Let X be a simply connected compact Riemannian symmetric space, let U be the universal covering group of the identity component of the isometry group of X, and let g denote the complexification of the Lie algebra of U, g=u^C. Each u-compatible triangular decomposition g= n_- + h + n_+ determines a Poisson Lie group structure pi_U on U. The Evens-Lu construction produces a (U, pi_U)-homogeneous Poisson structure on X. By choosing the basepoint in X appropriately, X is presented as U/K where K is the fixed point set of an involution which stabilizes the triangular decomposition of g. With this presentation, a connection is established between the symplectic foliation of the Evens-Lu Poisson structure and the Birkhoff decomposition of U/K. This is done through reinterpretation of results of Pickrell. Each symplectic leaf admits a natural torus action. It is shown that these actions are Hamiltonian and the momentum maps are computed using triangular factorization. Finally, local formulas for the Evens-Lu Poisson structure are displayed in several examples.

This thesis explores aspects of generalized geometry, a geometric framework introduced by Hitchin and Gualtieri in the early 2000s. In the first part, we introduce a new class of submanifolds in stable generalized complex manifolds, so-called Lagrangian branes with boundary. We establish a correspondence between stable generalized complex geometry and log symplectic geometry, which allows us to prove results on local neighbourhoods and small deformations of this new type of submanifold. We further investigate Lefschetz thimbles in stable generalized complex Lefschetz fibrations and show that Lagrangian branes with boundary arise in this context. Stable generalized complex geometry provides the simplest examples of generalized complex manifolds which are neither complex nor symplectic, but it is sufficiently similar to symplectic geometry for a multitude of symplectic results to generalize. Our results on Lefschetz thimbles in stable generalized complex geometry indicate that Lagrangian branes with boundary are part of a potential generalisation of the Wrapped Fukaya category to stable generalized complex manifolds. The work presented in this thesis should be seen as a first step towards the extension of Floer theory techniques to stable generalized complex geometry, which we hope to develop in future work. The second part of this thesis studies Dorfman brackets, a generalisation of the Courant- Dorfman bracket, within the framework of double vector bundles. We prove that every Dorfman bracket can be viewed as a restriction of the Courant-Dorfman bracket on the standard VB-Courant algebroid, which is in this sense universal. Dorfman brackets have previously not been considered in this context, but the results presented here are reminiscent of similar results on Lie and Dull algebroids: All three structures seem to fit into a more general duality between subspaces of sections of the standard VB-Courant algebroid and brackets on vector bundles of the form T M ⊕ E ∗ , E → M a vector bundle. We establish a correspondence between certain properties of the brackets on one, and the subspaces on the other side.

Cette thèse contient essentiellement deux chapitres principaux qui ont en commun de porter sur ce que l'on appelle en géométrie de Poisson les chemins cotangents. Dans le premier chapitre, nous introduisons pour chaque hamiltonien, un lagrangien sur les chemins à valeurs dans l'espace cotangent dont les points stationnaires indiquent si le champ de bivecteur est de Poisson ou au moins définit une distribution intégrable - une classe de champs de bivecteurs qui généralise les structures de Poisson tordus que nous étudions en détail. Nous traitons dans le deuxième chapitre d'un autre résultat classique à propos des chemins cotangents, dû à Klimčík, Strobl et étudiée par Cattaneo et Felder. Un bivecteur sur une variété $M$ est de Poisson si et seulement si l'ensemble $C_pi$ des chemins cotangents pour $pi$ est co"{i}sotrope dans la variété symplectique des chemins à valeurs dans $T^*M$. Notre but dans le deuxième chapitre est de reprendre la caractérisation des bivecteurs de Poisson, en travaillant avec des fonctions locales sur l'ensemble des chemins lisses, pour lesquels l'utilisation d'une variété de Banach peut être évitée. Ceci permet d'étendre au cas périodique / In this thesis, we study cotangents paths. In chapter 1 we introduce for every Hamiltonian a Lagrangian on paths valued in the cotangent space whose stationary points projects onto Hamiltonian vector fields. We show that the remaining components of those stationary points tell whether the bivector field is Poisson or at least defines an integrable distribution - a class of bivector fields generalizing twisted Poisson structures that we study in detail. In chapter 2, we establish a local function version of a result due to Klimčík and Strobl then Cattaneo and Felder claiming that a bivector field on a manifold $M$ is Poisson if and only if cotangent paths form a coisotropic submabifold of the infinite dimensional symplectic manifold of paths valued in $T^*M$. Our purpose in chapter 2 is to prove this result without using the Banach manifold setting used by Cattaneo and Felder, which fails in the periodic case because cotangent loops do not form a Banach sub-manifold. Instead, we use local functions on the path space, a point of view that allows to speak of a coisotropic set

Introduzimos o conceito de estrutura de Poisson não comutativa em álgebras associativas e mostra como este conceito se relaciona com o caso clássico, quando a álgebra em questão é a álgebra de funções em uma variedade de Poisson. Mostramos como quocientes simpléticos, não necessariamente suaves, fornecem exemplos de estruturas de Poisson não comutativas. / We introduce the concept of noncommutative Poisson structure on associative algebras and shows how this concept is related to the classical case, that is, the algebra under study is the algebra of functions on a Poisson manifold. We also show how symplectic quotients, not necessarily smooth, provides examples of noncommutative Poisson structures.

We study a number of local and global classification problems in generalized complex geometry. Generalized complex geometry is a relatively new type of geometry which has applications to string theory and mirror symmetry. Symplectic and complex geometry are special cases. In the first topic, we characterize the local structure of generalized complex manifolds by proving that a generalized complex structure near a complex point arises from a holomorphic Poisson structure. In the proof we use a smoothed Newton’s method along the lines of Nash, Moser and Conn. In the second topic, we consider whether a given regular Poisson structure and transverse complex structure come from a generalized complex structure. We give cohomological criteria, and we find some counterexamples and some unexpected examples, including a compact, regular generalized complex manifold for which nearby symplectic leaves are not symplectomorphic. In the third topic, we consider generalized complex structures with nondegenerate type change; we describe a generalized Calabi-Yau structure induced on the type change locus, and prove a local normal form theorem near this locus. Finally, in the fourth topic, we give a classification of generalized complex principal bundles satisfying a certain transversality condition; in this case, there is a generalized flat connection, and the classification involves a monodromy map to the Courant automorphism group.

We study a number of local and global classification problems in generalized complex geometry. Generalized complex geometry is a relatively new type of geometry which has applications to string theory and mirror symmetry. Symplectic and complex geometry are special cases. In the first topic, we characterize the local structure of generalized complex manifolds by proving that a generalized complex structure near a complex point arises from a holomorphic Poisson structure. In the proof we use a smoothed Newton’s method along the lines of Nash, Moser and Conn. In the second topic, we consider whether a given regular Poisson structure and transverse complex structure come from a generalized complex structure. We give cohomological criteria, and we find some counterexamples and some unexpected examples, including a compact, regular generalized complex manifold for which nearby symplectic leaves are not symplectomorphic. In the third topic, we consider generalized complex structures with nondegenerate type change; we describe a generalized Calabi-Yau structure induced on the type change locus, and prove a local normal form theorem near this locus. Finally, in the fourth topic, we give a classification of generalized complex principal bundles satisfying a certain transversality condition; in this case, there is a generalized flat connection, and the classification involves a monodromy map to the Courant automorphism group.

This thesis is devoted to the study of holomorphic Poisson structures and Lie algebroids, and their relationship with differential equations, singularity theory and noncommutative algebra. After reviewing and developing the basic theory of Lie algebroids in the framework of complex analytic and algebraic geometry, we focus on Lie algebroids over complex curves and their application to the study of meromorphic connections. We give concrete constructions of the corresponding Lie groupoids, using blowups and the uniformization theorem. These groupoids are complex surfaces that serve as the natural domains of definition for the fundamental solutions of ordinary differential equations with singularities. We explore the relationship between the convergent Taylor expansions of these fundamental solutions and the divergent asymptotic series that arise when one attempts to solve an ordinary differential equation at an irregular singular point. We then turn our attention to Poisson geometry. After discussing the basic structure of Poisson brackets and Poisson modules on analytic spaces, we study the geometry of the degeneracy loci---where the dimension of the symplectic leaves drops. We explain that Poisson structures have natural residues along their degeneracy loci, analogous to the Poincar\'e residue of a meromorphic volume form. We discuss the local structure of degeneracy loci that have small codimensions, and place strong constraints on the singularities of the degeneracy hypersurfaces of log symplectic manifolds. We use these results to give new evidence for a conjecture of Bondal. Finally, we discuss the problem of quantization in noncommutative projective geometry. Using Cerveau and Lins Neto's classification of degree-two foliations of projective space, we give normal forms for unimodular quadratic Poisson structures in four dimensions, and describe the quantizations of these Poisson structures to noncommutative graded algebras. As a result, we obtain a (conjecturally complete) list of families of quantum deformations of projective three-space. Among these algebras is an ``exceptional'' one, associated with a twisted cubic curve. This algebra has a number of remarkable properties: for example, it supports a family of bimodules that serve as quantum analogues of the classical Schwarzenberger bundles.

This thesis is devoted to the study of holomorphic Poisson structures and Lie algebroids, and their relationship with differential equations, singularity theory and noncommutative algebra. After reviewing and developing the basic theory of Lie algebroids in the framework of complex analytic and algebraic geometry, we focus on Lie algebroids over complex curves and their application to the study of meromorphic connections. We give concrete constructions of the corresponding Lie groupoids, using blowups and the uniformization theorem. These groupoids are complex surfaces that serve as the natural domains of definition for the fundamental solutions of ordinary differential equations with singularities. We explore the relationship between the convergent Taylor expansions of these fundamental solutions and the divergent asymptotic series that arise when one attempts to solve an ordinary differential equation at an irregular singular point. We then turn our attention to Poisson geometry. After discussing the basic structure of Poisson brackets and Poisson modules on analytic spaces, we study the geometry of the degeneracy loci---where the dimension of the symplectic leaves drops. We explain that Poisson structures have natural residues along their degeneracy loci, analogous to the Poincar\'e residue of a meromorphic volume form. We discuss the local structure of degeneracy loci that have small codimensions, and place strong constraints on the singularities of the degeneracy hypersurfaces of log symplectic manifolds. We use these results to give new evidence for a conjecture of Bondal. Finally, we discuss the problem of quantization in noncommutative projective geometry. Using Cerveau and Lins Neto's classification of degree-two foliations of projective space, we give normal forms for unimodular quadratic Poisson structures in four dimensions, and describe the quantizations of these Poisson structures to noncommutative graded algebras. As a result, we obtain a (conjecturally complete) list of families of quantum deformations of projective three-space. Among these algebras is an ``exceptional'' one, associated with a twisted cubic curve. This algebra has a number of remarkable properties: for example, it supports a family of bimodules that serve as quantum analogues of the classical Schwarzenberger bundles.

Made available in DSpace on 2016-12-23T14:34:49Z (GMT). No. of bitstreams: 1 Carla Rodrigues Almeida.pdf: 592195 bytes, checksum: 070fca888db010e772db2fafedfd378d (MD5) Previous issue date: 2012-08-13 / O objetivo deste trabalho é estudar as folheações que surgem a partir de estruturas de Nambu e apresentar a relação entre formas diferenciais e algumas destas estruturas. Mais precisamente, fazer um estudo da geometria de Poisson e de folheações singulares, enfatizando o caso da folheação simplética que surge da estrutura de Poisson e, em seguida, apresentar a geometria de Nambu, estudando o caso das folheações que surgem destas estruturas de ordem maiores ou iguais a três. Neste caso particular, vamos mostrar como tais estruturas de Nambu se relacionam com formas diferenciais e, por esta relação, classificar as estruturas de Nambu lineares através de um resultado de classificação de p-formas integráveis / The aim of this work is to study the foliations that arise from Nambu structures and present the relationship between differential forms and some of this structures. More specifically, to make a study of the Poisson geometry and of singular foliations, emphasiz-ing the case of the simplectic foliation that arises from the Poisson structure and then, to present the Nambu geometry, studying the case of the foliations that arise from the this structures of order grater than or equal to three. In this particular case, we shall show how this Nambu structures are related with differential formas and, by this relationship, classify linear Nambu structure through a result of classification of integrable differential p-forms

Le réseau de Toeplitz est un système hamiltonien dont la structure de Poisson est connue. Dans cette thèse, nous donnons l'origine de cette structure de Poisson et nous en déduisons des équations de Lax associées au réseau de Toeplitz. Nous construisons tout d'abord une sous-variété de Poisson Hn de GLn(C), ce dernier étant vu comme un groupe de Lie-Poisson réel ou complexe dont la structure de Poisson provient d'un R-crochet quadratique sur gln(C) pour une R-matrice fixée. L'existence d'hamiltoniens associés au réseau de Toeplitz pour la structure de Poisson sur Hn ainsi que les propriétés du R-crochet quadratique permettent alors d'expliciter des équations de Lax du système. On en déduit alors l'intégrabilité au sens de Liouville du réseau de Toeplitz. Dans le point de vue réel, nous pouvons ensuite construire une sous-variété de Poisson Han du groupe Un qui est lui-même une sous-variété de Poisson-Dirac de GLR n(C). Nous construisons alors un hamiltonien, pour la structure de Poisson induite sur Han, correspondant à un autre système déduit du réseau de Toeplitz : le réseau de Schur modifié. Grâce aux propriétés des sous-variétés de Poisson-Dirac, nous explicitons une équation de Lax pour ce nouveau système et nous en déduisons une équation de Lax pour le réseau de Schur. On en déduit également l'intégrabilité au sens de Liouville du réseau de Schur modifié. / The Toeplitz lattice is a Hamiltonian system whose Poisson structure is known. In this thesis, we reveil the origins of this Poisson structure and we derive from it the associated Lax equations for this lattice. We first construct a Poisson subvariety Hn of GLn(C), which we view as a real or complex Poisson-Lie group whose Poisson structure comes from a quadratic R-bracket on gln(C) for a fixed R-matrix. The existence of Hamiltonians, associated to the Toeplitz lattice for the Poisson structure on Hn, combined with the properties of the quadratic R-bracket allow us to give explicit formulas for the Lax equation. Then, we derive from it the integrability in the sense of Liouville of the Toeplitz lattice. When we view the lattice as being defined over R, we can construct a Poisson subvariety Han of Un which is itself a Poisson-Dirac subvariety of GLR n(C). We then construct a Hamiltonian for the Poisson structure induced on Han, corresponding to another system which derives from the Toeplitz lattice : the modified Schur lattice. Thanks to the properties of Poisson-Dirac subvarieties, we give an explicit Lax equation for the new system and derive from it a Lax equation for the Schur lattice. We also deduce the integrability in the sense of Liouville of the modified Schur lattice.