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.

Gives a complete classification of Moufang polygons, starting from first principles Includes a totally new classification of the spherical buildings of rank 3 at least J. Tits is one of the best and most influential algebraists of the past 50 years R. Weiss is one of the world's leading researchers in the field of combinatorial group theory The book will become a classic in the field

Klappentext

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.

Inhalt
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.