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Moufang Polygons

  • Jacques Tits
  • Richard M. Weiss

Part of the Springer Monographs in Mathematics book series (SMM)

Table of contents

  1. Front Matter
    Pages i-x
  2. Preliminary Results

    1. Front Matter
      Pages 1-1
    2. Jacques Tits, Richard M. Weiss
      Pages 3-5
    3. Jacques Tits, Richard M. Weiss
      Pages 7-14
    4. Jacques Tits, Richard M. Weiss
      Pages 15-17
    5. Jacques Tits, Richard M. Weiss
      Pages 19-22
    6. Jacques Tits, Richard M. Weiss
      Pages 23-25
    7. Jacques Tits, Richard M. Weiss
      Pages 27-30
    8. Jacques Tits, Richard M. Weiss
      Pages 31-32
    9. Jacques Tits, Richard M. Weiss
      Pages 33-42
  3. Nine Families of Moufang Polygons

    1. Front Matter
      Pages 43-43
    2. Jacques Tits, Richard M. Weiss
      Pages 45-55
    3. Jacques Tits, Richard M. Weiss
      Pages 57-60
    4. Jacques Tits, Richard M. Weiss
      Pages 61-70
    5. Jacques Tits, Richard M. Weiss
      Pages 71-90
    6. Jacques Tits, Richard M. Weiss
      Pages 91-123
    7. Jacques Tits, Richard M. Weiss
      Pages 125-132
    8. Jacques Tits, Richard M. Weiss
      Pages 133-162
    9. Jacques Tits, Richard M. Weiss
      Pages 163-174
    10. Jacques Tits, Richard M. Weiss
      Pages 175-176
  4. The Classification of Moufang Polygons

    1. Front Matter
      Pages 177-177
    2. Jacques Tits, Richard M. Weiss
      Pages 179-184
    3. Jacques Tits, Richard M. Weiss
      Pages 185-190
    4. Jacques Tits, Richard M. Weiss
      Pages 191-202
    5. Jacques Tits, Richard M. Weiss
      Pages 203-213
    6. Jacques Tits, Richard M. Weiss
      Pages 215-228
    7. Jacques Tits, Richard M. Weiss
      Pages 229-238
    8. Jacques Tits, Richard M. Weiss
      Pages 239-242
    9. Jacques Tits, Richard M. Weiss
      Pages 243-250
    10. Jacques Tits, Richard M. Weiss
      Pages 251-273
    11. Jacques Tits, Richard M. Weiss
      Pages 275-283
    12. Jacques Tits, Richard M. Weiss
      Pages 285-300
    13. Jacques Tits, Richard M. Weiss
      Pages 301-318
    14. Jacques Tits, Richard M. Weiss
      Pages 319-338
    15. Jacques Tits, Richard M. Weiss
      Pages 339-351
    16. Jacques Tits, Richard M. Weiss
      Pages 353-364
  5. More Results on Moufang Polygons

    1. Front Matter
      Pages 365-365
    2. Jacques Tits, Richard M. Weiss
      Pages 367-374
    3. Jacques Tits, Richard M. Weiss
      Pages 375-380
    4. Jacques Tits, Richard M. Weiss
      Pages 381-389
    5. Jacques Tits, Richard M. Weiss
      Pages 391-395
    6. Jacques Tits, Richard M. Weiss
      Pages 397-418
    7. Jacques Tits, Richard M. Weiss
      Pages 419-424
  6. Moufang Polygons and Spherical Buildings

    1. Front Matter
      Pages 425-425
    2. Jacques Tits, Richard M. Weiss
      Pages 427-445
    3. Jacques Tits, Richard M. Weiss
      Pages 447-476
    4. Jacques Tits, Richard M. Weiss
      Pages 477-488
    5. Jacques Tits, Richard M. Weiss
      Pages 489-519
  7. Back Matter
    Pages 521-538

About this book

Introduction

Spherical buildings are certain combinatorial simplicial complexes intro­ duced, at first in the language of "incidence geometries," to provide a sys­ tematic geometric interpretation of the exceptional complex Lie groups. (The definition of a building in terms of chamber systems and definitions of the various related notions used in this introduction such as "thick," "residue," "rank," "spherical," etc. are given in Chapter 39. ) Via the notion of a BN-pair, the theory turned out to apply to simple algebraic groups over an arbitrary field. More precisely, to any absolutely simple algebraic group of positive rela­ tive rank £ is associated a thick irreducible spherical building of the same rank (these are the algebraic spherical buildings) and the main result of Buildings of Spherical Type and Finite BN-Pairs [101] is that the converse, for £ ::::: 3, is almost true: (1. 1) Theorem. Every thick irreducible spherical building of rank at least three is classical, algebraic' or mixed. Classical buildings are those defined in terms of the geometry of a classical group (e. g. unitary, orthogonal, etc. of finite Witt index or linear of finite dimension) over an arbitrary field or skew-field. (These are not algebraic if, for instance, the skew-field is of infinite dimension over its center. ) Mixed buildings are more exotic; they are related to groups which are in some sense algebraic groups defined over a pair of fields k and K of characteristic p, where KP eke K and p is two or (in one case) three.

Keywords

Buildings Graph algebra classification combinatorics generalized polygons graph theory incidence geometry polygon

Authors and affiliations

  • Jacques Tits
    • 1
  • Richard M. Weiss
    • 2
  1. 1.Collège de FranceParis Cedex 05France
  2. 2.Department of MathematicsTufts UniversityMedfordUSA

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-662-04689-0
  • Copyright Information Springer-Verlag Berlin Heidelberg 2002
  • Publisher Name Springer, Berlin, Heidelberg
  • eBook Packages Springer Book Archive
  • Print ISBN 978-3-642-07833-0
  • Online ISBN 978-3-662-04689-0
  • Series Print ISSN 1439-7382
  • Buy this book on publisher's site
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