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On Moment Maps and Jacobi Manifolds Alexander Leguizam´onRobayo Universidad Nacional de Colombia Facultad de Ciencias, Departamento de Matem´aticas Bogot´aD.C., Colombia 2021 On Moment Maps and Jacobi Manifolds Alexander Leguizam´onRobayo Tesis o trabajo de grado presentada(o) como requisito parcial para optar al t´ıtulode: Magister en Ciencias Matem´aticas Directores: Ph.D. Daniele Sepe Ph.D. Nicol´asMart´ınezAlba L´ıneade Investigaci´on: Geometr´ıaDiferencial Universidad Nacional de Colombia Facultad de Ciencias, Departamento de Matem´aticas Bogot´aD.C., Colombia 2021 A mis padres y mi hermana That is the first thing I know for sure: If the questions don't make sense, neither will the answers. Kurt Vonnegut, The Sirens of Titan Agradecimientos Primeramente, quisiera agradecer a mis supervisores Daniele Sepe y Nicol´asMartinez, por el apoyo durante este proyecto. Quisiera agradecerle a Nicol´aspor introducirme al mundo de la geometr´ıade Poisson y sus gen- eralizaciones. El curso de variedades de Nicol´asfue una fuerte base para iniciar con el estudio de este trabajo. Esto sin contar el apoyo de Nicol´asal motivarme a participar en distintos eventos y charlas y por siempre ser curioso respecto a los posibles resultados y aplicaciones de la geometr´ıa de Poisson. Estoy agradecido con Daniele por su paciencia para resolver mis dudas sin importar lo b´asicas que pudieran ser. El curso de geometr´ıasimplectica me ayud´omucho a aclarar dudas y tener una visi´onm´asaterrizada a la hora de aplicar conceptos del curso. Tambi´enpor la constante retroal- imentaci´on,esto me permiti´ocomprender la diferencia entre entender un concepto matem´atico superficialmente y realmente entenderlo a fondo, sabiendo mostrar todas las sutilezas que hay detr´asde un razonamiento riguroso. Agradezco a la Universidad Nacional de Colombia, por apoyar mis estudios a trav´esde la Beca de exenci´onde derechos acad´emicosdurante el primer a~no,y a trav´esde la Beca auxiliar docente, durante el segundo a~no. Tambi´enagradezco a todos mis amigos en el Departamento de Matem´aticas: a Arturo y Fabio, por su apoyo en el ejercicio de la docencia; a Haly y Jaime, por ser compa~nerosde viaje; a Lizeth y Yamid, por las discusiones sobre geometr´ıa;y a todos los dem´aspor las risas y charlas. Agradezco a Gia y a Cuore por darme una excusa para salir a caminar, despejarme, y llegar a trabajar con nuevas ideas. Agradezco a Adriana Isabel, por darme su mano y tener siempre sus brazos abiertos para abrazarme y escucharme. Tu cari~nome mantuvo siempre motivado a seguir luchando. Agradezco a mis padres, Yubby y Marvin, y a mi hermana Adriana por todo su apoyo mental y emocional, estando siempre ah´ıpara alentarme, motivarme, y hacerme ver que las cosas siempre pueden mejorar. ix Abstract The main goal of this work is to introduce the idea of a Hamiltonian action in the context of Jacobi structures on line bundles. This work aims to make these construction without relying on the "Poissonization trick". Our definition allows us to recover the notion of (weakly)Hamiltonian action in the context of Poisson, contact, and locally conformally symplectic geometry. Keywords: Jacobi structures; contact manifolds; locally conformally symplectic structures; Hamiltonian actions; moment maps. 2010 Mathematics Subject Classification: 53D10;53D17; 53D20, 37J15. Resumen En el siguiente trabajo introducimos la idea de acci´onHamiltoniana en el contexto de la geometr´ıa de Jacobi en fibrados de l´ıneagenerales. Esta construcci´onla realizamos de forma intr´ınsecasin necesidad de recurrir al "truco de Poissonizaci´on". El concepto de acci´onHamiltoniana en ge- ometr´ıade Jacobi nos permite recuperar resultados conocidos en geometr´ıade Poisson, contacto, y localmente conformemente simpl´ectica. Palabras clave: estructuras de Jacobi; variedades de contact; estructuras localmente con- formemente simpl´ecticas;acci´onHamiltoniana; aplicaci´onmomento . Contents Agradecimientos vii Abstract ix 1 Introduction 2 2 Vector Bundles 4 2.1 Vector bundles . .4 2.1.1 Morphisms . .7 2.1.2 Sections of a Vector Bundle . .8 2.1.3 Subobjects . 12 2.2 Operations on vector bundles . 13 2.2.1 Pullback bundles . 15 2.3 Algebraic preliminaries . 17 2.3.1 Differential Operators on Algebras . 17 2.3.2 Derivations and infinitesimal symmetries . 19 2.4 Derivations and infinitesimal automorphisms of vector bundles . 20 2.4.1 Infinitesimal Automorphisms . 23 2.5 Group actions and automorphisms . 30 2.5.1 Infinitesimal Actions on Vector Bundles . 31 3 Poisson and Symplectic Manifolds 35 3.1 First definitions and properties . 35 3.1.1 Multivector fields and the Poisson bivector . 37 3.1.2 Hamiltonian vector fields . 40 3.1.3 Poisson symmetries . 41 3.2 Nondegenerate Poisson manifolds . 42 3.2.1 Hamiltonian vector fields . 44 3.3 Poisson actions and Hamiltonian Poisson actions . 45 4 Jacobi Geometry 52 4.1 Local Lie algebras and Jacobi manifolds . 52 4.2 Jacobi morphisms . 56 4.3 Nondegenerate Jacobi manifolds . 62 4.3.1 Locally conformally symplectic manifolds . 63 4.3.2 Contact manifolds . 68 Contents 1 5 Group actions on Jacobi manifolds 78 5.1 Jacobi group action . 78 5.2 Weakly Hamiltonian action . 82 5.3 Hamiltonian actions . 85 6 Closing remarks 90 6.1 Further Work . 90 Bibliography 91 1 Introduction Consider a symplectic manifold (M; Ω). The symplectic form Ω allows us to associate a (Hamilto- 1 nian) vector field Xf to any smooth function f 2 C (M). The assignment of Hamiltonian vector 1 fields allows us to endow C (M) with a Lie algebra structure given by ff; gg = Ω(Xf ;Xg). This bracket is of key importance in many areas of mathematics and physics [21, 22]. A similar construction can be done for manifolds with other structures such as: Poisson structures [7, 22], contact structures [9, 3], and locally conformally symplectic (LCS) structures [25, 23]. All of these brackets are particular examples of Jacobi structures, which were defined by Lichnerowicz [15] as a pair (Λ;E) of a bivector Λ and a vector field E such that [Λ; Λ] = 2E ^ Λ; [Λ;E] = 0: The work of Lichnerowicz studied Jacobi structures in a setting, in which the pairs (Λ;E) are locally defined and are related on overlaps by a conformal transformation. This approach leads to the notion of a Jacobi bundle defined by Marle [17]. Earlier, Kirillov [11] studied Lie algebra structures {·; ·} : Γ(E) × Γ(E) ! Γ(E); on the space of sections of a vector bundle E, such that the bracket is local i.e. for any pair of sections u; v 2 Γ(E) supp(fu; vg) ⊂ supp(u) \ supp(v): Kirillov proved that when E is a vector bundle with one dimensional fibres (i.e. a line bundle), the Lie algebra structure on Γ(E) is given locally by a Lichnerowicz' Jacobi structures (Λ;E). On the works of Marle [17], and Vaisman [25], Hamiltonian vector fields are shown to be infinites- imal conformal symmetries of the Jacobi structures induces by the respective brackets. Then the definition of a Hamiltonian vector field must be given in terms of the Jacobi bundle. Here the main question of our work appears: How can the concept of Hamiltonian vector field be defined in the setting of Jacobi bundles? Recent works [24, 26, 28] have introduced a more algebraic language to the study of Jacobi bundles by studying the C1(M)−module structure of Γ(E). In this setting, the notion of an infinitesimal symmetry is replaced by that of a derivation. A derivation ∆ on Γ(E) is an R−linear operator such that for all u 2 Γ(E) and f 2 C1(M) ∆(fu) = f∆(u) + X∆(f)u; where X∆ 2 X(M) is the symbol of ∆. In this sense, Hamiltonian vector fields appear as the symbols of derivations associated to sections of Γ(E). In this context, this work has three aims: 3 1. to provide a more geometric interpretation of the algebraic language of [24] via examples. 2. to propose the main definitions and prove the main results related to (weakly) Hamilto- nian actions in Jacobi bundles in the spirit of [3] (i.e without resorting to the use of other structures). 3. to recover previous results on Hamiltonian actions, i.e., Stanciu's results [23] for the LCS case, and Loose's [16] for the contact case, and the relevant results in Poisson geometry [7, 21]. The structure of this work is given as follows: • In Chapter 2, we discuss the basic theory of vector bundles. We focus on the constructions that are instrumental to make sense of the theory of Jacobi manifolds. • In Chapter 3, we discuss the basic theory of Poisson manifolds and their Hamiltonian actions. We recover several results in symplectic geometry as an instance of the results on Poisson manifolds. • In Chapter 4, we present the theory of Jacobi manifolds. We pay special attention to the transitive cases (contact and locally conformally symplectic manifolds) and present important results in each case. • In Chapter 5, we introduce the idea of (weakly) Hamiltonian actions. We instantiate it in the trivial bundle case and recover previous results on contact [16] and locally conformally symplectic [23] manifolds. • In Chapter 6, we close this work by stating some of the main questions that could be explored as a result of this work. 2 Vector Bundles The following chapter contains a brief introduction to the study of vector bundles. We begin by introducing the basic definitions and constructions on vector bundles. The main references for this section are [12, 7, 2, 18]. In particular, we focus on pullback bundles as they are needed to make sense of morphisms of Jacobi manifolds.
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