Search Results

This chapter assembles a few standard definitions, fixes some notation, and reviews a few of the results about buildings and Moufang polygons. It also summarizes the basic facts about Coxeter groups and buildings, including the fundamental properties of roots, residues, apartments, and projection maps. The chapter defines a Moufang building as spherical, thick, irreducible and of rank at least 2, and a Bruhat-Tits building as a thick irreducible affine building whose building at infinity is Moufang. Furthermore, it presents a fundamental result of Tits: that an irreducible thick spherical building of rank at least 3 satisfies the Moufang condition as do all the irreducible residues of rank at least 2 of such a building. Finally, it considers a simplicial complex, the dimension of which is its cardinality minus one.

This chapter introduces some basic facts about Moufang polygons and root group sequences. For each root group sequence Ω‎, there is a unique Moufang polygon Δ‎ such that Ω‎ is isomorphic to a root group sequence of Δ‎. The classification of Moufang n-gons states that, up to isomorphism, there are no other Moufang polygons. The chapter also considers the notion of an isomorphism of root group sequences and the notion of an anti-isomorphism of root group sequences. It concludes with an example involving a non-trivial anisotropic quadratic space and a generalized quadrangle with a root group sequence.

This chapter presents various results about quadratic forms of type F₄. The Moufang quadrangles of type F₄ were discovered in the course of carrying out the classification of Moufang polygons and gave rise to the notion of a quadratic form of type F₄. The chapter begins with the notation stating that a quadratic space Λ‎ = (K, L, q) is of type F₄ if char(K) = 2, q is anisotropic and: for some separable quadratic extension E/K with norm N; for some subfield F of K containing K² viewed as a vector space over K with respect to the scalar multiplication (t, s) ↦ t²s for all (t, s) ∈ K x F; and for some α‎ ∈ F* and some β‎ ∈ K*. The chapter also considers a number of propositions regarding quadratic spaces and discrete valuations.

This chapter deals with the residues of a Bruhat-Tits building whose building at infinity is an exceptional quadrangle. It begins with the remark that if Λ‎ is an arbitrary quadratic space of type Eℓ for ℓ = 6, 7 or 8 or of typeF₄ over a field K that is complete with respect to a discrete valuation, and if in the F4-case the subfield F is closed with respect to this valuation and if Δ‎ is the corresponding Moufang quadrangle of type Eℓ or F₄, then there always exists a unique affine building Ξ‎ such that Δ‎ is the building at infinity of Ξ‎ with respect to its complete system of apartments. The chapter also considers the standard embedding of the apartment A in the Euclidean plane which takes the intersection of A and R to the set of eight triangles containing the origin. Finally, it describes a Moufang polygon with two root group sequences.

This chapter summarizes the different cases about Moufang quadrangles of type E⁶, E₇ and E₈. The first case is that the building at infinity of the Bruhat-Tits building Ξ‎ is an unramified quadrangle; the second, a semi-ramified quadrangle; and the third, a ramified quadrangle. The chapter considers a theorem that takes into account two root group sequences, both of which are either indifferent or the various dimensions, types, etc., are as indicated in exactly one of twenty-three cases. It also presents a number of propositions relating to a quaternion division algebra and a quadratic space of type Eℓ for ℓ = 6, 7 or 8. Finally, it emphasizes the fact that the quadrangles of type F₄ could have been overlooked in the classification of Moufang polygons.

This chapter deals with the case that the building at infinity of the Bruhat-Tits building Ξ‎ is a Moufang quadrangle of type E⁶, E₇, and E₈. It begins with a hypothesis that takes into account a quadratic space of type Eℓ for ℓ = 6, 7 or 8, K which is complete with respect to a discrete valuation, the two residues of Ξ‎, and the two root group sequences of a Moufang polygon. It then considers the case that Ξ‎ is an unramified quadrangle if the proposition δ‎_Ψ‎ = 2 holds. It also explains two other propositions: Ξ‎ is a semi-ramified quadrangle if δ‎_Λ‎ = 1 and δ‎_Ψ‎ = 2 holds, and a ramified quadrangle if δ‎_Λ‎ = δ‎_Ψ‎ = 1 holds.