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This book introduces a new class of non-associative algebras related to certain exceptional algebraic groups and their associated buildings. Richard Weiss develops a theory of these "quadrangular algebras" that opens the first purely algebraic approach to the exceptional Moufang quadrangles. These quadrangles include both those that arise as the spherical buildings associated to groups of type E6, E7, and E8 as well as the exotic quadrangles "of type F4" discovered earlier by Weiss. Based on their relationship to exceptional algebraic groups, quadrangular algebras belong in a series together with alternative and Jordan division algebras. Formally, the notion of a quadrangular algebra is derived from the notion of a pseudo-quadratic space (introduced by Jacques Tits in the study of classical groups) over a quaternion division ring. This book contains the complete classification of quadrangular algebras starting from first principles. It also shows how this classification can be made to yield the classification of exceptional Moufang quadrangles as a consequence. The book closes with a chapter on isotopes and the structure group of a quadrangular algebra. Quadrangular Algebras is intended for graduate students of mathematics as well as specialists in buildings, exceptional algebraic groups, and related algebraic structures including Jordan algebras and the algebraic theory of quadratic forms.

Forms, Quadratic. --- Algebra. --- Mathematics --- Mathematical analysis --- Quadratic forms --- Diophantine analysis --- Forms, Binary --- Number theory --- Algebra over a field. --- Algebraic group. --- Associative property. --- Axiom. --- Classical group. --- Clifford algebra. --- Commutator. --- Defective matrix. --- Division algebra. --- Fiber bundle. --- Geometry. --- Isotropic quadratic form. --- Jacques Tits. --- Jordan algebra. --- Moufang. --- Non-associative algebra. --- Polygon. --- Precalculus. --- Projective plane. --- Quadratic form. --- Simple Lie group. --- Subgroup. --- Theorem. --- Vector space.

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Mathematics

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"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"--

Moufang loops. --- Jordan algebras. --- Buildings (Group theory) --- Graph theory. --- Polygons. --- Nonassociative rings and algebras -- Jordan algebras (algebras, triples and pairs) -- Exceptional Jordan structures. --- Group theory and generalizations -- Structure and classification of infinite or finite groups -- Groups with a $BN$-pair; buildings. --- Geometry -- Finite geometry and special incidence structures -- Generalized quadrangles, generalized polygons. --- Geometry -- Finite geometry and special incidence structures -- Buildings and the geometry of diagrams.

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Descent in Buildings begins with the resolution of a major open question about the local structure of Bruhat-Tits buildings. The authors then put their algebraic solution into a geometric context by developing a general fixed point theory for groups acting on buildings of arbitrary type, giving necessary and sufficient conditions for the residues fixed by a group to form a kind of subbuilding or "form" of the original building. At the center of this theory is the notion of a Tits index, a combinatorial version of the notion of an index in the relative theory of algebraic groups. These results are combined at the end to show that every exceptional Bruhat-Tits building arises as a form of a "residually pseudo-split" Bruhat-Tits building. The book concludes with a display of the Tits indices associated with each of these exceptional forms.This is the third and final volume of a trilogy that began with Richard Weiss' The Structure of Spherical Buildings and The Structure of Affine Buildings.

Buildings (Group theory) --- Combinatorial geometry. --- Geometric combinatorics --- Geometrical combinatorics --- Combinatorial analysis --- Discrete geometry --- Theory of buildings (Group theory) --- Tits's theory of buildings (Group theory) --- Linear algebraic groups --- Bruhat-Tits building. --- Clifford invariant. --- Coxeter diagram. --- Coxeter group. --- Coxeter system. --- Euclidean plane. --- Fundamental Theorem of Descent. --- Moufang building. --- Moufang condition. --- Moufang polygon. --- Moufang quadrangle. --- Moufang set. --- Moufang structure. --- Pfister form. --- Structure Theorem. --- Tits index. --- abelian group. --- absolute Coxeter diagram. --- absolute Coxeter system. --- absolute rank. --- affine building. --- algebraic group. --- anisotropic pseudo-quadratic space. --- anisotropic quadratic space. --- anti-isomorphism. --- apartment. --- arctic region. --- automorphism. --- bilinear form. --- biquaternion division algebra. --- building. --- canonical isomorphism. --- chamber. --- compatible representation. --- descent group. --- descent. --- discrete valuation. --- exceptional Moufang quadrangle. --- exceptional quadrangle. --- finite dimension. --- fixed point building. --- fixed point theory. --- gem. --- generalized quadrangle. --- hyperbolic plane. --- hyperbolic quadratic module. --- hyperbolic quadratic space. --- involutory set. --- isomorphism. --- isotropic quadratic space. --- length function. --- non-abelian group. --- parallel residues. --- polar space. --- projection map. --- proper indifferent set. --- proper involutory set. --- pseudo-quadratic space. --- pseudo-split building. --- quadratic form. --- quadratic module. --- quadratic space. --- quaternion division algebra. --- ramified quadrangle. --- ramified quaternion division algebra. --- ramified separable quadratic extension. --- relative Coxeter diagram. --- relative Coxeter group. --- relative Coxeter system. --- relative rank. --- residual quadratic spaces. --- residue. --- root group sequence. --- root. --- round quadratic space. --- scalar multiplication. --- semi-ramified quadrangle. --- separable quadratic extension. --- simplicial complex. --- special vertex. --- spherical building. --- split quadratic space. --- standard involution. --- subbuilding of split type. --- subbuilding. --- tamely ramified division algebra. --- thick building. --- thin T-building. --- trace map. --- trace. --- unramified quadrangle. --- unramified quadratic space. --- unramified quaternion division algebra. --- unramified separable quadratic extension. --- vector space. --- vertex. --- weak isomorphism. --- wild quadratic space.