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Computational Topics in Lie Theory and Representation Theory Utah State University DigitalCommons@USU All Graduate Theses and Dissertations Graduate Studies 5-2014 Computational Topics in Lie Theory and Representation Theory Thomas J. Apedaile Utah State University Follow this and additional works at: https://digitalcommons.usu.edu/etd Part of the Applied Statistics Commons Recommended Citation Apedaile, Thomas J., "Computational Topics in Lie Theory and Representation Theory" (2014). All Graduate Theses and Dissertations. 2156. https://digitalcommons.usu.edu/etd/2156 This Thesis is brought to you for free and open access by the Graduate Studies at DigitalCommons@USU. It has been accepted for inclusion in All Graduate Theses and Dissertations by an authorized administrator of DigitalCommons@USU. For more information, please contact [email protected]. COMPUTATIONAL TOPICS IN LIE THEORY AND REPRESENTATION THEORY by Thomas J. Apedaile A thesis submitted in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE in Mathematics Approved: Ian Anderson Nathan Geer Major Professor Committee Member Zhaohu Nie Mark R McLellan Committee Member Vice President for Research and Dean of the School of Graduate Studies UTAH STATE UNIVERSITY Logan, Utah 2014 ii Copyright c Thomas J. Apedaile 2014 All Rights Reserved iii ABSTRACT Computational Topics in Lie Theory and Representation Theory by Thomas J. Apedaile, Master of Science Utah State University, 2014 Major Professor: Dr. Ian Anderson Department: Mathematics and Statistics The computer algebra system Maple contains a basic set of commands for work- ing with Lie algebras. The purpose of this thesis was to extend the functionality of these Maple packages in a number of important areas. First, programs for defining multiplication in several types of Cayley algebras, Jordan algebras and Clifford algebras were created to allow users to perform a variety of calculations. Second, commands were created for calculating some basic properties of finite-dimensional representations of complex semisimple Lie algebras. These commands allow one to identify a given representation as direct sum of irreducible subrepresentations, each one identified by an invariant highest weight. Third, creating an algorithm to calculate the Lie bracket for Vinberg's symmetric construction of Freudenthal's Magic Square allowed for a uniform construction of all five exceptional Lie algebras. Maple examples and tutorials are pro- vided to illustrate the implementation and use of the algebras now available in Maple as well as the tools for working with Lie algebra representations. (255 pages) iv PUBLIC ABSTRACT Computational Topics in Lie Theory and Representation Theory by Thomas J. Apedaile, Master of Science Utah State University, 2014 Major Professor: Dr. Ian Anderson Department: Mathematics and Statistics The computer algebra system Maple contains a basic set of commands for working with Lie algebras and matrices. The purpose of this thesis was to extend the functionality of these Maple packages in a number of important areas. First, programs for defining multiplication in several different types of algebras were created to allow users to perform a wider variety of calculations. Second, commands were created for calculating some basic properties of matrix representations of semisimple Lie algebras. This allows a user to identify a given matrix representation by a collection of integers which do not change when the basis of the representation is changed. These integers, called highest weights, uniquely identify the representation. Third, an algorithm was created to allow for a uniform construction of all five exceptional Lie algebras. Maple examples and tutorials are provided to illustrate the implementation and use of the algebras now available in Maple as well as the tools for working with Lie algebra representations. (255 pages) v ACKNOWLEDGMENTS I would like to thank Dr. Ian Anderson for his infinite patience, prodding and expertise. Without his help and encouragement, I doubt I would have learned so much. I would also like to thank Dr. Zhaohu Nie for the time he took out of his busy schedule to sit down with and brainstorm on problems that helped me understand my thesis better. Most of all, I am grateful for my wonderful family and amazing friends, for their support over the years and for all the times I would force them to listen to me talk about every breakthrough I encountered. I know by the looks on their faces that it was painful at times, but helped keep the fire alive. Thomas J. Apedaile vi CONTENTS Page ABSTRACT.................................... iii PUBLIC ABSTRACT.............................. iv ACKNOWLEDGMENTS............................ v LIST OF TABLES................................viii LIST OF FIGURES............................... ix SYMBOLS..................................... x 1 INTRODUCTION............................... 1 2 ALGEBRAS.................................. 6 2.1 Preliminaries..................................6 2.2 Cayley-Dickson Construction......................... 11 2.2.1 Quaternions............................... 12 2.2.2 Octonions................................ 15 2.2.3 A Brief Discussion of the Cayley Algebras A(−).......... 19 2.3 Jordan Algebras................................ 21 2.4 Clifford Algebras................................ 24 2.5 Lie Algebras................................... 30 3 DECOMPOSITION OF LIE ALGEBRA REPRESENTATIONS . 45 3.1 Introduction................................... 45 3.2 Overview of Theorem3 :1........................... 48 3.3 Overview of Theorem3 :2........................... 51 3.4 Overview of Theorem3 :3........................... 53 3.5 Representations of sl2(C)........................... 54 3.6 Representations of sl3(C)........................... 56 3.7 Detailed Discussion of Theorems3 :1 and3 :2................. 62 3.8 Examples.................................... 74 3.8.1 sl3(C) : Decomposing V ⊗ V ..................... 74 2 3.8.2 g2(C) : Decomposing ^ (der (O))................... 77 3.8.3 so5(C): Creating an \Optimal" Basis of V .............. 83 vii 4 MAGIC SQUARE LIE ALGEBRAS.................... 90 4.1 Introduction................................... 90 4.2 Derivations and the Derivation Algebra................... 91 4.3 Vinberg's Magic Square Construction.................... 103 4.4 Computing the Lie Bracket on M(K; M)................... 110 BIBLIOGRAPHY.................................133 APPENDIX....................................135 A Miscellaneous Results............................. 136 B Maple Code................................... 140 B.1 Algebras................................. 140 B.1.1 Quaternion Library Code.................. 140 B.1.2 Octonion Library Code................... 141 B.1.3 Jordan Algebra Library Code................ 144 B.1.4 Clifford Algebra Library Code............... 150 B.2 Representations Code......................... 159 B.2.1 Fundamental Weights Code................. 159 B.2.2 Highest Weight Vector Code................ 163 B.2.3 Decompose Representation Code.............. 170 B.3 Magic Square.............................. 180 C Maple Tutorials................................. 200 C.1 Building a Quaternion Algebra.................... 200 C.2 Building an Octonion Algebra..................... 202 C.3 Building a Jordan Algebra....................... 206 C.4 Building a Clifford Algebra...................... 210 C.5 Decomposing Lie Algebra Representations.............. 214 C.6 Derivations of the Octonions..................... 229 C.7 Magic Square Lie Algebras...................... 234 viii LIST OF TABLES Table Page 2.1 Multiplication Table - Quaternions...................... 13 2.2 Multiplication Table - Octonions....................... 16 2.3 Multiplication Table - Split-Quaternions................... 20 2.4 Multiplication Table - Split-Octonions.................... 21 2.5 Multiplication Table - Basis Elements for the Clifford Algebra C(Q)... 28 2.6 Lie Bracket Table for t3 ............................ 34 2.7 Lie Bracket Table for [Dt3; Dt3]....................... 34 2.8 Lie Bracket Table for b4 ............................ 34 2.9 Lie Bracket Table for [b4; Db4]........................ 35 2.10 Lie Bracket Table for e(3)........................... 40 2.11 Classification of semisimple Lie algebras over C ............... 41 4.1 Freudenthal's Magic Square Lie Algebras.................. 91 4.2 Decomposition of the Lie Bracket....................... 109 0 0 4.3 Symmetric Magic Square M(K ; M )..................... 110 0 4.4 Non-Symmetric Magic Square M(K ; M)................... 110 ix LIST OF FIGURES Figure Page 2.1 Multiplication for Imaginary Quaternion Basis Elements.......... 14 2.2 The Fano Plane: Octonion Multiplication.................. 17 3.1 Diagram for the eigenvalues corresponding to the decomposition (3:6).. 59 x SYMBOLS [X; Y ] Commutator/Lie bracket XY − YX (x; y; z) Associator (xy)z − x(yz) tr(X) X Trace free part of a matrix XX − · I 0 n n 1 x ◦ y Jordan product (xy + yx) 2 j Ei A square matrix with 1 in the ith row and jth column 1 I (A) The imaginary part of an element A (A − A) 2 1 R (A) The real part of an element A (A + A) 2 1 CHAPTER 1 INTRODUCTION The study of symmetries plays a prominent role in the study of mathematics. Lie groups provide a way to express the concept of a continuous family of symmetries for geometric objects. The fundamental theorems of Lie describe a connection between a Lie group and a corresponding Lie algebra. Given a Lie algebra, one can obtain con- nected Lie groups which are (at worst) locally isomorphic. Also,
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