Moufang Polygons

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.

List of contents

I Preliminary Results.- 1 Introduction.- 2 Some Definitions.- 3 Generalized Polygons.- 4 Moufang Polygons.- 5 Commutator Relations.- 6 Opposite Root Groups.- 7 A Uniqueness Lemma.- 8 A Construction.- II Nine Families of Moufang Polygons.- 9 Alternative Division Rings, I.- 10 Indifferent and Octagonal Sets.- 11 Involutory Sets and Pseudo-Quadratic Forms.- 12 Quadratic Forms of Type E6, E7 and E8, I.- 13 Quadratic Forms of Type E6, E7 and E8, II.- 14 Quadratic Forms of Type F4.- 15 Hexagonal Systems, I.- 16 An Inventory of Moufang Polygons.- 17 Main Results.- III The Classification of Moufang Polygons.- 18 A Bound on n.- 19 Triangles.- 20 Alternative Division Rings, II.- 21 Quadrangles.- 22 Quadrangles of Involution Type.- 23 Quadrangles of Quadratic Form Type.- 24 Quadrangles of Indifferent Type.- 25 Quadrangles of Pseudo-Quadratic Form Type, I.- 26 Quadrangles of Pseudo-Quadratic Form Type, II.- 27 Quadrangles of Type E6, E7 and E8.- 28 Quadrangles of Type F4.- 29 Hexagons.- 30 Hexagonal Systems, II.- 31 Octagons.- 32 Existence.- IV More Results on Moufang Polygons.- 33 BN-Pairs.- 34 Finite Moufang Polygons.- 35 Isotopes.- 36 Isomorphic Hexagonal Systems.- 37 Automorphisms.- 38 Isomorphic Quadrangles.- V Moufang Polygons and Spherical Buildings.- 39 Chamber Systems.- 40 Spherical Buildings.- 41 Classical, Algebraic and Mixed Buildings.- 42 Appendix.- Index of Notation.

Summary

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.

Additional text

From the reviews:

"In this excellently written book, the authors give a full classification for Moufang polygons. … The book is self contained … . the content of the book is accessible for motivated graduate students and researchers from every branch of mathematics. We recommend this book for everybody who is interested in the developments of the modern algebra, geometry or combinatorics." (Gábor P. Nagy, Acta Scientiarum Mathematicarum, Vol. 71, 2005)
"The publication of this long-awaited book is a major event for geometry in general, and for the theory of buildings in particular. … The classifications established in this book are splendid achievements of fundamental significance. The whole book is extremely well written, in a clear and concise style … . It is the definitive treatment and a standard reference." (Theo Grundhöfer, Mathematical Reviews, Issue 2003 m)
"This book contains the complete classification of all Moufang generalized polygons, including the full proof. … So, in conclusion, the book is a Bible for everyone interested in classification results related to spherical buildings. It is written in a very clear and concise way. It should be in the library of every mathematician as one of the major results in the theory of (Tits) buildings, (combinatorial) incidence geometry and (algebraic) group theory." (Hendrik Van Maldeghem, Bulletin of the Belgian Mathematical Society, Vol. 11 (3), 2005)

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From the reviews:

"In this excellently written book, the authors give a full classification for Moufang polygons. ... The book is self contained ... . the content of the book is accessible for motivated graduate students and researchers from every branch of mathematics. We recommend this book for everybody who is interested in the developments of the modern algebra, geometry or combinatorics." (Gábor P. Nagy, Acta Scientiarum Mathematicarum, Vol. 71, 2005)
"The publication of this long-awaited book is a major event for geometry in general, and for the theory of buildings in particular. ... The classifications established in this book are splendid achievements of fundamental significance. The whole book is extremely well written, in a clear and concise style ... . It is the definitive treatment and a standard reference." (Theo Grundhöfer, Mathematical Reviews, Issue 2003 m)
"This book contains the complete classification of all Moufang generalized polygons, including the full proof. ... So, in conclusion, the book is a Bible for everyone interested in classification results related to spherical buildings. It is written in a very clear and concise way. It should be in the library of every mathematician as one of the major results in the theory of (Tits) buildings, (combinatorial) incidence geometry and (algebraic) group theory." (Hendrik Van Maldeghem, Bulletin of the Belgian Mathematical Society, Vol. 11 (3), 2005)

Product details

• Authors: Jacque Tits, Jacques Tits, Richard Weiss, Richard M Weiss, Richard M. Weiss
• Publisher: Springer, Berlin
• Languages: English
• Product format: Hardback
• Released: 07.10.2002
• EAN: 9783540437147
• ISBN: 978-3-540-43714-7
• No. of pages: 535
• Dimensions: 157 mm x 243 mm x 37 mm
• Weight: 924 g
• Illustrations: X, 535 p.
• Series: Springer Monographs in Mathematics; Springer Monographs in Mathematics
• Subjects: Natural sciences, medicine, IT, technology > Mathematics > Geometry; Algebra, B, Diskrete Mathematik, Gruppen und Gruppentheorie, Algebraische Geometrie, geometry, Group Theory, Combinatorics, Mathematics and Statistics, Algebraic Geometry, Discrete Mathematics, Combinatorics & graph theory, Groups & group theory, Group Theory and Generalizations