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
Previous Page Next Page