چندضلعی‌های سینه

A generalized polygon is the same thing as a spherical building of rank 2. An arbitrary thick irreducible spherical building of rank at least 3 is uniquely determined by the substructure spanned by the irreducible rank 2 residues containing a single chamber. These residues all satisfy the Moufang condition. Moufang polygons (that is, generalized polygons satisfying the Moufang condition) were classified and this classification was used to give a new proof of the classification of irreducible spherical buildings of rank greater than 2. In the first few chapters, the study of Moufang polygons is reduced to the study of “root group sequences” and the commutator relations that define them. In the course of the classification, it is shown in each case that the root group sequence in question can be coordinatized by a suitable algebraic structure. In this way, composition algebras, quadratic spaces, pseudo-quadratic spaces, quadratic Jordan algebras of degree 3 (also known as cubic norm structures) and more exotic algebraic structures come into play. In every case, the algebraic structure that arises must be anisotropic in a suitable sense. "We introduce the notion of a Tits polygon, a generalization of the notion of a Moufang polygon, and show that Tits polygons arise in a natural way from certain configurations of parabolic subgroups in an arbitrary spherical buildings satisfying the Moufang condition. We establish numerous basic properties of Tits polygons and characterize a large class of Tits hexagons in terms of Jordan algebras. We apply this classification to give a "rank 2" presentation for the group of F-rational points of an arbitrary exceptional simple group of F-rank at least 4 and to determine defining relations for the group of F-rational points of an an arbitrary group of Frank 1 and absolute type D4, E6, E7 or E8 associated to the unique vertex of the Dynkin diagram that is not orthogonal to the highest root. All of these results are over a field of arbitrary characteristic". Sommario fornito dall'editore Cover 1 Title page 2 Introduction 10 Acknowledgment 12 Chapter 1. Tits polygons 14 1.1. Basic definitions 15 1.2. Examples 17 1.3. Commutator relations 26 1.4. Opposite roots 32 1.5. Uniqueness 36 1.6. A bound on n 43 Chapter 2. Tits hexagons 50 2.1. Cubic norm structures 51 2.2. Hexagons 55 2.3. Coordinates for Δ 58 2.4. Hexagons of polar type 62 2.5. The associated cubic norm structure 67 2.6. Automorphisms and classification 73 Chapter 3. Groups of relative rank 1 80 3.1. Descent 81 3.2. The subgraph Λ 82 3.3. The Galois involution ω 85 3.4. The Moufang set M(Δ,⟨ω⟩) 88 3.5. The structure map τ 91 3.6. The generic case 94 3.7. A formula for τ 96 3.8. Arbitrary Galois groups 101 Chapter 4. Appendix by Holger P. Petersson 106 4.1. Cubic norm structures 106 4.2. The cubic norm structure H(C,K) 114 4.3. Irreducibility of the structure group 118 Bibliography 124 Index 126 Back Cover 132 "We introduce the notion of a Tits polygon, a generalization of the notion of a Moufang polygon, and show that Tits polygons arise in a natural way from certain configurations of parabolic subgroups in an arbitrary spherical buildings satisfying the Moufang condition. We establish numerous basic properties of Tits polygons and characterize a large class of Tits hexagons in terms of Jordan algebras. We apply this classification to give a "rank 2" presentation for the group of F-rational points of an arbitrary exceptional simple group of F-rank at least 4 and to determine defining relations for the group of F-rational points of an an arbitrary group of Frank 1 and absolute type D4, E6, E7 or E8 associated to the unique vertex of the Dynkin diagram that is not orthogonal to the highest root. All of these results are over a field of arbitrary characteristic"-- Provided by publisher