Moufang Polygons

Spherical buildings are certain combinatorial simplicial complexes introduced, at first in the language of “incidence geometries,” to provide a systematic 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.

In this chapter, we assemble the few standard definitions and elementary results from graph theory, group theory and the theory of algebras which we will require in Parts I–IV.

A generalized polygon is a bipartite graph such that $$ {\text{grith = 2}} \times {\text{diameter < }}\infty {\text{.}} $$

Let Γ be a generalized n-gon for some n ≥ 3.

We continue to assume that Γ is a Moufang n-gon for some n ≥ 3 and that G is a subgroup of Aut(Γ) containing the root groups corresponding to every half-apartment of Γ. We now choose once and for all an apartment Σ and a labeling of the vertices of Σ by the integers as defined in (4.15). For each i, let U(i, i+1,..., i+n) be as in (4.1).

Two half-apartments will be called opposite if their union is an apartment. In this case, we say that the corresponding root groups are opposite. In Chapter 5, we proved the first few results about the subgroups U_{[i, i + n − 1]}. In this chapter, we turn our attention to pairs U _i , U _{i + n} of opposite root groups.

We continue to assume that Γ is a Moufang n-gon for some n ≥ 3, that G is a subgroup of Aut(Γ) containing all the root groups of Γ, that Σ is an apartment labeled by the integers and that the root groups U _i are as in (5.1).

The two propositions (5.5) and (5.6) govern the structure of U_[1,n].

At this point, we have assembled all the tools we will need for the classification of Moufang polygons, so that the enthusiastic reader can turn immediately to Chapter 18 and get started. The less impulsive reader, however, will want to have in advance a complete list of the Moufang polygons which come out of the classification.

In this chapter, we introduce two types of parameter systems which exist only in characteristic two; in Chapter 14, we will introduce a third.

In this chapter, we introduce involutory sets and pseudo-quadratic forms. pseudo-quadratic forms are closely related to hermitian and skew-hermitian forms. They were first introduced in 1968 in an earlier version of [101]; see also [18].

In this chapter, we introduce the quadratic forms which will be required in (16.6) to describe the quadrangles of type E₆, E₇ and E₈. For basic material on Clifford algebras, we follow [38]. In particular, (12.26), (12.27), (12.28) and the main ideas in the proof of (12.43) are taken from this source.

This chapter is a continuation of the previous one. Let (K, L₀,q) be a quadratic space of type E _k for k = 6, 7 or 8 as defined in (12.31) and let ϵ be an element of L₀*.

In this chapter, we introduce the class of quadratic forms which will be required in (16.7) to describe the quadrangles of type F₄. These quadrangles (and the corresponding quadratic forms) are thus named because of their connection to certain mixed groups of type F₄; see (41.20). The quadratic forms of type F₄ are defective (in fact, the defect can be of arbitrarily large dimension) and they exist only over certain imperfect fields of characteristic two.

In this chapter, we describe the algebraic systems which arise as parameters in the study of Moufang hexagons; see (15.14) below.

In this chapter, we describe nine families of Moufang polygons. In Part III of this book, we prove that every Moufang polygon is isomorphic to one of the Moufang polygons described in this chapter.

In Part III, we prove Theorems 17.2–17.8.

In this chapter, we prove Theorem 17.2. Our goal is to show that Moufang triangles are parametrized by alternative division rings. See (9.1) for the definition of an alternative division ring A and (16.1) for the definition of the Moufang triangle T(A).

In this chapter we prove Theorem 17.3. Our goal is to show that the Cayley-Dickson algebras defined in (9.8) are the only non-associative alternative division rings. This result was first proved in [17] and [56] by R. Bruck and E. Kleinfeld. See also [3], [74] and [87]. In the proof we give here, the characteristic does not play any role.

In this chapter, we begin the classification of Moufang quadrangles which was formulated in Theorem 17.4. The proof of Theorem 17.4 will be spread over this and the next seven chapters. We begin here with some preliminary results which enable us to break Theorem 17.4 up into several sub-theorems.

We continue to assume that n = 4. In this chapter, we prove Theorem 21.8. Our goal is to classify the quadrangles which are reduced but not normal; see (21.2) and (21.7) for the definitions of these terms.

We continue to assume that n = 4. In this chapter, we prove Theorem 21.9. Our goal is to show that normal quadrangles are parametrized by anisotropic quadratic spaces as defined in (12.2). For the definition of a normal quadrangle, see (21.7).

We continue to assume that n = 4. In this chapter, we prove Theorem 21.10 which asserts that indifferent quadrangles are of indifferent type.

We continue to assume that n = 4. In Chapters 22–24, we have given the proofs of Theorems 21.8–21.10. We can thus assume that Γ is wide as defined in (21.2). By (21.6), Γ is an extension of a reduced quadrangle Ω; see (21.5) for the definition of an extension.

In this chapter, we begin the proof of Theorem 21.12. Our goal is to give the classification of extensions of quadrangles of quadratic form type.

In this chapter, we continue with the proof of Theorem 21.12 begun in the previous chapter. We assume that m > 4, where m = dim _K L₀. In the chapter, we assume, too, that if char(K) = 2, then U₁ is not elementary abelian.

In this final chapter of the classification of Moufang quadrangles, we complete the proof of Theorem 21.12. By the results of the previous two chapters, it remains only to consider the case that char (K) = 2, U₁ is elementary abelian and m > 4, where m = dim _K L₀.

In this chapter, we prove Theorem 17.5. Our goal is to show that Moufang hexagons are parametrized by hexagonal systems as stated in [102]. See (15.15) for the definition of an hexagonal system (J, F, #) and (16.8) for the definition of the Moufang hexagon H(J, F, #).

In this chapter, we give the classification of hexagonal systems as formulated in Theorem 17.6. Our goal is to show that the list of hexagonal systems described in (15.14) and summarized in Figure 2 on page 148 is complete.

In this chapter, we prove Theorem 17.7. Our goal is to present the classification of Moufang octagons.

In this chapter, we show that the graphs defined in Chapter 16 are actually Moufang polygons. These graphs are described in terms of root group sequences. As was observed in the introduction to that chapter, it will suffice by Theorem 8.11 to show that in each case, the defining root group sequence satisfies the conditions M₃ and M₄ given on pages 35 and 37.

In Part III, we proved that every Moufang polygon belongs (up to isomorphism) to one of the nine families described in Chapter 16. In Part IV, we study the isomorphism problem (roughly speaking) for Moufang polygons. In particular, we examine the structure of the automorphism group of a Moufang polygon (family by family), we investigate the pairs of parameter systems of a given type which give rise to isomorphic polygons and we determine the Moufang quadrangles which are (up to isomorphism) of more than one type (i.e. belong to more than one family).

Both the statement and our proof of the classification of Moufang polygons can be simplified if we restrict our attention to finite Moufang polygons. In this chapter, we take a closer look at this case.

We have shown that every Moufang polygon belongs (up to isomorphism) to one of the nine families of Moufang polygons described in Chapter 16 and summarized in Figures 3 and 4 on pages 165 and 167. Each family is described in terms of systems of parameters. In this chapter, we examine the extent to which the parameter systems are invariants of the polygons constructed from them.

In Chapter 15, we described six families, or types, of hexagonal systems; they are summarized in Figure 2 on page 148. In Chapter 30, we showed that every hexagonal system belongs to one of these families and in (35.13), we showed that two hexagonal systems give rise to isomorphic Moufang hexagons if and only if they are similar as defined in (29.36). In this chapter, we investigate the extent to which the different families overlap and the extent to which non-isomorphic hexagonal systems can be similar.

In this chapter, we examine the structure of the automorphism group of a Moufang polygon.

Two generalized polygons isomorphic to each other must have the same girth and are thus generalized n-gons for the same value of n. For n = 3, 6 and 8, two Moufang n-gons are isomorphic if and only if they arise from isotopic or anti-isotopic parameter systems as defined in Chapter 35. There are, however, six different families, or types, of Moufang quadrangles. In this chapter, we determine their pairwise intersections.

Irreducible spherical buildings of rank at least three were classified in [101]. In Chapter 40, we give a proof of this result which is based on the classification of Moufang polygons. The project of carrying out such a proof was first proposed in the addenda of [101]. See (40.15), (40.17), (40.22) and (40.56) for a statement of the main results.

In this chapter, we give a proof of the classification of irreducible spherical buildings of rank at least three which is based on the classification of Moufang polygons.

All Moufang spherical buildings can be found “in nature” as geometries associated with classical groups, with algebraic groups and with the mixed groups introduced in (10.3.2) of [101]. In this last chapter, we describe (without proofs) the Moufang spherical buildings of rank at least two as given in (16.1)–(16.9), (40.22) and (40.63)–(40.64) in terms of these three kinds of groups. Our main purpose is merely to provide a kind of lexicon between spherical buildings as organized from the “local” point of view taken in this book and spherical buildings as they are found “in nature.” In the appendix, we take a much more detailed look at spherical buildings as they arise in the theory of algebraic groups.

Up until the last chapter, we have adopted throughout this book a purely algebraic and graph-theoretical point of view with prerequisites of the most elementary kind. In this appendix, we take a closer look at some of the results concerning algebraic buildings (especially those concerning existence and automorphisms) allowing now the use, without definitions or proofs except possibly for heuristic substitutes, of standard notions and results of algebraic group theory and Galois descent. Further details about those can be found, for instance, in [7], [8], II, §1, of [18], [26] and [95].