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The Structure of E6

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The Structure of E6 arXiv:0711.3447v2 [math.RA] 20 Dec 2007 AN ABSTRACT OF THE THESIS OF Aaron D. Wangberg for the degree of Doctor of Philosophy in Mathematics presented on August 8, 2007. Title: The Structure of E6 Abstract approved: Tevian Dray A fundamental question related to any Lie algebra is to know its subalgebras. This is particularly true in the case of E6, an algebra which seems just large enough to contain the algebras which describe the fundamental forces in the Standard Model of particle physics. In this situation, the question is not just to know which subalgebras exist in E6 but to know how the subalgebras fit inside the larger algebra and how they are related to each other. In this thesis, we present the subalgebra structure of sl(3, O), a particular real form of E6 chosen for its relevance to particle physics through the connection between its associated Lie group SL(3, O) and generalized Lorentz groups. Given the complications related to the non-associativity of the octonions O and the restriction to working with a real form of E6, we find that the traditional methods used to study Lie algebras must be modified for our purposes. We use an explicit representation of the Lie group SL(3, O) to produce the multiplication table of the corresponding algebra sl(3, O). Both the multiplication table and the group are then utilized to find subalgebras of sl(3, O). In particular, we identify various subalgebras of the form sl(n, F) and su(n, F) within sl(3, O) and we also find algebras corresponding to generalized Lorentz groups. Methods based upon automor- phisms of complex Lie algebras are developed to find less obvious subalgebras of sl(3, O). While we focus on the subalgebra structure of our real form of E6, these methods may also be used to study the subalgebra structure of any other real form of E6. A maximal set of simultaneously measurable observables in physics corresponds to a maximal set of Casimir operators in the Lie algebra. We not only identify six Casimir operators in E6, but produce a nested sequence of subalgebras and Casimir operators in E6 containing both su(3) su(2) u(1) corresponding to the Standard Model and the Lorentz group of ⊕ ⊕ special relativity. c Copyright by Aaron D. Wangberg August 8, 2007 All Rights Reserved The Structure of E6 by Aaron D. Wangberg A THESIS submitted to Oregon State University in partial fulfillment of the requirements for the degree of Doctor of Philosophy Presented August 8, 2007 Commencement June 2008 Doctor of Philosophy thesis of Aaron D. Wangberg presented on August 8, 2007 APPROVED: Major Professor, representing Mathematics Chair of the Department of Mathematics Dean of the Graduate School I understand that my thesis will become part of the permanent collection of Oregon State University libraries. My signature below authorizes release of my thesis to any reader upon request. Aaron D. Wangberg, Author ACKNOWLEDGEMENTS I wish to thank my advisor, Tevian Dray, and Corinne Manogue for their participation and advice as I worked to finish this thesis. I also want thank them for allowing me to pursue interesting questions and pursuits, whether they be related to mathematics, teaching, or exploring the Pacific Northwest. I also greatly appreciate the support and encouragement of my wife, Robyn. This would not have been completed without her. Thank you. TABLE OF CONTENTS Page 1 INTRODUCTION ...................................... ..................... 2 1.1 Motivation...................................... ....................... 2 1.2 Summary......................................... ...................... 4 1.2.1 Thesis Organization............................ ................. 6 2 LIE GROUPS AND LIE ALGEBRAS ........................... ............ 11 2.1 Basic Definitions................................. ...................... 11 2.1.1 An Example ..................................... ............... 11 2.1.2 Lie Groups ..................................... ................ 12 2.1.3 Lie Algebras ................................... ................. 13 2.1.4 Correspondence between Lie groups and Lie algebras . .......... 17 2.1.5 Regular Representation and Killing Form. ............. 23 2.1.6 Different Conventions used by Physicists and Mathematicians . 28 2.1.7 Classical Matrix Groups and their Algebras . ............ 28 2.2 Classifying Complex Lie Algebras................... .................... 31 2.2.1 Precise Definitions ............................. ................. 32 2.2.2 Introduction .................................. .................. 36 2.2.3 Root and Weight Diagrams of Lie Algebras . .......... 37 2.2.4 Subalgebras of Complex Lie Algebras . ............ 50 2.2.5 Applications to Algebras of Dimension Greater than 3 . .......... 59 TABLE OF CONTENTS (Continued) Page 2.2.6 Conclusion.................................... .................. 67 3 DIVISION ALGEBRAS AND APPLICATIONS ................... .......... 68 3.1 Normed Division Algebras.......................... .................... 69 3.1.1 Reals, Complexes, Quaternions and Octonions . ............ 69 3.1.2 Conjugation, Reflections, and Rotations . .............. 74 3.1.3 Triality ...................................... ................... 77 3.2 Lorentz Transformations.......................... ...................... 80 3.2.1 Spacetime and Lorentz Transformations . ............. 80 3.2.2 Spacetimes related to Division Algebras . .............. 82 3.2.3 Lorentz Transformations related to Division Algebras............ 85 3.3 Jordan Algebras and Albert Algebras . .................. 91 4 THE BASIC STRUCTURE OF E6 .......................................... 96 4.1 The Lie Group E6 = SL(3, O) ......................................... 97 4.1.1 Lorentz Transformations in 3 3 case........................... 98 × 4.1.2 A Basis for the Category 3 Transformations . ........... 104 4.2 The Lie Algebra sl(3, O) of E6 = SL(3, O).............................. 107 4.2.1 Constructing the algebra for SL(3, O)........................... 108 TABLE OF CONTENTS (Continued) Page 4.2.2 Linear Dependencies ............................ ................ 112 4.2.3 Triplets of Subgroup Chains...................... ............... 115 4.3 Triality........................................ ......................... 120 4.4 Type Transformation.............................. ..................... 126 4.4.1 Type Independent and Dependent Subgroups . .......... 127 4.5 Reduction of O to H, C, and R ......................................... 129 4.6 A Subalgebra fixing ℓ ................................................... 134 4.7 Gell-Mann Matrices and su(3) G .................................... 136 ⊂ 2 5 THE FURTHER STRUCTURE OF E6 ...................................... 138 5.1 Automorphisms of Lie Algebras...................... ...................139 5.1.1 Identification of Algebras . ................ 142 5.2 Three Important Involutive Automorphisms of sl(3, O)................. 150 5.3 Additional Subalgebras of sl(3, O) from Automorphisms................ 157 5.4 Chains of Subalgebras of sl(3, O) with Compatible Bases and Casimir Operators.......................................... .................... 162 6 OPEN QUESTIONS ..................................... .................... 168 TABLE OF CONTENTS (Continued) Page 7 CONCLUSION ........................................ ...................... 170 BIBLIOGRAPHY ....................................... ........................ 172 APPENDICES ......................................... ......................... 176 A Preferred Basis for sl(3, O)............................................. 177 B Preferred Basis for various subalgebras of E6 ........................... 178 C Basis for Small Subalgebras of sl(3, O).................................. 183 D Direct sums in sl(3, O)................................................. 187 LIST OF FIGURES Figure Page 2.1 Root diagram of A1 = su(2) ............................................ 40 2.2 Root and minimal weight diagrams of A2 = su(3) ...................... 42 2.3 Rank 2 Dynkin diagrams ............................. .................. 43 2.4 Rank 2 simple roots ................................ .................... 43 2.5 Generating an algebra’s full root system using Weyl reflections . 45 2.6 Root diagrams of simple rank 2 algebras ............... ................ 46 2.7 B3 = so(7) weight skeleton .................................. ........... 49 2.8 B3 = so(7) weight diagram ................................... .......... 49 2.9 Rank 3 root diagrams ............................... ................... 50 2.10 Rank 3 minimal weight diagrams ..................... .................. 51 2.11 B B ................................................... ............. 52 2 ⊂ 3 2.12 D B ................................................... ............ 52 2 ⊂ 3 2.13 A D B ................................................... ....... 52 2 ⊂ 3 ⊂ 3 2.14 B = so(5) B = so(7)................................................ 53 2 ⊂ 3 2.15 G B = so(7)................................................ ........ 53 2 ⊂ 3 1 2 2.16 Slicing of B3 = so(7) using root r , colored red, and root r , colored blue. 55 1 2 3 2.17 Slicing of B4 = so(9) using roots r (red), r (green), and r (blue). 55 LIST OF FIGURES (Continued) Figure Page 2.18 Projecting the C = sp(2 4) root diagram along simple root r1 ........ 58 4 · 2.19 Collapsing the slices of C = sp(2 4) defined by simple roots r2, r3, 4 · and r4 onto the origin...................................... ............ 58 2.20 F4 Dynkin diagram.....................................
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