UNIVERSIDADE FEDERAL DE MINAS GERAIS INSTITUTO DE CIENCIAS^ EXATAS DEPARTAMENTO DE MATEMATICA´ OCTONIONIC PLANES In´acio Augusto Rabelo Pinto Belo Horizonte - Minas Gerais - Brasil 2021 UNIVERSIDADE FEDERAL DE MINAS GERAIS INSTITUTO DE CIENCIAS^ EXATAS DEPARTAMENTO DE MATEMATICA´ OCTONIONIC PLANES Vers~aoFinal Inacio´ Augusto Rabelo Pinto* Orientador: Nikolai Goussevskii Disserta¸c~ao apresentada ao Programa de P´os-Gradua¸c~aoem Matem´aticada Univer- sidade Federal de Minas Gerais, como requi- sito parcial `aobten¸c~aodo t´ıtulode Mestre em Matem´atica. Belo Horizonte - Minas Gerais - Brasil 2021 *Durante os estudos de Mestrado o autor recebeu suporte financeiro do CNPq. © 2021, Inácio Augusto Rabelo Pinto. Todos os direitos reservados Pinto, Inácio Augusto Rabelo P659o Octonionic planes [manuscrito] / Inácio Augusto Rabelo Pinto. ­ 2021. ix, 90 f. il. Orientador: Nikolai Alexandrovitch Goussevskii. Dissertação (mestrado) ­ Universidade Federal de Minas Gerais, Instituto de Ciências Exatas, Departamento de Matemática. Referências: f.87­90 . 1. Matemática – Teses. 2. Octônios ­ Teses. 3.Espaços hiperbólicos – Teses. 4. Espaços simétricos – Teses. 5. Planos projetivos ­ Teses. 5. Geometria projetiva – Teses. I. Goussevskii, Nikolai Alexandrovitch. II. Universidade Federal de Minas Gerais; Instituto de Ciências Exatas, Departamento de Matemática. III.Título. CDU 51(043) Ficha catalográfica elaborada pela bibliotecária Belkiz Inez Rezende Costa CRB 6ª Região nº 1510 Universidade Federal de Minas Gerais Departamento de Matemática Programa de Pós-Graduação em Matemática FOLHA DE APROVAÇÃO Octonionic Planes INÁCIO AUGUSTO RABELO PINTO Dissertação defendida e aprovada pela banca examinadora constituída pelos Senhores: ___________________________________________ Prof. Nikolai Alexandrovitch Goussevskii UFMG ___________________________________________ Prof. Heleno da Silva Cunha UFMG ___________________________________________ Prof. Maurício Barros Corrêa Júnior UFMG Belo Horizonte, 09 de março de 2021. Av. Antônio Carlos, 6627 – Campus Pampulha - Caixa Postal: 702 CEP-31270-901 - Belo Horizonte – Minas Gerais - Fone (31) 3409-5963 e-mail: [email protected] - home page: http://www.mat.ufmg.br/pgmat (...) e nisso eu sou - como a Poesia ´e e eu sei que meu trabalho ´e´arduo e existe para si mesmo como no cemit´erioda cidade a ins^oniado vigia Guen´adiAigui Trecho de Sil^encio (1954-1956) - tradu¸c~aode Boris Schnaiderman. Abstract The octonionic projective and hyperbolic planes are studied. We present the construc- tions based on Jordan Algebras, the Riemannian metrics, the structure of homogeneous spaces, and the classification of totally geodesic submanifolds. We also develop a similar description of the real, complex, and quaternionic projective and hyperbolic spaces. Keywords: Octonions, Octonionic projective plane, Octonionic hyperbolic plane, Projective spaces, Hyperbolic spaces, Riemannian symmetric spaces of rank one. vi Resumo Os planos projetivo e hiperb´olicooctoni^onicoss~aoestudados. Apresentamos as con- stru¸c~oesbaseadas em ´algebrasde Jordan, as m´etricas Riemannianas, a estrutura de espa¸coshomog^eneose a classifica¸c~aodas subvariedades totalmente geod´esicas. Desen- volvemos tamb´emuma descri¸c~aosimilar dos espa¸cosprojetivos e hiperb´olicosreais, com- plexos e quaterni^onicos. Palavras-chave: Oct^onios,Plano projetivo octoni^onico,Plano hiperb´olicooctoni^onico, Espa¸cosprojetivos, Espa¸cos hiperb´olicos,Espa¸cossim´etricosRiemannianos de posto um. vii Agradecimentos A` minha fam´ılia,em particular minha m~ae,pelo constante e irrestrito apoio; A` Dindi, pelo suporte, companheirismo e paci^enciainabal´aveis; Ao Nikolai, pela forma¸c~aoacad^emicae apresenta¸c~aodo tema; Aos irm~aosBruno e Toni; Ao Maur´ıcioe o Heleno, por comporem a banca; Aos demais colegas e amigos de departamento, pelas impag´aveis conversas de corre- dor. viii Contents Introduction1 Introdu¸c~ao2 Index of Notations3 0 Preliminaries4 0.1 Division Algebras................................4 0.1.1 Quaternions...............................6 0.1.2 Octonions................................8 0.1.3 Further Results............................. 11 0.2 Lie groups and Lie Algebras.......................... 11 0.2.1 Basic Theory.............................. 11 0.2.2 Unitary Groups............................. 12 0.2.3 Killing Form............................... 14 0.2.4 Classification of Simple Lie Algebras................. 16 0.3 Spin Groups................................... 17 0.3.1 Clifford Algebras............................ 17 0.3.2 Spin Groups............................... 18 0.3.3 The Group Spin(8)........................... 20 0.3.4 The Group Spin(9)........................... 20 0.4 Principal Bundles................................ 21 0.5 Symmetric Spaces................................ 22 0.5.1 Introduction............................... 22 0.5.2 Symmetric Pairs............................ 23 0.5.3 Orthogonal Symmetric Lie Algebras.................. 24 0.5.4 Lie Triple Systems and Totally Geodesic Submanifolds....... 24 0.5.5 Duality and Rank............................ 25 1 F-Projective Spaces 28 1.1 Introduction................................... 28 1.2 Collineation Groups............................... 31 1.3 Riemannian Metric............................... 33 1.4 The Projective Spaces as Symmetric Spaces................. 36 1.4.1 The Real Case............................. 36 1.4.2 The Complex Case........................... 36 1.4.3 The Quaternionic Case......................... 37 1.5 The Matrix Model............................... 37 1.5.1 The Real Case............................. 38 ix CONTENTS x 1.5.2 The Complex and Quaternionic Cases................. 38 1.6 Totally Geodesic Submanifolds......................... 39 2 The Octonionic Projective Plane 41 2.1 The Octonionic Projective Line........................ 41 2.2 The Jordan Algebra H3(O).......................... 43 2.3 The Octonionic Projective Plane........................ 47 2.4 Riemannian Metric............................... 50 2 2.5 OP as a Homogeneous Space......................... 53 2.6 Totally Geodesic Submanifolds......................... 57 2.7 Non-existence of Higher-Dimensional Spaces................. 59 3 F-Hyperbolic Spaces 62 3.1 Introduction................................... 62 3.2 Riemannian Metric............................... 63 3.3 The Hyperbolic Spaces as Symmetric Spaces................. 65 3.3.1 The Real Case............................. 66 3.3.2 The Complex Case........................... 67 3.3.3 The Quaternionic Case......................... 68 3.4 Totally Geodesic Submanifolds......................... 69 4 The Octonionic Hyperbolic Plane 71 4.1 The Jordan Algebra H1;2(O).......................... 71 4.2 The Octonionic Hyperbolic Plane....................... 72 4.3 Riemannian Metric............................... 76 4.4 2 as a Homogeneous Space.......................... 78 HO 4.5 Totally Geodesic Submanifolds......................... 80 A Automorphisms of H3(F) and H1;2(F) 82 A.1 Compact Case.................................. 82 A.2 Non-compact Case............................... 84 Bibliography 87 Introduction The real numbers, complex numbers, quaternions, and octonions form the four normed division algebras. While the quaternions are not commutative, octonions are not even associative. Despite this peculiarity, these numbers are related to exceptional structures founded by E. Cartan and W. Killing at the end of the 20th century and have a growing interest in Physics. In this work, we study the octonionic projective and hyperbolic planes. They are also called Cayley projective and hyperbolic planes. In addition to containing all the geometry of the real, complex, and quaternionic planes, these spaces also represent the relations mentioned above. However, the construction using projectivization of a vector space is not applicable because of non-associativity, so new tools are introduced to play the role of a vector structure and describe the octonionic spaces. Chapter 0 introduces the preliminaries to the theme. It has written to be a reference, then several proofs have been omitted. The content of the following chapters is primarily: Construction of the spaces; Riemannian Metric; Structure of homogeneous and symmetric spaces; Classification and stabilizers of totally geodesic submanifolds; We highlight the different topics of each chapter. The matrix model in Section 1.5 is the motivation to the main model of the octonionic projective plane. Chapter 2 starts with the configuration Jordan algebra and the exceptional group F4. In particular, Section 2.7 intends to detail a complete argumentation about the non-existence of higher-dimensional projective spaces over octonions. Chapter 3 deals with classical hyperbolic spaces. It contains the proofs of duality and rank. In Chapter 4, the configuration algebra of the octonionic hyperbolic plane is similar to that of the projective case, so several results have the same proof. We spend some work to describe and give characterizations of this space. Further, the original proofs of classification of totally geodesic submanifolds are given in the last two chapters and applied to the compact cases by duality. We also add an appendix in which we obtain the automorphism groups of Jordan subalgebras related to the classical planes. Although the constructions are not simple due to the non-associativity, they follow the same lines or motivations of the classical spaces. Finally, we note that the spaces studied here,