FACULTY OF SCIENCE AND TECHNOLOGY Department of Mathematics and Statistics Symmetry transformation groups and differential invariants — Eivind Schneider MAT-3900 Master’s Thesis in Mathematics November 2014 Abstract There exists a local classification of finite-dimensional Lie algebras of vector fields on C2. We lift the Lie algebras from this classification to the bundle C2 × C and compute differential invariants of these lifts. Acknowledgements I am very grateful to Boris Kruglikov for coming up with an interesting assignment and for guiding me through the work on it. I also extend my sincere gratitude to Valentin Lychagin for his help through many insightful lectures and conversations. A heartfelt thanks to you both, for sharing your knowledge and for the inspiration you give me. I would like to thank Henrik Winther for many helpful discussions. Finally, I thank my family, and especially my wife, Susann, for their support and encouragement. ii Contents 1 Introduction 1 2 Preliminaries 3 2.1 Jet spaces and prolongation of vector fields . 3 2.2 Classification of Lie algebras of vector fields in one and two dimensions . 5 2.3 Lifts . 7 2.3.1 Cohomology . 8 2.3.2 Coordinate change . 9 2.4 Differential invariants . 9 2.4.1 Determining the number of differential invariants of order k ........................... 10 2.4.2 Invariant derivations . 11 2.4.3 Tresse derivatives . 12 2.4.4 The Lie-Tresse theorem . 13 3 Warm-up: Differential invariants of lifts of Lie algebras in D(C) 15 3.1 g1 = h@xi .............................. 15 3.2 g2 = h@x; x@xi ........................... 16 3.2.1 Lift of g2 to D(C × C) and invariants on C × C . 16 3.2.2 Differential invariants of first order . 16 3.2.3 Differential invariants of higher order . 17 3.2.4 Invariant derivations . 17 2 3.3 g3 = h@x; x@x; x @xi ........................ 18 3.3.1 Lift of g3 to D(C × C) and invariants on C × C . 18 3.3.2 Differential invariants of first and second order . 19 3.3.3 Differential invariants of higher order . 19 3.3.4 Invariant derivations . 19 3.4 Summary . 20 iii 4 Differential invariants of lifts of Lie algebras in D(C2) 21 4.1 Lifts to D(C2 × C) ........................ 21 4.1.1 Coordinate change . 21 4.1.2 Solving the differential equations . 22 4.1.3 List of lifts and cohomologies . 24 4.2 Counting differential invariants . 26 4.3 List of differential invariants . 32 4.3.1 g1; g2; g3 .......................... 32 4.3.2 g6; g7; g11; g12; g15; g16; g17; g18; g19; g20 . 34 4.3.3 g8; g9; g10; g13; g14 ..................... 35 4.3.4 g4 ............................. 43 4.3.5 g5 ............................. 46 4.3.6 g21; g22 ........................... 46 5 General remarks and applications 49 5.1 Algebraic actions . 49 5.2 Projectable Lie algebras of vector fields . 51 5.3 Differential equations and their symmetries . 53 6 Appendix 55 iv Chapter 1 Introduction Consider the problem of classifying all complex analytic scalar partial differ- ential equations of two independent variables with finite-dimensional sym- metry groups. The local action of a symmetry group can be described in terms of its Lie algebra of infinitesimal generators. We say that the vec- tor field X 2 D(C3(x; y; u)) is an (infinitesimal) symmetry for the equation F (x; y; u; ux; uy; :::; uyk ) = 0 if X(k)(F ) = λF where λ is a smooth function of x; y; :::; uyk . Given a differential equation F = 0, we can find its symmetries by solving for X. These symmetries form a Lie algebra. We can also go the other way: Given a Lie algebra of symmetries g ⊂ D(C3(x; y; u)), we can solve for F to find all differential equations with the given Lie algebra as its symmetry algebra. There exists a local classification of finite-dimensional Lie algebras of vector fields on C2. By using this, one can get a local description of all scalar ODEs with finite-dimensional Lie algebras of symmetries, up to point transformations. For C3 there exists no complete classification of Lie algebras of vector fields, and therefore we cannot classify scalar PDEs of two independent vari- ables in the same way. In this thesis we take the Lie algebras of vector fields on C2 from the classification, and lift them on the bundle C2 × C. This gives us a subset of all Lie algebras of vector fields on C3. A subproblem of finding all differential equations with a given Lie algebra g of symmetries, is to find functions F that satisfy X(k)(F ) = 0 for every X 2 g: We call such functions differential invariants (of order k). For each of the lifted vector fields in C2 × C, we will compute differential invariants. 1 Structure of the thesis We begin by describing the Jet space J k(π) for the bundle π : C2 ×C ! C2 in 2.1. In section 2.2 we recall a classification of Lie algebras of vector fields on C and C2. Then, in 2.3 we describe the procedure we use for lifting vector fields on C2 to C2 × C, and discuss how the lifts correspond to cohomology groups. In section 2.4 we define differential invariants and invariant derivations, and state the Lie-Tresse theorem. In chapter 3 we take the classification of Lie algebras of vector fields on C, and lift each Lie algebra into a Lie algebra of vector fields on C × C. We also find the differential invariants for the lifts. In this chapter the calculations are described in much more detail than in the later part of the thesis, and the chapter can therefore be considered as continuation of the introduction in 2.3 and 2.4. Chapter 4 contains the main results. Lists of the lifts and their differential invariants are given in 4.1 and 4.3, respectively. In section 4.2 we look at the dimension of a generic orbit of some of the lifts and their jet-prolongations, and we use this to count how many independent differential invariants we expect to find. Finally, in chapter 5, we look back on our computations and discuss possible applications of our results and some interesting properties of our lifts of Lie algebras. In 5.1 we discuss algebraicity of Lie algebra actions by looking at examples from our computations, and we see how this relates to the form of the differential invariants. In 5.2 we introduce the notion of projectable Lie algebra of vector fields, and discuss the surprising fact that all our lifts has at least one differential invariant of order two. In 5.3 we give an example of how our results can be usefull in the study of differential equations. The appendix contains a list of differential invariants and invariant deriva- tions that were to long to fit into 4.3 in a reasonable way. 2 Chapter 2 Preliminaries 2.1 Jet spaces and prolongation of vector fields We start by fixing some notations regarding Jet spaces. For a more general and detailed description, see for example [KL08], [KV99] or [ALV91]. Jet spaces Consider the trivial bundle π : C2 × C ! C2. Let x1; x2 be coordinates on C2 and let u be a coordinate on C. We call x1; x2 independent coordinates, and u the dependent coordinate.1 Let s: C2 ! C2 ×C be a section of the bundle. We can describe this section by a function f : C2 ! C in the following way: 1 2 1 2 1 2 s(x ; x ) = (x ; x ; f(x ; x )). We say that two sections s1; s2 are tangent up jσj jσj n @ f1 @ f2 to order k at a 2 C if @xσ (a) = @xσ (a) for 0 ≤ jσj ≤ k, where σ is a k multi-index. Denote by [s]a the equivalence class of all sections which are tangent up to order k to s at a 2 C2. We call this the k-jet of s at a. Let k 2 k 2 Ja (C × C) = Ja (π) be the set of k-jets of sections on π at the point a 2 C , k k and J (π) = [a2C2 Ja (π). This set is naturally equipped with the structure of a smooth manifold. This description of the k-jets of sections, gives a natural set of coordinates i k x ; u; uσ on J (π): @jσjf xi([s]k) = ai; u([s]k) = f(a); u ([s]k) = ; 1 ≤ jσj ≤ k a a σ a @xσ We will also use the notation u0 = u. 1We will for the most part be naming the coordinates x; y; u, but for general discussion it’s convenient to use indices. 3 0 0 2 2 The map (a; f(a)) 7! [s]a gives us the identification J (C ×C) = C ×C. k l k l The projections πk;l : J (π) ! J (π) defined by [s]a 7! [s]a for k ≥ l give a tower structure: 2 0 π1;0 1 π2;1 πk;k−1 k πk+1;k C × C = J (π) − J (π) − · · · − J (π) − · · · We denote the inverse limit of this system of maps by J 1(π). k Let Fk be the algebra of analytic functions on J (π). Through the pro- jections πk;k−1, we get a filtering of the function algebras: F0 ⊂ F1 ⊂ · · · ⊂ Fk ⊂ · · · ⊂ F1 Prolongation of vector fields Consider a diffeomorphism φ: C2 ×C ! C2 ×C. The kth prolongation of φ is (k) k k (k) k k the diffeomorphism φ : J (π) ! J (π) defined by φ ([s]a) = [φ ◦ s]φ(a).