Preface

The classical groups, for which there is no canonical definition, may be taken to be the linear groups on right vector spaces of finite dimension over a field or skew field and the subgroups of linear groups on right vector spaces of arbitrary dimension leaving invariant a non-degenerate hermitian or skew-hermitian pseudo-quadratic form of finite Witt index. The notion of a pseudo-quadratic form was introduced by Jacques Tits in Chapter 8 of [11] to allow a characteristic-free description of these groups. To each classical group (as defined here) there is associated, via the notion of a BN-pair, a spherical building.

The classification of irreducible spherical buildings of rank at least three in [11]¹ showed that the only such buildings are those arising from

(i) a classical group,

(ii) an isotropic K-form of an exceptional algebraic group or

(iii) a group of “mixed type.”

The existence of isotropic K-forms of exceptional algebraic groups (and the corresponding spherical buildings) is intimately connected with the existence of certain classes of non-associative algebras, in particular, alternative division rings and quadratic Jordan division algebras of degree three.

The definition of an alternative division ring can be obtained from the definition of a skew field by keeping certain identities which hold in a skew field and then deleting the associativity of multiplication.

The definition of a quadratic Jordan division algebra of degree three (called, more succinctly, an hexagonal system in [12]) can be obtained by taking a cubic separable field extension E/K, keeping certain identities involving the norm and trace as well as the structure of E as a vector space over K and then deleting the multiplication on E (and all the axioms which involve this multiplication).

In both cases, it is geometry (the connection to projective planes in the first case, to Moufang hexagons in the second) which guided the choice of identities to be kept.²

More precisely, a maximal unipotent subgroup of one of these forms is defined by commutator relations expressed in terms of a quadrangular algebra in the corresponding isotopy class, and the anisotropic kernel is precisely the structure group of . All of these things are explained in Chapters 10–12.

By [10] and Chapter 41 (see Figures 11–14) of [12], all the non-split forms of exceptional groups of K-rank greater than one, other than the three indicated above, are classified by either alternative division algebras³ or hexagonal systems.⁴ In this sense, quadrangular algebras belong in a series with alternative and Jordan division algebras.

The remaining quadrangular algebras exist only in characteristic two. Improper quadrangular algebras are of two types, one classified by degenerate anisotropic quadratic spaces and the other by indifferent sets, the structures which give rise to (and classify) the Moufang quadrangles of indifferent type.

The most exotic quadrangular algebras are those which are defective but not improper. These exist, it turns out, only in dimension four, not, however, over the original field K, but instead over a certain field F extending K (and which might even be of infinite dimension over K) such that F/K is purely inseparable. These quadrangular algebras classify the Moufang quadrangles of type F4 which were discovered in the course of working out the original proof of Theorem 21.12 in [12].

Those parts of the classification of quadrangular algebras presented here in Chapters 1–7 were obtained to a large extent by extracting the purely algebraic parts of Chapters 26–28 in [12] (i.e. the proof of Theorem 21.12), supplying algebraic arguments along the way to repair the numerous gaps left by this strategy. This distillation of the purely algebraic parts of the classification of the exceptional Moufang quadrangles yields a much clearer and more satisfying understanding of these remarkable geometrical structures.

By combining the classification of quadrangular algebras with just the first few pages of Chapter 26 of [12], we obtain a new version of the proof of Theorem 21.12. This is explained in Chapter 11.

We emphasize that the classification of quadrangular algebras given in Chapters 1–9 is completely elementary and (except for a few standard facts about Clifford algebras cited in Chapter 2) self-contained.

In [3] Tom De Medts introduced a class of algebras he called “quadrangular systems” which parametrize all six families of Moufang quadrangles, not just the exceptional quadrangles considered here, and gave a reorganization of the classification of all Moufang quadrangles (Chapters 21–28 in [12]) based on this unifying concept. Our notion of a quadrangular algebra, which focuses only on the exceptional quadrangles, is quite different.

As indicated above, we hope that quadrangular algebras will turn out to be of interest alongside alternative and Jordan algebras. A first step toward a more general theory might be to investigate quadrangular algebras which do not satisfy the last axiom, the one requiring a certain “quasi-pseudo quadratic form” (a frightening term which we will, in fact, avoid) to be anisotropic.

This book was written while the author was a guest of the University of Würzburg and while he was supported by a Research Prize from the Humboldt Foundation. He would like to express his gratitude to both of these institutions. It is also a pleasure to thank Yoav Segev, whose comments and suggestions led to many improvements in the text.

1 Subsequently extended in [12] to the case of spherical buildings of rank two, also called generalized polygons, satisfying the Moufang condition.

2 In fact, quadratic Jordan algebras were introduced by McCrimmon who was building on work of Jacobson and Albert going back to [7], i.e. to quantum mechanics rather than geometry. What is nevertheless true is that the notion of an hexagonal system would have turned up in Tits’s work on Moufang hexagons (done at a time when he was a frequent visitor at Yale) even if these other developments had never taken place.

3 More precisely, those which are quadratic in the sense of (20.1) and (20.3) in [12].

4 Classified by Racine and Petersson (building on years of work by Jacobson, McCrimmon and others) in [8] and [9]; see Chapters 15 and 30 in [12].