Moufang Polygons

by Tits, Jacques; Weiss, Richard M.

• ISBN13: 9783540437147
• ISBN10: 3540437142
• Format: Hardcover
• Copyright: 2002-12-01
• Publisher: Springer Nature

Summary

This book gives the complete classification of Moufang polygons, starting from first principles. In particular, it may serve as an introduction to the various important algebraic concepts which arise in this classification including alternative division rings, quadratic Jordan division algebras of degree three, pseudo-quadratic forms, BN-pairs and norm splittings of quadratic forms. This book also contains a new proof of the classification of irreducible spherical buildings of rank at least three based on the observation that all the irreducible rank two residues of such a building are Moufang polygons. In an appendix, the connection between spherical buildings and algebraic groups is recalled and used to describe an alternative existence proof for certain Moufang polygons.

Table of Contents

Part I Preliminary Results
Introduction	3	(4)
Some Definitions	7	(8)
Generalized Polygons	15	(4)
Moufang Polygons	19	(4)
Commutator Relations	23	(4)
Opposite Root Groups	27	(4)
A Uniqueness Lemma	31	(2)
A Construction	33	(12)
Part II Nine Families of Moufang Polygons
Alternative Division Rings, I	45	(12)
Indifferent and Octagonal Sets	57	(4)
Involutory Sets and Pseudo-Quadratic Forms	61	(10)
Quadratic Forms of Type E6, E7 and E8, I	71	(20)
Quadratic Forms of Type E6, E7 and E8, II	91	(34)
Quadratic Forms of Type F4	125	(8)
Hexagonal Systems, I	133	(30)
An Inventory of Moufang Polygons	163	(12)
Main Results	175	(4)
Part III The Classification of Moufang Polygons
A Bound on n	179	(6)
Triangles	185	(6)
Alternative Division Rings, II	191	(12)
Quadrangles	203	(12)
Quadrangles of Involution Type	215	(14)
Quadrangles of Quadratic Form Type	229	(10)
Quadrangles of Indifferent Type	239	(4)
Quadrangles of Pseudo-Quadratic Form Type, I	243	(8)
Quadrangles of Pseudo-Quadratic Form Type, II	251	(24)
Quadrangles of Type E6, E7 and E8	275	(10)
Quadrangles of Type F4	285	(16)
Hexagons	301	(18)
Hexagonal Systems, II	319	(20)
Octagons	339	(14)
Existence	353	(14)
Part IV More Results on Moufang Polygons
BN-Pairs	367	(8)
Finite Moufang Polygons	375	(6)
Isotopes	381	(10)
Isomorphic Hexagonal Systems	391	(6)
Automorphisms	397	(22)
Isomorphic Quadrangles	419	(8)
Part V Moufang Polygons and Spherical Buildings
Chamber Systems	427	(20)
Spherical Buildings	447	(30)
Classical, Algebraic and Mixed Buildings	477	(12)
Appendix	489	(32)
Bibliography	521	(6)
Index of Notation	527	(2)
Index	529