The main focus of this dissertation is to present an introduction to gradings of Lie algebras. The aim is twofold: to lay the necessary foundations to become (in the near future) an algebraist working in this area of research, and to tackle the problem of finding and classifying the Lie algebras arising as graded contractions of a specific (Z_2)^3-grading of the Lie algebra g_2. As a result, this dissertation consists of five chapters and six appendices which might appear very different, at first glance, but they are indeed connected since they are all very important tools for anyone interested in gradings of Lie algebras. The first chapter is devoted to introducing Lie algebras and the study of semisimple Lie algebras. This chapter will place into context much of the work of the second chapter. We begin by describing the basic notions of Lie algebras and their representations, and continue with the study of the Killing form of a Lie algebra, as well as, the root space decomposition. In the second chapter we study root systems and their bases. This leads to an investigation of the Weyl group associated to a root system. This work allows us to describe how one can uniquely extend isomorphisms between root systems to isomorphisms between Lie algebras to which those root systems correspond. We are then able to describe the special properties of Chevalley bases. Gradings make their first appearance in Chapter three. We quickly shift our focus to group gradings. We describe a process to obtain a universal grading group amongst equivalent gradings. We spend some time preparing and presenting an example of this process. The chapter ends with some results relating to the automorphisms of a grading. We present the construction of the exceptional Lie algebra g_2 in the fifth chapter. This chapter uses some definitions and results which are presented in Appendix B. We start by looking at useful results relating to alternative algebras. Then we introduce upper bounds to the dimension of g_2. Finally we show that g_2 is 14- dimensional and we construct an important (Z_2)^3-grading of g_2. In the fifth and final chapter we study graded contractions. This work continues into Appendix A, however this is the newest work and as such is still under revision. It is worth mentioning here that the bulk of Chapter 5 and Appendix A is original work under construction. It is the result of an ongoing collaboration with Dr Cristina Draper and Dr Juana Sánchez- Ortega. Although some of the proofs may be shortened in the future, we decided to include them as we are excited about the findings. After introducing the notions for general Lie algebras and gradings we look specifically at the grading on g2 which we constructed in the previous chapter. We are now in a position to attack the problem of finding and classifying the graded contractions relating to the Z3 2-grading of g2 presented in Chapter 4. The definitions in the first section of this chapter come from. The rest of Chapter 6 and Appendix A consist of original work completed for this dissertation. Tensor products of modules over a commutative ring R are the sole focus of Appendix B. We explicitly construct the tensor product of two R-modules, and see how all multi-linear maps filter through tensor products. This is followed by a collection of results chosen to help build intuition for the structure and workings of the tensor product. Lastly, we examine how tensor products interact with direct sums and how linear maps, between modules, may induce maps between the tensor products of those modules. Appendix C is centred around affine group schemes. We introduce the topic as familiarity with this area presents opportunities for future research problems and investigations. Our main aim in this chapter is to describe Hopf algebras. Appendix D is focussed on presenting a proof of Weyl's Theorem, used in the third chapter. Such a proof requires results about the Jordan canonical form of a matrix and the Casimir operator of a Lie algebra representation. The main goal of Appendix E is to describe the differential of a Lie group homomorphism. We make use of this work in Chapter 3. Before we can study the differential of a Lie group homomorphism we need to study matrix Lie groups and the exponential map. The last appendix, Appendix F, is a brief summary of important definitions and results related to the octonions. We need this work to accomplish our goals in Chapter 5.
| Identifer | oai:union.ndltd.org:netd.ac.za/oai:union.ndltd.org:uct/oai:localhost:11427/36614 |
| Date | 04 July 2022 |
| Creators | Meyer, Thomas Leenen |
| Contributors | Sanchez-Ortega, Juana |
| Publisher | Faculty of Science, Department of Mathematics and Applied Mathematics |
| Source Sets | South African National ETD Portal |
| Language | English |
| Detected Language | English |
| Type | Master Thesis, Masters, MSc |
| Format | application/pdf |
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