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Algorithms for Finite-Dimensional Lie Algebras

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Algorithms for Finite-Dimensional Lie Algebras Algorithms for finite-dimensional Lie algebras Citation for published version (APA): Graaf, de, W. A. (1997). Algorithms for finite-dimensional Lie algebras. Technische Universiteit Eindhoven. https://doi.org/10.6100/IR495936 DOI: 10.6100/IR495936 Document status and date: Published: 01/01/1997 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. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website. • The final author version and the galley proof are versions of the publication after peer review. • The final published version features the final layout of the paper including the volume, issue and page numbers. 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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: 28. Sep. 2021 Algorithms for Finite-Dimensional Lie Algebras PROEFSCHRIFT ter verkrijging van de graad van doctor aan de Technische Universiteit Eindhoven, op gezag van de Rector Magnificus, prof.dr. M. Rem, voor een commissie aangewezen door het College van Dekarren in het openbaar te verdedigen op dinsdag 10 juni 1997 om 16.00 uur door Willem A. de Graaf geboren te Gorinchem Dit proefschrift is goedgekeurd door de promotoren: prof.dr. A.M. Cohen en prof.dr. A.E. Brouwer Contents 1 Introduetion 1 1.1 Notation, definitions, and basic theory 2 1.2 Presentations of Lie algebras . 7 1.3 Complexity 8 1.4 Earlier work 9 1.4.1 Product spaces 9 1.4.2 The centre ... 9 1.4.3 The centraliser 10 1.4.4 The normaliser 10 1.4.5 The solvable radical . 10 1.4.6 The direct sum decomposition . 11 1.4.7 The Levi decomposition 13 1.4.8 Zeros of polynomials 14 2 The nilradical 17 2.1 Known algorithms 17 2.2 The Downward Method 20 2.3 Evaluation . 25 2.4 Acknow ledgements 26 3 Cartan subalgebras 27 3.1 Known algorithms 27 11 CONTENTS 3.2 Locally regular elements . 28 3.3 Finding a non-nilpotent element in a Lie algebra . 31 3.4 Cartan subalgebras 33 3.5 Evaluation . 34 3.6 Acknowledgement . 37 4 The decomposition of a semisimpte Lie algebra 39 4.1 The generalised Cartan decomposition . 39 4.2 Calculating a generalised Cartan decomposition 42 4.2.1 Splitting elements .... 42 4.2.2 Decomposable elements. 45 4.3 Examples 48 4.4 Evaluation 49 4.5 Acknowledgements 50 5 The type of a semisimple Lie algebra 51 5.1 Identifying a semisimpleLie algebra . 51 5.2 Isomorphism of semisimpte Lie algebras . 56 5.3 Examples . 59 5.4 Acknowledgements 60 6 Constructive Ado 63 6.1 Calculating a series of extensions 64 6.2 The extension space 65 6.3 Extending a representation . 66 6.4 An effective version of Ado's theorem 70 6.5 Examples, and practical experience 72 7 Practice 75 7.1 Isomorphism testing 75 7.2 Calculations in E8 79 CONTENTS 111 7.2.1 Preliminaries . 79 7.2.2 Centralisers of uilpotent elements in E8 • 81 A Manual of ELIAS 91 Index of Terminology 109 Index of Symbols 110 Acknowledgements 111 Samenvatting 112 Curriculum Vitae 113 IV CONTENTS Chapter 1 Introduetion Lie algebras arise naturally in various areas of rnathematics and physics. Examples are: representation theory of Lie groups ([20]) and of algebraic groups ([30]) and the theory of Lie point symmetriesof a differential equation ([9], [39]). The topic of the present work is the algorithmic treatment of finite dimensional Lie algebras. These Lie algebras live in an obscure world where they are only known by their multiplication table, that is by a faint shadow. Here we present algorithms for obtaining information about a Lie algebra. These allow us to shed rays of light in this world that make a Lie algebra cast more distinct shadows. In some cases, particularly when the Lie algebra is semisimpte and of charaderistic 0, this enables us to recognise it. In other cases we have to content ourselves with only partial information. The work described here is implemented in a package called ELIAS (for Eindhoven Lle Algebra System), that will be a part of GAP4. In this chapter we will first introduce some basic mathematica] concepts. Then in Section 1.2 we will deal with the first step of the process of the algorithmic identification of a Lie algebra: repreaenting ît on a computer. In the next section we briefly discuss some complexity issues. Finallyin Section 1.4, we will present a survey of algorithms known in the literature. The chapters 2 to 6 each deal with a partienlar algorithmic problem. In Chapter 2, this is the calculation of the nilradical. In Chapter 3 we describe how a Cartan subalgebra can be found. This is used in Chapter 4, where algorithms for decomposing a semisimple Lie algebra are given. In Chapter 5 the problem of determining the isomorphism type of a semisimpte Lie algebra is discussed. An effective version of Ado's theorem is given in Chapter 6. In Chapter 7, the system ELIAS is applied in two practical problems. Finally in Appendix A there is a manual of ELIAS. In the chapters 2 to 7 running times of calculations are presented. All computations were performed on a SUN SPARC classic workstation. 1 2 Introduetion 1.1 Notation, definitions, and basic theory In this section we describe the basic theoretica! tools that we use. For the proofs we refer to the standard monographs ([29],[32]). Definition 1.1 A Lie algebra is a vector space over a field F equipped with a bilinear map ( m ultiplication) [·, ·] : L x L ___, L satisfying (L1) [x,x] OforallxEL, {L 2) [[x, y], z] + [[y, z], x]+ [[z, x], y] = 0 Jor all x, y, zE L. The second condition (L2 ) is called the Jacobi identity. By applying (L1 ) to the element x+ y we see that the first condition implies [x, y] -[y, x]. lf the characteristic of the field is not 2, then this in turn implies (LI). Example 1.2 Let L be a 3-dimensional vector space over Qwith basis {x,y, h} and Lie product described by [x,y] = h, [h,x] 2x, [h,y] By using bilinearity and anticommutativity this defines the Lie product for all elementsof L. Example 1.3 Let L be the 3-dimensional subspace of M3(Q spanned by For two elements A, BEL we set [A, B] A·B B·A (where the ·stands for ordinary ma­ trix multiplica.tion). lt is seen that the spaceL is closedunder this operation. Furthermore the [·, ·] defined in this way satisfies the two requirements for being a Lie multiplica.tion. It follows that L is a Lie algebra. Definition 1.4 Let L be a Lie algebra over F and V a vector space over F. A represen­ tation of L on V is a linear map p : L ___, End(V) such that p([x,y]) = p(x)p(y)- p(y)p(x). 1,1 Notation, defînitions, and basic Example 1.5 Let L be the Lie algebra of Example 1.2 and let A~, A2, A3 be as in Example 1.3. Let p be the linear map from L into the space spanned by the A; given by Then it is seen that pis a representation of L. Furthermore we have that the kemel of p is 0, so that L is isomorphic to its image. Representations with this property are called faithful. Example 1.6 Let L be a Lie algebra. Define a map ad: L-+ End(L) by ad(x)(y) [x,y]. The fact that this is a Lie algebra representation is equivalent to the Jacobi identity. Themapad is called the adjoint representation. Toa representation pof L we associate a bilinearform fP defined by fp(x, y) Tr(p(x)p(y)). In the case of the adjoint representation this form is called the KilZing form and is denoted by !'i. Definition 1. 7 Let L be a Lie algebra. A subspace K of L is called a subalgebra if [x,y] EK for all x,y EK. Definition 1.8 Let L be a Lie algebra. A subspace I of L is called anideal if [x, y] E I for all x EL, y EI. Let L be a Lie algebra and let I be an ideal of L such that there is a subalgebra K of L with the property that L K EB I (direct sum of vector spaces).
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