory of grids including their classification and coordinization of their cover. Among the applications given are - classification of simple Jordan trip
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1280
Erhard Neher
Jordan Triple Systems by the Grid Approach
Spri nger-Verlag Berlin Heidelberg New York London Paris Tokyo
Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann
1280
Erhard Neher
Jordan Triple Systems by the Grid Approach
Spri nger-Verlag Berlin Heidelberg New York London Paris Tokyo
Author
Erhard Neher Department of Mathematics, University of Ottawa 585 King Edward, Ottawa, Ontario, Canada K1N 6N5
Mathematics Subject Classification (1980): 14J26, 17C10, 17C20, 17C40, 17C65,46K15,46L10,46L35 ISBN 3-540-18362-0 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-18362-0 Springer-Verlag New York Berlin Heidelberg
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. Duplication of this publication or parts thereof is only permitted under the provisions of the German Copyright Law of September 9, 1965, in its version of June 24, 1985, and a copyright fee must always be paid. Violations fall under the prosecution act of the German Copyright Law.
© Springer-Verlag Berlin Heidelberg 1987 Printed in Germany Printing and binding: Druckhaus Beltz, Hemsbach/Bergstr. 2146/3140-543210
TABLE OF CONTENTS INTRODUCTION CHAPTER I. SPECIAL FAMILIES OF COMPATIBLE TRIPOTENTS 1. Basic definitions, known results, examples 2. Elementary configurations: quadrangle, triangle and diamond 3. Cogs 4. Closed cogs and grids 5. Atomic cogs with minimal tripotents 6. Examples 7. Local, minimal and primitive tripotents
i v
1 1
16 25 33
43
49 54
CHAPTER II. CLASSIFICATION OF GRIDS 59 Non-ortho-collinear grids 61 2. Ortho-coll i near gri ds I (the s pe c i al cases) 66 3. Ortho-collinear grids II (the exceptional cases) 78 4. Construction of Peirce-dense atomic grids with minimal tripotents 90 5. The 27 lines upon a cubic surface and the Albert grid 100
1.
CHAPTER III. COORDINATIZATION THEOREMS 1. Coordinatization theorems for rectangular, symplectic and hermitian grids 2. Coordinatization theorems for quadratic form grids 3. Coordinatization theorems for the exceptional
107 108
CHAPTER IV. CLASSIFICATIONS Simple Jordan triple systems Hilbert triples JBW*-triples
136 136 147
REFERENCES
184
I NO EX
188
LIST OF SYMBOLS
193
1. 2. 3.
117 127
165
INTRODOCTION Some background and definitions In these notes we study Jordan triple systems. A prominent class of examples of Jordan triple systems are associative algebras. To consider an associative algebra A with a bilinear product (x,y} + xy as a Jordan triple system means to forget the bilinear product of A and its unit element and instead work just with the triple product (x,y) + P(x)y = xyx and its linearization (x,y,z) + {xyz} = xyz + zyx. There are good reasons for doing this, some of which are indicated below. Of course, the associative law has to be rephrased in terms of the triple product, leading to the
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