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2015 Analysis of Functions of Split-Complex, Multicomplex, and Split-Quaternionic Variables and Their Associated Conformal John Anthony Emanuello

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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 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 and the Multicomplex 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 and Miscellaneous Topics ...... 16 2.3.5 TheSplit-ComplexNumbers ...... 18 2.3.6 TheSplit- ...... 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 ...... 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 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 ...... 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 . Points with the same numeric label are identified. . . . . 67

viii ABSTRACT

The connections between , 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 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 = i ∂y + ∂y

and the C-R equations are obtained:

∂u ∂v ∂v ∂u and . ∂x = ∂y ∂x = −∂y

The minus in the second equation occurs because 1 i, which is a direct consequence i = − of i2 1. Thus, when a function of a complex variable with C1 components is holomorphic = − if and only if the C-R equations are satisfied. One may also consider functions of a complex variable which are annihilated by the operator 1 ∂ ∂ ∂z¯ i . ∶= 2 ‹∂x + ∂y 

Indeed, a function is holomorphic if and only if it is annihilated by ∂z¯ and its complex

derivative is given by ∂zf, where

1 ∂ ∂ ∂z i . ∶= 2 ‹∂x − ∂y 

1 Holomorphic functions are also (locally) conformal mappings of the plane if ∂zf does not vanish. Additionally, the algebraic properties of i make it possible to write the conformal mappings of the (as the compactification of the plane) using only the usual operations on C. For decades, many mathematicians have been interested in extending complex analysis to new settings involving higher dimensional analogues of C. This rich area of mathematics is commonly referred to as and has spawned thousands of articles and scores of books, all the while inspiring the research of countless scientists in mathematics, , and even meteorology. We cannot see now where future work in this area may lead. The current work is intended to give only a flavor of hypercomplex analysis and those results which the author and his major professor found over the last three years; there are many works whose results are not summarized here. Indeed, the bibliography of this dissertation is a good place for interested readers to further their knowledge.

1.1 Outline of the Dissertation

The dissertation is organized as follows. In Chapter 2, two kinds of analogues of the complex numbers, namely the real Clifford algebras and the algebra of multicomplex num- bers, are introduced and their basic properties are discussed. That the Clifford algebras are related to certain quadratic forms play an interesting role in later chapters (see also [45]). More specifically, we associate the split-complex numbers and the split-quaternions as the 1 1 2 2 semi-Riemmanian manifolds R , and R , , respectively, in a the same manner that we asso- 2 ciate C with R . These associations play an important role in later chapters. The latter is 1 1 2 particularly important as it contains both R , and R as embeddings. Also a careful study of the multicomplex numbers provides the background to study a simple analogue of the M¨obius geometry of the complex plane. Chapter 3 gives a flavor of the resulting analysis obtained by considering these higher dimensional analogues as domains for functions. Over the last few decades there has been a great deal of work in this area, and some of these results are summarized in this chapter.

2 Indeed, there are analogues of the operator ∂z¯, which give a function theory akin to the theory of holomorphic functions, as in [5, 15, 53, 49, 62]. Chapter 4 contains preliminaries from conformal geometry which play an important role in the remaining chapters. For example we define semi-Riemannian manifolds and discuss the basics properties of conformal mappings as outlined in [47, 11, 12]. We also introduce the notion of a conformal compactification, which is the analogue of the Riemann sphere for general semi-Riemannian manifolds [55]. Next in Chapter 5, the split-complex numbers are discussed as an analogue of C along with the resulting function theory and conformal geometry. This chapter contains both previously published results, e.g. [14, 16, 39, 34], and original research. For example, we have not seen work in the literature which carefully shows that the M¨obius transformations of the split-complex plane are direct sums of M¨obius transformations on the which extends to its compactification. We also have not seen a careful treatment of the fixed points of these transformations, especially as they pertain to the eigenvalues of the associated matrices and properties of the cross ratio. Chapters 6 and 7 can be viewed as a sequels of Chapter 5, in the context of the mul- ticomplex numbers and split-quaternions, respectfully. These chapters also contain original work, some of which are reformulations of other works in these contexts. For example, the Cauchy-Kowalewski extensions discussed in Chapter 7 are analogues of ideas which were proved earlier and whose proofs directly translate to the new setting. Additionally, we show 2 2 that the M¨obius transformations of R , may be realized as linear fractional transformations over the split quaternions. This was conjectured in our previous joint work [45] and, to the 2 2 best of our knowledge, this is the first work to think of the M¨obius transformations of R , in this way.

1.2 Topics for Further Research

There are several directions for further research on topics introduced in the dissertation. Indeed, the areas of split-complex and split-quaternionic variables seem quite promising.

3 One important analogue of complex analysis which has not been fully developed is the theory of meromorphic functions. Defining Laurent-like for split-complex variables would be quite simple and would decompose into two real series. This might make it possible to define different singularities (although they would no longer be isolated) like we can in the complex plane. Because the algebra is more complicated, similar analogues would be more difficult to explore in the split-quaternionic case, but it might still be a worthwhile endeavor. Chapter 5 contains a fully developed fix theory for split-complex M¨obius transfor- mations, but such analogues must be explored in the split-quaternionic case. It would be interesting to explore if eigenvalues of an associated in PSL 4, R say anything about ( ) fixed points of a given M¨obius transformation, like we see in the complex and split-complex cases. It would also be interesting to study subgroups which leave embeddings of the com- plex and split-complex planes invariant in the split-quaternions and their presentations in PSL 4, R . ( )

4 CHAPTER 2

CLIFFORD ALGEBRAS AND THE MULTICOMPLEX NUMBERS

Following the discovery of the complex numbers, mathematicians have sought to generalize this notion to higher . During the latter half of the nineteenth century, works by Hamilton, Peirce, Grassmann, Clifford, and others led to a proliferation of algebras [15, 49]. To this day, these objects continue to offer rich results in analysis and geometry. This chapter will summarize the basic construction and results of two families of algebras: Clifford algebras and the Multicomplex numbers.

2.1 Clifford’s Geometric Algebras

Informally speaking, Clifford algebras are associative algebras with unit and an embedded vector (normally, this is a euclidean or pseudo-euclidean space[24]). Under certain circumstances, a Clifford algebra satisfies an important universal property and can be used to assure that the algebras we are most interested in are unique up to an .

2.1.1 A Brief History

Although usually credited to Sir , the famous Irish mathemati- cian, the quaternions were actually discovered in 1840 by Benjamin Olinde Rodrigues, a Frenchman who never received credit at the time (and rarely receives credit today) [4]. Three years later, Hamilton would discover independently the algebra after years of trying to find higher dimensional analogues of C [25]. Thirty-five years later, William Kingdon Clifford, an English Geometer and Philosopher, combined the work of Hamilton (there is no evidence he was aware of Rodrigues’s work) and Grassmann to create what he called geometric algebras [10]. They were since renamed Clifford algebras and have been used by physicists and mathematicians alike. Of notable

5 importance is P. Dirac’s use of the γ- matrices (see the original work in [17]), which are the generators of the Clifford algebra Cℓ1,3, to linearize the Klein-Gordan equation [15].

2.1.2 Basic Examples

We shall assume throughout that the underlying fields of these algebras are of a charac- teristic other than 2. While the constructions we are about to see can be done over fields of any characteristic, some of the formulas we see will not be valid when 2 0 in the underlying = field. First, we define a quadratic space.

Definition 2.1.1. Let V be a finite dimensional over a field F. A is a map Q V → F ∶ such that Q λv λ2Q v for all λ F and v V . The associated to Q is the ( ) = ( ) ∈ ∈ map B V V → F defined by ∶ × 1 B v, w Q v w Q v Q w . ( ) ∶= 2 ( ( + ) − ( ) − ( ))

When the subspace V – w B v, w 0 v V ∶= { ∶ ( ) = ∀ ∈ } is precisely 0 , we say V,Q is non-degenerate; otherwise V,Q is degenerate. { } ( ) ( ) Next we define a special kind of for V , which is important for the construction of Clifford algebras.

Definition 2.1.2. Let e1,..., en be a basis for V . It constitutes a normalized basis when { }

1. B ei, ej 0 whenever i j, ( ) = ≠ – 2. ei Q ei 0 is a basis of V , { ∶ ( ) = } – – 3. ei Q ei 0 is a basis for V and Q ei 1 if F R, while Q ei 1 if F C. { ∶ ( ) ≠ } ( ) ( ) = ± = ( ) = = We define Clifford algebras over quadratic in a very natural way.

6 Definition 2.1.3. Let V,Q be a quadratic space over a field F, A an ( ) over F with identity 1, and ν V →A a linear embedding. ∶ The pair A,ν is a called a Clifford algebra for V,Q if ( ) ( ) (i) the algebra A is generated by ν v v V λ ⋅ 1 ∶ λ F and { ( ) ∶ ∈ } ∪ { ∈ } 2 (ii) ν v −Q v ⋅ 1, ∀v V . ( ( )) = ( ) ∈ Typically, we will refrain from writing ν v and write v for an element in V and its ( ) image in A. Note that we are using bold-faced letters to denote vectors, to distinguish these elements from the others in the algebra.

Example 2.1.4. Let V R and Q x x2. Then if we let A C and ν ∶ y ↦ yi, it is clear = ( ) = = that the C is a Clifford algebra, which is also called Cℓ0,1 for reasons which will be clear soon.

It is this example which provides the motivation for studying these structures. We would like to view these algebras as higher dimensional analogues of C. We will soon see, however, that C is also a distinguished example (i.e. it is a field) and that in general Clifford algebras are neither commutative nor rings.

2 2 2 Example 2.1.5. Let V R and Q ∶ x1,x2 ↦ x1 + x2. Then if we let A H and ν ∶ = ( ) = x1,x2 ↦ x1i + x2j, it is clear that H is a Clifford algebra. Unlike C, however, H is not ( ) commutative.

To help us see an example of a Clifford algebra that is not a division , it will be helpful to note that the basis elements of the algebra satisfy a nice rule.

Proposition 2.1.6. Let A,ν be a Clifford algebra for V,Q and let e1,..., en be an ( ) ( ) { } normalized (with respect to Q) basis of V . Then

ejek + ekej −2Q ej δj,k. = ( )

Proof. This is essentially the proof in [24]. First, if j k, then the claim is trivial. =

7 Now, note that the normalized condition says

1 0 B ej, ek Q ej ek Q ej Q ek , = ( ) ∶= 2 ( ( + ) − ( ) − ( ))

whenever j k. Also, by definition ≠

2 2 2 Q ej ek Q ej Q ek ej ek ej ek ( + ) − ( ) − ( ) = −( + ) + ( ) + ( ) 2 2 2 2 e ejek ekej e e e = −( j + + + k) + j + k

ejek ekej . = −( + )

Hence when j k, we have ejek ekej 0. ≠ + =

3 Example 2.1.7. Let e1, e2, e3 be an normalized basis for V R and let Q be the euclidean { } = quadratic form. The algebra Cℓ0,3 defined by the generating set

1, e1, e2, e3, e1e2, e1e3, e2e3, e1e2e3 { } is a Clifford algebra when ν is defined in the most obvious way.

The Clifford algebra Cℓ0,3 is not a . For

2 e1 e2e3 e1 e2e3 1 e1e2e3 e2e3e1 e2e3 ( + )(− + ) = + + (− ) + ( ) 1 e1e2e3 e1e2e3 1 = + − − 0, = i.e. e1 e2e3 and e1 e2e3 are zero divisors. + − + We are also concerned with the of Clifford algebras, for it will play a role in determining when a Clifford algebra satisfies a universal property.

Theorem 2.1.8. Let A,ν be a Clifford algebra for V,Q then A is of dimension at most ( ) ( ) 2dim(V ).

The following will make the proof of the previous much simpler.

8 Lemma 2.1.9. Let A,ν be a Clifford algebra for V,Q and let e1,..., en be an normal- ( ) ( ) { } ized basis of V . Then A is spanned by all reduced products of the form

m1 m2 mn e1 e2 e , where mj 0, 1. ⋯ n =

m1 m2 mn Proof. The inclusion Span e1 e2 en , where mj 0, 1 A is obvious. { ⋯ = } ⊆ If we apply Proposition 2.1.6 enough times to any x A, it may be written as linear ∈ combinations of reduced products.

m1 m2 mn Hence, Span e1 e2 en , where mj 0, 1 A. { ⋯ = } = Proof of the Theorem. Since we know how large a spanning set of A can be, it must be the case that dim(V ) dim V dim dim A ( ) 2 (V ). ( ) ≤ kQ0 ‹ k  = =

As evidenced by all of our examples so far, we are particularly concerned with those Clifford algebras which are of dimension exactly 2dim(V ). We shall see why in the next subsection.

2.2 Clifford Algebras 2.2.1 A Categorical Construction

In to say anything about universal properties, we must understand the categorical confines we face. Chief among these is to define a desirable category for which every Clifford algebra over the quadratic space V,Q is an object. ( ) Definition 2.2.1 (Category Q-alg1). Let V,Q be a quadratic space over a field F. The ( ) objects of the category Q-alg are pairs A,ν , which are Clifford algebras in the sense of ( ) Definition 2.1.3. The morphisms between objects A,ν and B,µ are those algebra ϕ ( ) ( ) which give rise to commutative diagrams:

1This is the definition given in [42]. Some of the literature discusses the universal properties without expressly defining a category.

9 ϕ / A_ ? B

ν µ

V

The category Q-alg has initial and final objects, but not zero objects in general[42].

Theorem 2.2.2. The trivial algebra 0 ,ν0 is an initial object in Q-alg. There exists a ({ } ) Clifford algebra Cℓ V,Q which is a final object in Q-alg. ( ) The proof first part is trivial and for the second part we shall prove a more intuitive corollary instead.

Corollary 2.2.3. A Clifford algebra A,ν in Q-alg is initial when dim A 2dim V . ( ) =

Proof. Let vi be an orthonormal basis of V and let B,µ be any Clifford algebra for { } ( ) V,Q . Also, let ej ν vj and fj µ vj . Then ( ) = ( ) = ( )

ejek ekej 2Q vj δj,k + = − ( )

fjfk fkfj 2Q vj δj,k. + = − ( )

Then the collections of reduced products E is a basis of A and the collection of reduced A products E span B, the map B ϕ ej fj ∶ ↦ gives a well-defined algebra of A onto B. Notice that ϕ ν µ. Thus A is initial, as required. ○ = Needless to say, all of the examples we have given are initial in their respective categories. We shall call any such Clifford algebra the universal Clifford algebra of the quadratic space V,Q , as it is unique up to an isomorphism in Q-alg. ( )

10 2.2.2 Analogues of the Complex Numbers

As previously mentioned, we view Clifford algebras as higher dimensional analogues of the complex numbers [32]. In fact, we may even define an analogue of the . n First, we need to address some issues of notation. Let ei i 1 be a basis for V . Given { } = a set A i1, i2, . . . , im 1,...,n such that 1 i1 i2 ... im n, we let eA denote = { } ⊂ { } ≤ < < < ≤ the basis element ei1 ei2 eim of Cℓ V,Q . For the case A , we have e 1. As such, any ⋯ ( ) = ∅ ∅ = x Cℓ V,Q may be written ∈ ( ) x xAeA, xA F. = QA ∈

Definition 2.2.4. We define the following involutions on eA and extend to Cℓ V,Q by ( ) linearity.

2 1. The main x x′ is defined by ↦ k e′ 1 eA, if A k. A = (− ) S S =

2. The reversion x x is defined by ↦ ̃

ei1 ei ei ei1 Ë⋯ k = k ⋯

3. The (Clifford) conjugation x x¯ is defined by ↦

eA eA ′ e′ = ( ̃ ) = ̃A

We get this lemma which will be useful later [12].

Lemma 2.2.5. Let x,y Cℓ V,Q . Then ∈ ( )

1. xy ′ x′y′, ( ) = 2. xy y˜x˜, and (Ê) = 3. xy y¯x¯. ( ) = 2This is sometimes called the principal involution or the grade involution.

11 Example 2.2.6. For the Clifford algebras C and H, the conjugation operation can be used to obtain the euclidean . Notice that the definition above on C coincides with the usual definition of conjugation.

For a Z x1 x2i x3j x4ij = + + +

Z x1 x2i x3j x4ij, = − − − by the above. Notice that 2 2 2 2 ZZ x1 x2 x3 x4, = + + + 4 which is the of the euclidean norm on R .

This is not the case in general, for the of an arbitrary element and its conjugate need not be a real . But it turns out that if we look only at the real part of the 2n product, we will always get a quadratic form over the vector space R (see Theorem 2.3.4 for more details).

2.3 Real Clifford Algebras

For now, we limit ourselves to the case where V Rn and = n 2 Q x xj . ( ) = jQ1 = In such a case we will denote the (universal) Clifford algebra by Cℓ0,n. We choose to investi-

gate this case in further depth for the analysis on Cℓ0,n-valued functions is well-known and will provide inspiration for studying analogous ideas for pseudo-euclidean spaces. There is also a great deal of work on Clifford algebras over C. Unlike the real case, there is only one complex Clifford algebra of a given dimension [5].

2.3.1 The Euclidean Case

We saw in examples 2.1.4 and 2.1.5 that C and H are the Clifford algebras Cℓ0,1 and

m Cℓ0,2 in the manner described at the beginning of this section. Because we may embed R in Rn for m 1, 2,...n 1, we get an embedding of the Clifford algebras = −

C H Cℓ0,3 Cℓ0,n. ⊂ ⊂ ⊂ ⋯ ⊂ 12 We would expect many of the properties of the complex numbers and the quaternions to have analogues in the more general case. We shall see that this is indeed the case.

Definition 2.3.1. Let k 1,...,n . We define the space of k-multivectors3 by ∈ { } ⎫ k ⎧ ⎪ Cℓ0(,n) ⎪⎨x Cℓ0,n x xAeA⎬ . = ⎪ ∈ ∶ = AQk ⎪ ⎩ S S= ⎭ For k 0, define = 0 Cℓ0(,n) {x Cℓ0,n x x0e } . = ∈ ∶ = ∅

Example 2.3.2. The Clifford algebra Cℓ0,3 may be decomposed as follows:

0 Cℓ0(,3) the scalars 1 Cℓ0(,3) the vectors 2 Cℓ0(,3) the 3 Cℓ0(,3) the volume elements These the , vector, bi-vectors terminologies carry over in higher dimensions. We also have that 0 1 Cℓ0(,n) ⊕ Cℓ(0,n) is called the collection of [48]. It is now clear that as a vector space,

n k Cℓ0,n Cℓ0(,n). = ?k 0 = Hence, we may view this decomposition as an analogue of the decomposition of a into its real and imaginary parts. In fact, the decomposition also comes with a set of projections.

Remark 2.3.3. For each k 0, 1,...,n and each x Cℓ0,n, let x denote projection of x = ∈ k k on Cℓ0(,n). [ ]

By way of the conjugation we get from Definition 2.2.4, we may define a norm on Cℓ0,n.

3 (k) This is the terminology we find in [24], but it may not be standard. In most literature, Cℓn is not given a name (as in [15] and [42] ).

13 Theorem 2.3.4. Let x Cℓ0,n then the Clifford norm of x denoted

1 2 ∈ 1 2 2 ~ x xx¯ 0 ~ xA = = A S S ([ ] ) ŒQ S S ‘ is a norm in the usual sense.

We will not prove this now. Instead, we will see in the next section that x x,x for = some inner product on Cℓn and the needed result will follow immediately. S S ⟨ ⟩

2.3.2 Relation with the

Given a quadratic space V,Q over field F a one can construct the exterior algebra

∗ V independent of Q. It( turns) out that there is a vector space isomorphism between V Cℓ V,Q ⋀∗( ) and . This is particularly easy to see in the Euclidean case.

⋀Theorem( ) 2.3.5.( If) we extend the map

k k ϕ  V → Cℓ0(,n)

eα1 eα2 ∶ (eα)k ↦ eα1 eα2 eαk ⋯ ∧ ⋯

n ∧ ∧ by linearity to all of ⋀∗ R , then this gives an isomorphism of vector spaces between Rn ⋀∗ and Cℓ0,n. Further,( this) isomorphism is completely independent of a choice of basis.

(We) omit the proof, as it may found in [24].

The isomorphism induces an inner product on Cℓ0,n. For if we define

ξ1 ξk,η1 ηk det ξi ⋅ ηj ∧ ⋯ ∧ ∧ ⋯ ∧ = ij

⟨ ⟩ k ‰[ – ] Žj k n n n n on ⋀ R and extend it to all of ⋀∗ R so that  R  R . The inner product = j k ≠ extends( to) Cℓ0,n via the isomorphism( and) inducesŒ a( Hilbert)‘ space structure[24( ) ].

Proposition 2.3.6. The inner product on Cℓ0,n induces the Clifford norm defined above.

The proof of the proposition, will follow directly after we prove a lemma.

14 Lemma 2.3.7. Let x xAeA and y yBeB, then = A = B Q Q x,y xAyA. = A ⟨ ⟩ Q Proof. For every x,y Cℓn we have ∈

x,y xAeA, yBeB, = A B

⟨ ⟩ dQ xAyBQeA,eB i xAyB eA,eB = A B A B S S=S S S S≠S S ⟨ ⟩ + ⟨ ⟩ j k Q xAyB eA,eB , byQ the orthogonality of Cℓ(n ) and Cℓn( ),j k . = A B ≠ S S=S S Q ⟨ ⟩ ( ) Recall, eA,eB det em ⋅ em . Then if A B we see that em ⋅ em I and = i j ij = i j ij = eA,eB 1. If A B, we find that em ⋅ em is singular and so eA,eB 0. = ⟨ ≠⟩ Š  i j ij  =  ⟨ Thus,⟩   ⟨ ⟩

x,y xAyA. = A ⟨ ⟩ Q

2.3.3 The Pseudo-Euclidean Case

We now consider those Clifford algebras which are formed from the quadratic space

p q 4 R + ,Qp,q , where p p q 2 − + 2 ( ) Qp,q x xi xi . = i 1 i p 1 = = + We shall denote the (universal) Clifford( algebra) Q whichQ arises by Cℓp,q. Given an orthonormal p q Rp q basis ei i +1 of + , Definition 2.1.3 shows us that the multiplication rules in Cℓp,q are = slightly{ different} from the Euclidean case:

2 −1 if i 1,...,p ei = = ⎧1 if i p + 1,...,p + q. ⎪ = ⎨ ⎪ A calculation similar to that in⎩⎪ the proof of Proposition 2.1.6 shows that we still have

eiej −ejei. = 4This is usually denoted Rp,q. Of course the usual Euclidean space Rn is just Rn,0 in this context.

15 For these Clifford algebras, we may define an analogous k- structure and ob- tain an indefinite inner product which is precisely the positive-definite inner product obtained in Lemma 2.3.7 when q 0. =

Example 2.3.8. The split-complex numbers, which are denoted Cℓ1,0, represent the simplest of these algebras. We shall revisit this example in further detail below.

2.3.4 Clifford Group and Miscellaneous Topics

A Clifford algebra is not a multiplicative group, for 0 has no inverse. We can, however, define a subset of Cℓp,q which is a group. Since every vector squares to a (by Definition 2.1.3), these are natural choices of invertible elements.

2 Example 2.3.9. Let x Cℓp,q be a vector such that x 0, then x has a multiplicative ∈ ≠ inverse [24]. Indeed, x x2 x 1 x2 = x2 = Definition 2.3.10. The set ‹ 

2 Γp,q x Cℓp,q x x1 ⋅ x2⋯xm, x 0 = ∈ = k ≠ ∶ ™ ∶ ž forms a group under Clifford multiplication, and is called the Clifford (or Lipschitz)

group. Γp,q

The subgroup of Γp,q

2 Pin p,q ∶ x Cℓp,q ∶ x x1 ⋅ x2⋯xm, x ±1 = ∈ = k = ( ) ™ ž is called the Pin group.

Indeed, we can easily check when x Cℓp,q is in the Clifford group [12]. ∈ Proposition 2.3.11.

x Γp,q xx R ∖ 0 . ∈ ∈ The following is a result which will prove⇐⇒ to be useful{ } later [11].

16 Lemma 2.3.12. The group Pin p,q gives a double covering of the O p,q .

Recall, that O p,q is the group( ) of p q p q matrices such that for all A O( p,q)

( ) ( + ) × ( + ) ∈ ( ) Qp,q Ax Qp,q x , = ( ) ( ) and for all x Rp,q. ∈ We can write Cℓp 1,q 1 as a combination of elements in Cℓp,q. We just need two more + + basis elements e , e which satisfy the multiplication rules above. Then, every x Cℓp 1,q 1 + − ∈ + + may be decomposed: x A Be Ce De e . = − + + − This observation leads to an important+ algebraic+ fact.+

Proposition 2.3.13. The Clifford algebra Cℓp 1,q 1 is isomorphic to the algebra of 2 2 + + matrices over Cℓp,q. ×

Proof. We construct the isomorphism found in [12]. Define ϕ Cℓp 1,q 1 → M 2,Cℓp,q by + + A D B∶ C ( ) A Be Ce De e z→ . − + + − B′ C′ A′ D′ + − + + + + Œ ‘ It is easy to check that φ is an .+ − Also, φ has a two-way inverse:

A B 1 z→ A D′ B C′ e B C′ e A D′ e e . CD 2 − + + − Œ ‘ [( + ) + (− + ) + ( + ) + ( − ) ]

We take this opportunity to introduce an analogue of , which will be useful later. A B Definition 2.3.14. Let . The pseudo-determinant of A is given by = CD A Œ ‘ λ AD BC = (A) ̃ − ̃ The pseudo-determinant is well behaved in some helpful ways.

17 Proposition 2.3.15.

1. λ λ λ = 2. If(BA)represents(A) a(B) vector v in Rp,q then λ v2. = 3. If A Pin p,q then λ 1. (A) ∈ = 2.3.5A The( Split-Complex) (A) ± Numbers

We have already seen that the split-complex numbers are the Clifford algebra Cℓ1,0. Every split-complex number is written

z x yj, where j2 1. = = + The name is somewhat suggestive, indicating that even though this algebra resembles C, there are some key differences. Indeed, we shall soon see why the word split is the appropriate qualifier. The conjugation we get from Definition 2.2.4 for the split-complex numbers is the same as we have in C: z x jy z¯ x jy. = =

Unlike in C, conjugation does not induce+ ↦ a norm− in Cℓ1,0. For the positivity condition fails:

1 j 1 j 1 j j j2 = ( + ) ( − ) 0.− + − =

2 2 Remark 2.3.16. The map z zz¯ x y is the quadratic form Q1,1. Hence we may = = R1,1 associate Cℓ1,0 with (see⟨ our⟩ previous− joint works in [20, 45]). This seemingly small fact (and it’s analogue for Cℓ1,1) is the linchpin of the results in Chapters 5 and 7.

Definition 2.3.17. The light cone is the subset L L L of Cℓ1,0 where = + − ∪ L a 1 j a R and L a 1 j a R . + = ∈ − = ∈ { ( + ) ∶ } { ( − ) ∶ } 18 It is immediately clear that the light cone contains precisely those elements which are annihilated by . It also contains precisely those elements which are not invertible; for

⟨ ⟩ 1 z¯ z− . = z

We would like to define a new basis for Cℓ⟨ 1⟩,0 in terms of the light cone. To see this, define 1 j 1 j j and j . + = 2 − = 2 + − Notice that for any z x jy, we may write = + z uj vj , = + − + where u x y and v x y. We call this the light cone basis. This basis makes our = = calculations easier!+ −

Proposition 2.3.18. Let j ,j be the light cone basis for Cℓ1,0 and z u1j v1j , + − = + − w u2j v2j . Then = + − { } + 1. j and+ j are idempotents, + −

2. z v1j u1j = + −

3. zw u1u+2j v1v2j = + − Proof. For the first+ part, we see that

1 j 2 1 j 1 j + = 4 1 ( ) (2 + 2)(j + ) = 4 j (. + ) = +

A similar calculation shows j 2 j , as required. − = − Since conjugation changes( the) sign of the coefficient of j, it is clear that in the new basis conjugation swaps the components. Hence, the second statement is proved.

19 For the last part, we again proceed by direct calculation. By the above argument and an earlier observation,

2 2 zw u1u2 j u1v2j j v1u2j j v1v2 j = + + − − + − u1u2j( )v1+v2j . + + ( ) = + − +

Hence, the split-complex numbers is (algebra) isomorphic to R⊕R. This is why the word split is used to name the algebra.

Remark 2.3.19. When defined, the inverse is given by

1 1 1 z− u− j + v− j . = + − It is also be useful to invert only one of the components:

1 1 z− u− j + vj and + = + − 1 1 z− uj + v− j . − = + − 2.3.6 The Split-Quaternions

The split-quaternions are the real Clifford algebra

Cℓ1,1 ∶ Z x0 + x1i + x2j + x3ij ∶ x0,x1,x2,x3 R . = = ∈ Functions of a split-quaternionic{ variable and notions of regularity have} been the subject of interest in the literature [40, 44]. It is worth noting that the split-quaternions contain both the complex and split-complex numbers as subalgebras. In a manner similar to the split-complex case, we may obtain the indefinite quadratic form Q2,2 by 2 2 2 2 ZZ x0 + x1 − x2 − x3. = Hence we shall identify the split-quaternions with R2,2. There are a number of ways to express the split-quaternions as2 × 2 matrices over R and C.

20 Lemma 2.3.20. As algebras, the split-quaternions and real 2 2 matrices are isomorphic.

Proof. If we identify × ⎤ ⎡ ⎤ ⎡ ⎤ 1 0 ⎥ ⎢ 0 1 ⎥ ⎢ 0 1 ⎥ 1 ∼ ⎥ , i ∼ ⎢ ⎥ , j ∼ ⎢ ⎥ , ⎡ 0 1 ⎥ ⎢ 1 0 ⎥ ⎢ 1 0 ⎥ ⎢ ⎥ ⎢ − ⎥ ⎢ ⎥ ⎢⎣ ⎦ ⎣ ⎦ ⎣ ⎦ Cℓ1 1 ⎢ then we may map , to the⎢ real 2 2 matrices by ⎡ ⎤ ⎢ x0 x3 x1 x2 ⎥ x0 x1i x2j ×x3ij ⎢ ⎥ . → ⎢ ⎥ ⎢ x1 x2 x0 x3 ⎥ ⎣ + − + ⎦ Notice that + + + z ⎡ ⎤ + − ⎢ x0 x3 x1 x2 ⎥ 2 2 2 2 ⎢ ⎥ x x x x , det ⎢ ⎥ 0 1 2 3 ⎢ x1 x2 x0 x3 ⎥ = ⎣ + − + ⎦ which is the form Q2 2. It’s easy to check that this gives+ an− algebra− homomorphism. Further, , + − ⎡ ⎤ ⎢ y1 y2 ⎥ 1 ⎢ ⎥ y1 y4 y3 y2 i y3 y2 j y1 y4 ij ⎢ ⎥ z→ ⎢ y3 y4 ⎥ 2 ⎣ ⎦ gives a two-sided inverse, so that[( the+ above) + ( is− an algebra) + ( + isomorphism.) + ( − ) ]

2.4 The Multicomplex Numbers

The other family of algebras we are interested in is the multicomplex numbers. Initially conceived by Segre in 1892, these are commutative algebras with unity, whose dimensions are powers of two [57]. We will first study a low dimensional example, the bicomplex numbers, and use it to generate the infinite family. There is not a unique way to realize the multicomplex numbers. In this work, we define the bicomplex numbers as the complex Clifford algebra obtained by complexifying the complex numbers: BC C C, = so that the complex numbers and split-complex∶ numbers⊗ are subalgebras. Higher dimensional members of the family are obtained iteratively by tensoring with C. One may also think of these as a commutativization of a Clifford algebra of the appropriate dimension. One can easily show that these algebras are isomorphic to direct sums of C. With the appropriate euclidean norm, Segre’s algebras become Banach algebras, making them appropriate spaces on which to do analysis.

21 2.4.1 The Bicomplex Numbers

The bicomplex numbers,

2 BC ζ ξ ωi2 ξ x yi1, ω s ti1 C, i2 1 , = = = = ∈ = ∶ ™ + S + + ( ) − ž is a commutative algebra with unity generated over C by 1 and i2 (and over R by 1, i1, i2, 2 and i1i2). Notice that the complex numbers is a subalgebra of BC. Further, i1i2 1 so = that the split-complex numbers is also a subalgebra. Indeed, BC C Cℓ1,0. ≅ ( ) BC And since the split-complex numbers are not a , neither⊗ is , for there are zero divisors: 2 1 i1i2 1 i1i2 1 i1i2 0. = =

As in Cℓ1,0, these zero divisors( + can be)( used− to) define− ( a useful) basis of BC. We define

1 i1i2 1 i1i2 e+ and e− , = 2 = 2 ( + ) ( − ) which are idempotent, annihilate each other under multiplication, and sum to 1. These properties afford us a useful basis for BC:

ζ ze+ we−, = + where z ξ i1ω,w ξ i1ω [49]. This basis makes BC C ⊕ C as algebras, since we may = = ≅ add and multiply− elements+ component-wise:

ζ1 ⋅ ζ2 z1e+ + w1e− z2e+ + w2e− z1z2e+ + w1w2e−. = = ( )( ) When it exists, the inverse of ζ is given by

1 1 1 ζ− e+ + e−. = z w

Clearly we have that ζ is a if and only if ζ ze+ or ζ we−. Collectively, we call = = these points the light cone L .

22 We also get some involutions which serve as analogues of complex conjugation in the plane:

ζ ze+ we− = ❸ ζ ∶ we++ ze− = ❸ ❸ ζ ζ ∶ we+ + ze− = = ∶ + These give rise to three moduli with different ranges:

2 2 ζ 1 ζζ z e+ w e− Cℓ1,0 = = ∈ ❸ C SζS2 ∶ ζζ S wzS + S S = = ∈ ❸ BC SζS3 ∶ ζζ zwe+ wze− . = = ∈ S S ∶ + Of course, the usual Euclidean norm is given by

2 2 z w ζ . = ¾ 2 S S + S S Y Y The euclidean norm gives us the following [49].

Theorem 2.4.1. BC is a .

2.4.2 The Multicomplex Numbers

n 5 n 1 The multicomplex numbers, MCn isa2 -dimensional commutative algebra over R (2 − - dimensional over C) defined recursively as follows

MCn ξ ωin ξ, ω MCn 1 , = ∈ − ∶ { + S } where MC0 R, MC1 C, and MC2 BC. Thus, the algebra is generated over R by = = = 1, i1, . . . , in ∶ subject to∶ the following multiplication∶ rules:

{ } 2 ik 1 and ikiℓ iℓik, = = 5 MC R By viewing n as an algebra over( )it becomes− clear that it is indeed a commutativization of Cℓ0,n.

23 for all k,ℓ. That is, elements of MCn are of the form

ζ ζAiA, = A Q where ζA C, A ℓ1,...,ℓk S 2,...,n , and ∈ = ⊆ = { } { } iA iℓ1 iℓ . = ⋯ n By convention i 1. ∅ = Similarly, MCn is also not a division algebra. Furthermore, we also have

2 2 1 ikiℓ 1 ikiℓ 1 ik iℓ 0, + − = − = for any k, ℓ. ( )( ) ( ) ( )

In a manner analogous to the bicomplex case, we obtain an idempotent basis for MCn

[62]. That is, for any ζ MCn we may write ∈

ζ ζ1e1 ζ2e2 ζ2n−1 e2n−1 , = + + ⋯

where ζk xk yki1 C, ekeℓ δk,ℓek, and the ek is some product of n 1 elements of the = + ∈ = − form

1 ik1 ik2 ± . 2 ( ) Also, e1 e2 e2n−1 1. + + ⋯ = 2n−1 Thus, this new basis shows MCn C as algebras. ≅ k 1 4 = For example in MC3 C has a basis? given by ≅ k 1 = ? 1 i1i2 1 i2i3 1 i1i2 1 i2i3 e1 + + ,e2 + − = 4 = 4 (1 i1i2) (1 i2i3) (1 i1i2) (1 i2i3) e3 − + ,e4 − − . = 4 = 4 ( ) ( ) ( ) ( ) It is clear that these algebras can be embedded as follow:

C MC1 MC2 MC3 . = ⊂ ⊂ ⊂ ⋯ Of course, there are numerous ways in which these embeddings can occur.

24 When it exists, the inverse of ζ MCn is given by

2n−1 ∈ 1 1 ζ− zk− ek. = k 1 = Q To avoid notational confusion, we shall write Ln for the light cone in MCn. Then, ζ Ln ∈ if and only if zk 0 for some k. = Additionally, the usual Euclidean norm is given by

1 2 2n−1 ~ 1 2 ζ n 1 zk . = 2 − k 1 ⎛ = ⎞ Y Y Q S S Thus, we have the following [49]. ⎝ ⎠

Theorem 2.4.2. MCn is a Banach Algebra.

25 CHAPTER 3

CLIFFORD ANALYSIS AND MULTICOMPLEX ANALYSIS

3.1 Clifford Analysis

Clifford analysis usually involves studying differential operators acting on functions

Rn F U → Cℓ0,n, ∶ ⊆ although there is a more general theory. These differential operators are usually called Dirac operators, and they are Clifford valued operators which are factors of the Laplacian. One can also find extensive work in the literature on complex Clifford analysis, which is the parallel theory involving Clifford algebras over C (for details see [53, 32, 58]).

3.1.1 Dirac Operators

Roughly speaking, a Dirac operator is any first order differential operator which factorizes the Laplacian ∆. We shall define a Dirac operator on smooth functions which take values in a universal Clifford algebra. Let Cℓ Cℓ V,Q , where V,Q is a real, non-degenerate quadratic space with a nor- = n malized basis ei i 1. We shall denote the space of smooth Cℓ-valued functions on an open ( = ) ( ) set U V by C∞ U,Cℓ . This is a Cℓ-module under point-wise multiplication, a fact which ⊂ { } R is more easily seen( once) we write f C∞ U,Cℓ as f fAeA, where fA U → . ∈ = A Since V is embedded in Cℓ, we may define a vector field ∂v for every v V such that ( ) Q ∶ ∈ d ∂vf x f x tv , = dt t 0 = ( ) ( + )V for every f C∞ U,Cℓ and every x U. ∈ ∈ The following( are results) about ∂v [24, 38].

26 Theorem 3.1.1. Let α, β F and v, w V . Then

1. The map v → ∂v is linear:∈ ∈ ∂αv βw α∂v β∂w. + =

2. The vector field ∂v acts linearly on C∞ U,Cℓ +:

∂v αf βg( α∂)vf β∂vvg. = ( + ) + n With these properties, we may now define a Dirac operator on C∞ U,Cℓ . Let ei i 1 = be an orthonormal basis (with respect to Q) of V and for each i, define(∂i to) be the{ vector} n field of ei. When V R and the standard basis ei , we have = ∂{ } ∂i . = ∂xi

Definition 3.1.2. The Dirac operator DQ associated to V,Q is defined by

n ( ) DQ Q ei ei∂i. = i 1 = Q ( ) The Laplacian operator ∆Q associated to V,Q is defined by

n ( ) 2 ∆Q Q ei ∂i . = i 1 = Q ( ) Notice that if V Rp,q, this is precisely the Laplacian obtained by the metric of = signature p,q (see the next chapter for details). A couple of simple calculations yield the

following. ( ) 2 Theorem 3.1.3. 1. D ∆Q Q = R 2. If f x A fA x eA, where− fA U → , satisfies DQf 0 then ∆QfA 0, for every = = = A. ( ) ∑ ( ) ∶

While this is interesting all on its own, what is truly profound is that DQ and ∆Q are independent of the normalized basis we choose on V .

n n Theorem 3.1.4. Let ej j 1 and fj j 1 be normalized bases of V such that Q ej Q fj , = = = n n and let ∂j j 1 and δj j 1 be the associated vector fields. Then = { =} { } ( ) ( ) { } { } 27 1. Q ej ej∂j Q fj fjδj. j = j

Q ( ) 2 Q ( )2 2. Q ej ∂j Q fj δj . j = j ( ) ( ) ForQ the proof, findQ an orthogonal transformation between the bases and use the linearity of the vector fields[24]. Now we can talk about an analogue of holomorphic for Cℓ-valued functions.

Definition 3.1.5. Let f C∞ U,Cℓ . We say f is left(right) Clifford holomorphic if ∈ ( ) DQf 0 or if fDQ 0 . = = ( ) In the literature, there is a preference for studying left Clifford holomorphic functions. For that reason, we shall drop the qualifying “left” unless there is ambiguity. The study of Clifford analysis is primarily concerned with investigating the consequences

Clifford holomorphic Most of the known results are about Cℓn-valued functions.

3.2 Clifford Analysis on Cℓ0,n-valued Functions

Rn Let U be an open subset. A function f U → Cℓ0,n defined by f x fA x eA, ⊂ = A where fA C∞ U , is called Cℓ0,n-valued. We denote the Cℓ0,n-valued functions on U by ∈ ∶ ( ) Q ( ) C∞ U,Cℓ0,n . ( ) The( differential) operator which we will use is the corresponding Dirac operator we dis- cussed in definition 3.1.2, namely:

Definition 3.2.1. Let f x C∞ U,Cℓ0,n and let D denote the differential operator ∈ ) ( ) ( n ∂ D ei . = i 1 ∂xi = Q We say f x is left Clifford-holomorphic on U (with respect to the Dirac operator D) if Df x 0. We say f x is right Clifford-holomorphic on U if f x D 0. = ( ) = (So,) If f happens to( ) be Rn-valued,the property that f is left( Clifford-holomorphic) with respect to the Dirac operator D is equivalent to satisfying the following system of partial

28 differential equations

n ∂f i 0 i 1 ∂xi = = ∂f ∂f iQ j 0. ∂xj ∂xi = − This system is the classical Riesz system. When n 2, this is nothing more than the = Cauchy-Riemann equations in the plane.

Rn 1 Remark 3.2.2. One may also consider functions f U + → Cℓ0,n and a ⊆ operator ∶ ∂ D, ∂x0 n 1 which is a factor of the Laplacian in R + . In fact+ to get the usual Cauchy-Riemann operator in C, one must use this operator. A similar theory of Clifford holomorphic functions is obtained with this operator. See [15, 54].

Remark 3.2.3. Note that one may also consider operators which are not vector or paravector valued. Indeed, analogues of the Cauchy Integral formula (see [22, 23]) can be obtained in the quaternion context using the operator

∂ ∂ ∂ ∂ e1 e2 e3 . ∂x0 ∂x1 ∂x2 ∂x3 + + + In some ways, this is, perhaps, a more accurate analogue of the C-R operator. Indeed, when we explore functions of a split-quaternionic variable, this is the kind of operator we will study.

3.2.1 Examples and Non-examples of Clifford Holomorphic Functions

Before we delve further into the theory, let us observe some examples and non-examples of Clifford holomorphic functions. x Example 3.2.4. Let G x n . Then G is left and right Clifford holomorphic on U = x = Rn 0 . ( ) S S ∖ { } 29 Proof.

n ∂ n x DG x e j e i 2 2 2 j = i 1 ∂xi j 1 x1 x2 x = ⎛ = ⋯ n ⎞ ( ) ŒQn ‘ Q » ∂ ⎝ xi+ + ⎠ n . = − 1 ∂xi 2 2 2 i ⎛ x1 x2 xn ⎞ = + + ⋯ Q ⎜ » ⎟ Now for each i, ⎝Š  ⎠ n 2 2 ∂ xi j 1 xj nxi n = − . 2 2 n 2 1 ∂xi 2 = n 2 ~ + ⎛ x1 x2 xn ⎞ j 1 xj + + ⋯ ‰∑ = Ž ⎜ » ⎟ Summing over all i, we get DG x 0. Similar calculations yield the other assertion. ⎝Š =  ⎠ ‰∑ Ž The function G is the generalized( ) Cauchy kernel, and will be used to prove an analogue of the Cauchy Integral Formula. It’s usefulness is due primarily to the fact that it is also right Clifford holomorphic. There are some functions which we would want to be Clifford holomorphic, but sadly are not.

2 Example 3.2.5. Let n 4 and f x x be the squaring map on Cℓ4. Then f is not = = Clifford holomorphic on U R4. = ( ) 2 2 2 2 Proof. Let x e1x e2y e3z e4w. Then x x y z w (by definition 2.1.3). So, = + + + = − + + + ∂ x2 y2 z2 w2 ∂ x2 y2 z2 w2 Df x e1 + + + e2 +( + + ) = − ∂x − ∂y ( ) ( 2 2 2 2) 2 2 2 2 ( ) ∂ x y z w ∂ x y z w e3 + + + e4 + + + − ∂z − ∂w ( ) ( ) e1 2x e2 2y e3 2z e4 2w = − + − + − + − 2(x. ) ( ) ( ) ( ) = − Assuming x 0, we are done. ≠ It is possible to build Clifford holomorphic functions from another Clifford holomorphic function in interesting ways.

Theorem 3.2.6. Let f be Clifford holomorphic on U Rn. Then g x Dk xkf x is also ⊆ = Clifford holomorphic for every k N. ∈ ( ) ( ( )) 30 The proof is a simple induction argument for which the basis step is proved below.

Lemma 3.2.7. Let f be Clifford holomorphic on U Rn. Then g x xf x is harmonic = (i.e. D2g 0). = ⊆ ( ) ( ) Proof. First, note that

g x xf x eieAxifA. = = i,A Then we have ( ) ( ) Q n ∂ xifA Dg x ejeieA = i,A j 1 ∂xj = n ( ) ( ) Q Q ∂ xifA ∂ xifA eieieA ejeieA = i,A i 1 ∂xi j,i,A ∂xj = ( ) i j ( ) ≠ Q Qn + Q ∂ xifA ∂ xifA 1 eA ejeieA = i,A i 1 ∂xi j,i,A ∂xj = ( ) i j ( ) ≠ Q Qn(− ) + Q ∂fA ∂xi ∂fA ∂xi eA xi fA ejeieA xi fA = i,A i 1 ∂xi ∂xi j,i,A ∂xj ∂xj = i j ≠ − Q Qn ‹ + n  + Q Œ + ‘ ∂fA ∂fA eAfA xieA ejeieAxi = A i 1 A i 1 ∂xi j,i A ∂xj = = i j − Q Q + − Q Q + Q≠ Q n ∂f x ∂f x nf x xi eiejxi . = i 1 ∂xi j,i ∂xj = ( ) i j ( ) − ( ) − Q − Q≠ After an appropriate amount of simplifying and using the fact that f is Clifford holo- morphic, we get ∂f x n ∂f x ejeixi xi . j,i ∂xj = i 1 ∂xi i j ( ) = ( ) Q≠ Q Then it is clear that n ∂f x D xi 0. i 1 ∂xi = = ( ) Thus, ŒQ ‘ n n 2 ∂f x ∂f x D g x D nf x xi xi = i 1 ∂xi i 1 ∂xi = ( ) = ( ) ( ) (− ( ) − Q n ∂f x− Q n )∂f x nDf x D xi D xi = i 1 ∂xi i 1 ∂xi = ( ) = ( ) −0. ( ) − ŒQ ‘ − ŒQ ‘ = 31 There are other ways to generate Clifford holomorphic functions and which do not involve having a Clifford holomorphic function a priori. For example there are at least two ways in

the literature to generate such a function, given a Cℓ0,n-valued function whose components are analytic (see [5, 54]). Below is one of these Cauchy-Kowalewski extension theorems.

n 1 Theorem 3.2.8 (Cauchy-Kowalewski Extension). Let U be an open set of R − with the

basis e2,..., e2. Suppose that g is Cℓ0,n-valued function whose components are analytic on U. Then the function

∞ 1 k k f x1,...,xn x1 e1D′ g x2,...xn , k 0 k! = ∂ ( ) Q (− ) ( ) n where D′ D e1, is left Clifford holomorphic on an open neighborhood of U in R . = x1 3.2.2 (Some)− Analogues of Results from Complex Analysis

Perhaps the two most important theorems in all of complex analysis are Cauchy’s Theo- rem and Cauchy’s Integral Formula. These theorems have analogues in the Clifford setting and are worth mentioning. But first we will need a lemma which is a direct consequence of Stokes’ Theorem.

n Lemma 3.2.9. Let f,g C∞ U , (smooth and scalar valued) for some U R . ∈ ⊆ Then for each compact, orientable sub domain C U which has a piecewise differentiable ( ) ⊆ boundary, g x dσf x g x D f x g x Df x dµ, S∂C = SC n j 1( ) ( ) [( ( ) ) ( ) + ( )( ( ))] where dσ ∑j 1 1 − dx1 dxj dxnej, and m denotes n-dimensional Lebesgue = = ⋯ ∧ ∧ ⋯ ∧ measure. (− ) ∧ Ã

Proof. We follow the proof found in [24]. Let

n j 1 ω g x dσf x g x f x dσ Q 1 − g x f x dx1 dxj dxnej = = = j 1 − ∧ ⋯ ∧ ∧ ⋯ ∧ = ( ) ( ) ( ) ( ) ( ) ( ) ( ) Ã 32 Then the exterior derivative

n j 1 ∂ dω 1 − g x f x dxj dx1 dxj dxnej = j 1 ∂xj ⋯ ∧ ∧ ⋯ ∧ = Qn (− ) ∂f x( ( ) ( ∂g)) x ∧ ∧ à g x f x ejdµ = j 1 ∂xj + ∂xj = ( ) ( ) QgŒx(D) f x g x( )Df x ‘ dµ. = + [( ( ) ) ( ) ( )( ( ))] By Stokes’ Theorem, we know ω dω. S∂C = SC Thus the claim follows.

Lemma 3.2.10. Let f,g C∞ U,Cℓ0,n and C as above. Then ∈ ( ) g x dσf x g x D f x g x Df x dµ. S∂C = SC + ( ) ( ) [( ( ) ) ( ) ( )( ( ))] The proof follows by applying the previous lemma component-wise. We shall use this to prove an analogue of Cauchy’s Theorem.

Theorem 3.2.11 (Clifford-Cauchy Theorem). Let f,g C∞ U,Cℓ0,n and suppose C U is ∈ ⊆ bounded, orientable sub domain with a piecewise differentiable( boundary.) If f is left Clifford holomorphic on U and g is right Clifford holomorphic on U, then

1. dσf 0, and S∂C =

2. gdσ 0 S∂C = Proof. To prove item 1, let g 1. Then by the lemma, ≡

1 ⋅ dσf x 1 ⋅ D f + 1 ⋅ Df dµ 0, S∂C = SC = ( ) [( ) ( )] since 1 and f are both right and left Clifford holomorphic, respectively. A similar argument works for item 2.

33 Theorem 3.2.12 (Cauchy Integral Formula). Let U, C, f and g be as in Theorem 3.2.11. Then for every y C 1 ∈ f y G x y dσ x f x = ωn S∂C and ( ) ( − ) ( ) ( ) 1 g y g x dσ x G x y , = ωn S∂C n 1 where G is the Cauchy kernel( defined) in example( ) ( 3.2.4) ( and− ω)n is the surface area of S −

(i.e. ωn n−1 dS). = ∫S We follow the proof in [54, 24], which is roughly the same proof as in complex analysis:

Proof. A quick calculation shows that dσ x η x dS x , where η x is the outwardly = pointing normal to C at x and (dS) is the( scalar-valued) ( ) surface( ) measure. Let r 0 be sufficiently small so that the closed ball B y,r centered at y of radius r > lies in C. Then by the Clifford-Cauchy Theorem, ( )

G x y dσ x f x G x y dσ x f x . S∂C S∂B y,r = ( ) ( − ) ( ) ( ) x y ( − ) ( ) ( ) 1 On ∂B y,r , the unit normal vector n x x−y . Then G x y n x rn−1 and = S − S = ( ) ( ) ( − ) ( ) G x y dσ x f x G x y η x f x dS x S∂B y,r S∂B y,r ( ) = ( ) ( − ) ( ) ( ) 1( − ) ( ) ( ) ( ) 1 f x dS x S∂B y,r rn = ( ) − f x f y f y ( ) (dS) x dS x S n 1 S n 1 = ∂B y,r r − ∂B y,r r − ( ) ( ) − ( ) ( ) ( ) f x f y ( ) + ( ) 1 dS x f y dS x S∂B y,r rn SSn−1 = ( ) − f(x) − f(y) dS(x) + ω(f)y . ( ) S n 1 n = ∂B y,r r − ( ) ( ) − ( ) ( ) + ( ) By continuity of f, f x f y lim dS x 0. 0 S n 1 r→ ∂B y,r r − = ( ) ( ) − ( ) The necessary result follows after we divide by ωn. ( ) The formula for g is proven by a similar argument.

34 We also have an analogue of Morera’s Theorem.

n Theorem 3.2.13. Let f be a continuous Cℓn-valued function on a domain U R such that

⊆ dσf 0 S∂C =

for every compact, orientable C sub domain with a piecewise differentiable boundary. Then f is Clifford holomorphic.

Again, the proof is just like the corresponding proof in complex analysis. There are also analogues of The Mean Value and Liouville Theorems [15, 54]. However, the latter requires knowledge of Taylor expansions, which is not terribly relevant to the rest of the material.

3.3 Multicomplex Analysis

There is a great deal of literature on the analysis of regular functions of bicomplex and multicomplex variables [51, 53, 62, 13, 49, 7, 8, 43, 9]. Due to the algebraic structure, it can be easily shown that a regular function is merely a direct sum of holomorphic functions of distinct single complex variables. In fact a theory of elementary functions for the bicomplex case may be found in [43]. Additionally, we find that much of single variable complex analysis can be extended to this new setting [49].

3.3.1 Bicomplex Analytic Functions

The natural analogue of analytic is well known and is discussed thoroughly in the liter- ature.

Definition 3.3.1. Let U BC be open and ζ0 U. A function f U → BC is said to be ⊂ ∈ bicomplex analytic at ζ0 if ∶ f ζ f ζ0 lim ζ→ζ0 ζ ζ0 ζ L ∉ ( ) − ( ) exists. When it exists, the limit is the bicomplex− derivative and is denoted f ′ ζ0 .

( ) 35 Via a similar argument from our previous work (and in Chapter 5) and in other places in the literature, we have the following [20]. BC Theorem 3.3.2. A function f U → such that f f1 z,w e+ f2 z,w e− is bicomplex = C2 analytic if and only if f1, f2 are∶ analytic functions of and( ) + ( ) f 0, = ∂ ∂ where e+ e−. This second condition∇ means f1 f1 z and f2 f2 w . = ∂w ∂z = = Example∇ 3.3.3.+ Because of the idempotent basis and the( way) it simplifies( multiplication,) polynomials are bicomplex analytic. We shall explore these functions in greater detail later in this section. Additionally, any function which has a convergent power series will be bicomplex analytic. For example we define ζn exp ζ ∞ = n 1 n! = n n ( ) ∶ Q∞ z ∞ w e+ e− = n 1 n! n 1 n! = = expQ z e+ Qexp w e−, = which is clearly bicomplex analytic. ( ) + ( )

∂ ∂ Defining the conjugate operator e+ e−, so that = ∂z ∂w ∂2 ∇ + , = = ∂z∂w which is a complexified wave equation.∇∇ ∇∇

3.3.2 Multicomplex Analytic and Meromorphic Functions

The notion of a multicomplex analytic function is a natural generalization of what we saw for the bicomplex case [51, 49, 62].

C2n−1 MC Definition 3.3.4. Let U be open. A function f U → n is said to be multicomplex ⊂ analytic on U if 2n−1 ∶ f ζ fk zk ek, = k 1 = where fk is complex analytic for every( k). Q ( )

36 This definition along with some basic results from complex analysis we get the following [43].

C2n−1 MC Theorem 3.3.5. Let U be open. A function f U → n is said to be multicomplex ⊂ analytic on U if and only if f ζ can be represented by∶ a convergent power series of the form

( ) ∞ k Akζ . k 1 = We wish to extend the definition of a meromorphicQ function.

Cn MC Definition 3.3.6. Let U be open. A function f U → n is said to be multicomplex ⊂ meromorphic on U if it is analytic everywhere on U∶ except for an isolated set of points, 1 called the poles of f, whereby the function f , defined to be zero at the poles, is multicomplex analytic in a neighborhood of each pole.

Cn MC Theorem 3.3.7. Let U be open. A function f U → n is multicomplex meromorphic ⊂ on U if 2n−1 ∶ f ζ fk zk ek, = k 1 = where fk is complex meromorphic (or( analytic)) Q for( every) k.

The proof is a simple extension of the analogous fact from complex analysis. With this theorem, it is clear that bicomplex rational functions are meromorphic. An- other example is the multicomplex Riemann zeta function [52, 50].

3.3.3 Polynomials and Rational Maps

Multicomplex polynomials are quite different objects from their complex counterparts. Though the maps m j p ζ Ajζ = i 0 = physically resemble polynomials over C,( the) numberQ of roots a general polynomial of degree m can have does vary pretty widely (the bicomplex case is covered quite nicely in [43]). Because of the light cone basis, we can write

2n−1 p ζ pk zk ek = k 1 = ( ) ( ) Q37 where the pk are complex polynomials of degree m (here we are using the convention that ≤ the zero polynomial is of degree ). Clearly, ζ0 zk0 ek is a root of p if and only if = −∞ ∑ pk zk0 0, = ( ) for all k. Thus, we get the following.

2n−1 Proposition 3.3.8. Consider the degree n polynomial p ζ pk zk ek. Then the number = k 1 p = of roots has is precisely equal to ( ) Q ( )

deg pk . k From the above, it is clear that a generalM S degree( )Sn polynomial over the bicomplex numbers

c n 1 can have its number of roots equal to km , where 0 k m and 0 c 2 − or even an ≤ < ≤ ≤ infinite number of roots. In the same way as above, we can decompose multicomplex rational functions:

2n−1 p ζ pk zk ek. q ζ = k 1 qk zk ( ) = ( ) Q Proposition 3.3.9. A rational function( ) (in lowest( terms))

2n−1 pk zk ek k 1 qk zk = ( ) Q has a pole whenever qk 0 for some k. ( ) =

38 CHAPTER 4

PRELIMINARIES FROM CONFORMAL GEOMETRY

Conformal geometry is the study of the conformal mappings of a semi-Riemannian manifold. 2 0 There are rich results for Rp,q, most notably that of R , C, and its compactification, the

Riemann sphere. Indeed, one of the most beautiful theorems≅ of complex analysis is that the M¨obius transformations are conformal mappings of the Riemann sphere [1, 33, 59]. We outline the basic theoretical constructs which give rise to the conformal group of the compactification of Rp,q.

4.1 Smooth and Semi-Riemannian Manifolds

The objects of interest are semi-Riemannian manifolds, which are smooth manifolds with a metric tensor [47]. Recall, that a smooth manifold is essentially a topological space that is locally homeomorphic to an open set of Rn. Below, we summarize the basic terminologies found in [47, 38]

Definition 4.1.1. 1. A (of dimension n) is Hausdorff, second-countable topological space M where every v M is contained in an open set U which is homo- n morphic to an open set U ′ of R . ∈ Rn 2. A coordinate chart is a pair U, φ where U M and φ U → U ′ is a homeomor- phism. ( ) ⊆ ∶ ⊆

3. An atlas on M is a collection A Uα,φα of charts such that Uα constitutes an = open cover of M and whenever Uα Uβ the transition map {( ≠ )} { } 1 ∩ ∅ φβ φα− φα Uα Uβ → φβ Uα Uβ

is a smooth map in Rn. ○ ∶ ( ∩ ) ( ∩ )

39 4. A smooth manifold is a pair M, A , where M is a topological manifold and A is a maximal(with respect to inclusion) atlas. ( ) There is some ambiguity in the word smooth. Unless otherwise stated, we will always

mean C∞(i.e. having continuous partial derivatives of all orders) in this context.

Definition 4.1.2. Let M and N be smooth manifolds. We say F M → N is smooth if for all v M there exists a chart U, φ containing v and a chart V,ψ∶ containing F v such 1 that F U V and ψ F φ− is a smooth map. When N R we denote the collection of ∈ ( ) = ( ) ( ) R smooth( maps) ⊆ from M ○to ○by C∞ M .

Before we can define a semi-Riemannian( ) structure on a smooth manifold, we need a metric tensor, which is a symmetric non-degenerate form acting on pairs of tangent vectors. Although tangency is a well understood notion in Rn, it is not immediately clear how this notion translates to manifolds.

4.1.1 Tangent Vectors and Metric

R Definition 4.1.3. Let M be a smooth manifold and v M. A linear map χ C∞ M → ∈ is called a derivation at v if ∶ ( )

χ fg f v ⋅ χ g + g v ⋅ χ f = ( ) ( ) ( ) ( ) ( ) for all f,g C∞ M . The set of all derivations at v is called the tangent space at v and ∈ denoted by TvM (and) we call an element of TvM a tangent vector at v

Of note is that in local coordinates, tangent vectors have a nice form [38].

Proposition 4.1.4. Let v M and U, φ a chart containing v. If we denote the i − th ∈ coordinate of φ by xi, then in these coordinates a χ TvM may be written ( ) ∈ n ∂ χ ai . = i 1 ∂xi v = Definition 4.1.5. The tangent bundle Q V

TM ∶ TvM = v M ∈ 40O can be realized as a collection of pairs v,X where X TvM and equipped with a projection map π TM → M defined by π v,X v, which we use along with the smooth structure on =( ) ∈ M to define∶ a smooth atlas (and( hence) a smooth structure).

Definition 4.1.6. Let M be smooth. The continuous map χ M → TM is called a vector Rn field if π χ idM . Given a local chart ϕ U → with ϕ v a1 v , . . . an v then = =∶ ○ ∶ n ∂ ( ) ( ( ) ( )) χ v ai v . = i 1 ∂xi v = ( ) ( ) V The collection of vector fields on M is denotedQ by X M .

Definition 4.1.7. A metric tensor g on a smooth manifold( ) is a symmetric, non-degenerate tensor which smoothly assigns to each v M a scalar product on TvM for which the index ∈ does not depend on v

Definition 4.1.8. A semi-Riemannian manifold M is a smooth manifold with a metric tensor g. If gv is positive definite, that is it has index 0, then M is called Riemannian.

In local coordinates x1,...,xn around v, we write

∂ ∂ gij g , , = ∂xi ∂xj Œ ‘ so that n gv χ,γ gijaibj = i,j 1 = n ∂ ( ) Qn ∂ for any tangent vectors χ ai and γ bi at v [47]. That is, as a tensor, = i 1 ∂xi v = i 1 ∂xi v = = Q V n Q V g gijdxi dxj. = i,j 1 = Q ⊗ p,q p q Example 4.1.9. The semi-Riemannian manifold R is the smooth manifold R + with the metric tensor g defined by

1 if i j and 1 i, j p = ≤ ≤ gij 1 if i j and p 1 i, j p q . = ⎪⎧ = ≤ ≤ ⎪0 otherwise ⎨− + + ⎪ ⎩⎪ 41 Notice when q 0 this is the standard inner-product on Rp. We shall be very interested in = Rp,q it represents a natural analogue of the complex plane to study conformal geometry. We Rp,q shall denote the metric tensor on by p,q or just when the context is clear. For a Rp,q tangent vector χ, χ shall denote χ,χ .⟨ We ⟩ also extend⟨ ⟩ to and obtain the usual quadratic form Qp,q⟨. ⟩ ⟨ ⟩ ⟨ ⟩

4.1.2 Operators

On a semi-Riemannian manifold there are operators which are generalizations of the common differential operators one encounters in .

Definition 4.1.10. The gradient of a smooth function f M → R, denoted grad f is a vector field such that for all χ X M , ∈ ∶ ( ) g grad f,χ χ f . = ( ) ( ) In local coordinates, n ∂f ∂ grad f gij , = i,j 1 ∂xi ∂xj = ij where g denotes the ij-entry of the inverseQ of the matrix with entries gij.

Example 4.1.11. For Rp,q, the gradient is given in local coordinates by

p p q ∂f ∂ + ∂f ∂ grad f . = i 1 ∂xi ∂xi i p 1 ∂xi ∂xi = = + Q −3 Q Thus, we get the familiar gradient formula for R . n ∂ Definition 4.1.12. The divergence of a vector field χ ai X M is given in local = i 1 ∂xi ∈ coordinates by = n n Q ( ) ∂ai i div χ Γijaj , = i 1 ∂xi j 1 = = i Q Œ + Q ‘ where Γij denote the Christoffel symbols of g.

Remark 4.1.13. The Christoffel symbols play an important role in the coordinate expres- sion of the Levi-Civita connection . Indeed, these describe the parallel transport of the

∇ 42 semi-Riemmanian manifold. Thus, they are also related to curvature and along with their derivatives can define curvature in local coordinates [19]. Since the divergence is (point-wise)

also the trace of the linear map γ v → γχ v [19]. Though this is very beautiful mathematics,( ) ∇ ( ) it is beyond the scope of the dissertation. The reader is encouraged to see [19] or [47] for the full details.

Example 4.1.14. For Rp,q, the divergence is given by

p q + ∂a div χ i . = i 1 ∂xi = 3 Thus, we get the familiar divergence formula forQ R .

Definition 4.1.15. The Laplacian of a smooth function f M → R, denoted ∆f is the divergence of its gradient, i.e. ∶ ∆f div grad f . = Example 4.1.16. For Rp,q, the Laplacian is given( by)

p 2 p q 2 ∂ f + ∂ f 2 2 . i 1 ∂xi i p 1 ∂xi = = + Thus, we get the familiar formula forQ the Laplacian− Q in Rn.

4.1.3 Conformal Mappings

The notion of a between euclidean spaces is most simply defined as a map which preserves . But in euclidean space we have that the θ between two vectors is given by a, b cos θ . = a b ⟨ ⟩ This is really a quantity involving of the usual metric tensor in Rn. Thus, we can extend the S S S S definition of conformal more generally to smooth maps between semi-Riemannian manifolds.

Definition 4.1.17. Let M,g and N,g′ be semi-Riemannian manifolds and F M → N R be a smooth map. We say( F is) conformal( ) if there exists a smooth function Ω M →∶ 0 such that ∶ ∖ { }

g′ dF χ ,dF γ Ωg χ,γ , = ( ( ) (43)) ( ) where χ,γ X M and dF denotes the derivative of F . That is in local coordinates, ∂F ∈ ( ) dF i . = ∂xj ij Œ ‘ The function Ω is called the conformal factor of F .

p,q p,q In the coming chapters, we will be interested in conformal mappings F R → R for the 1, 1 and 2, 2 cases. Thus, the next lemma will be very useful. ∶

p,q p,q Lemma( ) 4.1.18.( A) smooth map F R → R is conformal precisely when p p q ∂F ∂F + ∂F ∂F ℓ ℓ ∶ ℓ ℓ 0 i k ℓ 1 ∂xi ∂xk ℓ p 1 ∂xi ∂xk = ∀ ≠ = = + Qp ∂F 2 − pQ∂F 2 ℓ ℓ Ω if i 1,...,p ℓ 1 ∂xi − ℓ 1 ∂xi = = = = p 2 p 2 Q ‹∂Fℓ  Q ‹∂Fℓ  Ω if i p 1,...,p q ℓ 1 ∂xi − ℓ 1 ∂xi = = + + = = Proof. The author makesQ ‹ no claims Q of‹ originality of this proof, although he could not find a source where it is written down. Now, the conformal condition means that for tangent vectors at a fixed v Rp,q: ∈ p q p q + ∂ + ∂ p,q χ ai , γ bj Tv R , = i 1 ∂xi = j 1 ∂xj ∈ = = we must have that Q Q

p p q + g dF χ ,dF γ Ωg χ,γ Ω aibi aibi . = = i 1 − i p 1 = = + Notice that ( ( ) ( )) ( ) ŒQ Q ‘

p q p q p q p q + + ∂Fk ∂ + + ∂Fℓ ∂ dF χ ai and dF γ bj , = k 1 i 1 ∂xi ∂xk = ℓ 1 j 1 ∂xj ∂xℓ = = = = and ( ) Q ŒQ ‘ ( ) Q ŒQ ‘

p p ∂Fm ∂Fm ∂Fm ∂Fm g dF X ,dF Y ai bj ai bj = m 1 i ∂xi j ∂xj − m 1 i ∂xi j ∂xj = = p q p p ( ( ) ( )) Q+ ŒQ ∂Fm ∂F‘ ŒmQ ∂F‘m ∂FQmŒQ ‘ ŒQ ‘ aibk. = i,k 1 m 1 ∂xi ∂xk − m 1 ∂xi ∂xk = = = Compare this with the aboveQ Œ Q and claim followsQ immediately.‘

44 4.1.4 Conformal Compactification

There are many advantages of using the Riemann sphere instead of the complex plane 2 0 (i.e. R , ) as a domain for functions of a complex variable. As a compact space, it possesses some desirable topological properties. In terms of the geometry, one finds that a conformal map in the plane may be extended to one on the sphere. Are there higher dimensional analogues of this idea for general Rp,q? This question may be answered in the affirmative and one model for the compactification of Rp,q may be found in [55]. But first, let us be more precise about what we mean by conformal compactification.

Definition 4.1.19. A conformal compactification, up to isomorphism, of Rp,q is a compact semi-Riemannian manifold M with a conformal embedding ι such that ι Rp,q is dense in

M. ( )

2 0 Remark 4.1.20. In this context, the sphere S2 is the conformal compactification of R , C

with the stereographic projection as a conformal embedding. ≅

We shall denote the conformal compactification of Rp,q by N p,q, which is a subspace of real projective space. Before we can give a careful description of it, we must show how to embed Rp,q in to real projective space. Rp,q R n 1 Let n p q 2 and define ι → P + by = ≥ + ∶ 1 x 1 x x x1,...,xn z→ x1 xn + . = 2 ⋯ ∶ ∶ 2  − ⟨ ⟩ ⟨ ⟩ ( )  ∶ ∶ As proved in [55], N p,q is the topological closure of ι Rp,q and a quadric hyper-surface ( ) n 1 in R P + . Indeed, p,q R n 1 N X P + X p 1,q 1 0 , ∶= ™ ∈ ∶ + + = with the metric tensor p 1,q 1. A careful proof⟨ of⟩ this assertionž can be found in the next + + R1,1 chapter for . That proof⟨⟩ can be easily reworked for the general dimension. The critical fact is that any conformal map on Rp,q can be extended uniquely to N p,q in the following sense [55].

45 p,q p,q Lemma 4.1.21. Let ϕ U R → R be a conformal map on an open connected subset U. If p q 2 then there exists a unique conformal diffeomorphism ϕ N p,q → N p,q such that > ∶ ⊆ ϕ ι ι φ on U. = + ̂ ∶ 0 ̂ ○ Now,○ this continuation is very simple in the case of Rn, , since the conformal compact- ification is merely the one point compactification Rn . So one may use the original map and extend by taking limits. Because of the indefinite∪ {∞} nature of the metric tensor the case of q 0 has some complications, which explains the need to develop more complicated > machinery.

4.2 M¨obius Transformations of Rp,q

The theory of M¨obius transformations in Rn has been treated in various ways. The first is as those mappings which map to spheres, or in terms of the matrix group O n 1, 1 Rp,q [2]. These ideas extend to , as outlined below. ( + ) That the M¨obius transformations of Rn may be realized as linear fractional transforma-

tions with coefficients in the Clifford algebra Cℓ0,n was initially conceived by Vahlen in 1902 [61]. The paper, which was written in German, seems to have been forgotten until Ahlfors endorsed the idea in [3, 2]. This led to the analogues for the pseudo-euclidean space Rp,q by Fillmore et al. and in Cnops’s works [21, 11, 12, 41]. The following are the primary results as presented in the later works.

4.2.1 The Traditional Construction

The M¨obius transformations of Rn may also be defined as functions which are finite compositions of reflections in spheresà or planes. They form a group under composition. Indeed, the composition of any two M¨obius transformations is again a M¨obius transformation and any reflection composed with itself is the identity. Hence, we have closure under the

operation, an identity, and closure under inverses, since if ϕ ϕ1 ϕ2 ϕk is a composition = ⋯○ of reflections, then ○ ○ 1 ϕ− ϕk ϕk 1 ϕ1. = ○ − ○ ⋯ ○ 46 As such, we also have some important subgroups [55, 12]: ❼ Translations: ϕ x x v v Rp,q ; = ∈ { ( ) + ∶ } x x v ❼ Special Transformations: ϕ x v Rp,q ; = 1 2 x, v x v ∈ − ⟨ ⟩ œ ( ) ∶ ¡ ❼ Dilations: ϕ x λx λ 0 ; − ⟨ ⟩ + ⟨ ⟩ ⟨ ⟩ = > { ( ) ∶ } ❼ Orthogonal Maps: ϕ x Ax A O p,q . = ∈ { ( ) ∶ ( )} It can be shown that any M¨obius transformation is a composition of translations, special transformations, dilations, orthogonal maps, and an inversion (i.e. a reflection through the ). These subgroup definitions extend to Rp,q using the appropriate inner product. This will seem more reasonable once we understand the meaning of sphere in Rp,q.

4.2.2 The Vahlen Construction

Essentially, the collection of conformal mappings which map “spheres” to “spheres” are called the M¨obius transformations of Rp,q. These are denoted M p,q . The use of the quo-

tations is intentional, because we are using the word in a different( sense) than the reader may have seen. That is, hyperbolas and their higher dimensional analogues are also considered spheres” as indicated in the definition below [12, 41].

Definition 4.2.1. A sphere centered at c Rp,q of radius r is a set of points of the form ∈ x Rp,q x c 2 r2 . ∈ = Here, we require r2 to be a scalar,™ but not necessarily∶ ( − ) positive.ž

Definition 4.2.2. A map f Rp,q → Rp,q is conformal if

∶ g df χ , df γ Ωg χ,γ , = where χ,γ are vectors in the appropriate( ( ) tangent( )) space,( and) ∂f df i . = ∂xj ij Œ ‘ 47 Of course when p q 2, every conformal map is a M¨obius transformation [11]. > Lemma 4.2.3. The group+ M p,q has two connected components and the orthogonal group M O p 1,q 1 gives a double cover( ) of p,q .

( Combined+ + ) with Lemma 2.3.12, this tells( us) that we may express a M¨obius transformation by an element of Pin p 1,q 1 . A B Hence, given Pin p 1,q 1 , then acts on x Rp,q as follows = (CD+ +∈ ) ∈

A Œ ‘ ( + + ) A 1 x Ax B Cx D − . = A( ) ( + )( + ) This construction can also be done to define conformal mappings of the so-called par- avectors of Cℓp,q which are R Rp,q ⊕ Cℓp,q. ⊂ In a manner analogous to the situation from Vahlen’s work on euclidean spaces, we get the following [11, 12]. A B Theorem 4.2.4. The group Pin p + 1,q + 1 consists of matrices of the form = CD where ( ) A Œ ‘ 1. A, B, C, D are products of vectors in Rp,q;

2. BD,AC, AB, CD Rp,q; ∈ 3. λ ̃ ̃1.̃ ̃ = Hence,(A) we get a way to describe the M¨obius transformations. Notice the similarity to the case of complex M¨obius transformations. Indeed, we also have the classical subgroups expressed a matrix groups:

1 v p,q ❼ Translations: v v R ; = 0 1 ∈ œT Œ ‘ ∶ ¡

1 0 p,q ❼ Special Transformations: v v R ; = v 1 ∈ œK Œ ‘ ∶ ¡

48 λ 0 ❼ Dilations: λ 1 λ 0 ; = 0 λ− > œD Œ ‘ ∶ ¡ α 0 ❼ Orthogonal Maps: α α Γp,q . = 0 α′ ∈ œR Œ ‘ ∶ ¡ It is also possible to classify the M¨obius transformations as compositions of these [11].

Theorem 4.2.5. Let Pin p 1,q 1 . Then at most, may be decomposed as products ∈ of the above transformations.A ( + + ) A 1. If all the entries of are invertible then

A ⋅ u ⋅ v ⋅ λ ⋅ α ⋅ , = A E K T D 0R 1 F where and are either the identity or . = 1 0 2. OtherwiseE F J Œ ‘

v ⋅ v ⋅ ⋅ u ⋅ w ⋅ λ ⋅ α. = − 4.2.3 When p + q 2 A K T J K T D R > When p + q 2, Liouville’s Theorem tells us the conformal mappings of a domain to > another in Rp,q is a M¨obius transformation. In fact, every M¨obius transformation on Rp,q may be extended to a conformal map on its conformal compactification N p,q. The proof of theorem below, which can be found in [55] shows precisely how one may do that.

Theorem 4.2.6. The subgroup M + p,q is isomorphic to SO+ p+1,q+1 (or SO+ p+1,q+ ± − + + M 1 I if I is in the same component( ) as I in O p 1,q 1 ),( where )+ p,q denotes( the M + + connected)~ { } component of p,q which contains the( identity) and SO+ p (1,q) 1 denotes + + the connected component of(SO )p 1,q 1 containing I. ( )

Proof. As this proof can already( be found in) Schottenloher’s book, we repeat only the inter- esting part, which is the isomorphism. Let n p + q. To define the isomorphism, it is enough to show where the generators of = M p,q are mapped.

( ) 49 1 ❼ Orthogonal maps. As every x αx α′ − may be realized as n n A O p,q , we associate this map with the n 1 n 1 matrix ↦ ( ) × ∈ ( ) 1 0 0+ × + 0 A 0 , ⎛0 0 1⎞ ⎜ ⎟ which clearly lies in SO p 1,q 1 . ⎝ ⎠

❼ Translations. A translation( +x +x )b gets mapped to the matrix

1 T 1 1 ↦ 2 +b ηb 2 b b In b , 1 T 1 ⎛ −2 b⟨ ⟩ −(ηb ) 1− 2⟨ b⟩ ⎞ ⎜ ⎟ I 0 ⎝ ⟨ ⟩ ( ) + ⟨ ⟩⎠ where η p . = 0 Iq Œ ‘ ❼ Dilations. The map− x λx is associated with

1 λ2 1 λ2 ↦ +2λ 0 −2λ 0 In 0 . 1 λ2 1 λ2 ⎛ −2λ 0 +2λ ⎞ ⎜ ⎟ ⎝ ⎠ ❼ Special transformations. Every special transformation gets mapped to

1 T 1 1 2 b ηb 2 b b In b . 1 T 1 ⎛ −2 b⟨ ⟩ −(ηb) 1− 2⟨ b⟩ ⎞ ⎜ − ⎟ ⎝ ⟨ ⟩ −( ) + ⟨ ⟩⎠

50 CHAPTER 5

SPLIT-COMPLEX ANALYSIS AND THE MOBIUS¨ TRANSFORMATIONS OF R1,1

In this chapter (which is essentially our joint work in [20]), we consider the split-complex numbers and explore the analogous theory of holomorphic functions. Recall, just as we can 2 associate C with R and it’s induced Riemannian geometry, we can also associate Cℓ1,0 with 1 1 R , , which is just the semi-Riemannian manifold R2 with the indefinite metric

g χ,γ a1b1 a2b2. = ( ) − A natural question rises: Can we find a better domain for functions of a split-complex vari- able, like we do in complex analysis? This is a question which has been addressed over many years and in many places in the lit- erature, although not always directly, but always in the affirmative [63, 56]. M. Schottenloher linked the previous question with questions of conformality [55]. In particular, Schottenlo- her describes a model for compactifying Rp,q with an induced conformal structure, building on the work of Dirac (see [18]), who used an analogous model for the compactification of four-dimensional space-time several decades earlier. Schottenloher also discusses the notion of conformal group on R1,1 and identifies it as a product of circle diffeomorphisms. In the works of Cnops, one sees that the linear fractional 1,1 transformations with coefficients in Cℓ2,2 acting on R as a subspace of Cℓ2,2 does give a collection of conformal mappings. A review of the literature has not yielded a work which clearly identifies the collection of linear fractional transformations with coefficients in Cℓ1,0 acting on Cℓ1,0 as a subgroup of the conformal group, even though this is the case. This chapter does make this clear and develops a M¨obius transformation theory for R1,1.

51 Yaglom also defined a cross ratio on R1,1 in [63], which seems to have been largely overlooked or forgotten until [6]. This cross ratio has been previously shown to be well behaved with respect to the M¨obius transformations, but this also seems to have been lost. The purpose of this chapter is threefold. First, we wish to carefully construct the com- pactification of R1,1 in a conformal setting. Second, we use holomorphic and conformal conditions in R1,1 to extend these notions to the compactification in a simple way. Last, we use algebraic properties of the split-complex numbers to show that the M¨obius transforma- tions are a direct product of real M¨obius transformations and to discuss their conformality, transitivity, and fixed points carefully (something which has also not been found in a rather thorough review of the literature).

5.1 Obtaining “Cauchy-Riemann” Equations and Operators

Now, we consider functions of a split-complex variable:

1,1 1,1 f U R → R ,

where U is an open subset of R1,1. Such∶ functions⊆ have been examined in numerous places [14, 16, 39].

These functions may be written f z f1 x,y jf2 x,y . Throughout this work, we = 1 shall assume that f1,f2 C U . Now just as in complex analysis, we must understand ∈ ( ) ( ) + ( ) what it means for the limit of( the) difference quotient f z0 h f z0 lim h→0 h h L ∉ ( + ) − ( ) to exist.

Now suppose this limit exists and assume that h hx is real. Then = f z0 h f z0 lim x 0 hx→ hx f1( x0+ h),y −0 ( f)1 x0,y0 f2 x0 h ,y0 f2 x0,y0 lim x j x 0 = hx→ hx hx ∂f1 ( + ∂f2) − ( ) ( + ) − ( ) x0,y0 j x0,y0 . + = ∂x ∂x ( ) + ( ) 52 Similarly, if we assume h jhy is purely imaginary then = f z0 jhy f z0 ∂f1 ∂f2 lim j x0,y0 x0,y0 . 0 jhy→ jhy = ∂y ∂x ( + ) − ( ) Remark 5.1.1. This gives the Cauchy-Riemann equations( ) + in the( split-complex) plane [14]:

∂f1 ∂f2 ∂f1 ∂f2 and . ∂x = ∂y ∂y = ∂x Conversely, if these equations are satisfied and the partial derivatives are continuous, then the limit of the difference quotient exists (just as in complex analysis). If either of

these conditions are satisfied at a point z0, then we say that f is Cℓ1,0-differentiable at z0.

If this is the case for all z0 U we say f is Cℓ1,0-differentiable on U. ∈ Now consider the differential operator 1 ∂ ∂ ∂ j , = 2 ∂x ∂y 1 1 which is a the gradient1 in R , times a constant,‹ − and its conjugate 1 ∂ ∂ ∂ j = 2 ∂x ∂y which clearly satisfies ‹ +  1 ∂2 ∂2 ∂∂ ∂∂ , = = 4 ∂x2 ∂y2 which is the wave operator in the plane. Functions‹ − which are annihilated by the wave operator are called ultrahyperbolic.

Theorem 5.1.2. Let f be a split-complex valued function of a split-complex variable.

∂f z0 0 f is Cℓ1,0-differentiable at z0. = Proof. Notice that ( ) ⇐⇒

1 ∂f1 ∂f2 ∂f1 ∂f2 ∂ f1 jf2 j j = 2 ∂x ∂x ∂y ∂y  1 ∂f1 ∂f2 ∂f2 ∂f1 ( + )  + − j − . = 2 ‹ ∂x ∂y  ‹ ∂x ∂y  − + − Thus ∂ f1 jf2 0 if and only if f is Cℓ1,0-differentiable. ( ) = 1 C ∂ 1 ∂ ∂ R2 Notice that+ the corresponding operator in , ∂z ∶= 2 Š ∂x + i ∂y , is the gradient in times a constant.

53 Remark 5.1.3. Since Cℓ1,0 is commutative, we can apply the operators on any side and obtain the same results. This is in stark contrast with the case for Cℓ0,n or Cℓ1,1 which we shall see in a later chapter.

Remark 5.1.4. The above theorem tells us that the components of a Cℓ1,0-differentiable function satisfy the wave equation in the plane [39]. Complex holomorphic functions have components which satisfy Laplace’s equation [1].

Definition 5.1.5. We say a split-complex valued function of a split-complex variable f is

Cℓ1,0-antidifferentiable at z0 if

∂f z0 0 = ( ) 2 Corollary 5.1.6. Given an ultrahyperbolic function g U R → R, the function f ∂g is ⊆ = Cℓ1,0-differentiable and h ∂g is Cℓ1,0-antidifferentiable. = ∶ If we use our alternative basis for split-complex plane, we may rewrite

1 ∂ ∂ ∂ j = 2 ∂x ∂y 1 ∂u ∂ ∂v ∂ ∂u ∂ ∂v ∂ ‹ −  j = 2 ∂x ∂u ∂x ∂v ∂y ∂u ∂y ∂v  1 ∂ ∂ ∂ ∂ ‹ + j j − ‹ + = 2 ∂u ∂v ∂u ∂v  ∂ ∂ j + j −. + = ∂v + ∂u − It is easy to see that + ∂ ∂ ∂ j j , = ∂u + ∂v − so that + ∂2 ∂∂ ∂∂ , = = ∂u∂v for reasonably behaved f u,v . ( ) This gives us the following corollary[39].

Corollary 5.1.7. The split-complex valued function f is Cℓ1,0-differentiable if and only if

f f1 u j f2 v j . = ( ) + ( ) − 54+ Also, f is Cℓ1,0-antidifferentiable if and only if

f f1 v j f2 u j . = + − ( ) + ( ) With this new method for checking differentiability, we are able to check that analogues

1,1 of some holomorphic functions in C are Cℓ1,0-differentiable in R .

m Example 5.1.8. Let f z z for a positive m. Then f is Cℓ1,0-differentiable, and = hence so are split-complex( ) polynomials.

Proof. With the simplified multiplication in the coordinates z uj vj , we have that = + − + zm umj vmj . = + − +

It will be useful to have a notion of meromorphic functions.

Definition 5.1.9. Let f be a split-complex valued function of a split-complex variable. We say f is Cℓ1,0-meromorphic if

f f1 u j f2 v j , = + −

and f1, f2 are real meromorphic functions.( ) We+ say( f) is Cℓ1,0-antimeromorphic if

f f1 v j f2 u j , = + − ( ) + ( ) and f1, f2 are real meromorphic functions.

5.2 Analogues and Non-Analogues from Complex Analysis

The notion of Cℓ1,0-differentiablity yields some analogues of theorems from complex anal- ysis. In particular, we have an analogue of Cauchy’s Theorem.

55 2 Proposition 5.2.1. Let U Cℓ1,0 be open (in the Euclidean topology of R ). Suppose

S U is bounded, orientable⊆ subdomain with a piecewise differentiable boundary. If f is Cℓ⊆1,0-differentiable on U , then fdz 0, S∂S = where dz dx jdy = The proof,+ which can be found in [39], uses Stoke’s Theorem. There is also an analogue of the Cauchy Integral formula, which is presented in [39].

However, we find that because Cℓ1,0 is not a field, we do not use a Cℓ1,0-valued kernel.

Rather, we will use a kernel which takes values in Cℓ1,0 C.

Lemma 5.2.2. Define K Cℓ1,0 × → Cℓ1,0 × by ⊗

1 z¯ 1 1 ∶ ( K )z (z− ) j j . = = z = u + v − ( ) + Then ∂K z 0 for every z Cℓ1,0 ×. ⟨ ⟩ = ∈ To obtain( ) the kernel we seek,( we) simply “complexify” K. Define 1 1 Kǫ z j + j . = u iǫ ⋅ sign v + v + iǫ ⋅ sign u −

Proposition 5.2.3 (Libine’s( ) Integral Formula). Let R 0 and define S z Cℓ1,0 ∶ N z R . + ( ) > ( ) = ∈ < ∶ Let U be an open neighborhood of S. Suppose f U → Cℓ1,0 is smooth and ∂f 0. Then for { = S ( )S } any ζ S, ∈ 1 f ζ lim Kǫ z − ζ f z dz. 2πi ǫ→0 S z R = {S⟨ ⟩S= } The proof is found in [39].( ) ( ) ( ) A direct consequence of Example 5.1.8 is that any polynomial is holomorphic. Then the following is not so surprising [16].

∶ Proposition 5.2.4. Suppose that f Cℓ1,0 → Cℓ1,0 has a power series representation in the following sense: for each z0 Cℓ1,0, there is an r 0 such that ∈ > ∞ − n f z Q cn z z0 , an Cℓ1,0 = n 1 ∈ = for every z z ∶ x − x0 + y − y0( ) r . Then( f is)Cℓ1,0-differentiable. ∈ < { S S S S } 56 Example 5.2.5. We may obtain the analogues of trigonometric functions this way. For example, define 1 nz2n 1 sin z ∞ + = n 1 2n 1 ! = (− )n 2n 1 n 2n 1 ( ) ∶ Q∞ 1 u + ∞ 1 v + ( + ) j j = n 1 2n 1 ! + n 1 2n 1 ! − = (− ) = (− ) sinQ u j sin v j+ .Q = ( + + ) − ( + ) The converse, however, is not true.( ) This+ is in( stark) contrast with the situation in complex analysis.

Example 5.2.6. Let f z Ψ u j vj , where Ψ is the bump function = + − −1 ( ) ( ) + e u if u 0 Ψ u > . ⎧0 if u 0 ⎪ ≤ ( ) ⎨ 1 Then f is clearly Cℓ1,0-differentiable (since⎪ both components are C in u and v resp.). How- ⎩⎪ ever, since Ψ does not have a power series expansion at u 0 (i.e. it is not analytic), then = neither does f.

We also have an analogue of Liouville’s Theorem which is not true[16].

Example 5.2.7. The function f z sin z is bounded (in the sense of , in the euclidean = sense, and in the sense of the above( ) proposition) and entire but not constant.⟨⟩

5.3 Conformal Mappings

Recall, a smooth map f U X → X on a semi-Riemannian manifold X,g of maximal ⊂ rank (that is, f is a local diffeomorphism)∶ is called conformal on U if there( is a) smooth map Ω U → R 0 such that

f ∗g Ωg, ∶ ∖ { } = where f ∗g χ,γ g df χ , df γ , and df is the tangent map of f [47]. = R1,1 For (, the) conformal( ( ) factor( ) has a simple form [55]: 2 2 2 2 ∂f1 ∂f2 ∂f2 ∂f1 Ω . = ∂x ∂x = ∂y ∂y Moreover, we get a simple condition‹  to− ‹ check for‹ conformalit − ‹ y. 

57 1 1 1 1 1 Theorem 5.3.1. A one-to-one C map f R , → R , , where f f1 x,y jf2 x,y , is = (locally) conformal when either ∶ ( ) + ( ) 2 2 ∂f1 ∂f2 1. ∂f 0 and 0, or = ∂x ∂x ≠ 2 2 ‹∂f1  − ‹∂f2  2. ∂f 0 and 0, = ∂x ∂x ≠ everywhere on R1‹,1.  − ‹  If we use the alternative basis, these conditions are

∂g1 ∂g2 1. ∂f 0 and 0, or = ∂u ∂v ≠

‹∂g1  ‹∂g2  2. ∂f 0 and 0, = ∂v ∂u ≠ ‹  ‹  where f g1 u,v j g2 u,v j . = + − Proof. The proof( ) in+ the( original) basis for positive conformal factor can be found in [55]. Requiring a non-vanishing conformal factor does not change the proof. Thus, it only remains to show that the first set of conditions (that is those in the original basis) are equivalent to the second set of conditions. Suppose ∂f 0. Thus, =

f z f1 x,y f2 x,y j g1 u j g2 v j , = = + − ( ) ( ) + ( ) ( ) + ( ) so that g1 u g2 v g1 u g2 v f1 and f2 . = 2 = 2 ( ) + ( ) ( ) − ( ) Recall, ∂ ∂ ∂ . ∂x = ∂u ∂v Then, +

∂f1 1 ∂ ∂ g1 u g2 v g1 u g2 v ∂x = 2 ∂u ∂v )) 1 ∂g1 ∂g2  ( ( ) +. ( )) + ( ( ) − ( = 2  ∂u ∂v  + 58 Similarly, ∂f2 1 ∂g1 ∂g2 . ∂x = 2 ∂u ∂v  Thus,  − 2 2 ∂f1 ∂f2 ∂g1 ∂g2 . ‹ ∂x  ‹ ∂x  = ‹ ∂u  ‹ ∂v  That is, the first conditions in each set− are equivalent. A similar argument shows the equiv- alence of the second ones.

Soon, we will see that M¨obius transformations of R1,1 are a large, but not exhaustive class of conformal mappings.

5.4 Conformal Compactifcation of R1,1

The notion of compactifying R1,1 is well known (see [56, 35]) and is of some interest to physicists [55, 29, 36]. One finds that this problem has been explained in numerous places in the literature, though there are some differences in the models used. For example, Kisil and others use a model in extended 3-space [37, 16], in addition to the quadric model N 1,1 that we saw in the previous chapter. We shall explicitly construct N 1,1 via the largely forgotten method presented in Segal’s book (see[56]): we embed R1,1 in a torus and then quotient by a projection.

5.4.1 The Torus in R4

Our study of the torus is motivated by our desire to find a compactification of R1,1. That is, we are looking for a space where we can embed R1,1 and the analysis, topology, and geometry are analogous to the Riemann sphere. The torus does not provide the model we need for the compactifcation, but it is close to what we need. We denote the torus S1 S1 embedded in R4 as follows:

1,1 × 2 2 2 2 T x0,x1,x2,x3 x0 x1 1,x2 x3 1 . = {( )S = = } + + The torus T 1,1 is the of two disjoint open sets along with their common boundary:

59 1,1 1,1 1,1 1,1 T T T T0 = + − 1,1 1,1 1,1 1,1 1,1 1,1 where T x T x0 x3 0 ,T x T∪ x0∪ x3 0 and T0 x T x0 x3 0 . + = ∈ > − = ∈ < = ∈ = 1,1 We define{ some involutionsS + } of T {which areS + also bijective} diffeomorphisms{ S + and are} related to the inverses we discussed in Remark 2.3.19. They will also play an important role in developing the compactification of R1,1.

Definition 5.4.1. a. We define some involutions of T 1,1 :

i. Left Inversion: J x x2,x3,x0,x1 ; + =

ii. Right Inversion: J( )x ( x2,x3, x)0,x1 ; − =

iii. Inversion: J x (x)0,x1(−, x2,x3 − J x) J x . = = + − ( ) (− − ) ( ) ○ ( ) 1,1 b. The following involutions preserve T0 :

iv. Left Negation: N x x3, x2, x1,x0 ; + =

v. Right Negation: N( )x ( x3−,x2,x−1,x0 ;) − =

vi. Negation: N x x(0,) x1(, x2,x3 N) x N x ; = = + −

vii. Conjugation:(C) x( x−0,x1−, x2,x)3 ; ( ) ○ ( ) =

viii. Reflection: R x( ) (x0, x1,− x2, x)3 J N J N x . = = + + 5.4.2 Embedding( ) (− of R−1,1 onto− −N)1,1 ○ ○ ○ ( )

A preliminary step to constructing the conformal compactification is to find a way to 1 1 embed R1,1 in the torus. This embedding τ R1,1 T , is given by → + 1 uv,u v,u v, 1 uv τ z τ u,v ∶ , = = 1 u2 1 v2 ( − + − + ) ( ) ( ) » 1,1 and is a bijective diffeomorphism of R1,1 onto T( +. Later,)( + we) shall see that τ is a conformal + mapping with respect to the semi-Riemannian metric on R1,1 and N 1,1. We remark the 1 1 R τ z is a bijective diffeomorphism of R1,1 onto T , . − ( ( )) 60 Notice that for z R1,1, when the inverses exist,

∈ τ z¯ C τ z , = 1 τ(z)− ( J( τ))z , = 1 τ(z− ) ±J( τ( z)) , + = + 1 τ(z− ) ±J (τ(z)). − = − Similar formulas hold for N,N ,N( : ) ± ( ( )) + −

τ z N τ z , = τ(−uj) vj( ( ))N τ z , + − = + τ(−uj +vj ) N τ( z( )). + − = − Also notice that J and N do not( commute− ) on(T(1,1)): +

N J x x2, x3, x0,x1 J N x x2,x3,x0, x1 , + = ≠ + = even though the corresponding○ ( ) ( − involutions− ) on R1○,1 commute.( ) (− As such− we) descend to the projective space of the torus, which turns out to be the conformal compactification we seek (and we shall see why this is true in the subsequent subsections).

Definition 5.4.2. We define a quadric surface N 1,1 in the projective space P3 by

1,1 1,1 N T ∼ = under the equivalence ~

x ∼ y if and only if x y. = Remark 5.4.3. We denote this identification by π. ±

1 1 Notice that this equivalence identifies x and R x and hence points in T , with points + 1,1 1,1 1,1 in T and points in T0 with points in T0 . These identified points are pairs of antipodal − ( ) points on the 3-sphere containing T 1,1.

61 1 1 We denote points in N , by x0 x1 x2 x3 . As a matter of notation, if x0 x1 x2

1,1 x3 N , then we assume that (λx0∶ λx∶1 λx∶ 2 )λx3 represents this point for( all∶ non-zero∶ ∶ R λ ) ∈. ( ∶ ∶ ∶ ) ∈

1 1,1 1,1 The inverse mapping τ − T R is given by + → ∶ x1 x2 u , = x0 x3 x1 + x2 v + . = x0 x3 1 1,1 1,1 − 1,1 Notice that τ − extends to T T as a two to one cover of R . + − + For notational convenience,∪ we shall denote the composition π τ by

1,1 1,1 ○ η R → N .

We shall also extend our definition of inversion∶ to N 1,1:

x0 x1 x2 x3 ↦ x0 x1 x2 x3 .

For simplicity we shall also( denote∶ ∶ this∶ J )(there(− should∶ ∶ − be no∶ ambiguity) since π J τ J η.) = 5.4.3○ ○ Added○ Points

The conformal compactification of C has one more point than the plane, namely the point at infinity. For R1,1, we must add an additional point for every point in the light cone plus two additional points which compactify the light cone. We calculate the coordinates of these additional points. Suppose v 0. Then = η z 1 u u 1 , = which goes to 0 1 1 0 as u → ,( and) ( ∶ ∶ ∶ )

( ∶ ∶ ∶ ) ∞Jη z 1 u u 1 , = ( ) (−62∶ ∶ − ∶ ) goes to 0 1 1 0 as u → . When u 0, = ( ∶ ∶ − ∶ ) ∞ η z 1 v v 1 , = ( ) ( ∶ ∶ − ∶ ) goes to 0 1 1 0 as v → , and ( ∶ ∶ − ∶ ) ∞ Jη z 1 v v 1 , = ( ) (− ∶ ∶ ∶ ) tends to 0 1 1 0 as v → . Throughout α , define ( ∶ ∶ ∶ )< < ∞ −∞ ∞ η L 1 α α 1 and η L 1 α α 1 . + = + = − = − = L ( ) {( ∶ ∶ ∶ )} L ( ) {( ∶ ∶ − ∶ )} The above observations give us the following.

Remark 5.4.4. If we define

1 − Jη L 1 α α 1 α R 0 1 1 0 and + = + = ∈ 1 L− ∶ Jη(L ) {(−1 ∶ α ∶α − 1∶ ) ∶α R } ∪0 ( 1∶ 1∶ −0 ∶. ) − = − = ∈

L ∶ ( ) {(− ∶ ∶ ∶ ) ∶ } ∪ ( ∶ ∶ ∶ ) 1,1 1 1 Then these intersect at πJ τ 0 1 0 0 1 and we shall define N0 − − . = = + − Remark 5.4.5. To avoid an( ( unnecessarily)) (− ∶ ∶ cumbersome∶ ) notation, we shall∶ adoptL ∪L the follow- ing:

1. 1 j j shall denote 1 α α 1 . α + − 2. j + ∞1 j shall denote (−1 ∶ α ∶α − 1∶ .) + α − 3. ∞j +shall denote 0 1 (−1 0∶ . ∶ ∶ ) + 4. ∞j shall denote (0 ∶ 1 ∶ 1∶ )0 . − 1 5. ∞ shall denote (j ∶ ∶ −j ∶ 0), the inversion of zero. + − = In this∞ manner, it is∞ clear+ that ∞ elements of N 1,1 can be regarded as elements R R, where 1 1 R R . In fact, N , is locally isomorphic to R R (in the sense of conformal̂ geometry)̂ = × [55].̂ ∶ These∪{∞} are sometimes referred to as the “extended̂×̂ double numbers” [16]. 63 Other rays through the origin embed as follows. If u βv,β 0, then = ≠

η z 1 βu2 1 β u 1 β u 1 βu2 , = ( ) ( − ∶ ( + ) ∶ ( − ) ∶ + ) which goes to 1 0 0 1 as u → . Hyperbolas, uv R, R R, 0 have the following embeddings : (− ∶= ∶ ∶ )∈ ≠ ∞

1 R u R u u R u 1 R .

( − ∶ + ~ ∶ − ~ ∶ + ) Notice that as u → , this goes to 0 1 1 0 . Alternatively, we can write ±∞ ( ∶ ∶ ∶ ) 1 R R v v R v v 1 R ,

( − ∶ ~ + ∶ ~ − ∶ + ) which goes to 0 1 1 0 as v → . 1,1 The figures( below∶ ∶ − give∶ ) a nice picture±∞ of N and show what the above curves look like in R1,1 and what there embeddings look like in N 1,1.

5.4.4 N 1,1 as a Conformal Compactification

We claim that N 1,1 is the conformal compactifcation we seek[55]. Recall that this means we have two things to check:

1. η R1,1 is dense;

2. η (is a conformal) map.

The first is true by construction and the second is rather simple:

1 1 1 1 Lemma 5.4.6. The quotient map π T , → N , is conformal.

Proof. Since T 1,1 R2,2, it inherits the∶ semi-Riemannian metric g2,2 of signature 2, 2 . We ⊆ 1,1 also pass this indefinite metric to N . ( ) It is clear that since π is surjective and a local diffeomorphism, it is also a local isometry, and hence conformal.

We also have the following [55].

64 1 1 Figure 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= = of N 1,1. The= plus and= minus signs= indicate the≤ signs≤ of the cosines and{( sines in)S the parametrization. − }

65 Figure 5.2: The R1,1 plane and some curves.

1 1 1 1 Proposition 5.4.7. The map η π τ R , → N , is conformal. = Remark 5.4.8. By Lemma 4.1.21, we○ can∶ take a conformal map

1,1 1,1 f R → R ∶ and extend it to a conformal map

1,1 1,1 fˆ N → N .

ˆ 1,1 ∶ ˆ 1 In particular, when z, f z η R , f z η f η− . In such a case we continue to ∈ = ˆ write f z for f 1 uv u (v) u (v 1 ) uv(. ) ○ ○

( ) ( − ∶ + ∶ − ∶ + )66 Figure 5.3: The embedded curves. Points with the same numeric label are identified.

67 1,1 At N0 , fˆ is continuous, and this means that the values of such points can be defined as limits. For example, fˆ 0 1 1 0 lim fˆ 1 u u 1 . u→ = ∞ ( ∶ ∶ ∶ ) ( ∶ ∶ ∶ ) 5.4.5 Differentiable Functions and Conformal Mappings on N 1,1

We want to define a notion of differentiability on N 1,1 which is consistent the notion of differentiability on R1,1 defined in Section 5.1. This has been explored on the equivalent space of extended double numbers in [16]. We shall proceed in a similar fashion and borrow some ideas from complex analysis (as in [1]). In complex analysis one adds a point at infinity to invert the origin and discusses the behavior of functions at this added point using the inversion. In a similar way, we invert the added points to investigate functions at these added points. We define the following sets and maps for this purpose. They do not serve as an atlas.

1,1 1 A1 η R , φ1 η− = = 1 A2 ‰ ,Ž φ2 η− J = = 1 1 A3 {∞}− , φ3 ○ η− J = + = + 1 1 A4 L− ∖ {∞} , φ4 η− ○ J = − = − L ∖ {∞} ○ Definition 5.4.9. Consider the function

1,1 1,1 f N → N ,

1,1 ∶ 1,1 and z0 N . Suppose z0 Ai and f z0 Aj. We say f is N -differentiable at z0 if ∈ ∈ ∈

( ) 1 φj f φi−

1 ○ 1,○1 is Cℓ1,0-differentiable at η− z0 . If f is N -differentiable everywhere, we will say f is 1,1 N -differentiable. ( )

68 We also define N 1,1-meromorphic (and N 1,1-antimeromorphic) as functions of the form

f1 u j f2 v j (resp. f1 v j f2 u j ), where f1 and f2 are meromorphic functions on + − + − R .( ) + ( ) ( ) + ( ) 1 1 2 2 ̂ We recall that N , inherits the semi-Riemannian metric g , . Hence, a notion of confor- mal map is defined. By using the conformality of η, J, and J on their respective domains ± 1 1 1 1 to get a shortcut definition for conformality of maps f N , → N , , :

Theorem 5.4.10. A function ∶ 1,1 1,1 f N → N is (globally) conformal iff the maps ∶ 1 φj f φi− are conformal (in the sense of Theorem 5.3.1○ )○ everywhere they are defined.

First we need some lemmas:

Lemma 5.4.11. The maps J, J , J N 1,1 N 1,1 are conformal. + − → ∶ Proof. Write p x0 x1 x2 x3 . Then J p x0 x1 x2 x3 , and = = ⎤ ( ∶ ∶ ∶ ) (10) (− 0∶ 0 ⎥∶ − ∶ ) ⎥ ⎡ 0 1 0 0 ⎥ ⎢ − ⎥ dJ ⎢ ⎥ . = ⎢ 0 0 1 0 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 0 0 0 1 ⎥ ⎢⎣ − ⎦ ⎢ Let χ ∂ . Then ⎢ i ∂xi = ⎧ ⎪ χi if i 0, 2 dJ(χi) ⎨ = = ⎪χi if i 1, 3. ⎩⎪− = Thus,

J ∗g(χi,χj) g(χi,χj). = This means J is conformal. Similar calculations show

(J )∗g(χi,χj) g(χi,χj) + = 69 − and

J ∗g χi,χj g χi,χj , − = i, j 0,..., 3. So, J and J are also( ) conformal.( ) − ( ) = + − The next lemma is trivial.

Lemma 5.4.12. Let M,g be a semi-Riemannian manifold. Let f M → M and h M → M be smooth maps of maximal( ) rank. ∶ ∶

(i) Suppose f and h f are conformal maps with non-vanishing factors Ω1 and Ω2, respect- fully. Then h is conformal. ○

(ii) Suppose h and h f are conformal maps with non-vanishing factors Ω1 and Ω2, respect- fully. Then f is conformal. ○ 1 1 1 1 Proof of Theorem 5.4.10. Let f N , → N , be a smooth map of maximal rank. Suppose f is conformal. Then∶ by lemma 5.4.11, we conclude that

1 φj f φi− ,

○ ○ is conformal for every i, j. 1 1 Conversely, suppose the φj f φi− are conformal. Then φj f and f φi− are conformal.

By Lemma 5.4.12, this means ○f is○ conformal. ○ ○

In the next section, we shall see that the M¨obius transformations form a large class of (globally) conformal mappings. However, these are not the only such mappings.

Example 5.4.13. Consider

f u,v arctan u u j vj . = + − ( ) ( ( ) + ) + It is easy to check that f is conformal everywhere. Yet, it is not a M¨obiustransformation.

70 5.5 M¨obius Geometry

With A, B, C, D R1,1, we consider the actions of M¨obius transformations expressed as Az B ∈ z . M = Cz D ( + ) Also referred to as linear fractional transformations,( ) these have been the subject of recent ( + ) works on analogues of theorems in complex analysis [16, 34]. As is done in the complex plane, we can represent such functions via a matrix A B PGL 2,Cℓ1,0 . = CD ∈

Notice that if we write A a1jA aŒ2j ,B ‘b1j b2(j ,C c)1j c2j ,D d1j d2j , then = + − = + − = + − = + − a1 b1 a2 b2 + j + j 1j +2j , + = c1 d1 + c2 d2 − = + − It is easy to check thatA Œ ‘ + Œ ‘ A +A

det det 1j det 2j and N det det 1 det 2. = + − = Interestingly, the literatureA A does+ not containA such a( decomA)positionA for M¨obiusA transforma- tions. We feel that this makes proving interesting facts about these functions much easier.

Notice that N det 0 if and only if det 1 0 and det 2 0. Thus, is invertible if ≠ ≠ ≠ and only if 1 and( 2A)are invertible so that A A A 1 1 1 A A − 1− j 2− j . = + − That is, A A +A

PGL 2,Cℓ1,0 PGL 2, R PGL 2, R . ≅ We also have the decomposition( ) ( ) × ( ) Az B z 1 u j 2 v j . M = Cz D =M + M − + We call a M¨obius transformation( ) real when(A,) B,+ C, D ( )R. In this case we have a1 + ∈ = a2,b1 b2,c1 c2,d1 d2. = = = We also consider conjugate M¨obius transformations of the form

1 a1v b1 a2u b2 Az B Cz D − j j . = c1v d1 + c2u d2 − + + ( + )( + ) + 71 + + Proposition 5.5.1. M¨obiusTransformations of the form Az B Az B z or z M = Cz D M = Cz D + + such that N AD BC 0 are( conformal) on N 1,1.( ) ≠ + + Az B A1u B1 A2v B2 Proof. Let ( z − ) j j . It is clear that is N 1,1-differentiable M = Cz D = C1u D1 + C2v D2 − M + + + since combinations( ) of compositions of J and+ J on either side of are differentiable at ap- + + ±+ M propriate places.

Notice that N AD BC A1D1 B1C1 A2D2 B2C2 0. Also notice that the = ≠ compositions of J and J on either side of correspond to a possible swapping of rows and ( −± ) ( − M ) ( − ) or columns of the corresponding matrices of 1 and 2, and hence change at most the sign M M of N AD BC . Thus, the corresponding partial derivatives at appropriate points are still

non-zero.( − Hence,) we have conformality everywhere. Az B A similar argument works for z , using ∂. M = Cz D + ( ) 5.5.1 Fixed Points and Transitivity+

The notion of a fixed point under now makes sense everywhere, since we understand M what it means for a real M¨obius transformation to have a fixed point in R . As in complex analysis (see [1]), we find an association between conjugacy classes∪ of {∞} M¨obius transformations and their fixed points. However, the situation is a little more complicated.

Theorem 5.5.2. Let be a M¨obiustransformation on N 1,1 where neither component is M the identity. Then has either zero fixed points, one fixed point, two fixed points or four M fixed points. When the M¨obiustransformation is real, there cannot be two fixed points.

Proof. The fixed points of the M¨obius transformation are determined by those of the M component transformations 1 and 2. If one component has zero fixed points, then M M M has zero fixed points. If both have one fixed point, the has one fixed point. If one M component has one fixed point and the other two, then has two fixed points. Finally, if M both components have two fixed points, then has four. M If is a real M¨obius transformation, then both components are the same and so have M the same number of fixed points. Hence in this case two fixed points cannot occur.

72 Remark 5.5.3. Fixed points at an infinity occur when one or both components have infinity as a fixed point.

Remark 5.5.4. Consider the first component of a M¨obiustransformation:

au b cu d + where a, b, c, d R. +

Case 1. Infinity∈ is a fixed point if and only if c 0. In this case, when a d, infinity is the = = only fixed point.

Case 2. If c 0 and a d, there is a second fixed point namely b d a . = ≠ Case 3. When c 0 solutions to ~( − ) ≠ au b u cu d = have the form + a +d ∆ u . = 2c √ where ∆ a d 2 4bc. If ∆ 0, then( there− ) ± are no fixed points. When ∆ 0, there = < = is one fixed point. If ∆ 0, then there are two fixed points. ( − ) + >

Similar calculations hold for 2. M Remark 5.5.5. We can actually rewrite ∆ in terms of Tr and det:

∆ ∆ 4ad 4ad = 2 2 a + 2ad− d 4ad 4ad 4bc = 2 a − d +4 ad+ bc− + = 2 (Tr + ) 1 − (4det− )1 . = M M ( ) − ( ) Given the above discussion and the fact that Tr and det completely determine eigenvalues (see [31]), we get a link between the number of eigenvalues of the component matrices and the number of fixed points a M¨obius transformation has. a b Proposition 5.5.6. Let PGL 2, R be a non-. M = c d ∈ Œ ‘ ( ) 73 a. If has one (real) eigenvalue, then u has one fixed point. M M b. If has two real eigenvalues, then (u )has two fixed points. M M c. If has two complex eigenvalues, then( ) u has no fixed points. M M Proof. The ∆ defined above is precisely the discriminant( ) of the characteristic polynomial of . M Example 5.5.7. a. The mapping 1 z has four fixed points, 1, j,.

b. The mapping 1 z has zero fixed~ points, ± ± c. The mapping z− ~1 has one fixed point, . d. The mapping 2z+ 1 has four fixed points,∞ , 1, j j and j j . + − + − 1 e. The mapping j+ v 1 j has two fixed∞ points,− −j + ∞j and ∞j − j . u + − + − + − We can also have+ attracting ( + ) and repelling fixed points.+ ∞ − + ∞

Example 5.5.8. Consider z αuj βvj with α, β 0. Of course, then is confor- M = + − ≠ M mal. ( ) + Notice that fixes 0, , j , j . Now, n z αnuj βnvj and we have four M + − M = + − cases: ∞ ∞ ∞ ( ) + Case 1 Let α , β 1. Then for all u 0, , > ≠ n S S S S α u ∞→ as n → ,

and v 0, , ∞ ∞ ≠ n β v → as n → . ∞ Thus is an attracting fixed point and 0, j , j are repelling. ∞ + ∞− Case 2 Let∞α 1, β 1. Then for all u 0, ∞ ∞ > < ≠ n S S S S α u → ∞as n → ,

and v 0, , ∞ ∞ ≠ n β v → 0 as n → 0. ∞ Thus j is an attracting fixed point and 0, , j are repelling. + − ∞ 74 ∞ ∞ Case 3 Let α 1, β 1. Then for all u 0, , < > ≠ n S S S S α u → ∞ as n → 0,

and v 0, , ∞ ≠ n β v → as n → . ∞ Thus j is an attracting fixed point and 0, , j are repelling. − ∞ ∞+ Case 4 Let∞α 1, β 1. Then for all u 0, , ∞ ∞ < < ≠ n S S S S α u → ∞ as n → 0,

and v 0, , ∞ ≠ n β v → as n → 0. ∞ Thus 0 is an attracting fixed point and , j , j are repelling. ∞ + −

Now we see that the transitivity of the∞ M¨obius∞ ∞ transformations of N 1,1 are only one transitive, a property which is quite different from what we find on the Riemann sphere, where the M¨obius transformations are three transitive [1].

Theorem 5.5.9. The M¨obiusgroup acts transitively on N 1,1. It is not two transitive in general.

Proof. Given z1 u1j v1j and z2 u2j v2j , we know that by transitivity of M¨obius = + − = + − transformations of R that there is PGL 2, R such that u1 u2 and PSL 2, R + ∈ + = P ∈ such that v1 v̂2. Therefore, z u j v j maps z1 to z2. P = MN =N ( +) P − N ( ) ( ) To see that( ) M¨obius group is not( ) two-transitive,( ) + ( we) give a proof by counterexample. Consider j , 2j and j , 2j . Notice that in order for j j and 2j 2j , + + − − M + = − M + = − we would{ need a} real M¨obius{ } transformation which maps 0( to) 1 and 0 to 2,( an) obvious contradiction.

Definition 5.5.10. For our purposes, we shall define a hyperbola to be a subset H of N 1,1 which is M¨obiusequivalent to the closure of the set

uj vj uv 1 . + − = { + 75∶ } That is there exists real M¨obiustransformations 1, 2 such that M M

1 u 2 v 1, M M =

1 1 ( ) ( ) for every u,v H R , . ∈ 1,1 In a similar fashion,∩ we define a degenerate hyperbola to be a subset D of N that is M¨obiusequivalent to the closure of the light cone

L uj vj uv 0 . = + − = { + ∶ } Remark 5.5.11. By definition, the closure of the light cone L uv 0 is a degenerate = = hyperbola. It contains two branches, namely vj v R and uj u R . It is clear that − ∈ ∶ + { ∈ } the same is true for any degenerate hyperbola.™ ∶ ̂ž ™ ∶ ̂ž Notice that conjugation z vj uj maps a degenerate hyperbola to itself, interchanging + − branches. ↦ + In a similar way, u 0 consists of two branches: vj v R and uj j u R . v = − ∈ + − ∈ Again, conjugation interchanges™ ž these branches. ™ ∶ ̂ž ™ + ∞ ∶ ̂ž

Remark 5.5.12. Let α,β,µ,R R with µ,R 0. The closure of curves of the form u ∈ ≠ α v β R, u µv β, v µu β are hyperbolae. They are also the only hyperbolae. = = = ( − Let γ,δ R. Then it is clear that the degenerate hyperbolae are of the form u γ )( − ) ∈ + + = v δ . ̂ = { } ∪ { The} following is clearly true.

Proposition 5.5.13. The M¨obiusgroups act transitively on hyperbolae. That is, a M¨obius transformation will map any hyperbola to another. The M¨obiusgroups also acts transitively on degenerate hyperbolae.

5.5.2 Cross Ratio

Yaglom defines a cross ratio on R1,1, and shows that it can be used to define hyperbola and lines [63]. Recent works have brought further analogues of the cross ratio in the complex plane to new spaces [26, 6].

76 We shall extend Yaglom’s ratio to N 1,1 and use the j , j basis to understand it as a + − direct product of real cross ratios. This gives more natural proofs of geometric ideas brought forth in [63].

1,1 Four points in R , zi uij vij , i 1, 2, 3, 4, with distinct uis and distinct vis, are = + − = called completely distinct. + The following lemmas, though rather trivial, are useful.

Lemma 5.5.14. The image of a set of completely distinct points under a M¨obiustransfor- mation is a set of completely distinct points.

Lemma 5.5.15. Hyperbolae contain infinite sets of completely distinct points. In particular, they contain sets of four completely distinct points.

Remark 5.5.16. Because of their form, degenerate hyperbolae cannot contain three com- pletely distinct points; they may have at most two and they must lie on separate branches. This implies that hyperbolae and degenerate hyperbolae are not M¨obiusequivalent, since M¨obiustransformations are one-to-one maps.

Given a 4- of completely distinct points, we define the cross ratio as follows:

z1 z3 z2 z4 λ z1,z2; z3,z4 . = = z2 z3 z1 z4 ( − ) ( − ) [ ] We then have ( − ) ( − )

u1 u3 u2 u4 v1 v3 v2 v4 z1,z2; z3,z4 j j = u2 u3 u1 u4 + v2 v3 v1 v4 − ( − ) ( − ) ( − ) ( − ) [ ] u1,u2; u3,u4 j v1+,v2; v3,v4 j = ( − ) ( − + ) ( − ) (− − ) λ[ 1j λ2j . ] + [ ] = + − + By taking limits, this is defined when one of the uis or vis is infinite. The following two theorems and proposition are mentioned in Yaglom’s book without proof[63].

Theorem 5.5.17. Let z1,z2,z3,z4 and ξ1,ξ2,ξ3,ξ4 be 4- of completely distinct points.

There is a M¨obiustransformation sending z1,z2,z3,z4 to ξ1,ξ2,ξ3,ξ4 if and only if their cross

77 ratios are equal. As such, the cross ratio is a bijection on orbits of 4-tuples of completely distinct points.

Proof. The above lemma implies that for any M¨obius transformation , the cross ratio M z1 , z2 ; z3 , z4 is defined if and only if z1,z2; z3,z4 is defined. M M M M Suppose there exists such a M¨obius transformation 1j 2j . By a simple [ ( ) ( ) ( ) ( )] [M=M +] M − calculation, we have + ui uj a1d1 b1c1 1 ui 1 uj . M M = c1ui d1 c1uj d1 ( − )( − ) Then, ( ) − ( ) ( + )( + )

u1 u3 a1d1 b1c1 u2 u4 a1d1 b1c1 ( − )( − ) ( − )( − ) c1u1 d1 c1u3 d1 c1u2 d1 c1u4 d1 1 u1 , 1 u2 ; 1 u3 , 1 u4 ( + )( + ) ( + )( + ) u2 u3 a1d1 b1c1 u1 u4 a1d1 b1c1 M M M M = ( − )( − ) ( − )( − ) c1u2 d1 c1u3 d1 c1u1 d1 c1u4 d1 ( + )( + ) ( + )( + ) [ ( ) ( ) ( ) ( )] u1 u3 u2 u4 . = u2 u3 u1 u4 ( − ) ( − ) Similar calculations work for 2. ( − ) ( − ) M Thus, M¨obius transformations preserve cross ratios. Conversely, assume that the points are finite and suppose the cross ratios are equal. Then, z z3 z2 z4 S z = z2 z3 z z4 ( − ) ( − ) maps z1,z2,z3,z4 to z1,z2; z3,z4 , 0(, 1), and ( − ) ( − ) [ ] ∞ ξ ξ3 ξ2 ξ4 T ξ = ξ2 ξ3 ξ ξ4 ( − ) ( − ) ( ) maps ξ1,ξ2,ξ3,ξ4 to ξ1,ξ2; ξ3,ξ4 , 0, 1, . ( − ) ( − ) 1 Hence T − S is[ a M¨obius transformation] ∞ sending z1,z2,z3,z4 to ξ1,ξ2,ξ3,ξ4. A similar calculation holds○ when points are an infinity.

Proposition 5.5.18. If z1,z2,z3, are completely distinct points on some hyperbola H, then the cross ratio z1,z2; z3,z4 is in R if and only if z4 is another point on H.

[ ] ̂

78 Proof. Suppose that z4 also lies on H, then by definition of hyperbola and M¨obius invariance of cross ratios, we may assume that H is the hyperbola uv 1. = Hence,

u1 u3 u2 u4 v1 v3 v2 v4 z1,z2; z3,z4 j j = u2 u3 u1 u4 + v2 v3 v1 v4 − ( 1 − 1 ) ( 1 − 1 ) ( − ) ( − ) [ ] v1 v3 v2 v4 + v1 v3 v2 v4 ( 1 − 1 ) ( 1 − 1 )j ( − ) ( − )j = + v2 v3 v1 v4 − ‰ v2 − v3 Ž ‰ v1 − v4 Ž ( − ) ( − ) 1 + v1v2v3v4 v3 v1 v4 ( v2− ) ( v1− v)3 v2 v4 ‰ −1 Ž ‰ − Ž j j = v3 v2 v4 v1 + v2 v3 v1 v4 − v1v2v3v4 ( − ) ( − ) ( − ) ( − ) Œv1 v3 ‘v2 v4 + ( − ) (R. − ) ( − ) ( − ) = v2 v3 v1 v4 ∈ ( − ) ( − ) ̂ For the converse, it suffices( − to show) ( − that) if uivi 1 for i 1, 2, 3 and z1,z2; z3,z4 R, = = ∈ then u4v4 1. ̂ = [ ] Then by hypothesis,

1 1 1 4 v1 v3 v2 u v1 v3 v2 v4 1 1 1 . u4 = v2 v3 v1 v4 ‰ v2 − v3 Ž ‰ v1 − Ž ( − ) ( − ) After some algebra and‰ using− theŽ ‰ fact− thatŽ none( − of) the ( factors− ) vanish, we see that

1 u4v2 v2 v4 , 1 u4v1 = v1 v4 ( − ) ( − ) which immediately implies that ( − ) ( − )

v1 1 u4v4 v2 1 u4v4 . = ( − ) ( − ) But since v1 v2, this must mean that ≠

1 u4v4 0. = −

Theorem 5.5.19. If the cross ratio z1,z2; z3,z4 R, then there exists a M¨obiustransfor- ∈ mation sending z1,z2,z3,z4 to any hyperbola.[ ] ̂

79 Proof. By assumption we have that u1,u2; u3,u4 v1,v2; v3,v4 λ R. = = ∈ Define M¨obius transformations by[ ] [ ] ̂

u u3 u2 u4 v v3 v2 v4 U u and V v . = u2 u3 u u4 = v2 v3 v v4 ( − ) ( − ) ( − ) ( − ) ( ) ∶ ( ) Then ( − ) ( − ) ( − ) ( − )

U u1 V v1 λ = = U(u2) V (v2) 1 = = U(u3) V (v3) 0 = = U(u4) V (v4) . = = ( ) ( ) ∞ Thus, z U u j V v j maps z1,z2,z3,z4 to a line u v , which can be mapped by M = + − = way of a M¨obius transformation (namely J or J ) to a hyperbola. ( ) ( ) + ( ) + − { }

80 CHAPTER 6

THE MOBIUS¨ GROUP OF THE EXTENDED MULTICOMPLEX NUMBERS

We can apply many of the results of the previous chapter to a compactification of MCn, called the extended multicomplex numbers, which turns out to be a direct sum of copies of C. The results of the Riemann sphere give similar outcomes to the extended multicomplex numbers.̂

6.1 The Extended Complex Numbers

The extended complex numbers is the one point compactification of C:

C C . = ̂ As a topological space, the open sets are∶ the∪ open {∞} sets of C combined with those subsets containing and whose compliment is compact in C[33]. C There are∞ several advantages to using as a domain for functions, including that it is 2 a compact space. Via stereographic projection,̂ we can regard C as the Riemann sphere S (which is a homeomorphism). Below are basic results found in manŷ complex analysis books (such as [33]).

C C Definition 6.1.1. A function f → is analytic (meromorphic) if f C is analytic (mero- f 1 morphic) and z is analytic (meromorphic)∶ ̂ ̂ at 0. S

Example 6.1.2.( The) function 1 f z = z is meromorphic on C and on C. The function( )

̂ ez g z = z ( )81 C 1 is meromorphic on the complex plane, but not on , for g z has an essential singularity at 0. ̂ ‰ Ž

Unlike C, the meromorphic functions on C are simple [33]. ̂ Theorem 6.1.3. The meromorphic functions on C are precisely the rational functions. ̂ Definition 6.1.4. The of C, denoted by Aut C , are the meromorphic bi- C C jective functions f → . ̂ (̂) ̂ ̂ Remark 6.1.5. The∶ automorphisms of C are also homeomorphisms of C. ̂ ̂ Clearly, Aut C must be a subclass of the rational functions. In fact, they are the M¨obius

transformations.(̂)

Theorem 6.1.6.

az b Aut C f z a, b, c, d C, ad bc 0 . = = cz d ∈ ≠ + (̂) › ( ) ∶ − 6.2 The Extended+ Multicomplex Numbers

We want to construct to extended multicomplex numbers MCn to provide desired topo- logical and analytic properties. The precise construction for theÅ bicomplex case can be found in [9]. Essentially,

BC C ⊕ C. = To get the general case, we proceed inà a manner̂ ̂ similar to our previous work [20]. That

n 1 is, we need to add one additional point for every point on the light cone, and 2 − -points which compactify the light cone. Thus, 2n−1 MCn C. = k 1 = Å ∶ ̂ That is, MCn is compact and is the most natural? analogue of the extended complex numbers in this setting.Å

82 Remark 6.2.1. We reserve the symbol for the element of MCn with in every compo- nent. ∞ Å ∞

The following is now clear from results from complex analysis.

MC MC Definition 6.2.2. A function f n → n is called analytic (meromorphic) if

n−1 ∶ Å Å2 f ζ fk zk ek, = k 1 = ( ) Q ( ) where fk zk is analytic (meromorphic) in the sense of Definition 6.1.1.

( ) Proposition 6.2.3. The meromorphic functions on MCn are precisely the functions 2n−1 Å f ζ fk zk ek, = k 1 = ( ) Q ( ) where fk zk is a rational function.

( ) n 1 Proof. The proof is essentially the one found in [33] extended to 2 − complex dimensions.

Definition 6.2.4. The automorphisms of MCn, denoted by Aut MCn , are the homeomor- MC MC phisms f n → n which are meromorphic.Å (Å) Å Å We also∶ have the following:

Proposition 6.2.5. 2n−1 MCn Aut C . = k 1 = Å ? (̂) Corollary 6.2.6. The group of automorphisms Aut MCn is the collection of M¨obiustrans- formations (Å) Aζ B ζ . M = Cζ D ( + ) ( ) Remark 6.2.7. It is worth noting that the M¨obiustransformations( + ) are not the only home- omorphisms on MCn, rather the meromorphic and sense preserving ones. In fact,

2n−1 Å Akζk Bk f ζ eσ k = k 1 Ckζk Dk ( ) = ( + ) ( ) Q n 1 is a homeomorphism of MCn for any permutation( + σ of) 1, 2,..., 2 − . Å 83 { } 6.3 M¨obius Transformations

With A, B, C, D MCn, we consider the actions of M¨obius transformations expressed as

∈ Aζ B ζ . M = Cζ D ( + ) ( ) As is done in the complex plane, we can represent( + such) functions via a matrix

A B PGL 2, MCn . = CD ∈ A Œ ‘ ( ) Notice that if we use the light cone basis, we have

2n−1 2n−1 ak bk ek kek. = k 1 ck dk = k 1 = = A Q Œ ‘ Q A It is easy to check that 2n−1 det det kek. = k 1 = It then follows that det L if and onlyA ifQ det kA 0 for every k. Thus, is invertible if ∉ ≠ and only if for each k, Ak is invertible so that A A 2n−1 A 1 1 − k− ek. = k 1 = A Q A That is, 2n−1 2n−1 PGL 2, MCn PGL 2, C PSL 2, C . ≅ k 1 ≅ k 1 = = We also have the decomposition( ) M ( ) M ( )

−1 Aζ B 2n ζ k zk ek, M = Cζ D = k 1 M + = ( ) Q ( ) where each k zk is a M¨obius transformations+ of C. We call a M¨obius transformation M complex when A, B, C, D C. In this case we have ak̂ aℓ, bk bℓ, ck cℓ, dk dℓ, for each ( ) ∈ = = = = k,ℓ.

84 6.3.1 Fixed Points and Transitivity

We use elementary facts about the M¨obius transformations of C to prove analogous facts about M¨obius transformations of the extended multicomplex numbers.̂

2n−1 Theorem 6.3.1. Let ζ k 1 k zk ek, Then the number of fixed points of ζ is M = = M M a power of 2 or is infinite. Further, there are an infinite number of fixed points if and only ( ) ∑ ( ) ( )

if k is the identity for some k. M 2n−1 Proof. It is clear that the point ζ0 k 1 zk0 ek, is a fixed point of if and only if zk0 is a = = M fixed point of k for each k. Since k can each have precisely one, two, or infinitely many M M∑ m fixed points (if and only if k is the identity), we conclude that has precisely 2 fixed M M points, or infinitely many fixed points.

2n−1 Corollary 6.3.2. Let ζ k 1 k zk ek. Then M = = M ( ) ∑ ( ) 1 1 C i. ζ has precisely one fixed point if and only if k ∼ inside PSL 2, for M M 0 1 every( )k; Œ ‘ ( )

m λ 0 ii. ζ has precisely 2 fixed points if and only if k ∼ 1 for exactly m of the k M M 0 λ M ( ) 1 1 n 1 Œ ‘ C and k ∼ for exactly 2 − m of the k inside PSL 2, , where λ ≠ 1; M 0 1 M We find thatŒ transitivity‘ is not what− we have in complex analys( is. )

Theorem 6.3.3. The M¨obiustransformations are transitive, but not 2-transitive.

2n−1 2n−1 Proof. Given ζ1 zkek, and ζ2 wkek, we know that by transitivity of complex = k 1 = k 1 = = M¨obius transformations that for each k there is k PSL 2, C such that k zk wk. Q Q M ∈ M = 2n−1 Therefore, ζ k 1 k zk ek maps ζ1 to ζ2. M = = M ( ) ( ) To see that 2-transitivity fails, we consider a bicomplex example. Now, consider e , 2e ( ) ∑ ( ) + +

and e−, 2e− . Notice that in order for e+ e− and 2e+ 2e−, we would need a M = M = { } complex{ M¨obius} transformation which maps( 0 to) 1 and 0 to 2,( an obvious) contradiction.

We can get a transitivity theorem closer to what we encounter in complex analysis if we use a stronger notion of distinct points.

85 2n−1 Definition 6.3.4. A finite set ζ1,...,ζk ζℓ k 1 zkℓ ek, of points in the extended bicom- = = plex numbers is said to be completely distinct if zk zk for every k and every ℓ m. š ∶ ∑ ℓ ≠ m Ÿ ≠

Theorem 6.3.5. If ζ1,ζ2,ζ3 and υ1,υ2,υ3 are two sets of completely distinct points then

there is a unique M¨obiustransformation such that ζℓ υℓ for each ℓ. { } { M } M = n−1 n−1 2 2 ( ) Proof. Let ζℓ zkℓ ek and υℓ wkℓ ek. Then for each k there is a unique complex = k 1 = k 1 = = M¨obius transformations k such that Q M Q

k zk wk M ℓ = ℓ 2n−1 ( ) for every ℓ. Thus k 1 k zkℓ ek is the unique M¨obius transformation such that M = = M ζℓ υℓ for each ℓ. M = ∑ ( ) (As) before, we want to define a class of hypersurfaces on which the M¨obius transformations act transitively.

Definition 6.3.6. We shall define a M¨obius hypersurface to be a subset H of MCn which is M¨obiusequivalent to the closure of the set Å

ζ z1z2 z2n−1 1 . ⋯ = { ∶ } That is there exists real M¨obiustransformations 1, 2 ..., 2n−1 such that M M M

1 z1 2 z2 2n−1 z2n−1 1, M M ⋯M = ( ) ( ) ( ) for every ζ H R1,1. ∈ ∩ In a similar fashion, we define a degenerate M¨obius hypersurface to be a subset D

of MCn which is M¨obiusequivalent to the closure of the light cone Ln. Å 6.3.2 Cross Ratio

In the previous chapter (initially published in [20]), we defined a cross ratio over the extended split-complex numbers. This cross ratio extends to the extended multicomplex numbers in a natural way.

86 That is, given four completely distinct points ζ1,ζ2,ζ3,ζ4 MCn, their cross ratio is

ζ1 ζ3 ζ2 ζ4 ∈ ζ1,ζ2; ζ3,ζ4 . = ζ2 ζ3 ζ1 ζ4 ( − ) ( − ) [ ] This of course decomposes so that ( − ) ( − )

2n−1 zk1 zk3 zk2 zk4 ζ1,ζ2; ζ3,ζ4 ek = k 1 zk2 zk3 zk1 zk4 =−1 ( − ) ( − ) [ ] 2Qn (zk1 ,z−k2 ; z)k3 (,zk4−ek ) = k 1 = Q [ ] is just a combination of more familiar complex cross ratios. We remark that S4 acts on the order of the points just as it does in the complex case. Now, it is also clear that the cross ratio is M¨obius invariant. That is

ζ1 , ζ2 ; ζ3 , ζ4 ζ1,ζ2; ζ3,ζ4 . M M M M = [ ( ) ( ) ( ) ( )] [ ] Theorem 6.3.7. Let ζ1,ζ2,ζ3,ζ4 and ω1, ω2, ω3, ω4 be 4-tuples of completely distinct points.

There is a M¨obiustransformation sending ζ1,ζ2,ζ3,ζ4 to ω1, ω2, ω3, ω4 if and only if their cross ratios are equal. As such, the cross ratio is a bijection on orbits of 4-tuples of completely distinct points.

Proof. Since M¨obius transformations preserve cross ratios, only the converse must be proved.

Assume ζi, ωi MCn i, and suppose the cross ratios are equal. Then, ∈ ∀ ζ ζ3 ζ2 ζ4 S ζ − − = ζ2 ζ3 ζ ζ4 ( − ) ( − ) ( ) maps ζ1,ζ2,ζ3,ζ4 to ζ1,ζ2; ζ3,ζ4 , 0, 1, and( ) ( ) ∞ [ ] ω ω3 ω2 ω4 T ω − − = ω2 ω3 ω ω4 ( − ) ( − ) ( ) maps ω1, ω2, ω3, ω4 to ω1, ω2; ω3, ω4 , 0, 1,( . ) ( ) ∞ 1 Hence T − S is a M¨obius transformation sending ζ1,ζ2,ζ3,ζ4 to ω1, ω2, ω3, ω4. A similar ○ [ ] calculation holds when points are an infinity.

87 For the general multicomplex case, that is when n 2, there is not analogous statement > of Proposition 5.5.18 which is true. Indeed, we can find 4 completely distinct points on a M¨obius hypersurface whose cross ratio is not C-valued. ̂ Example 6.3.8. Let

iπ i9π iπ i8π ζ1 exp e1 exp e2 exp e3 exp e4 = 10 10 9 9 iπ i7π iπ i6π ζ2 exp ‹ e1 + exp ‹ e2 exp ‹ e3 + exp ‹ e4 = 8 8 7 7 iπ i5π iπ i4π ζ3 exp ‹  e1 + exp ‹  e2 exp ‹  e3 + exp ‹  e4 = 6 6 5 5 iπ i3π iπ i2π ζ4 exp ‹  e1 + exp ‹  e2 exp ‹  e3 + exp ‹  e4 = 4 4 3 3 ‹  + ‹  ‹  + ‹  be four points in MC4. Notice that these are completely distinct points and that each lies on M¨obiushypersurface

z1z2z3z4 1. = Also notice that

iπ iπ iπ iπ iπ √3 1 exp 10 exp 6 exp 8 exp 4 exp 10 2 2 i , iπ iπ iπ iπ iπ √2 √2 exp 10 exp 4 exp 8 exp 6 = i ‰ ‰ Ž − ‰ ŽŽ ‰ ‰ Ž − ‰ ŽŽ exp ‰10 Ž − 2 − 2 while ‰ ‰ Ž − ‰ ŽŽ ‰ ‰ Ž − ‰ ŽŽ ‰ Ž − − iπ iπ iπ iπ iπ iπ exp 9 exp 5 exp 7 exp 3 exp 9 exp 5 . iπ iπ iπ iπ iπ 1 √3 exp 9 exp 3 exp 7 exp 5 = i ‰ ‰ Ž − ‰ ŽŽ ‰ ‰ Ž − ‰ ŽŽ exp‰ 9Ž − 2 ‰2 Ž Since these two components of the cross ratio are not equal, the cross ration cannot be ‰ ‰ Ž − ‰ ŽŽ ‰ ‰ Ž − ‰ ŽŽ ‰ Ž − − C-valued. ̂ There is an analogous statement in BC and the proof is the same in the Cℓ1,0 case.

88 CHAPTER 7

SPLIT-QUATERNIOINC ANALYSIS AND THE MOBIUS¨ TRANSFORMATIONS OF R2,2

Complex analysis and the (Riemannian) geometry of the plane are inextricably linked due to the seemingly coincidental fact that the (sense-preserving) conformal mappings of the plane are precisely the biholomorphic mappings [59]. Indeed, we saw in Chapter 5 that an analogous relationship exists for a distinguished class of functions on of a split-complex variable and the Minkowski plane. Additionally, one may find a similar theory for functions of a quaternionic variable [60]. The purpose of this chapter is to explore the analogous notions for the split-quaternions. Recall, the split-quaternions are the real Clifford algebra

Cℓ1,1 Z x0 x1i x2j x3ij x0,x1,x2,x3 R , = = ∈ ∶ { + + + ∶ } satisfying the following multiplication rules:

i2 1 j2 1 = = − ij ji. = − Also recall that 2 2 2 2 ZZ x0 x1 x2 x3. = Hence, we shall identify the split-quaternions+ with− the− semi-Riemannian manifold R2,2. As

2,2 such, we can extend the notions of inner product on R to Cℓ1,1, so that Z,W and Z make sense. With this, we immediately get a theory of conformal functions⟨ on Cℓ⟩1,1 ⟨ ⟩ A theory of holomorphic functions has been introduced recently in two different ways[40, 44]. Both cases are consistent with the analogues of certain geometric notions of the complex

89 case. The one in [40] stands out as the more natural analogue because it gives rise to a (relatively) large class of functions to be studied. Indeed, the analogue defined in [44] can be shown (by adopting a proof of an analogous statement in Sudbery’s paper [60]) to describe only affine functions, which is a (relatively) small class of functions.

7.1 Notions of Holomorphic

The functions we are concerned with are

R2,2 f U → Cℓ1,1, where U is open (in the euclidean sense).∶ ⊆ As higher dimensional analogues of functions of a complex variable, we are interested in obtaining an analogous definition for holomorphic. As we shall see, there are various ways of doing this in the literature. The first, and most interesting, way is through split quaternionic valued differential operators [40]. The second is more recent and less-interesting and is obtained by considering a difference quotient [44].

7.1.1 Analogues of the Cauchy-Riemann Operator

Recall that in complex analysis, one considers the Dirac operators

1 ∂ ∂ 1 ∂ ∂ ∂z i and ∂z i , = 2 ∂x ∂y = 2 ∂x ∂y whose product (in either order)∶ ‹ gives+ the Laplacian∶ for‹R2,− usually denoted by ∆. Also, f

is called holomorphic if ∂zf 0 and its (complex) derivative is given by ∂zf. Additionally, = the real and imaginary parts of f are harmonic functions, and the Dirichlet problem is well-posed.

The question asked in the literature is: Can we define operators valued in Cℓ1,1 which resemble ∂z and ∂z? This question has been answered in the affirmative, although with little mention of the differential geometry which lies just below the surface. However the question we are really asking is: can we factorize the Laplacian in R2,2 with linear first order operators over Cℓ1,1? In this semi-Riemannian manifold, the Laplacian,

90 which is understood to be the derivative of the gradient, is given by [47]:

∂2 ∂2 ∂2 ∂2 ∆2,2 2 2 2 2 . = ∂x0 ∂x1 ∂x2 ∂x3 + − − It is easy to check that the linear operators

∂ ∂ ∂ ∂ ∂ i j ij and = ∂x0 ∂x1 ∂x2 ∂x3 ∂ ∂ ∂ ∂ ∂ ∶ + i − j − ij = ∂x0 ∂x1 ∂x2 ∂x3 ∶ − + + are factors of ∆2,2. Due to the non-commutativity of Cℓ1,1, these operators may be applied to functions on either the left or right and with different results, in general.

Remark 7.1.1. There are other factorizations of ∆2,2 inside Cℓ1,1. Our choice of ∂ is deliberate– it is the gradient inside the semi-Riemannian manifold R2,2. For alternative interpretation of this idea, see [46].

R2,2 1 Definition 7.1.2. Let U Cℓ1,1 and let F U → Cℓ1,1 be C U . We say F is left ⊂ ≅ regular if ∶ ( ) ∂F 0 = for every Z U. Similarly, we say F is right regular if ∈

F ∂ 0 = for every Z U. ∈ We have adopted the above definition from [40], which contains a nice proof of a Cauchy- like integral formula. By multiplying arbitrary F with ∂ we obtain the following conditions which make it easier to check left and right regularity.

91 1 Proposition 7.1.3. Let F U → Cℓ1,1 be C U . Then F is left regular if and only if it satisfies the system of PDEs:∶ ( )

∂f0 ∂f1 ∂f2 ∂f3 0 ∂x0 ∂x1 ∂x2 ∂x3 = ⎪⎧ ⎪ − − − ⎪ ∂f1 ∂f0 ∂f3 ∂f2 ⎪ 0 ⎪∂x0 ∂x1 ∂x2 ∂x3 ⎪ = ⎪ ⎪ + + − ⎪ ∂f2 ∂f3 ∂f0 ∂f1 ⎨ 0 ⎪∂x0 ∂x1 ∂x2 ∂x3 = ⎪ ⎪ − − − ⎪ ∂f3 ∂f2 ∂f1 ∂f0 ⎪ 0. ⎪∂x0 ∂x1 ∂x2 ∂x3 = ⎪ ⎪ + + − Proof. The proof follows directly⎩⎪ from the definition. Simply multiply in the proper order, collect like components together, and equate them to zero to obtain the desired system.

1 Proposition 7.1.4. Let F U → Cℓ1,1 be C U . Then F is right regular if and only if it satisfies the system of PDEs:∶ ( )

∂f0 ∂f1 ∂f2 ∂f3 0 ∂x0 ∂x1 ∂x2 ∂x3 = ⎪⎧ ⎪ − − − ⎪ ∂f1 ∂f0 ∂f3 ∂f2 ⎪ 0 ⎪∂x0 ∂x1 ∂x2 ∂x3 ⎪ = ⎪ ⎪ + − + ⎪ ∂f2 ∂f3 ∂f0 ∂f1 ⎨ 0 ⎪∂x0 ∂x1 ∂x2 ∂x3 = ⎪ ⎪ + − + ⎪ ∂f3 ∂f2 ∂f1 ∂f0 ⎪ 0. ⎪∂x0 ∂x1 ∂x2 ∂x3 = ⎪ ⎪ − − − However, this notion of regularity⎩⎪ is also some what unsatisfying, for simple analogues of holomorphic functions in the complex plane are not regular.

Example 7.1.5. Let A a ib jc ijd Cℓ1,1. Then = ∈ + + + AZ ax0 bx1 cx2 dx3 i bx0 ax1 dx2 cx3 = j(cx0 −dx1 + ax2 + bx3) + ij( dx0+ cx1+ bx2− ax)3 .

+ ( + + − ) + ( − + + ) 92 Thus,

∂ AZ a ib jc dij i b ai dj cij = ( ) j( c+ id+ aj+ bij) + (−ij d+ ic+ bj− aij)

− (2a+ i2+b j+2c ij) −2d ( − − + ) = −2A,+ + + = − A similar calculation shows that AZ ∂ 2A. = Other calculations show that the function( ) ZA−is also neither left-regular nor right-regular.

We obtain similar systems of PDEs if we consider the equations ∂F 0 and F∂ 0: = = ∂f0 ∂f1 ∂f2 ∂f3 0 ∂x0 ∂x1 ∂x2 ∂x3 = ⎪⎧ ⎪ + + + ⎪ ∂f1 ∂f0 ∂f3 ∂f2 ⎪ 0 ⎪∂x0 ∂x1 ∂x2 ∂x3 ⎪ = ⎪ ⎪ − − + ⎪ ∂f2 ∂f3 ∂f0 ∂f1 ⎨ 0 ⎪∂x0 ∂x1 ∂x2 ∂x3 = ⎪ ⎪ + − + ⎪ ∂f3 ∂f2 ∂f1 ∂f0 ⎪ 0 ⎪∂x0 ∂x1 ∂x2 ∂x3 = ⎪ and ⎪ − − + ⎩∂f0 ∂f1 ∂f2 ∂f3 0 ∂x0 ∂x1 ∂x2 ∂x3 = ⎪⎧ ⎪ + + + ⎪ ∂f1 ∂f0 ∂f3 ∂f2 ⎪ 0 ⎪∂x0 ∂x1 ∂x2 ∂x3 ⎪ = ⎪ ⎪ − + − ⎪ ∂f2 ∂f3 ∂f0 ∂f1 ⎨ 0 ⎪∂x0 ∂x1 ∂x2 ∂x3 = ⎪ ⎪ − + − ⎪ ∂f3 ∂f2 ∂f1 ∂f0 ⎪ 0. ⎪∂x0 ∂x1 ∂x2 ∂x3 = ⎪ These also produce unsatisfying⎪ + analogues+ of+ holomorphic since linear functions, again, ⎩⎪ fail these conditions.

93 Example 7.1.6. Let A a ib jc ijd Cℓ1,1. Then, = ∈ + + + ∂ AZ a ib jc dij i b ai dj cij = ( ) j( c+ id+ aj+ bij) − (−ij d+ ic+ bj− aij)

+ 4(a. + + + ) + ( − − + ) =

A similar calculation shows that AZ ∂ 4A. = Other calculations show that the function( ZA) is not annihilated by ∂ on either side.

7.1.2 Difference

Recall, another (and probably primary) way to define holomorphic functions is via the limit of a difference quotient: f z ∆z f z lim . ∆z→0 ∆z ( + ) − ( ) One obtains the Cauchy-Riemann equations by allowing ∆z to approach 0 along the real axis and again along the imaginary axis and then setting the results equal to each other. In Masouri et. al., a similar method is used to produce another analogue of holomorphic [44]. However, since the split-quaternions are not commutative, so there are two ways to construct an analogue of the difference quotient. In Masouri the “quotient” is defined by

1 lim f Z ∆Z f Z ∆Z − . ∆Z→0 ( ( + ) − ( )) ( ) When this limit exists, such functions are called right Cℓ1,1-differentiable. By setting ∆Z

equal to ∆x0, i∆x1, j∆x2, and ij∆x3, taking the limit in each instance, we get four ways to take the “derivative” [44]. That is,

1 ∂f0 ∂f1 ∂f2 ∂f3 lim f ζ ∆x0 f ζ ∆x0 − i j ij , ∆x0→0 = ∂x0 ∂x0 ∂x0 ∂x0 ( ( + ) − ( )) ( ) + + + 1 ∂f0 ∂f1 ∂f2 ∂f3 lim f ζ ∆x0 f ζ i∆x1 − i ij j , i∆x1→0 = ∂x1 ∂x1 ∂x1 ∂x1 ( ( + ) − ( )) ( ) − + + − 94 1 ∂f0 ∂f1 ∂f2 ∂f3 lim f ζ ∆x0 f ζ j∆x2 − j ij i , j∆x2→0 = ∂x2 ∂x2 ∂x2 ∂x2 and ( ( + ) − ( )) ( ) + + + 1 ∂f0 ∂f1 ∂f2 ∂f3 lim f ζ ∆x0 f ζ ij∆x3 − ij j i . ij∆x3→0 = ∂x3 ∂x3 ∂x3 ∂x3 Equating the four( ( results,+ we) − obtain( )) ( the system) of PDEs− [44]: − +

∂f0 ∂f1 ∂f2 ∂f3 ∂x0 = ∂x1 = ∂x2 = ∂x3 ⎪⎧ ⎪ ⎪ ∂f1 ∂f0 ∂f3 ∂f2 ⎪ ⎪∂x0 ∂x1 ∂x2 ∂x3 ⎪ = = = ⎪ ⎪ − − ⎪ ∂f2 ∂f3 ∂f0 ∂f1 ⎨ ⎪∂x0 = ∂x1 = ∂x2 = ∂x3 ⎪ ⎪ − − ⎪ ∂f3 ∂f2 ∂f1 ∂f0 ⎪ . ⎪∂x0 = ∂x1 = ∂x2 = ∂x3 ⎪ Although the work which introduces⎪ this notion of differentiability, [44], does not mention ⎩⎪ any specific examples of functions of right Cℓ1,1-differentiable functions, an entire class of functions can be easily shown to have this property.

Example 7.1.7. Recall that

AZ K ax0 bx1 cx2 dx3 k i bx0 ax1 dx2 cx3 ℓ = + j(cx0 −dx1 + ax2 + bx3 + m) + ij( dx+0 cx+1 bx−2 ax+3 )n .

+ ( + + − + ) + ( − + + + ) Notice f Z AZ K is right Cℓ1,1-differentiable. Indeed, the “derivative” is = ( ) + 1 1 lim A Z ∆Z K AZ K ∆Z − lim A∆Z ∆Z − A. ∆Z→0 = ∆Z→0 = ( ( + ) +R2,2− − ) ( ) ( ) ( ) Theorem 7.1.8. Let F U → Cℓ1,1. Then F is right Cℓ1,1-differentiable if and only ⊆ F Z AZ K, where A, K Cℓ1,1. That is, the right Cℓ1,1-differentiable functions must be = ∶ ∈ affine( ) mappings.+

95 Proof. As similar fact is true for functions of a quaternionic variable and so we follow a similar proof from Sudbery’s paper1 [60].

First notice that Z x0 x1i x2 x3i j z wj. As such we may write f Z = = = g z,w h z,w j, where g z,w f0 z,w if1 z,w and h z,w f2 z,w if3 z,w . ( + =) + ( + ) + = ( ) ( Now,) + the( above) system of( PDEs) gives( us) that + g( is holomorphic) ( with) respect( ) to+ the( complex) variables z and w. Similarly, h is holomorphic with respect to the complex variables w and z. Additionally,

∂g ∂f0 ∂f1 ∂f2 ∂f3 ∂h i i , ∂z = ∂x0 ∂x0 = ∂x2 ∂x2 = ∂w ∂g ∂f1 ∂f0 ∂f3 ∂f2 ∂h + i + i . ∂w = ∂x3 ∂x3 = ∂x1 ∂x1 = ∂z − + − + Now, g and h have continuous partial derivatives of all orders. Thus, we must have

∂2g ∂ ∂h ∂ ∂h 0, ∂z2 = ∂z ∂w = ∂w ∂z = ∂2h ∂ ∂g ∂ ∂g ‹  ‹  0, ∂w2 = ∂w ∂z = ∂z ∂w = 2 ∂ g ∂ ‹∂h ∂ ‹ ∂h 2 0, ∂w = ∂w ∂z = ∂z ∂w = 2 ∂ h ∂ ‹∂g  ∂ ‹∂g  2 0. ∂z = ∂z ∂w = ∂w ∂z = ‹  ‹  W.L.O.G. we may assume that U is connected and convex (since each connected component may be covered by convex sets, which overlap pair-wise on convex sets). Thus integrating on line segments allows us to conclude that g and h are linear:

g z,w α βz γw δzw, = h(z,w) ǫ +ηz +θw +νzw. = 1 We are very grateful to Professor( Uwe) K¨ahler+ of University+ + of Aveiro for bringing this paper to our attention.

96 Since ∂g ∂h , we must have that β θ and δ ν 0. Also since ∂g ∂h , it is the case that ∂z = ∂w = = = ∂w = ∂z γ η. Thus, = f Z g z,w h z,w j = ( ) α( βz) + γ(w ) ǫ γz βw j = (β + γj +z wj) + ( +α +ǫj ) = (AZ+ K,)( + ) + ( + ) = as required. +

Remark 7.1.9. The above theorem proves that right Cℓ1,1-differentiable functions are not left or right regular and conversely (except for when A 0). Indeed, they are also not = annihilated by ∂ on either side.

As an alternative to the definition found in [44], one may reverse the multiplication in the difference quotient to obtain

1 lim ∆Z − f Z ∆Z f Z . ∆Z→0

When this limit exists, such functions( ) are( ( called+ left) − Cℓ(1,1))-differentiable. Proceeding as above, a slightly different system of PDEs than the one found in [44] is obtained:

∂f0 ∂f1 ∂f2 ∂f3 ∂x0 = ∂x1 = ∂x2 = ∂x3 ⎪⎧ ⎪ ⎪ ∂f1 ∂f0 ∂f3 ∂f2 ⎪ ⎪∂x0 ∂x1 ∂x2 ∂x3 ⎪ = = = ⎪ ⎪ − − ⎪ ∂f2 ∂f3 ∂f0 ∂f1 ⎨ ⎪∂x0 = ∂x1 = ∂x2 = ∂x3 ⎪ ⎪ ⎪ ∂f3 ∂f2 ∂f1 ∂f0 ⎪ . ⎪∂x0 = ∂x1 = ∂x2 = ∂x3 ⎪ Example 7.1.10. Recall that ⎪ − − ⎩⎪

AZ ax0 bx1 cx2 dx3 i bx0 ax1 dx2 cx3 = j(cx0 −dx1 + ax2 + bx3) + ij( dx0+ cx1+ bx2− ax)3 .

+ ( + + − 97) + ( − + + ) Notice f Z AZ K is not left Cℓ1,1-differentiable. = However, the map F Z ZA K is left Cℓ1,1-differentiable. Indeed, the “derivative” is ( ) + = ( ) + 1 1 lim ∆Z − Z ∆Z A K ZA K lim ∆Z − ∆ZA A. ∆Z→0 = ∆Z→0 = R2,2 Theorem 7.1.11.( Let) F(( U+ ) →+Cℓ1−,1. Then− )F is left(Cℓ1,)1-differentiable( ) if and only ⊆ F Z ZA K, where A, K Cℓ1,1. That is, the left Cℓ1,1-differentiable functions must be = ∶ ∈ affine( ) mappings.+

Proof. We can make a few adjustments to the proof of the right Cℓ1,1-differentiable case.

First note that if we write F Z g z,w jh z,w , where g z,w f0 z,w if1 z,w , = = h z,w f2 z,w if3 z,w , and the complex variables z,w as above. = ( ) ( )+ ( ) ( ) ( )+ ( ) ( The) system( of) − PDEs( above) assures that g is holomorphic with respect to z and w, while h is holomorphic with respect to z and w. Additionally, we get that ∂g ∂h ∂z = ∂w ∂g ∂h . ∂w = ∂z We also have that g and h have partial derivatives of all orders and similarly to the “right” case the second partials vanish: ∂2g ∂2h ∂2g ∂2h 2 2 0. ∂z2 = ∂w = ∂w2 = ∂z =

Thus, by the same argument for the right Cℓ1,1-differentiable proof, we conclude that g and h are linear:

g z,w α βz γw δzw, = h(z,w) ǫ +ηz +θw +νzw. = Since ∂g ∂h and ∂g ∂h , we( must) have+ that+ β +θ, γ η, and δ ν 0. Thus, ∂z = ∂w ∂w = ∂z = = = = f Z g z,w jh z,w = ( ) α( βz) + γw( j) ǫ γz βw = (z +wj +β )γj + ( +α ǫj+ ) = ZA( + K,)( + ) + ( + ) = + 98 as required.

Remark 7.1.12. Thus, right Cℓ1,1-differentiability is perhaps not a good analogue of holo- morphic. Even though these are equivalent notions in the complex setting, in the split quater- nionic setting there are more directions in which to take the limit and this requires much stronger conditions. For this reason we are justified in studying functions in the kernels of the operators, and not the Cℓ1,1-differentiable functions.

7.1.3 Regularity and John’s Equation

2 Given a F U → Cℓ1,1 whose components are at least C and which satisfies at least one of the following:∶ ∂F 0, F ∂ 0, ∂F 0, or F∂ 0, = = = = must have components which satisfy John’s equation [40]:

∆2,2u 0. =

Such functions are said to be ultra-hyperbolic. In fact, we can use ultra-hyperbolic functions to build regular functions.

Theorem 7.1.13. Let f U → R be ultra-hyperbolic, then ∂f is both left and right regular.

Proof. Write F ∂f. Then∶ clearly =

∂F ∆2,2f 0 ∂f ∂ F ∂. = = = = ( )

It turns out that left and right differentiable functions also have components which are ultra-hyperbolic.

2 Theorem 7.1.14. Let F U → Cℓ1,1, with components which are at least C , be left- differentiable or right-differentiable.∶ Then the components of F are ultra-hyperbolic.

99 Proof. Suppose F x0,x1,x2,x3 f0 f1i f2j f3ij is right-differentiable. Then notice that = 2 2 2 2 ∂ (f0 ∂ f0 ∂) f0 +∂ f0+ + 2 2 2 2 ∂x0 ∂x1 ∂x2 ∂x3 + ∂ −∂f1 − ∂ ∂f1 ∂ ∂f1 ∂ ∂f1 = ∂x0 ∂x1 ∂x1 ∂x0 ∂x2 ∂x3 ∂x3 ∂x2 0. ‹  + ‹−  − ‹−  − ‹  =

A similar argument works for the other fi and for the left-differentiable case.

Ultra-hyperbolic functions also satisfy an important mean value property [30, 27].

4 Theorem 7.1.15 (Asgeirsson’s Mean Value). Let f R → R be ultra-hyperbolic. Then for 0 0 0 0 4 each r 0 and each point Z0 x0,x1,x2,x3 R , the spherical integrals are equivalent: > = ∈ ∶

0 (0 ) 0 0 f x0,x1,x ,x dω x0,x1 f x ,x ,x2,x3 dω x2,x3 . 0 0 2 3 0 0 0 1 SSr x ,x SSr x ,x ( 0 1) = ( 2 3) ( ) ( ) ( ) ( ) 7.2 A Theory of Left-Regular Functions

With all of these notions of holomorphic functions, it becomes necessary to choose one and deem it the “canonical” one. Since the difference quotients do not yield an extensive class of functions, we believe use of an operator to be the be the best place to start. Given

2,2 the association between Cℓ1,1 and R , it seems ∂ is the operator for our purposes,

2,2 1 since it is also the gradient in R (and since it is an analogue of ∂z¯, which is 2 times the 2 gradient of R ). Given the overwhelming convention of applying operators on the left of functions, we choose left-regular to be the canonical notion of holomorphic. Indeed, this is the one chosen by Libine [40]. In his work, he shows that left-regular functions satisfy a Cauchy-like integral formula.

Theorem 7.2.1 (Libine’s Integral Formula). Let U Cℓ1,1 be a bounded open (in the ⊆ Euclidean topology) region with smooth boundary ∂U. Let f U → Cℓ1,1 be a function which extends to a real-differentiable function on an open neighborhood V Cℓ1,1 of U such ∶ ⊆ that ∂f 0. Then for any Z0 Cℓ1,1 such that the boundary of U intersects the cone = ∈

100 C Z Cℓ1,1 Z Z0 Z Z0 0 transversally, we have = ∈ =

š 1∶ ( − )( − )Z ZŸ0 f Z0 if Z0 U lim ⋅ dZ ⋅ f Z , 0 2 S 2 ∈ ǫ→ 2π ∂U Z Z0 Z Z0 iǫ Z Z0 = ⎧0 else − ( − ) ⎪ ( ) ( ) ⎨ where the three form dZ is given by ⎪ ( − )( − ) + Y − Y ⎩⎪

dZ dx1 ∧ dx2 ∧ dx3 − dx0 ∧ dx2 ∧ dx3 i + dx0 ∧ dx1 ∧ dx3 j − dx0 ∧ dx1 ∧ dx2 ij. = Given this interesting property( has been) proven,( it is some) what( surprising that) a more detailed description of left-regular functions has not been given in the literature. So we close by showing that some left regular functions have a simple description.

7.2.1 A Class of Left Regular Functions

To date, the author has not been able to find a description for left regular functions in a manner similar to the split-complex case [20, 39]. It may be the case that no such description exists in general. However, it is possible to give a large class of left-regular functions a simple description.

∶ R2,2 Theorem 7.2.2. Let F U → Cℓ1,1 have the form ⊆

F x0,x1,x2,x3 g1 x0 + x2,x1 + x3 + g2 x0 − x2,x1 − x3 = ( ) + (g3(x0 − x2,x1 − x3) + g4(x0 + x2,x1 + x3))i

+ (g1(x0 + x2,x1 + x3) − g2(x0 − x2,x1 − x3)) j

+ (g3(x0 − x2,x1 − x3) − g4(x0 + x2,x1 + x3)) ij,

1 where gi C U . Then ∂F 0. ( ( ) ( )) ∈ = Proof. We can( easily) check that such an F satisfies the necessary system of PDEs. However, it is far more enlightening to see how one can arrive at such a solution.

Write F f0 + f1i + f2j + f3ij. Using an argument from [20], we have that = ∂ ∂ ∂ ∂ ∂ − j + i − j = ∂x0 ∂x2 ∂x1 ∂x3 ∂ ∂ ∂ ∂ 2‹ j + j +‹ 2i j +  j = ∂v0 + ∂u0 − ∂v1 + ∂u1 − ∶ ∂‹1 + i∂2,  ‹  = 101 1 j 1 j where u0 x0 x2, v0 x0 x2, u1 x1 x3, u1 x1 x3, j , and j . = = = = + = 2 − = 2 + − + − + − The key fact is that j and j are idempotents and annihilate each other. Also, notice + − that ij j i and ij j i. + = − − = + Similarly, we may write

F F0j F1j i F2j F3j . = + − + − Now, one way in which ∂F ( 0 is if+ ) + ( + ) =

∂1 F0j F1j ∂2 F0j F1j + − = + − ( + ) ∂1 (i F2j+ F3)j ∂2 i F2j F3j 0. = + − = + − = Using the above facts about j and( ( j , we+ see)) that the( ( conditions+ implies)) that + − ∂F0 ∂F0 0 ∂v0 = ∂v1 = ⎪⎧ ⎪ ⎪∂F1 ∂F1 ⎪ 0 ⎪ ∂u0 ∂u1 ⎪ = = ⎪ ⎪ ⎪∂F2 ∂F2 ⎨ 0 ⎪ ∂u0 = ∂u1 = ⎪ ⎪ ⎪∂F3 ∂F3 ⎪ 0 ⎪ ∂v0 = ∂v1 = ⎪ This, of course, means that ⎪ ⎩⎪

F0 F0 u0,u1 , F1 F1 v0,v1 , F2 F2 v0,v1 , F3 F3 u0,u1 . = = = = Translating back to( the) original coordinates,( ) we see( F has) the desired( form.)

The converse is not true, in general. Here is a simple counter-example.

Example 7.2.3. Consider the Cℓ1,1-valued function

f x0,x1,x2,x3 x1x2x3 x0x2x3i x0x1x3j x0x1x2ij. = ( ) −102 + + It is easy to check that f satisfies the necessary system of PDEs so that ∂f 0. However, = notice that if we write f as in the above proof, then

2 2 2 2 u1 v1 u0v0 f u0j v0j i u0j v0j . = 4 + − 4 + − ( − ) ( ) ( − ) + (− + ) Now, 2 2 u1 v1 ∂2 ( ) u0j v0j 2v1u0j 2u1v0 0.  4 ( + −) = + ~≡ − Thus, f is not of the form as prescribed− in Theorem− 7.2.2. −

7.2.2 Generating Left Regular Functions

In a manner similar to the Cℓ0,n case, we can also take a Cℓ1,1-valued function whose

components are real analytic and generate a left regular function valued in Cℓ1,1. In fact, there are two ways to do this. The first borrows heavily from a result found in Brackx, Delanghe, and Sommen’s book [5].

2 Theorem 7.2.4. Let g x2,x3 be a Cℓ1,1-valued function on U R with real-analytic com- ( ) ⊆ ponents. Then the function

2k 1 2k 1 ∞ x0 + x1 + k f Z ∂ ∆ g x2,x3 , ( ) = k 0 Œ 2k 1 ! ‘ ( ) = ( + ) Q where ∆ is the Laplace operator in the x2x3-plane,+ is left-regular in an open neighborhood of 2,2 0, 0 U in R and f 0, 0,x2,x3 g x2,x3 ig x2x3 . {( )} ( ) = ( ) ( ) Proof. We× proceed by a similar proof found in [5].−

Let g x2,x3 g0 x2,x3 g1 x2,x3 i g2 x2,x3 j g3 x2,x3 ij. Since gℓ is analytic, then ( ) = ( ) ( ) ( ) ( ) on every compact set K U there are constants cK and λK , depending on K, such that ⊂ + + +

k k ∂ k k sup ∆ g x2,x3 2k !cK λK and sup ∆ g x2,x3 2k 1 !cK λK , x2,x3 K T ( )T ≤ ( ) x2,x3 K V∂xℓ ( )V ≤ ( ) ( )∈ ( )∈ 4 + where ⋅ denotes the euclidean norm in R . S S

103 Thus,

2k 1 2k 1 x0 + x1 + k sup ∂ ∆ g x2,x3 x2,x3 K Œ 2k 1 ! ‘ ( ) W ( )∈ ( + ) W x2k x2k x2k 1 x2k 1 ∂ ∂ sup + 0 i 1 ∆kg 0 + 1 + j ∆kg ij ∆kg = x2,x3 K WŒ 2k ! 2k !‘ 2k 1 ! ‹ ∂x2 ∂x3 W ( )∈ ( ) ( ) ( + ) x2k + x2k − + sup 0 ∆kg 1 ∆kg + ≤ x0,x1 K ŒW 2k !W T T W 2k !W T T ( )∈ ( ) ( ) x2k 1 x+2k 1 ∂ x2k 1 x2k 1 ∂ 0 + 1 + ∆kg 0 + 1 + ∆kg W 2k 1 ! W V∂x2 V W 2k 1 ! W V∂x3 V‘ 2k( k + ) 2k k ( + ) + cK 1 2 x0 x0 λ 1 2 x1 x1 λ + , ≤ K + K + ( + S S) + ( + S S)  so that f converges uniformly on

1 1 1 1 , , K˚ . K U Œ √λK √λK ‘ Œ √λK √λK ‘ ⊆ − × − × Now,

2k 1 2k 1 ∞ x0 + x1 + k ∂f ∆2,2 ∆ g x2,x3 = k 0 Œ 2k 1 ! ‘ ( ) = 2k(1 + 2k) 1 2k 1 2k 1 Q∞ x0 + x1 + k ∞ x0 + x1 + k ∆2,2 + ∆ g x2,x3 ∆2,2 ∆ g x2,x3 = k 0 Œ 2k 1 ! ‘ ( ) k 0 Œ 2k 1 ! ‘ ‰ ( )Ž = = 2k 1 ( 2k+1 ) 2k 1 ( 2k+1 ) Q∞ x0 − x1 − k ∞ +x0Q+ x1 + k 1 + ∆ g x2,x3 + ∆ + g x2,x3 = k 1 Œ 2k 1 ! ‘ ( ) k 0 Œ 2k 1 ! ‘ ( ) = ( + ) = ( + ) 0Q, − Q = − +

as needed.

Example 7.2.5. Let g x2,x3 x2x3. Then ∆g 0 and the formula above gives ( ) = =

f Z ∂ x0 x1 x2x3 ( ) = x2[(x3 +x2x3)(i x0)]x3 x1x3 j x0x2 x1x2 . = − + ( + ) + ( + ) A less trivial example demonstrates that the more complicated g is the more complicated f is.

4 3 2 2 Example 7.2.6. Let g x2,x3 x2 x2x3. Thus, ∆g 12x2 6x2x3 and ∆ g 24. = = = ( ) + 104 + Then from the formula, we get

4 2 2 4 3 4 2 2 4 3 f Z x0 3x0 2x2 x2x3 x2 x2x3 x1 3x1 2x2 x2x3 x2 x2x3 i = 4 4 2 2 2 2 2 x2 2 2 2 2 2 x3 ( )  +x0x1 ( x2+x0 x)1 + + j  +x0x1+ x3( x0 +x1 ) + ij.+  ( ) ( ) 3 ( ) ( ) 3 + + + + + + + + We can define a true extension of an analytic function which is left regular and closely resembles the Cauchy-Kowalewski extension found in [15, 54]. Again, we are again grateful

to Brackx et. al for their proof in the Cℓ0,n case, which again gives the convergence of the series.

Theorem 7.2.7 (Cauchy-Kowalewski Extension in Cℓ1,1). Let g x1,x2,x3 be a Cℓ1,1-valued ( ) 3 function whose components are real-analytic functions on U R . Then the function ⊆ k ∞ x0 k f x0,x1,x2,x3 ( ) D g x1,x2,x3 , ( ) = k 0 k! ( ) = − ∂ Q where D ∂ , is left-regular in an open neighborhood of 0 U, and f 0,x1,x2,x3 = ∂x0 { } ( ) = g x1,x2,x3 . ( ) − × The following lemma will be useful in demonstrating the convergence of f in an open neighborhood of U.

Lemma 7.2.8. Let g x1,x2,x3 be a Cℓ1,1-valued function whose components are real- ( ) 3 analytic functions on U R . Then on a compact set K, there are constants cK and λK ⊆ such that k k k D g x1,x2,x3 3 cK k! λ . T ( )T ≤ ( ) K Proof of the Lemma. Also from [5], it is clear that on a compact set K, there are constants cK and λK such that k ∂ k g x1,x2,x3 c k λ . k1 k2 k3 K ! K W∂x1 ∂x2 ∂x3 ( )W ≤ ( ) Notice that when k is even, Dk is a scalar operator. So suppose k is even. Then by the trinomial theorem, k! ∂k Dk . k1 k2 k3 = k1 k2 k3 k k1 ! k2 ! k3 ! ∂x1 ∂x2 ∂x3 + + = ( ) ( ) ( ) Q 105 Then,

k k k! ∂ D g x1,x2,x3 g x1,x2,x3 k1 k2 k3 T ( )T ≤ k1 k2 k3 k k1 ! k2 ! k3 ! W∂x1 ∂x2 ∂x3 ( )W + + = ( ) ( ) ( ) Q k! cK k! λK ≤ ( ) k1 k2 k3 k k1 ! k2 ! k3 ! + + = ( ) ( ) ( ) k k 3 cK k! λ . Q = ( ) K

Now suppose k is odd. Then we have

k 1 ! ∂k Dk i ( ) k1 1 k2 k3 = k1 k2 k3 k1 ! k2 ! k3 ! Œ ∂x1 + ∂x2 ∂x3 +k +1 = ( ) ( − ) ( ) Q− ∂k ∂k j ij . k1 k2 1 k3 k1 k2 k3 1 ∂x1 ∂x2 + ∂x3 ∂x1 ∂x2 ∂x3 + ‘ − − This means that

k k k 1 ! ∂ g x1,x2,x3 D g x1,x2,x3 i ( ) k1( 1 k2 k)3 T ( )T ≤ k1 k2 k3 k1 ! k2 ! k3 ! W ∂x1 + ∂x2 ∂x3 +k +1 = ( ) ( − ) ( ) Q− ∂kg x1,x2,x3 ∂kg x1,x2,x3 j ij k1( k2 1 k)3 k1( k2 k3 )1 ∂x1 ∂x2 + ∂x3 ∂x1 ∂x2 ∂x3 + W − k 1 ! − ∂kg x1,x2,x3 ( ) k1( 1 k2 k)3 ≤ k1 k2 k3 k1 ! k2 ! k3 ! ŒW∂x1 + ∂x2 ∂x3 W +k +1 = ( ) ( − ) ( ) Q− ∂kg x1,x2,x3 ∂kg x1,x2,x3 k1( k2 1 k)3 k1( k2 k3 )1 W∂x1 ∂x2 + ∂x3 W W∂x1 ∂x2 ∂x3 + W‘

+ k 1 ! + k ( ) 3cK k! λK = k1 k2 k3 k1 ! k2 ! k3 ! ‰ ( ) Ž +k +1 = ( ) ( − ) ( ) − k Q k 3 cK k! λ , = ( ) K as required.

Proof of the Theorem. The above lemma gives us that on a compact set K U there are ⊂ constants cK and λK , depending on K, such that

k x0 k k k ( ) D g x1,x2,x3 cK k! 3λK x0. W k! ( )W ≤ ( ) ( ) −

106 Thus, f converges uniformly on

1 1 , K˚ . K U 3λK 3λK  ⊆ ‹−  × The essential calculation is

k ∞ x0 k ∂f ∂ ( ) D g x1,x2,x3 = k 0 Œ k! ( )‘ = − k k Q∞ x0 k ∞ x0 k ∂ ( ) D g x1,x2,x3 ( ) ∂ D g x1,x2,x3 = k 0 Œ k! ‘ ( ) k 0 k! ‰ ( )Ž = = − k 1 − k Q∞ x0 − k +∞Q x0 k 1 ( ) D g x1,x2,x3 ( ) D + g x1,x2,x3 = k 1 k 1 ! ( ) k 0 k! ( ) = (− ) = − 0−,Q + Q = − as required.

Remark 7.2.9. We may think of the above extension as a solution to the boundary value problem: ⎧ ∂f x0,x1,x2,x3 0 ⎪ = . f 0,x1,x2,x3 g x1,x2,x3 ( =) ⎨ Example 7.2.10. Consider the⎪ homogeneous polynomial of degree 2 ⎩⎪ ( ) ( )

2 2 2 g x1,x2,x3 x1 x2 x3 x1x2 x1x3 x2x3. = ( ) + + + + + Now,

Dg x1,x2,x3 2x1 x2 x3 i 2x2 x1 x3 j 2x3 x2 x1 ij = 2 D g( x1,x2,x)3 ( 6 + + ) − ( + + ) − ( + + ) = 3 D x(1,x2,x3 ) 0.− = ( ) Thus,

2 2 2 2 f Z 3x0 x1 x2 x3 x1x2 x1x3 x2x3 = ( ) ‰−x0 2+x1 +x2 +x3 i+ x0 2+x2 x1+ x3 Žj x0 2x3 x1 x2 ij.

− ( + + ) − ( + + ) − ( + + ) is the left- regular function obtained by the above theorem.

107 Remark 7.2.11. In both of these formulas, a polynomial g will be transformed to a Clifford valued function where every component is a polynomial. This is the case because polynomials have partial derivatives of 0 after a certain order. That is, Dkg and ∆kg will be uniformly 0 for all k M for some finite M. > 7.3 Linear Fractional Transformations

7.3.1 M¨obius Transformations of Cℓ1,1

The Clifford algebra Cℓp,q is, in many ways, a higher dimensional analogue of the complex numbers. In fact, much of the literature examines the analogous holomorphic function theory. Therefore, it is reasonable to study the analogous theory of linear fractional transformations

1 1 Z AZ B CZ D − and Z AZ B CZ D − ,

↦ ( + ) ( + ) ↦ ‰ + Ž ‰ + Ž defined on the Clifford algebra Cℓp,q and with the parameters A, B, C, D Cℓp,q. This can ∈ be seen in the case of the quaternions in [26]. To close this chapter, we wish to show that this may be done for the split-quaternions, as first suggested in our joint work [45]. It should be noted that we have not seen this carefully done in the literature for this precise context. But first, we must understand how the Vahlen construction from Section 4.2.2 (and in [11, 12]) for the M¨obius transformations of R2 can be reformulated as the more familiar complex meromorphic functions.

7.3.2 Low Dimensional Example: The Complex Plane

Recall from Section 4.2.2 that the domain of the transformations are vectors in the

Clifford algebra Cℓ0,2, which are of the form

x xe1 ye2. = Algebraically, these are quite different from C, as+ the product of vectors need not be a vector. This indicates that there is a certain complication in trying to view vectors and complex numbers as one in the same.

108 Rather, the more appropriate course of action is to take an arbitrary M¨obius transforma- 2 tion, in the sense of Section 4.2.2, acting on a vector x xe1 ye2 in R and find a complex = M¨obius transformation of the variable z x iy. Of course from the above we know that any = + R2 M¨obius transformation on is a composition+ of translations, dilations, orthogonal trans- formations, and special transformations. Thus it will suffice to find corresponding complex M¨obius transformations for each of these. We would like to note that we have not seen the equality of these cases carefully written out in the literature. The case for translations and dilations is completely trivial and hardly worth mentioning. As for orthogonal transformations, there is some work to be done. The following will prove useful in this endeavor.

Lemma 7.3.1. Every α Γ0,2 is either a vector or of the form α a be1e2. ∈ = In particular, a product of an odd number of non-zero vectors is a+ vector and a product of an even number of non-zero vectors is of the latter form. This is a simple calculation to check. 1 Now, orthogonal transformations are of the form x αx α′ − . If α ae1 be2, then = 1 1 ↦ ( ) + αx α′ − ae1 be2 xe1 ye2 ae1 be2 = a2 b2 b2 a2 x 2aby a2 b2 y 2abx ( ) ( + )( e1 + )( + ) e2. = + a2 b2 a2 b2 2 2 [( − ) − ] [( − ) − ] b a 2ab + 1 If we write A and B + , then x αx α′ − may+ be realized as z A Bi z. = a2 b2 = a2 b2 ( − ) − However, if α a be1e2, then ↦ ( ) ↦ ( + ) =+ + + 1 1 αx α′ − a be1e2 xe1 ye2 a be1e2 = a2 b2 a2 b2 x 2aby a2 b2 y 2abx ( ) ( + )( e1 + )( − ) e2. = + a2 b2 a2 b2 2 2 [( − ) − ] [( − ) + ] a b 2ab + 1 If we write C and D + , then x αx α+′ − may be realized as z = a2 b2 = a2 b2 ( − ) C Di z. ↦ ( ) ↦ + + R2 ( +For the) inversion, we see that a vector in inverts as follows

1 xe1 ye2 xe1 ye2 − , = x2 y2 −( + ) ( + ) 109 + which corresponds to the complex map

1 z z − .

Lastly, we examine the case of the special↦ −( transformation.) We saw that in Section 4.2.1, this kind of function has the following form

x x v ϕ x . = 1 2 x, v x v − ⟨ ⟩ 2 ( ) So in R we have − ⟨ ⟩ + ⟨ ⟩ ⟨ ⟩ 2 2 2 2 x x y v1 e1 y x y v2 e2 φ x,y 2 2 2 2 . = 1 2 xv1 yv2 x y v1 v2 ( − ( + ) ) + ( − ( + ) ) ( ) It is easy to check that this map− ( corresponds+ ) + ( to+ the)( following+ ) function of a complex variable:

z zzυ z z → , 1 υz 1 υz = υz 1 − where υ v1 v2i. z = ( − ) ( − ) − + R1,1 A similar+ relationship exists between the M¨obius transformations of and the M¨obius transformations of the split-complex plane, as we described in the previous chapter.

7.3.3 M¨obius Transformations of Cℓ1,1

2,2 We have shown that one can identify Cℓ1,1 with R . Now we shall show that the M¨obius transformations may be realized as linear fractional transformations with coefficients in Cℓ1,1, which is the truest analogue of the complex case. Again we want to make clear that the next four lemmas (and the resulting corollary) are original ideas to the best of our knowledge. As above, translations and dilations can be trivially shown to be expressed in this manner. And again, the orthogonal maps, inversion, and special transformations require light work.

Lemma 7.3.2. Let A a bi cj dij be an invertible split quaternion. Then the affine = 1 map Z ↦ AZ A − is given+ by + +

f1 Z f2 Z i f3 Z j f4 Z ij ( ) , a2 b2 c2 d2 ( ) + ( ) + ( ) + ( ) + 110− − where

2 2 2 2 f1 a b c d x0 2abx1 2acx2 2adx3 = 2 2 2 2 f2 (b −a + c + d )x1 − 2abx0 + 2bcx2 +2bdx3 = 2 2 2 2 f3 (a − b − c − d )x2 + 2acx0 + 2bcx1 + 2cdx3 = 2 2 2 2 f4 (a + b + c − d )x3 + 2adx0 + 2bdx1+ 2cdx2. = ( + − + ) + − + Proof. Recall that multiplying z by A on the left yields

ax0 bx1 cx2 dx3 ax1 bx0 cx3 dx2 i

( − ax2+ bx3+ cx)0 + (dx1 j+ ax−3 bx+2 cx)1 dx0 ij.

1 A+ ( − + + ) + ( + − + ) 1 Now, A − . A similar calculation shows that multiplying AZ by A − = a2 b2 c2 d2 yields( the) desired result. ( ) + − − 2 2 Lemma 7.3.3. Let a ae1 be2 ce3 de4 R , be invertible and let x x0e1 x1e2 x2e3 = ∈ = 2 2 x3e4 R , . Then the orthogonal map ∈ + + + + + +

1 x ax a′ −

↦ ( ) may be realized as the split quaternionic map

1 Z AZ A − ,

↦ − ( ) where A a bi cj dij. = + + + Proof. A calculation in Cℓ2,2 shows that

2 2 2 2 1 a b c d x0 2abx1 2acx2 2adx3 ax a′ − e1 = a2 b2 c2 d2 2 2 2 2 −(a −b +c +d x)1 −2abx0 +2bcx2 +2bdx3 ( ) e2 a2 +b2 −c2 −d2 2 2 2 2 (a − b − c − d )x2 − 2acx0 + 2bcx1 + 2cdx3 + e3 a2 + b2 − c2 − d2 2 2 2 2 (a + b + c − d )x3 − 2adx0 − 2bdx1+ 2cdx2 + e4. a2 + b2 − c2 − d2 ( + − + ) − − + + 111+ − − 1 From Lemma 7.3.2, we know that G Z AZ A − must have the form

g1 Z g∶ 2 Z↦i − g3(Z)j g4 Z ij , a2 b2 c2 d2 ( ) + ( ) + ( ) + ( ) where + − −

2 2 2 2 g1 a b c d x0 2abx1 2acx2 2adx3 = 2 2 2 2 g2 −(a −b +c +d x)1 −2abx0 +2bcx2 +2bdx3 = 2 2 2 2 g3 (a − b − c − d )x2 − 2acx0 + 2bcx1 + 2cdx3 = 2 2 2 2 g4 (a + b + c − d )x3 − 2adx0 − 2bdx1+ 2cdx2. = ( + − + ) − − + Hence the desired result is obtained.

Example 7.3.4. The above construction gives us that split quaternionic conjugation may be 2 2 realized as the orthogonal map on R , :

1 x e1x e1′ − .

↦ ( ) The lemma is the first step of the proof of following corollary.

Corollary 7.3.5. Let α anan 1 a2a1 Γ2,2, where ak ake1 bke2 cke3 dke4 Then the = − ⋯ ∈ = + + + orthogonal map 1 x αx α′ − may be realized as the split quaternionic map↦ ( )

1 1 1 1 1 Z AnAn− 1An 2 A2− A1 Z A1A−2 A3 An− 1An − , − − − ⋯ ⋯ − ↦ ( ) ( ) if n is odd, or 1 1 1 1 1 1 Z AnAn− 1An 2 A2A1− Z A−1 A2A−3 An− 1An − , − − ⋯ ⋯ − if n is even, where A↦k (ak bki ck dkij. ) ( ) = + + + Proof. First suppose n 2, then =

1 1 1 1 1 a2a1 x a2a1 ′ − a2a1 x a1′ − a2 − a2 a1x a1′ − a2 − . = = ( ) (( ) ) ( ) ( 112) ( ) [ ( ) ]( ) Thus the associated map on Cℓ1, 1 is

1 1 1 1 A2 A1Z A1 − A2 − A2 A1 − ZA1 A2 − = 1 1  ( ) ( ) A(2A1−) ZA1 (A2 )− = 1 1 1 (A2A1− )Z A(2 A1)− − = 1 1 1 (A2A1− Z)(A2A−1 −), = ( ) ( ) where the equalities follow from properties of conjugation (see Definition 2.2.4) and inverses. The proof of the general case follows by induction.

As in the complex case, we have the following.

Lemma 7.3.6. The inversion map on R2,2

1 x x−

↦ may be realized as the split quaternionic map

1 Z Z − .

↦ −( ) Lastly, a special transformation can be shown to have the alternative form below.

2,2 Lemma 7.3.7. Let v v1e1 v2e2 v3e3 v4e4 R then the special transformation ϕ = ∈ R2,2 R2,2 → defined by + + + ∶ x x v ϕ x = 1 2 x, v x v − ⟨ ⟩ may be realized as the split quaternionic( ) map − ⟨ ⟩ + ⟨ ⟩ ⟨ ⟩

1 Z z→ Z ΥZ 1 − , ‰− + Ž where Υ v1 v2i v3j v4ij Cℓ1,1. = ∈ + + +

113 Proof. First we have

1 Z ΥZ 1 Z ΥZ 1 − = 1 Υ‰−Z 1+ Վ Z ‰− + Ž Z 1 ZΥ ‰ − Ž ‰ − Ž = 1 ΥZ ‰ Υ−Z ՎZZΥ Z Z Υ − ( + + , ) = 1 2 Z, Υ Υ Z − ⟨ ⟩ where the second equality follows from Lemma− ⟨ 2.2.5.⟩ +This ⟨ ⟩ is ⟨ p⟩recisely the desired form.

Now it is sufficiently clear from the preceding and Theorem 4.2.5 that a conformal map-

ping on Cℓ1,1 is some linear fractional transformation. Additionally, these functions corre-

spond to matrices which act on Cℓ1,1 in the obvious way: ⎤ A B ⎥ 1 1 ⎥ (AZ B) (CZ D)− or ‰AZ BŽ ‰CZ DŽ− . ⎡ CD ⎥ ≅ ⎢ ⎥ ⎢⎣ ⎦ ⎢ + + + + It is easy to check⎢ that corresponds to function composition. Thus, the matrix group generated by these will be conformal. Of course since the matrix ⎡ ⎤ ⎢ 1 0 ⎥ 1 ⎢ ⎥ Z − Z, ⎢ ⎥ ( 1) ⎢ 0 1 ⎥ ≅ = ⎣ − ⎦ − − we consider all such matrices modulo −I, and denote the quotient group by . M The next step is to determine certain properties of . From Theorem 4.2.6, we know that − M M +(1, 1) is isomorphic to SO+(3, 3), the connected component of SO(3, 3) which contains R the identity. It is also know that SO+(3, 3) PSL(4, ). Hence the group of M¨obius ≅ transformations is a two to one cover of PSL(4, R). To see this we define the algebra homomorphism as follows:

⎡ a0 a3 a1 a2 b0 b3 b1 b2 ⎤ ⎢ ⎥ ⎢ ⎥ ⎡ ⎤ ⎢ ⎥ ⎢ A B ⎥ ⎢ a1 a2 a0 a3 b1 b2 b0 b3 ⎥ ⎢ ⎥ ⎢ + − + + − + ⎥ . ⎢ ⎥ ⎢ ⎥ ⎢ CD ⎥ ⎢ c0 c3 c1 c2 d0 d3 d1 d2 ⎥ ⎣ ⎦ ⎢ + − + − ⎥ ⎢ ⎥ ↦ ⎢ c1 c2 c0 c3 d1 d2 d0 d3 ⎥ ⎣ + − + + − + ⎦ Notice that the right-hand matrix may+ be realized− as a+ block ma−trix obtained by the 2 2 real matrices (see Lemma 2.3.20) of the entries in left-hand matrix. × 114 Proposition 7.3.8. The matrix groups are isomorphic:

PSL 4, R . M ≅ ( ) Proof. Let ϕ → PSL 4, R be the map defined above. Breaking the image matrix into M four 2 2 blocks∶ as suggested( above) makes it easy to see that ϕ preserves multiplication. It is easy to check that ker ϕ is precisely the identity in . The two-sided inverse defined × M in Lemma 2.3.20 induces a two-sided inverse of ϕ. Thus we have an isomorphism.

Remark 7.3.9. This isomorphism along with the inverse of the isomorphism in Theorem

4.2.6 gives an isomorphism between SO+ 3, 3 and PSL 4, R Indeed, the proof that SO 3, 3 R and PSL 4, are locally isomorphic can( be) found in [28].( ) ( ) ( ) Example 7.3.10. The realization of Z ↦ Z B and Z ↦ Z B as a matrix in PSL 4, R is + + ( ) b0 b3 b1 b2 ⎤ 1 0 ⎥ ⎥ ⎡ 0 1 b1 b2 b0 b3 ⎥ ⎢ + − + ⎥ ⎢ ⎥ , ⎢ 00 1 0 ⎥ ⎢ + − ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 00 0 1 ⎥ ⎢⎣ ⎦ 1⎢ 1 while the matrix for Z ↦ Z− ⎢and Z ↦ (Z)− is ⎡ ⎤ ⎢ 0010 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 0001 ⎥ ⎢ ⎥ . ⎢ ⎥ ⎢ 1000 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ 0100 ⎦

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120 BIOGRAPHICAL SKETCH

Education

❼ Ph.D. Mathematics, The Florida State University, May 2015.

❼ M.S. Mathematics, The Florida State University, 2012.

❼ B.A. Mathematics, New College of Florida, 2010.

Experience

❼ The Florida State University, Department of Mathematics, Teaching Assistant, August 2010–present.

❼ The Florida State University, Department of Mathematics, GAANN Fellow, January 2012–December 2013.

Teaching at The Florida State University

Instructor of Record with Full Classroom Responsibilities

❼ Discrete Mathematics II Fall 2014

❼ Calculus with Analytic Geometry III Spring 2014

❼ Calculus with Analytic Geometry I Spring 2013

❼ Precalculus Fall 2011, Spring 2012, Spring 2015.

Recitation Instructor

❼ Precalculus Fall 2011, Summer 2014

❼ Trigonometry Fall 2013

❼ College Algebra Fall 2013

❼ Mathematics for Liberal Arts Fall 2010, Spring 2011.

❼ Practical Finite Mathematics Fall 2010, Spring 2011.

121 Research Interests

I work in the field of Clifford analysis (also known as hypercomplex analysis), which is concerned with generalizing the major results of complex analysis. More specifically, I in- vestigated the theory of functions of a split-complex variable and the conformal mappings of their compactification. This led to some analogues of the M¨obius geometry on the Riemann sphere. Currently, I am trying to see if there is a parallel theory for the split-quaternions.

Research Papers

Published Papers

❼ Projective Compactification of R1,1 and its M¨obiusGeometry (with Craig A. Nolder) in Complex Analysis and Operator Theory, Volume 9 Issue 2, pp. 329-354, 2015.

❼ Clifford Algebras with Induced (Semi)-Riemannian Structures and Their Compactifi- cation (with Craig A. Nolder) in Vladimir V. Mityushev and Michael Ruzhansky, ed- itors Current Trends in Analysis and Its Applications: Proceedings of the 9th ISAAC Congress, Krak´ow, Birkhauser Boston, 2015

Papers in Preparation

❼ Notions of Regularity for Functions of a Split-Quaternion Variable.

❼ The M¨obiusGroup of the Extended Multicomplex Numbers.

❼ Linear Fractional Transformations of the split-quaternions.

Honors, Awards, & Fellowships

❼ Selected Attendee, Summer Graduate School in Geometry and Analysis, Mathematical Sciences Research Institute, Berkley, CA, Summer 2014.

❼ Distinguished Teaching Assistant Award, Florida State University Department of Math- ematics, April 2014

❼ Senate Ethics Award, Florida State University Student Government Association, March 2014

122 ❼ Certificate, Program for Instructional Excellence, The Florida State University, August 2012.

❼ Pi Mu Epsilon, The Florida State University, April 2012–present.

❼ Travel Grant for AMS Sectional Meeting, The American Mathematical Society, March 2012.

❼ GAANN Fellowship (funded by the U.S. Department of Education), The Florida State University, January 2012–December 2013.

Talks and Presentations

Invited Talks

❼ Projective Compactification of R1,1 and its M¨obiusGeometry, Math Seminar, Chapman University, Orange, CA, November 7, 2013.

❼ The Split-Complex Numbers and their Holomorphic Functions, Natural Science Semi- nar, New College of Florida, Sarasota, FL, September 20, 2013.

Contributed Conference Presentations

❼ Projective Compactification of R1,1 and its M¨obiusGeometry, AMS Session on Complex and Geometric Analysis, AMS-MAA Joint Meetings, Baltimore, MD, January 17, 2014

Seminar Presentations at The Florida State University

❼ M¨obiusTransformations of Rp,q, April 3, 2014 at the Complex Analysis Seminar.

❼ M¨obiusTransformations of Two Hypercomplex Spaces, September 11 and 18, 2013 at the Complex Analysis Seminar.

❼ Conformal Transformations of the Euclidean and Minkowski Planes, February 21, 2013 at the Complex Analysis Seminar.

❼ Complex M¨obiusTransformations II, September 18, 2012 at the Complex Analysis Seminar.

❼ Complex M¨obiusTransformations I, September 11, 2012 at the Complex Analysis Sem- inar.

123 ❼ Clifford Algebras and Clifford Analysis, September 7, 2012 at the Graduate Student Seminar.

❼ Hyperbolic Geometry, February 2012 at the Complex Analysis Seminar.

Service

Department Service

❼ Student Assistant to the organizers of Clifford Analysis and Related Topics: A Con- ference in Honor of Paul A.M. Dirac, December 2014.

❼ Organizer, Graduate Student Seminar, The Florida State University, Fall 2013.

❼ Volunteer, Math Fun Day at FSU, October 12, 2013.

❼ Department Representative for prospective graduate students, Spring 2012

University Service

❼ Senator, Student Government Association, The Florida State University, 2013–2014.

– Chair of the Internal Affairs Committee April 2014- August 2014.

❼ Member, Student Health Insurance Committee, The Florida State University, 2011– 2014.

❼ Graduate Student Representative, Bookstore Advisory Council, The Florida State Uni- versity, 2014.

Service to the Profession

❼ Chief Negotiator for United Faculty of Florida-Florida State University-Graduate As- sistants United, 2013–2015.

❼ Co-president of United Faculty of Florida-Florida State University-Graduate Assistants United, 2011–2013.

124