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Analysis of Functions of Split-Complex, Multicomplex, and Split-Quaternionic Variables and Their Associated Conformal Geometries John Anthony Emanuello

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Analysis of Functions of Split-Complex, Multicomplex, and Split-Quaternionic Variables and Their Associated Conformal Geometries John Anthony Emanuello Florida State University Libraries Electronic Theses, Treatises and Dissertations The Graduate School 2015 Analysis of Functions of Split-Complex, Multicomplex, and Split-Quaternionic Variables and Their Associated Conformal Geometries John Anthony Emanuello Follow this and additional works at the FSU Digital Library. For more information, please contact [email protected] FLORIDA STATE UNIVERSITY COLLEGE OF ARTS AND SCIENCES ANALYSIS OF FUNCTIONS OF SPLIT-COMPLEX, MULTICOMPLEX, AND SPLIT-QUATERNIONIC VARIABLES AND THEIR ASSOCIATED CONFORMAL GEOMETRIES By JOHN ANTHONY EMANUELLO A Dissertation submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy Degree Awarded: Spring Semester, 2015 Copyright ➞ 2015 John Anthony Emanuello. All Rights Reserved. John Anthony Emanuello defended this dissertation on March 27, 2015. The members of the supervisory committee were: Craig A. Nolder Professor Directing Dissertation Samuel Tabor University Representative Bettye Anne Case Committee Member John R. Quine Committee Member The Graduate School has verified and approved the above-named committee members, and certifies that the dissertation has been approved in accordance with university requirements. ii For my lovely wife Krisztina, whose love and support made this all worthwhile. iii ACKNOWLEDGMENTS First and foremost, I would like to thank my major professor, Dr. Craig Nolder. I have appreciated his assistance, patience, and collaboration more than I could possibly convey in words. Thank you for helping me become the mathematician I am today. Dr. Bettye Anne Case has gone above and beyond the call of duty as a committee member. Her advice and guidance in all areas of my career have far exceeded any expectation one could have for a professor and administrator as busy as her. I would like to thank my other committee members: Dr. John R. Quine and Dr. Samuel Tabor. Thank you for all your time, assistance, and encouragement throughout my graduate career. To the many faculty members at FSU who taught me, including Drs. Van Hoeij, Fenley, Aluffi, and Aldrovandi, I am very thankful for the quality education I received from you. I am also very thankful to the late Dr. Mika Sepp¨al¨a. Your encouragement and kind words were far more helpful to me than you ever knew. Dr. Penelope Kirby deserves special recognition for helping me become the educator that I am today. Thanks for giving me so many challenging classes to teach. I would like to thank Pam Andrews, Karmel Hawkins, and the rest of the FSU Math Department staff for lending a helping hand throughout my time as a graduate student. I would like to thank Kerr Ballenger for being the best friend I could ever want. We came in together and through these years we shared a multitude of experiences and emotions, which I will never forget. You will be my brother for life. Most importantly, I would like to thank my amazing wife, Krisztina, for her incredible love, unwaivering support, and for putting up with me during my nearly five years at FSU. You inspire me to be a much greater man than I thought I could ever be. I cannot wait to see what adventure fate has in store for us. iv TABLE OF CONTENTS ListofFigures....................................... viii Abstract ............................................ ... ix 1 Introduction 1 1.1 OutlineoftheDissertation . ....... 2 1.2 TopicsforFurtherResearch . ...... 3 2 Clifford Algebras and the Multicomplex Numbers 5 2.1 Clifford’sGeometricAlgebras . ...... 5 2.1.1 ABriefHistory ................................. 5 2.1.2 BasicExamples ................................. 6 2.2 CliffordAlgebras .................................. ... 9 2.2.1 ACategoricalConstruction . .... 9 2.2.2 AnaloguesoftheComplexNumbers . 11 2.3 RealCliffordAlgebras.............................. .... 12 2.3.1 TheEuclideanCase.............................. 12 2.3.2 RelationwiththeExteriorAlgebra. ..... 14 2.3.3 ThePseudo-EuclideanCase . ... 15 2.3.4 Clifford Group and Miscellaneous Topics . ...... 16 2.3.5 TheSplit-ComplexNumbers . 18 2.3.6 TheSplit-Quaternions . ... 20 2.4 TheMulticomplexNumbers ........................... ... 21 2.4.1 TheBicomplexNumbers............................ 22 2.4.2 TheMulticomplexNumbers. 23 3 Clifford Analysis and Multicomplex Analysis 26 3.1 CliffordAnalysis .................................. ... 26 3.1.1 DiracOperators ................................ 26 3.2 Clifford Analysis on Cℓ0,n-valuedFunctions . 28 3.2.1 Examples and Non-examples of Clifford Holomorphic Functions .... 29 3.2.2 (Some) Analogues of Results from Complex Analysis . ...... 32 3.3 MulticomplexAnalysis ............................. .... 35 3.3.1 BicomplexAnalyticFunctions. .... 35 3.3.2 Multicomplex Analytic and Meromorphic Functions . ....... 36 3.3.3 PolynomialsandRationalMaps . ... 37 4 Preliminaries from Conformal Geometry 39 4.1 SmoothandSemi-RiemannianManifolds . ....... 39 4.1.1 TangentVectorsandMetricTensors . ..... 40 v 4.1.2 Operators..................................... 42 4.1.3 ConformalMappings ............................. 43 4.1.4 ConformalCompactification. .... 45 4.2 M¨obius Transformations of Rp,q ............................ 46 4.2.1 TheTraditionalConstruction . ..... 46 4.2.2 TheVahlenConstruction . .. 47 4.2.3 When p q 2.................................. 49 + > 1 1 5 Split-Complex Analysis and The M¨obius Transformations of R , 51 5.1 Obtaining “Cauchy-Riemann” Equations and Operators . ........... 52 5.2 Analogues and Non-Analogues from Complex Analysis . ...... 55 5.3 ConformalMappings............................... .... 57 1 1 5.4 Conformal Compactifcation of R , .......................... 59 4 5.4.1 The Torus in R ................................. 59 5.4.2 Embedding of R1,1 onto N 1,1 ......................... 60 5.4.3 AddedPoints................................... 62 5.4.4 N 1,1 asaConformalCompactification . 64 5.4.5 Differentiable Functions and Conformal Mappings on N 1,1 ....... 68 5.5 M¨obiusGeometry ................................. ... 71 5.5.1 FixedPointsandTransitivity . .... 72 5.5.2 CrossRatio.................................... 76 6 The M¨obius Group of the Extended Multicomplex Numbers 81 6.1 TheExtendedComplexNumbers . ... 81 6.2 TheExtendedMulticomplexNumbers . ..... 82 6.3 M¨obiusTransformations . ...... 84 6.3.1 FixedPointsandTransitivity . .... 85 6.3.2 CrossRatio.................................... 86 2 2 7 Split-Quaternioinc Analysis and The M¨obius Transformations of R , 89 7.1 NotionsofHolomorphic.............................. ... 90 7.1.1 Analogues of the Cauchy-Riemann Operator . ..... 90 7.1.2 DifferenceQuotients ............................ .. 94 7.1.3 RegularityandJohn’sEquation . ... 99 7.2 ATheoryofLeft-RegularFunctions . .......100 7.2.1 AClassofLeftRegularFunctions . 101 7.2.2 GeneratingLeftRegularFunctions . ......103 7.3 LinearFractionalTransformations . .........108 7.3.1 M¨obius Transformations of Cℓ1,1 .......................108 7.3.2 Low Dimensional Example: The Complex Plane. .....108 7.3.3 M¨obius Transformations of Cℓ1,1 .......................110 vi Bibliography....................................... ......116 BiographicalSketch ................................. .......121 vii LIST OF FIGURES 1 1 5.1 We parametrize T , θ,φ x0 cos θ, x1 sin θ, x2 sin φ, x3 cos φ, π θ,φ π . This give a parametrization= {( )S = of N 1,1.= The plus and= minus signs= indicate− ≤ the signs≤ } of the cosines and sines in the parametrization. ............ 65 5.2 The R1,1 planeandsomecurves. ............................ 66 5.3 The embedded curves. Points with the same numeric label are identified. 67 viii ABSTRACT The connections between algebra, geometry, and analysis have led the way for numerous results in many areas of mathematics, especially complex analysis. Considerable effort has been made to develop higher dimensional analogues of the complex numbers, such as Clifford algebras and Multicomplex numbers. These rely heavily on geometric notions, and we explore the analysis which results. This is what is called hyper-complex analysis. This dissertation explores the most prominent of these higher dimensional analogues and highlights a many of the relevant results which have appeared in the last four decades, and introduces new ideas which can be used to further the research of this discipline. Indeed, the objects of interest are Clifford algebras, the algebra of the multicomplex numbers, and functions which are valued in these algebras and lie in the kernels of linear operators. These lead to prominent results in Clifford analysis and multicomplex analysis which can be viewed as analogues of complex analysis. Additionally, we explain the link between Clifford algebras and conformal geometry. We explore two low dimensional exam- ples, namely the split-complex numbers and split-quaternions, and demonstrate how linear fractional transformations are conformal mappings in these settings. ix CHAPTER 1 INTRODUCTION One need look no further than a text on complex analysis, such as [1] or especially [33], to know that algebraic properties of C play a major role in the analysis and geometry of the plane. The simple fact that i2 1 gives rise to the Cauchy-Riemann equations, which is the = − foundation of the theory of holomorphic functions, which are those functions of a complex variable which are differentiable in a complex sense. Indeed, the existence of the limit of the difference quotient f z ∆z f z lim ( + ) − ( ) ∆z→0 ∆z means that the limit is the same whether ∆z ∆x or ∆z i∆y. That is, = = ∂u ∂v 1 ∂u ∂v i , ∂x + ∂x
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