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AuthorTits, Jacques. author
TitleMoufang Polygons [electronic resource] / by Jacques Tits, Richard M. Weiss
ImprintBerlin, Heidelberg : Springer Berlin Heidelberg : Imprint: Springer, 2002
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Descript X, 535 p. online resource


Spherical buildings are certain combinatorial simplicial complexes introยญ duced, at first in the language of "incidence geometries," to provide a sysยญ 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 relaยญ tive rank ยฃ 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 ยฃ ::::: 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


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

Mathematics Algebra Algebraic geometry Group theory Geometry Discrete mathematics Combinatorics Mathematics Geometry Algebra Discrete Mathematics Algebraic Geometry Group Theory and Generalizations Combinatorics


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