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Moufang loops. --- Tits, Jacques.

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Abelian p-groups --- Finite generalized quadrangles --- Moufang loops --- Mathematical models --- Abelian p-groups - Mathematical models --- Finite generalized quadrangles - Mathematical models --- Moufang loops - Mathematical models

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Ordered algebraic structures. --- Lie groups. --- Lie algebras. --- Lie, Algèbres de. --- Lie, Groupes de. --- Structures algébriques ordonnées. --- Moufang loops. --- Cayley numbers (Algebra). --- Cayley numbers (Algebra) --- Moufang loops --- Cayley octave (Algebra) --- Cayley's numbers (Algebra) --- Cayley's octave (Algebra) --- Octonions --- Cayley algebras --- Loops, Moufang --- Loops (Group theory)

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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.

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In The Structure of Affine Buildings, Richard Weiss gives a detailed presentation of the complete proof of the classification of Bruhat-Tits buildings first completed by Jacques Tits in 1986. The book includes numerous results about automorphisms, completions, and residues of these buildings. It also includes tables correlating the results in the locally finite case with the results of Tits's classification of absolutely simple algebraic groups defined over a local field. A companion to Weiss's The Structure of Spherical Buildings, The Structure of Affine Buildings is organized around the classification of spherical buildings and their root data as it is carried out in Tits and Weiss's Moufang Polygons.

Buildings (Group theory) --- Moufang loops --- Automorphisms --- Affine algebraic groups --- Moufang loops. --- Automorphisms. --- Affine algebraic groups. --- Algebraic groups, Affine --- Loops, Moufang --- Theory of buildings (Group theory) --- Tits's theory of buildings (Group theory) --- Group schemes (Mathematics) --- Group theory --- Symmetry (Mathematics) --- Loops (Group theory) --- Linear algebraic groups --- Buildings (Group theory). --- Addition. --- Additive group. --- Additive inverse. --- Algebraic group. --- Algebraic structure. --- Ambient space. --- Associative property. --- Automorphism. --- Big O notation. --- Bijection. --- Bilinear form. --- Bounded set (topological vector space). --- Bounded set. --- Calculation. --- Cardinality. --- Cauchy sequence. --- Commutative property. --- Complete graph. --- Complete metric space. --- Composition algebra. --- Connected component (graph theory). --- Consistency. --- Continuous function. --- Coordinate system. --- Corollary. --- Coxeter group. --- Coxeter–Dynkin diagram. --- Diagram (category theory). --- Diameter. --- Dimension. --- Discrete valuation. --- Division algebra. --- Dot product. --- Dynkin diagram. --- E6 (mathematics). --- E7 (mathematics). --- E8 (mathematics). --- Empty set. --- Equipollence (geometry). --- Equivalence class. --- Equivalence relation. --- Euclidean geometry. --- Euclidean space. --- Existential quantification. --- Free monoid. --- Fundamental domain. --- Hyperplane. --- Infimum and supremum. --- Jacques Tits. --- K0. --- Linear combination. --- Mathematical induction. --- Metric space. --- Multiple edges. --- Multiplicative inverse. --- Number theory. --- Octonion. --- Parameter. --- Permutation group. --- Permutation. --- Pointwise. --- Polygon. --- Projective line. --- Quadratic form. --- Quaternion. --- Remainder. --- Root datum. --- Root system. --- Scientific notation. --- Sphere. --- Subgroup. --- Subring. --- Subset. --- Substructure. --- Theorem. --- Topology of uniform convergence. --- Topology. --- Torus. --- Tree (data structure). --- Tree structure. --- Two-dimensional space. --- Uniform continuity. --- Valuation (algebra). --- Vector space. --- Without loss of generality.