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An Introduction to Orthogonal Geometry

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An Introduction to Orthogonal Geometry An Introduction to Orthogonal Geometry Emil Eriksson Bachelor’s Thesis, 15 ECTS Bachelor’s degree in Mathematics, 180 ECTS Spring 2021 Department of Mathematics and Mathematical Statistics Abstract In this thesis, we use algebraic methods to study geometry. We give an introduction to orthogonal geometry, and prove a connection to the Clifford algebra. Sammanfattning I denna uppsats använder vi algebraiska metoder för att studera geometri. Vi ger en introduktion till ortogonal geometri och visar en koppling till Cliffordalgebran. Acknowledgments First, I wish to thank my supervisor and mathematical mentor Per Åhag, who gen- erously shared both his time and wisdom to support me through the development of this thesis. I would also like to thank my examiner Olow Sande, and peer reviewer Stig Lundgren for reading, and providing invaluable feedback. Finally, I wish to thank my partner Vania, for the proofreading of this text, and for her endless support. Contents Acknowledgments 1. Introduction 3 2. Vector Spaces 5 2.1. Homomorphisms and Matrices 8 3. Orthogonal Geometry 13 3.1. Metric Structures 13 3.2. Quadratic Forms 16 3.3. Orthogonal Geometry 17 4. The Clifford Algebra 27 4.1. The Lipschitz Group 32 5. References 35 1. Introduction In the 1820s, Nikolai Lobachevsky and János Bolyai started a revolution. Indepen- dently from each other, they had both discovered the existence of non-Euclidean ge- ometry. Following this, the first consistent non-Euclidean geometry was constructed by Eugenio Beltrami in 1868. Thereby, Beltrami had solved a problem that had remained unsolved for over 2000 years, providing definite proof that the fifth postulate of Euclid’s Elementa [8] is independent from the other four. Another crucial part in this revolution was played by Bernhard Riemann, who in- troduced spherical geometry as well as geometries of higher dimensions. As a result, it became clear that there are many different kinds of non-Euclidean geometries. In an attempt to bring some internal structure to these new geometries, Felix Klein published what would be known as the Erlanger Programm [13] in 1893. Therein, a method for classifying geometries using group theory was presented. In this method, each geometry is associated with a group of transformations on the whole space. Should the space have a metric structure, we can speak of the group of distance-preserving transformations called the isometry group. The structural relation- ships between these groups are what determine the relationships between the different geometries. We are going to use these ideas to study orthogonal geometry, which is a general geometry including both Euclidean and hyperbolic geometry. Our final goal is to show an important connection between orthogonal geometry and a certain associative algebra, called the Clifford algebra. This algebra was introduced in 1878 by William Clifford in his paper Applications of Grassmann’s Extensive Algebra [4]. In this paper, Clifford introduces what he calls a geometric algebra that manages to generalize the quaternions discovered by William Hamilton in 1843 [11]. As the title suggests, this algebra is built upon the idea of inner and outer products, introduced in Hermann Grassmann’s Ausdehnungslehre [10]. The Clifford algebra was essential to Élie Cartan’s discovery of the mathematical the- ory of spinors in 1913 [16]. The spin group, described mathematically as a subgroup of the Clifford algebra, shortly became fundamental in quantum mechanics. The the- ory of spinors in mathematical physics was developed by Wolfgang Pauli [14], and Paul Dirac [6], among others. In more recent years, the Clifford algebra, under its original name of geometric algebra, has been proposed as a unified language for mathematics and physics by David Hestenes and Garret Sobczyk [12], being influenced by Marcel Riesz [15]. These theories have also generated numerous applications in computer sci- ence [7]. The traditional applications of Clifford algebra in physics are based on its relationships with the isometry group of physical space. Since this space is usually described as having a Euclidean, or in certain cases hyperbolic geometry, and sometimes extended to higher dimensions, it is meaningful to study the relationships with the more general orthogonal geometry. In the present thesis we study vector spaces V over general fields F , where V has an orthogonal geometry. In this case, the isometry group of V is the orthogonal group O(V ). In the final section we construct the Clifford algebra, and our final result will be 3 a proof of Theorem 4.26, where we connect O(V ) to a subgroup of the Clifford algebra called the Lipschitz group, Γ(V ). We achieve this by showing that the factor group SΓ(V )/F ∗ is isomorphic to the special orthogonal group, SO(V ): Theorem 4.26. Let SΓ(V ) be all α ∈ C+(V ) that have an inverse, and for which α ◦ X ◦ α−1 ∈ V for all X ∈ V . Then SΓ(V )/F ∗ is isomorphic to SO(V ), where C+(V ) is an important subalgebra of the Clifford algebra C(V ), called the even Clifford algebra. The proof of this theorem, which is given as the conclusion of this thesis, relies heav- ily on the Cartan–Dieudonné Theorem (Theorem 3.44). It states that, in orthogonal geometry, all elements of O(V ) are products of symmetries. This was first proven for vector spaces over R or C by Élie Cartan in the beginning of the twentieth century, and a version of the theorem is included in his book The Theory of Spinors [17]. Jean Dieudonné would later provide a generalized proof for vector spaces over an arbitrary field [5]. In this thesis, however, we will follow the proof given by Emil Artin in Geometric Algebra [1]. In order to arrive at Theorem 4.26, we begin by introducing vector spaces and showing some important properties needed to study orthogonal geometry in Section 2. We then move on to Section 3, which is dedicated to the study of orthogonal geometry, and ends with a proof of the Cartan–Dieudonné Theorem. Finally, in Section 4, we introduce the Clifford algebra and prove Theorem 4.26. This thesis relies heavily on the excellent book Geometric Algebra [1] by famous math- ematician Emil Artin. A few definitions have been updated to modern standard using Juliusz Brzezinski’s Linjär och multilinjär algebra [3], and John B. Fraleigh’s A First Course In Abstract Algebra [9]. The last theorem has been rephrased in terms of the Lipschitz group using [3]. The reader of this thesis should be familiar with abstract and linear algebra at the undergraduate level. Should there be any uncertainty regarding the meaning of funda- mental concepts, the reader may consult [9] or any other introductory textbook in the subject. 4 2. Vector Spaces We begin by introducing vector spaces over fields and showing some important prop- erties that we will need later on in our study of orthogonal geometry. Definition 2.1. A right vector space V over a field F is an additive, abelian group together with a composition Aa of an element A ∈ V and a ∈ F such that Aa ∈ V , and such that the following rules hold: (1) (A + B)a = Aa + Ba, (2) A(a + b) = Aa + Ab, (3) (Aa)b = A(ab), (4) A · 1 = A, where A, B ∈ V , a, b ∈ F , and 1 is the multiplicative identity element of F .A left vector space can be defined in a similar way by using the composition aA. We should clarify some of the symbols we are going to use right away. Definition 2.2. If S is a set of vectors in V , we denote the space spanned by S by hSi. Definition 2.3. By the dimension of V we mean the number of elements n in a basis of V , and we write dim V = n. A detailed description of the span and basis of a vector space can be found in Section 30 of [9]. If we have a subspace U of V , we can consider the factor space V/U. Its elements are the cosets A + U, meaning that for each coset A + U, each vector A ∈ V is added to all vectors Ui of U, or in symbols: A + U = {A + Ui | Ui ∈ U}. Theorem 2.4. Let U be a subspace of V . The factor space V/U is also a vector space. Proof. Since V is abelian we can consider the additive factor group V/U whose elements are the cosets A + U, addition is explained by (A1 + U) + (A2 + U) = (A1 + A2) + U and is obviously abelian. We now define the composition of an element A + U of V/U and an element a ∈ F by: (A + U) · a = Aa + U This composition is well defined because if A + U = B + U, then A − B ∈ U, hence (A − B)a ∈ U, which shows Aa + U = Ba + U. The rules of Definition 2.1 are easily checked. We should move towards a definition of homomorphisms of vector spaces. From group theory, we recall that the natural homomorphism and isomorphism of groups have a special meaning. They are characterized by the following important theorem. Theorem 2.5 (The Fundamental Homomorphism Theorem). Let G, and G0 be groups, and let g be an element of G. Let φ : G → G0 be a group homomorphism with kernel K. φ(G) is a group, and the map µ : G/K → φ(G) given by µ(gK) = φ(g) is an 5 isomorphism. If γ : G → G/K is the homomorphism given by γ(g) = gK, then φ(g) = µγ(g) for each g ∈ G. Proof. See [9, p. 140]. The isomorphism µ and the homomorphism γ are referred to as the natural isomor- phism and homomorphism, respectively.
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