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From the Moufang World to the Structurable World and Back Again

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From the Moufang World to the Structurable World and Back Again Faculteit Wetenschappen Vakgroep Wiskunde 27 juni 2013 From the Moufang world to the structurable world and back again Lien Boelaert Promotor: Hendrik Van Maldeghem Proefschrift voorgelegd aan de Faculteit Wetenschappen tot het behalen van de graad van Doctor in de Wetenschappen richting wiskunde. Contents Introduction vii 1 General preliminaries 1 1.1 Quadratic forms and forms of higher degree . 1 1.2 Hermitian and skew-hermitian spaces . 4 1.3 Composition algebras . 5 1.4 Jordan algebras . 6 1.4.1 Definition and basic properties . 6 1.4.2 Peirce decomposition . 8 1.5 Lie algebras and linear algebraic groups . 12 2 The Moufang world 15 2.1 Definition and basic properties of Moufang polygons . 15 2.2 Quadrangular algebras . 18 2.2.1 Definition and basic properties . 19 2.2.2 Pseudo-quadratic spaces . 21 2.2.3 Quadrangular algebras of type E6;E7 and E8 . 23 2.3 Description of Moufang quadrangles . 27 2.4 Moufang sets . 31 2.4.1 Definitions and basic properties . 31 2.4.2 Examples of Moufang sets . 35 3 The structurable world 43 3.1 Definitions and basic properties of structurable algebras . 44 3.2 Conjugate invertibility in structurable algebras . 46 3.3 Examples of structurable algebras . 47 3.3.1 Associative algebras with involution . 47 3.3.2 Jordan algebras . 48 3.3.3 Hermitian structurable algebras . 48 3.3.4 Structurable algebras of skew-dimension one . 50 3.3.5 Forms of the tensor product of two composition algebras 53 3.4 Construction of Lie algebras from structurable algebras . 57 3.5 Isotopies of structurable algebras . 61 3.6 Freudenthal triple systems and structurable algebras . 64 iii iv Contents 3.7 J-ternary algebras . 67 4 Structurable algebras on quadrangular algebras 69 4.1 Quadrangular algebras are Freudenthal triple systems . 70 4.2 Structurable algebras on general quadrangular algebras . 73 4.3 Structurable algebras on pseudo-quadratic spaces . 75 4.4 Structurable algebras on quadrangular algebras of type E6, E7 and E8 ............................. 77 4.4.1 The quadrangular algebra of type E8 . 78 4.4.2 The pseudo-quadratic spaces on E6 and E7 . 81 5 A coordinate-free construction of quadrangular algebras 83 5.1 Special J-modules . 85 5.2 Construction of general quadrangular algebras . 87 5.3 Construction of pseudo-quadratic spaces . 90 5.4 Construction of quadrangular algebras of type E6;E7;E8 . 92 5.4.1 A characterization of quadratic forms of type E6;E7;E8 92 5.4.2 The construction . 94 5.5 Construction of arbitrary Moufang quadrangles from special Jordan modules . 103 6 A construction of Moufang sets from structurable division algebras 109 6.1 One-invertibility in A × S . 110 6.2 Structurable division algebras are one-invertible . 118 6.3 The construction of Moufang sets . 123 6.4 Examples . 128 Appendices A Explicit construction of a structurable algebra associated to a quadrangular algebra 137 A.1 The construction . 138 A.2 Structurable algebras on quadrangular algebras of type E6, E7 and E8 .............................144 B Implementations in Sage 149 C Dutch summary 155 Introduction The main theme of this thesis is to study some algebraic structures related to exceptional linear algebraic groups of relative rank 1 or 2. More precisely, the goal of this thesis is to connect the Moufang world and the structurable world, which seem to have been two isolated islands up to now. Some history Since the beginning of the 20th century mathematicians try to get hold on linear algebraic groups. These are certain structures that arise in different areas of mathematics and gave rise to the development of a lot of interesting theories. The theory of linear algebraic groups is a very active area of research, that contains a lot of open questions. Amongst others it is not clear how certain forms of exceptional linear algebraic groups can be described explicitly. From classification results we know they exist, but for certain forms of exceptional linear algebraic groups no explicit description is known. In order to get a deeper understanding of the isotropic linear algebraic groups, Jacques Tits introduced the notion of a building, which is a geomet- rical structure built up from the parabolics of the linear algebraic group. On the other hand, these geometries can be defined axiomatically, giving rise to a more general notion, where more examples occur. If a spherical building arises from a linear algebraic group, then the linear algebraic group has a natural action on its building. Therefore studying buildings could be useful to get a better understanding of those groups. In 1974, Jacques Tits published his lecture notes \Buildings of Spheri- cal Type and Finite BN-Pairs" [Tit74], in which he classified all spherical buildings of rank at least 3. The following quote is taken from loc. cit.: The origin of the notions of buildings and BN-pairs lies in an attempt to give a systematic procedure for the geometric inter- pretation of the semi-simple Lie groups and, in particular, the exceptional groups. vii viii Introduction There is no hope to classify spherical buildings of rank two. For example, the class of buildings of rank 2 contains the class of all projective planes, and these are impossible to classify. This is why it is necessary to impose an extra condition: the Moufang condition. It is only 26 years later that Jacques Tits and Richard Weiss in [TW02] finished the classification of spherical buildings of rank 2 satisfying the Mou- fang property, also called Moufang polygons; see Chapter 2. There is no doubt that the hardest part in the whole classification is precisely where the quadrangles coming from exceptional linear algebraic groups of relative rank 2 turn up. All the Moufang quadrangles can in turn be described using an algebraic structure, that is well understood when the Moufang polygon is coming from an (infinite dimensional) classical linear algebraic group of rank 2. However the algebraic structure that determines Moufang quadrangles of type E6, E7 and E8, has a very artificial coordinate-based definition, that came up during the classification, containing (in the case of E8) a 12-dimensional quadratic space, a 32-dimensional vector space, and several maps between those spaces. In fact, there had been several attempts to try to understand the struc- ture of these exceptional quadrangles. For example, Tom De Medts in- troduced the notion of quadrangular systems [DM05], a uniform algebraic structure describing all Moufang quadrangles. From a different point of view, Richard Weiss defined quadrangular algebras [Wei06b], which are a set of algebraic data coordinatizing the exceptional Moufang quadrangles. If we exclude the characteristic 2 case, then a quadrangular algebra is either obtained from a pseudo-quadratic space, or it is the structure that coordi- natizes Moufang quadrangles of type E6, E7 and E8. Spherical Moufang buildings of rank one are called Moufang sets, as they can be described without using any geometry only using a set and a collection of groups that acts on this set; see Section 2.4. One is far from giving a classification of Moufang sets, it is even not clear if it will be ever possible to give such a classification. In view of this, it is of interest to find descriptions of Moufang sets. In particular (and this is one of the main motivations for studying Moufang sets) it is interesting to try to describe Moufang sets obtained from exceptional linear algebraic groups of relative rank one, and we hope that they will give new insights in the study of those linear algebraic groups. As an example, we mention that the Kneser{Tits 66 problem for groups of type E8;2 has recently been solved using the theory of Moufang quadrangles [PTW12], and for many of the exceptional groups of relative rank one, the Kneser{Tits problem is still open. In this thesis we take a new point of view towards these not-so-well- understood-Moufang-objects. We apply the theory of structurable algebras to give new interpretations of these Moufang structures. Structurable al- Introduction ix gebras (see Chapter 3) are certain non-associative algebras with involution, this class of algebras contains, amongst others, Jordan algebras (with trivial involution) and associative algebras with involution. In order to provide explicit constructions of exceptional Lie algebras, sev- eral authors (I.L. Kantor, B. Allison, J. Tits, J. Faulkner, W. Hein and many others) have introduced various algebraic structures. In each case, the start- ing point of those constructions is a non-associative binary or ternary alge- bra, or a pair of them. One of the earliest is the Tits{Kantor{Koecher1 con- struction of Lie algebras from Jordan algebras (see [Jac68, Section VIII.5]); its main purpose was to provide models of Lie algebras of type E7. Structurable algebras were introduced by Bruce Allison, he gives a very successful generalization of the Tits{Kantor{Koecher construction starting from structurable algebras. They provide a construction of all isotropic simple Lie algebras in characteristic 0. Structurable algebras can only be defined over fields of characteristic different from 2 and 3. If char(k) > 3, all simple Lie algebras satisfying an additional (more technical) condition can be obtained. Central simple structurable algebras have been classified (over fields of characteristic bigger than five) and consist of six different classes. New progress In Chapters 4 and 5, we relate quadrangular algebras with structurable algebras in various ways. Since quadrangular algebras of type E8 are the most interesting and least understood case, we will focus on this case in this introduction. A quadrangular algebra of type E8 is totally determined by a so-called quadratic form of type E8.
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