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Lie Algebras, Extremal Elements, and Geometries Lie algebras, extremal elements, and geometries Citation for published version (APA): Panhuis, in 't, J. C. H. W. (2009). Lie algebras, extremal elements, and geometries. Technische Universiteit Eindhoven. https://doi.org/10.6100/IR643504 DOI: 10.6100/IR643504 Document status and date: Published: 01/01/2009 Document Version: Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers) Please check the document version of this publication: • A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. 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If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, please follow below link for the End User Agreement: www.tue.nl/taverne Take down policy If you believe that this document breaches copyright please contact us at: [email protected] providing details and we will investigate your claim. Download date: 29. Sep. 2021 Lie Algebras, Extremal Elements, and Geometries Copyright c 2009 by J.C.H.W. in ’t panhuis, Eindhoven, The Netherlands. Unmodified copies can be freely distributed. Printed by Printservice Technische Universiteit Eindhoven. Cover: dual affine plane of order two. Design by Oranje Vormgevers, Eindhoven. A catalogue record is available from the Eindhoven University of Technology Library. ISBN: 978-90-386-1912-5 Lie Algebras, Extremal Elements, and Geometries PROEFSCHRIFT ter verkrijging van de graad van doctor aan de Technische Universiteit Eindhoven, op gezag van de rector magnificus, prof.dr.ir. C.J. van Duijn, voor een commissie aangewezen door het College voor Promoties in het openbaar te verdedigen op maandag 12 oktober 2009 om 16.00 uur door Jozef Clemens Hubertus Wilhelmus in ’t panhuis geboren te Roermond Dit proefschrift is goedgekeurd door de promotor: prof.dr. A.M. Cohen Copromotor: dr. F.G.M.T. Cuypers This research was financially supported by NWO (Netherlands Organisation for Scien- tific Research) in the framework of the Free Competition, grant number 613.000.437. Preface Lie algebras, extremal elements, and geometries This thesis is about Lie algebras generated by extremal elements and geometries whose points correspond to extremal points, that is, projective points corresponding to extremal elements. Inside a Lie algebra g over a field F of characteristic not two, extremal ele- ments are those nonzero elements x for which [x; [x; g]] ⊆ Fx. Extremal elements for which [x; [x; g]] = f0g are called sandwich elements. The definitions of extremal ele- ments and sandwich elements in characteristic two are somewhat more involved. Sandwich elements were originally introduced in relation with the restricted Burn- side problem. An important insight for the resolution of this problem is the fact that a Lie algebra generated by finitely many sandwich elements is necessarily finite-dimensional. While this fact was first only proved under extra assumptions, later it was proved in full generality. Extremal elements play important roles in both classical and modern Lie algebra theory. In complex simple Lie algebras, or their analogues over other fields, extremal elements are precisely the elements that are long-root vectors relative to some maximal torus. In the classication of simple Lie algebras in small characteristics extremal ele- ments are also useful: they occur in non-classical Lie algebras such as the Witt algebras. In the first chapter we give some definitions and basic results regarding Lie algebras, extremal elements, and the different geometries which are the subject of this thesis. Also we will already give a hint of how a Lie algebra can be related to a geometry using its extremal points: the points of the geometry are the extremal points in the Lie algebra and the lines are the projective lines all of whose points are extremal. Cohen and Ivanyos proved that the resulting geometry is a so-called root filtration space. Moreover, they showed that a root filtration space with a non-empty line set is the shadow space of a building. These buildings are geometrical and combinatorial structures introduced by Tits in order to obtain a better understanding of the semi-simple algebraic groups. If we are dealing with a Lie algebra for which no projective line consists entirely of extremal points, then the results of Cohen and Ivanyos are no longer applicable. There- fore, in that situation, the question is whether a non-trivial geometric structure can be associated to the extremal points in the Lie algebra. This is the subject of the second and third chapter. First, for Lie algebras generated by two or three extremal elements, we vi Preface find the isomorphism type of the corresponding Lie algebra and give a description of the extremal elements. Then, for an arbitrary number of generators, we construct a geometry whose point set is the set of extremal points. As lines we take the hyperbolic lines: sets of extremal points corresponding to the extremal elements in a Lie subalgebra generated by two non-commuting extremal elements. If the field contains precisely two elements, then the resulting geometry is a connected Fischer space. This is a connected geometry in which each plane is isomorphic to a dual affine plane of order two or an affine plane of order three. Connected meaning that the collinearity graph of the geometry is con- nected. If the field contains more than two elements, then we take as lines the singular lines: sets of all extremal points commuting with all extremal points commuting with two distinct commuting extremal points. Using a result by Cuypers we prove that the resulting geometry is a polar space. This is a geometry in which each point not on a line is collinear with either one or all points of that line. In fact, the polar space we construct is non-degenerate, that is, no point is collinear with all other points. It was proven by Buekenhout and Shult that such a non-degenerate polar space is also the shadow space of a building. Then, in the fourth chapter, we consider the problem of describing all Lie algebras generated by a finite number of extremal elements over a field of characteristic not two. Cohen et al. proved that the Chevalley algebra of type A2 is the generic Lie algebra in case of three extremal generators. Moreover, in ’t panhuis et al. extended this result to more generators. There, starting from a graph, they constructed an affine variety whose points parametrize Lie algebras generated by extremal elements, corresponding to the vertices of the graph, with prescribed commutation relations, corresponding to the non-edges. In addition, for each Chevalley algebra of classical type they found a finite graph such that all points in some open dense subset of the corresponding variety parametrize Lie algebras isomorphic to this Chevalley algebra. We take a different view point. Starting from a connected simply laced Dynkin diagram of finite or affine type, we prove that the variety is an affine space and, assuming the Dynkin diagram is of affine type, we prove that the points in some open dense subset parametrize Lie algebras isomorphic to the Chevalley algebra corresponding to the associated Dynkin diagram of finite type. In the fifth chapter, we take a closer look at one type of geometry whose points cor- respond to extremal elements inside a Lie algebra: the class of finite irreducible cotri- angular spaces. Each such cotriangular space is an example of a Fischer space in which each plane is isomorphic to a dual affine plane of order two. Hall and Shult proved that each irreducible cotriangular space is of three possible types, that is, triangular, symplec- tic, or orthogonal type. We use this fact to classify the polarized embeddings of a finite irreducible cotriangular space. Here, a polarized embedding is an injective map from the point set of the cotriangular space into the point set of a projective space satisfying certain properties. For instance, lines are mapped into lines and hyperplanes are mapped into hyperplanes. For the spaces of symplectic or orthogonal type we can describe, if the characteristic is not two, the polarized embeddings using the associated symplectic and Preface vii quadratic forms. For other characteristics the polarized embeddings can be described using the root systems of type E6, E7, and E8. For the spaces of triangular type the polarized embeddings can be described using the root systems of type An, n > 4. All this is an extension of the work by Hall who classified the polarized embeddings over the field with two elements. Finally, in the appendix, we give some of the basic terminology used throughout this thesis. viii Preface Contents Prefacev Contents ix 1 Preliminaries1 1.1 Lie algebras................................1 1.1.1 Linear Lie algebras........................2 1.1.2 Chevalley algebras........................4 1.1.3 Kac-Moody algebras.......................5 1.2 Extremal elements............................6 1.3 Geometries...............................
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