Lie Pseudogroups À La Cartan from a Modern Perspective Thesis Committee: Prof

Lie Pseudogroups à la Cartan from a Modern Perspective Thesis committee: Prof. dr. Erik P. van den Ban, Utrecht University Prof. dr. Christian Blohmann, Max Planck Institute for Mathematics Prof. dr. Rui Loja Fernandes, University of Illinois at Urbana-Champaign Prof. dr. Niky Kamran, McGill University Prof. dr. Shlomo Sternberg, Harvard University ISBN: 978-90-393-6611-0 Gedrukt door CPI Koninklijke Wöhrmann (CPI-Thesis) Copyright c 2016 by Ori Yudilevich Cover design c 2016 by Gali Yudilevich Lie Pseudogroups à la Cartan from a Modern Perspective Lie pseudogroepen à la Cartan vanuit een modern perspectief (met een samenvatting in het Nederlands) Proefschrift ter verkrijging van de graad van doctor aan de Universiteit Utrecht op gezag van de rector magnificus, prof. dr. G. J. van der Zwaan, ingevolge het besluit van het college van promoties in het openbaar te verdedigen op woensdag 14 september 2016 des middags te 12.45 uur door Ori Yudilevich geboren op 12 juli 1980 te Haifa, Israël Promotor: Prof. dr. M. N. Crainic This thesis was accomplished with financial support from the ERC Starting Grant number 279729 of the European Research Council. לזכרM של סבתי האהובה זלדה (2012-1928) וסבי האהוב יהושע (2014-1917) Table of Contents Introduction 1 1 Partial Differential Equations 15 1.1 Affine Bundles . 16 1.2 Jet Bundles . 18 1.3 The Cartan Form on Jet Bundles . 23 1.4 PDEs . 28 1.5 Formal Integrability of PDEs and the Spencer Cohomology . 34 1.6 A Proof of Goldschmidt’s Formal Integrability Criterion . 39 1.7 PDEs of Finite Type . 41 1.8 Linear PDEs and the Spencer Operator . 42 1.9 The Abstract Approach to Linear/Non-Linear PDEs: Pfaffian Bundles . 48 2 Jet Groupoids and Jet Algebroids 53 2.1 Lie Groupoids and Lie Algebroids . 53 2.2 Jet Groupoids . 61 2.3 Jet Algebroids . 63 2.4 The Cartan Form on a Jet Groupoid . 65 2.5 The Spencer Operator on a Jet Algebroid . 69 2.6 Jet Groupoids and Jet Algebroids in Local Coordinates . 71 2.7 Pfaffian Groupoids and Pfaffian Algebroids . 74 3 Lie Pseudogroups 77 3.1 Pseudogroups . 77 3.2 Haefliger’s Approach to Pseudogroups . 78 3.3 Lie Pseudogroups . 79 3.4 The Pfaffian Groupoids of a Lie Pseudogroup Γ .............. 80 3.5 Towers of Jet Groupoids of Γ ........................ 87 3.6 Generalized Pseudogroups . 90 4 The First Fundamental Theorem 93 4.1 Cartan’s Formulation . 93 4.2 Cartan’s Notion of Equivalence . 95 4.3 The First Fundamental Theorem . 96 5 The Second Fundamental Theorem 103 5.1 Cartan’s Formulation . 103 5.2 The Modern Formulation of Structure Equations: pre-Cartan Algebroids and their Realizations . 112 5.3 The Second Fundamental Theorem . 121 5.4 Cartan’s Examples Revisited . 128 6 The Third Fundamental Theorem 135 6.1 Cartan’s Formulation . 135 6.2 Cartan Algebroids . 137 6.3 The Need of Cartan Algebroids for the Existence of Realizations . 140 6.4 The Realization Problem . 142 6.5 Examples of Cartan Algebroids . 144 6.6 An Alternative Approach to Cartan Algebroids: Cartan Pairs . 148 7 The Role of the Jacobi Identity in the Realization Problem 155 7.1 An Outline of our Approach to the Realization Problem . 156 7.2 The Lie Algebra Case . 158 7.3 The Case of Local Symplectic Realizations of Poisson Structures . 162 7.4 The Lie Algebroid Case . 165 7.5 Appendix: The Maurer-Cartan Equation on a Lie Algebroid . 174 8 The Systatic Space and Reduction 179 8.1 Cartan’s Formulation . 180 8.2 The Systatic Space . 182 8.3 The Systatic Action on Realizations . 184 8.4 Intermezzo: The Pfaffian Groupoid of a Realization . 187 8.5 The Systatic Action on the Pfaffian Groupoid . 189 8.6 The Reduction Theorem . 190 8.7 Examples of Reduction . 195 8.8 Appendix: Basic Forms . 198 9 Prolongations 203 9.1 Cartan’s Formulation . 203 9.2 Prolongation of a Realization . 204 9.3 Prolongation a Cartan Algebroid . 211 9.4 Higher Prolongations and Formal Integrability . 214 Bibliography 217 Index 223 A Summary for the Non-Mathematician 225 Een samenvatting voor de niet-wiskundige 227 Acknowledgments 230 Curriculum Vitae 231 Introduction In the years 1904-05, Élie Cartan published two pioneering papers ([5, 6], and presented more concisely in [7, 8]) in which he introduced a structure theory for Lie pseudogroups. In this thesis, we present a modern formulation of Cartan’s theory. This work is the result of our endeavor to read Cartan’s writings from a modern perspective and understand his ideas and constructions in a more conceptual, global and coordinate-free fashion. In contrast to a great deal of the literature in this field, we have placed an emphasis here on remaining as close and faithful as possible to Cartan’s original ideas. In this introductory chapter, we present an overview of the thesis and state some of the main definitions and theorems (some in simplified form) with the aim of helping the reader navigate his way through. At the end of this chapter, we give a summary of the main results and an outline of the thesis. Lie Pseudogroups: from Lie to Cartan In a three volume monograph [43]1 (1888-93), Sophus Lie, in collaboration with Friedrich Engel, laid the foundations of the theory of “continuous transformation groups”. Lie’s notion of a “transformation group” evolved into the modern notion of a pseudogroup (see Definition 3.1.1). A pseudogroup on a manifold is, roughly speaking, a set of local transformations of the manifold that is of a group-like and sheaf-like nature. Intuitively, one can think of a pseudogroup as a set of local symmetries. One of Lie’s important insights, that served as the starting point of his theory, was that pseudogroups of local symmetries of geometric structures and differential equations have the property that they are characterized as the set of solutions of a system of partial differential equations (or a PDE in short). Lie referred to such pseudogroups as “continuous transformation groups”, a notion that evolved into the modern notion of a Lie pseudogroup (see Definition 3.3.1). The work of Lie, and a large part of the literature that followed (including Cartan), was restricted to the local study of Lie pseudogroups, i.e. Lie pseudogroups on open subsets of Euclidean spaces. In this case, given a Lie pseudogroup Γ on Rn or an open subset thereof, one typically introduces coordinates (x; y; :::) on the copy of Rn on which the elements of Γ are applied, coordinates (X; Y; :::) on the copy of Rn in which the elements of Γ take value, and, with respect to these coordinates, every element of Γ is represented by its component functions X = X(x; y; :::);Y = Y (x; y; :::); :::. Let us look at three examples cited from [7]: Example 1. The diffeomorphisms of R of the form X = ax + b; parametrized by a pair of real numbers a; b 2 R, generate a pseudogroup (generate here means that one should impose the sheaf-like axioms of a pseudogroup, see Remark 3.1.3). 1the first volume was recently translated into English [52] 1 2 This pseudogroup is characterized as the set of local solutions of the ordinary differential equation @2X = 0: @x2 Example 2. The diffeomorphisms of R2nfy = 0g of the form X = x + ay; Y = y; parametrized by a real number a 2 R, generate a pseudogroup. This pseudogroup is characterized as the set of local solutions of the system of partial differential equations @X @X X − x = 1; = ;Y = y: @x @y y Example 3. The locally defined diffeomorphisms of R2nfy = 0g of the form y X = f(x);Y = ; f 0(x) parametrized by a function f 2 Diffloc(R) (locally defined diffeomorphisms of R), generate a pseudogroup. This pseudogroup is characterized as the set of local solutions of the system of partial differential equations @X y @X @Y Y = ; = 0; = : @x Y @y @y y The above examples illustrate the general phenomenon that Lie observed – Lie pseudogroups are parametrized by a finite number of “continuous” parameters (hence the name “continuous transformation groups”), which he divided into two types: the “finite” parameters, i.e. real2 variables; and the “infinite” parameters, i.e. real-valued functions. He divided the Lie pseudogroups accordingly into two classes: the “finite” Lie pseudogroups, those parameterized only by finite parameters; and the “infinite” Lie pseudogroups, those parametrized by at least one infinite parameter. Examples 1 and 2 are finite, whereas Example 3 is infinite. In his celebrated three volume monograph [43], Lie developed an infinitesimal approach to the study of Lie pseudogroups in which he studied these objects by means of their infinitesimal transformations, i.e. their generating vector fields. Lie’s work is mainly concentrated on the case of finite Lie pseudogroups (although, it is worth mentioning that he also published two relatively short papers [41, 42] in 1891 that proposed a generalization of his theory to the infinite case). His work on the finite “case”, as is well known, formed the basis of what evolved into the modern notions of a Lie group and, at the infinitesimal level, a Lie algebra (see [2] for a historical account). 2Lie also considered complex parameters. Here we restrict ourselves to the real case. Introduction 3 In 1904-05, Élie Cartan, unsatisfied with the state of the theory of Lie pseudogroups in the infinite case, presented a new structure theory for Lie pseudogroups. Cartan writes [7]: “The generalization of the structure theory of the finite pseudogroups of Lie to the infinite pseudogroups on the basis of infinitesimal transformations proved to be very difficult, if not to say impossible, despite the hard work of S.

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