CHAPTER 1

Introduction

The irreducible spherical buildings of rank at least three have been classified

by J. Tits in 1974 [15]. The irreducible spherical buildings of rank two - which

are called generalized polygons - are too numerous to classify, but in the addenda

of [15], the Moufang condition for spherical buildings was introduced, and it was

observed that every thick irreducible spherical building of rank at least three as well

as every irreducible residue of such a building satisfies the Moufang condition. In

this sense, the Moufang polygons are the "building bricks" of any spherical building

of rank at least three.

Very recently, the classification of Moufang polygons has been completed by J.

Tits and R. Weiss in [20]. It was first shown by J. Tits (see [17] and [18]) that

Moufang n-gons exist for n G {3,4,6,8} only; see also [21]. For n 6 {3,6,8}, the

proof is divided into two parts, namely (A) it is shown that a Moufang n-gon can

be parametrized by a certain algebraic structure, and (B) these algebraic structures

are classified.

More precisely, it was already shown in 1933 (but in a slightly different form;

see [2] or [5]) by R. Moufang (see [11]) that all Moufang triangles can be described

by an alternative division ring, a notion which had been introduced by M. Zorn (see

[22]). These alternative division rings were classified by R. Bruck and E. Kleinfeld

in 1951; see [3]. The Moufang hexagons are described by unital quadratic Jordan

division algebras of degree three, also known as anisotropic cubic norm structures

(see [16]). These structures have been classified in its full generality in 1986 by

H. Petersson and M. Racine (see [13] and [14]), whose proof is built on earlier

work by A. Albert [1], F.D. Jacobson and N. Jacobson [6], N. Jacobson [7], [8] and

K. McCrimmon [9], [10]. The Moufang octagons, finally, can be described by a

so-called octagonal system, as was shown by J. Tits in 1983 (see [19]); since these

systems have a very simple description, there is no need for part (B) in this case.

The classification of Moufang quadrangles (n=4) in [20] is not organized in

this way due to the absence of a suitable algebraic structure. Instead, there are

six different parameter systems, and even then, the division of the proof into parts

(A) and (B) is missing in the two cases which lead to the exceptional quadrangles.

Surprisingly, one of these classes, namely the exceptional quadrangles of type F4,

had only recently been discovered by R. Weiss during the classification process; see

also [12].

The goal of this article is to present a uniform algebraic structure for Mou-

fang quadrangles. These "quadrangular systems" reveal some of the structure of

Moufang quadrangles which is hard to see without them. For example, we have

successfully used them to answer a basic question about the automorphism group

of the Moufang quadrangles of type F4 left open in (37.38) of [20]; see [4]. More-

over, it is possible to classify these structures without referring back to the original