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Algorithms for Lie Algebras of Algebraic Groups Copyright C 2010 by Dan Roozemond Algorithms for Lie Algebras of Algebraic Groups Copyright c 2010 by Dan Roozemond. Unmodified copies may be freely distributed. A catalogue record is available from the Eindhoven University of Technology Library. ISBN: 978-90-386-2176-0 Printed by Printservice Technische Universiteit Eindhoven. Cover: The extended Dynkin diagram of type E7 and its automorphism, and the root system of type G2. Design in cooperation with Verspaget & Bruinink. Algorithms for Lie Algebras of Algebraic Groups 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 donderdag 18 maart 2010 om 16.00 uur door Danker Adriaan Roozemond geboren te Leiden Dit proefschrift is goedgekeurd door de promotor: prof.dr. A.M. Cohen Distance-Transitive6 Graphs Recognition 5of Lie Algebras Chevalley Bases Computing 2 4 of Lie Type 3 Twisted Groups Split Toral Split Toral Subalgebras Preliminaries1 Contents Introduction9 1 Preliminaries 13 1.1 Root systems.................................. 13 1.2 Coxeter systems and Dynkin diagrams.................. 16 1.3 Root data.................................... 17 1.4 Lie algebras.................................. 21 1.5 Algebraic groups............................... 26 1.6 The Lie algebra of an algebraic group................... 33 1.7 Tori and toral subalgebras.......................... 38 1.8 Algebraic groups and root data....................... 41 1.9 Chevalley Lie algebras............................ 42 1.10 The Steinberg presentation......................... 44 1.11 Tori and conjugacy classes of the Weyl group............... 49 1.12 Classification of finite simple groups.................... 51 1.13 Algorithms................................... 52 2 Twisted Groups of Lie Type 55 2.1 Definition of the twisted groups...................... 55 2 2 2 2.2 Definition of B2, F4, and G2 ........................ 57 2.3 The Clifford algebra............................. 63 short 2.4 Identifying Aut(L) and Aut(L )k .................... 65 2.5 Two isomorphic Lie algebras........................ 67 2.6 Viewing t as endomorphism of Aut(L) .................. 68 3 Split Toral Subalgebras 75 3.1 A characteristic 2 curiosity.......................... 75 3.2 Regular semisimple elements........................ 77 3.3 A heuristic algorithm............................. 79 3.4 Notes on the implementation........................ 84 4 Computing Chevalley Bases 89 4.1 Some difficulties................................ 90 4.2 Roots...................................... 90 4.3 Outline of the algorithm........................... 91 4.4 Multidimensional root spaces........................ 92 8 Contents 4.5 Finding frames................................ 102 4.6 Root identification.............................. 109 4.7 Conclusion................................... 117 4.8 Notes on the implementation........................ 118 5 Recognition of Lie Algebras 125 5.1 Lie algebras of simple algebraic groups.................. 125 5.2 Simple Lie algebras of algebraic groups.................. 127 5.3 Twisted Lie algebras............................. 130 5.4 Notes on the implementation........................ 133 6 Distance-Transitive Graphs 141 6.1 Distance transitivity............................. 141 6.2 From groups to graphs............................ 144 2 2 6.3 A7(2 ) < E7(2) ................................ 147 Samenvatting 155 Abstract 157 Acknowledgements 159 Curriculum Vitae 161 Bibliography 163 Index 167 Introduction Lie algebras are called after Sophus Lie (1842 – 1899), a Norwegian nineteenth century mathematician who realized that continuous transformation groups could be studied by linearizing them, obtaining what he called the infinitesimal group. These objects are what we now call Lie algebras. Independently, Wilhelm Killing (1847 – 1923) introduced Lie algebras, and he proved that (at least over the complex numbers) only certain finite-dimensional simple Lie algebras could exist: the four infinite series and the five exceptional Lie algebras that are well known today. To this end, he introduced the concepts of root system, Cartan subalgebra, and Cartan matrix. These last two concepts now carry the name of Élie Cartan (1869 – 1951), whose major contribution was to prove that the five exceptional Lie algebras Killing had found actually exist. A later major contributor to this area was Claude Chevalley (1909–1984), who wrote the Theory of Lie Groups, a book in three volumes that systematically treats the theory of groups of Lie type and Lie algebras. (The biographical information presented here may be found in the excellent MacTutor History of Mathematics archive [OR09].) Work by Chevalley and Leonard Dickson showed that the Lie algebras that Killing and Cartan found, commonly called the classical Lie algebras, also exist over finite fields, but there is more. Research by Nathan Jacobson, Aleksei Kostrikin, Ernst Witt, Igor Šafareviˇc,and Hans Zassenhaus produced the so-called Cartan type Lie algebras, and Hayk Melikyan found a new family of simple Lie algebras over fields of characteristic 5. Over the past 15 years, Alexander Premet and Helmut Strade have shown that over algebraically closed fields of characteristic at least 5 every simple Lie algebra belongs to one of these three classes. For characteristic 3 such a result has not been proved, and the characteristic 2 case is still far from set- tled: as recently as 2006 Michael Vaughan-Lee found two new simple Lie algebras over the field with two elements. A brief overview of the classification of the simple Lie algebras over finite fields can be found in an unpublished note by Strade [Str06]. The existence of several classes of simple Lie algebras over finite fields leads to the problem of recognizing these: given a simple Lie algebra, find out which class it belongs to. In particular: decide whether a given simple Lie algebra is classical or not. The new results in this thesis are set within the classical Lie algebras: the four infinite series An,Bn,Cn,Dn and the five exceptional Lie algebras E6,E7,E8,F4, G2. These Lie algebras occur in two ways: as Lie algebras of algebraic groups (in the manner that Lie himself envisioned) and as the main objects that the simple 10 Introduction groups of Lie type act on. The classification of finite simple groups, a major effort by the mathematical community in the twentieth century, shows that the simple groups of Lie type form a significant class of finite simple groups. A not too technical introduction to this classification is a short article by Ron Solomon [Sol95] that appeared in the notices of the AMS. In recent years significant progress has been made to effectively calculate with and in these groups and algebras on the computer, including implementations in, for example, the GAP and Magma computer algebra systems. This research is partly stimulated by the matrix group recognition project: an international project whose main aim is solving problems with matrix groups over finite fields. We build in particular on work by Arjeh Cohen, Willem de Graaf, Sergei Haller, Scott Murray, and Don Taylor. Many algorithms that have been previously developed in this branch of research, however, apply only to groups and algebras over fields of characteristic 0 or at least 5. In this thesis we focus mainly on the characteristic 2 and 3 cases. Reading guide Chapter1 covers the basic notions in the research area of Lie theory. Since this field has existed for quite some time now, the notions are rather numerous and the chapter accordingly elaborate. Chapter2 contains a digression to the twisted groups of Lie type. In particular we explicitly construct the automorphisms needed 2 2 2 to construct groups of type B2, F4, and G2, and we exhibit these automorphisms as endomorphisms of Lie algebras as well. In Chapter3 we investigate the compu- tation of split maximal toral subalgebras over fields of characteristic 2, show why existing methods will not always work, and present a heuristic algorithm for this purpose. Chapter4 shows how to construct Chevalley bases of the classical Lie algebras over any characteristic, including 2 and 3. We prove that the algorithm runs in time polynomial in the input. In Chapter5 the results of Chapters3 and4 are used to produce algorithms for recognition of Lie algebras. In Chapter6 we ap- ply the algorithms described and their implementation to obtain a computer aided proof that there is no graph on which a certain group acts distance transitively. If you are an expert in the subject area of this thesis, it is probably best to skip Chapter1, and start reading in Chapter2 (if you want to freshen up your knowledge of these extraordinary twisted groups) or Chapter3 (if you are primarily interested in the results). If you are no expert in this area, but you are a mathematician, it is probably best to simply start with Chapter1 and go from there. If you are not a mathematician or you have no desire to learn about Lie theory, skip to the abstract (or the samenvatting), possibly read the acknowledgements, and then get a copy of the excellent book Finding Moonshine (or Het Symmetriemonster) by Marcus du Sautoy to learn about the beauty of symmetry. 11 finite simple groups Classification of Tori and conjugacy 1.12 classes of the Weyl1.11 group Chevalley The Steinberg Lie1.9 algebras presentation1.10 Algebraic groups1.8 and root data Tori and toral 1.7subalgebras Root data Algorithms1.13
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