Thesis (Ph. D.)--University of Missouri-Columbia, 1996. / Typescript. Vita. Includes bibliographical references (leaves 114-118). Also available on the Internet.

Thesis (Ph. D.)--University of Missouri-Columbia, 1996. / Typescript. Vita. Includes bibliographical references (leaves 114-118). Also available on the Internet.

Les algébroïdes de Courant ont été introduits par T. J. Courant dans sa thèse portant sur l’intégrabilité des structures de Dirac. Ils sont devenus d’importants objets en géométrie différentielle depuis le travail de Z.-J. Liu, A. Weinstein et P. Xu sur les bigébroïdes de Lie. Ils jouent un rôle grandissant en physique théorique ainsi qu’en mathématiques. Dans cette thèse, on s’intéresse à décrire les automorphismes forts d’un algébroïde de Courant régulier. Dans une première partie des rappels sont faits sur les algébroïdes de Lie. Dans une seconde partie, on étudie les algébroïdes de Courant. Dans une troisième partie, après introduction de la notion de dissection, nous explicitons le groupe des automorphismes forts d’un algébroïde de Courant régulier relativement à une dissection, et calculons l’algèbre de Lie des automorphismes infinitésimaux relativement à cette dissection. De cette étude sont apparues de nouvelles symétries qui pourraient s’avérer utiles en physique théorique. / Courant algebroids have been introduced by T. J. Courant in his PhD thesis concerning the integrability of Dirac structures. They have become important objects in differential geometry since the seminal work of Z.-J. Liu, A. Weinstein and P. Xu on Lie bialgebroids. They play an increasing role in theoretical physics as well as inmathematics. In this thesis, we are interested by describing strong automorphisms of a regular Courant algebroid. In a first part, we review Lie algebroids. In a second part, we study Courant algebroids. In a third part, after introducing the notion of dissection, we compute the automorphism group of a regular Courant algebroid with respect to a dissection of it, and then compute the Lie algebra of infinitesimal automorphisms with respect to this dissection. From this work appeared new symmetries that could be useful in theoretical physics.

published_or_final_version / Mathematics / Master / Master of Philosophy

Thesis (M. Phil.)--University of Hong Kong, 2010. / Includes bibliographical references (p. 86-88). Also available in print.

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.

On dit qu'une variété est feuilletée lorsqu'il existe une partition de celle-ci en sous-variétés immergées. La théorie des feuilletages a des applications très profondes dans divers champs des Mathématiques et de la Physique, et il semble d'autant plus intéressant de pouvoir analyser le feuilletage à partir de ce qui semble être une donnée plus fondamentale : sa distribution de champs de vecteurs associée. C'est ainsi que nous avons observé que si le feuilletage est résolu par un fibré gradué, on peut relever le crochet de Lie des champs de vecteurs en une structure de Lie infini-algébroide sur ce fibré. D'autre part, cette structure est universelle dans le sens où toute autre résolution du feuilletage sera isomorphe à celle-ci dans un sens L_infini, mais seulement à homotopie près. Lorsqu'on se limite à l'étude au dessus d'un point, on observe que la cohomologie associée à la résolution devient potentiellement non triviale. La structure de Lie infini-algébroide universelle se réduit alors à une algèbre de Lie graduée sur cette cohomologie. Cette structure algébrique peut être transportée (non canoniquement) tout le long de la feuille, transformant la cohomologie au dessus d'une feuille en algébroide de Lie gradué. Cela nous permet de retrouver des résultats déjà connus par ailleurs et de déduire des avancées prometteuses / A smooth manifold is said to be foliated when it is partitioned into immersed submanifolds. Foliation Theory has profound applications in various fields of Mathematics and Physics, and it seems much more interesting to analyze the foliation from what seems to be a more fundamental point of view: its associated distribution of vector fields. Thus we have noticed that if the foliation is resolved by a graded fiber bundle, one can lift the Lie bracket of vector fields into a Lie infinity-algebroid structure on this fiber bundle. Moreover, this structure is universal in the sense that any other resolution of the foliation is isomorphic to it in the L_infinity setup, but only up to homotopy. When one restricts the analysis over a point, we observe that the cohomology associated to the resolution may become non trivial. The universal Lie infinity-algebroid structure hence reduces to a graded Lie algebra structure on this cohomology. This algebraic structure can be carried (non canonically) along the leaf, providing the cohomology over a leaf with a graded Lie algebroid structure. This enables us to retrieve former well-known results, as well as promising advances

Connus des mécaniciens de la géométrie de Poisson, les algébroïdes de Lie transitifs sont ici étudiés du point de vue de leurs sections afin de développer un formalisme algébrique plus proche de celui développé par les théories de jauge. Ici, les algébroïdes de Lie transitifs s'apparentent à une généralisation des champs de vecteurs sur la variété de base. Ce mémoire de thèse a pour objet l'étude des connexions généralisées sur les algébroïdes de Lie transitifs et la construction de théories de jauge. Les connexions ordinaires sur les algébroïdes de Lie transitifs sont définies par des 1-formes de connexion de l'algébroïde de Lie à valeurs dans son noyau et vérifiant une contrainte de normalisation sur ce noyau. En relâchant cette contrainte, on construit l'espace des 1-formes de connexions généralisées qui se décomposent, à l'aide d'une connexion ordinaire de fond, comme la somme d'une connexion ordinaire et d'un paramètre purement algébrique définit sur le noyau. Dans l'esprit des théories Yang-Mills, une action invariante de jauge est définie comme la “norme” de la courbure associée à une connexion généralisée. De cette action, il découle un lagrangien composé des termes des théories de jauge de type Yang-Mills-Higgs : le terme cinétique associé aux champs de jauge et le terme de couplage minimal pour un champ tensoriel scalaire plongé dans un potentiel quartique. La réduction du groupe de symétrie de la théorie s'effectue par une redistribution des degrés de liberté dans l'espace fonctionnel des champs de la théorie. Il résulte de ces manipulations la définition d'une théorie de type Yang-Mills dont les bosons vecteurs sont des champs massifs. / Transitive Lie algebroids are usually studied from the point of view of the geometry of Poisson. Here, they are preferentially defined in terms of sections of fiber bundle in order to get close to the formalism of the gauge field theory. Then, transitive Lie algebroids can be seen as a generalization of vector fields on the base manifold. This PhD thesis is concerned with the study of generalized connections on transitive Lie algebroids and the construction of gauge theories. Ordinary connections on transitive Lie algebroids are defined as the subset of 1-forms on Lie algebroids with values in its kernel which fulfill a normalization constraint on this kernel. By relaxing this constraint, we build the space of generalized connection 1- forms. Using a background connection, we show that any generalized connections can be decomposed as the sum of an ordinary connection and a purely algebraic parameter defined on the kernel. As in Yang-Mills theories, we define a gauge invariant functional action as the “norm” of the curvature associated to a generalized connection. Then, the Lagrangian associated to this action forms a Yang-Mills-Higgs type model composed with the field strength associated to gauge fields and a minimal coupling with a tensorial scalar field embedded into a quartic potential. In the case of Atiyah Lie algebroids, the symmetry group of the theory can be reduced by using an appropriate rearrangement of the degrees of freedom in the functional space of fields. We thus obtain a Yang-Mills type theory describing massive vector bosons.

Le mappe comomento omotopiche sono una generalizzazione della nozione di mappa momento introdotta al fine di estendere il concetto di azione hamiltoniana al contesto della geometria multisimplettica. L'obiettivo di questa tesi è fornire nuove costruzioni esplicite ed esempi concreti di azioni di gruppi di Lie su varietà multisimplettiche che ammettono delle mappe comomento. Il primo risultato è una classificazione completa delle azioni di gruppi compatti su sfere multisimplettiche. In questo caso, l'esistenza di mappe comomento omotopiche dipende dalla dimensione della sfera e dalla transitività dell'azione di gruppo. Il secondo risultato è la costruzione esplicita di un analogo multisimplettico dell’inclusione dell'algebra di Poisson di una varietà simplettica dentro il corrispondente algebroide di Lie twistato. E’ possibile dimostrare che questa inclusione soddisfa una relazione di compatibilità nel caso di varietà multisimplettiche gauge-correlate in presenza di un'azione di gruppo Hamiltoniana. Tale costruzione potrebbe giocare un ruolo nella formulazione di un analogo multisimplettico della procedura di quantizzazione geometrica. L’ultimo risultato è una costruzione concreta di una mappa comomento omotopica relativa all'azione multisimplettica del gruppo di diffeomorfismi che preservano la forma volume dello spazio Euclideo. Questa mappa ammette naturalmente un’interpretazione idrodinamica, nello specifico trasgredisce alla mappa comomento idrodinamica introdotta da Arnol'd, Marsden, Weinstein e altri. La mappa comomento così costruita può essere inoltre messa in relazione alla teoria dei nodi avvalendosi dell’approccio ai link nel formalismo dei vortici. Questo punto di apre la strada a un'interpretazione semiclassica del polinomio HOMFLYPT nel linguaggio della quantizzazione geometrica. / Homotopy comomentum maps are a higher generalization of the notion of moment map introduced to extend the concept of Hamiltonian actions to the framework of multisymplectic geometry. Loosely speaking, higher means passing from considering symplectic $2$-form to consider differential forms in higher degrees. The goal of this thesis is to provide new explicit constructions and concrete examples related to group actions on multisymplectic manifolds admitting homotopy comomentum maps. The first result is a complete classification of compact group actions on multisymplectic spheres. The existence of a homotopy comomentum maps pertaining to the latter depends on the dimension of the sphere and the transitivity of the group action. Several concrete examples of such actions are also provided. The second novel result is the explicit construction of the higher analogue of the embedding of the Poisson algebra of a given symplectic manifold into the corresponding twisted Lie algebroid. It is also proved a compatibility condition for such embedding for gauge-related multisymplectic manifolds in presence of a compatible Hamiltonian group action. The latter construction could play a role in determining the multisymplectic analogue of the geometric quantization procedure. Finally a concrete construction of a homotopy comomentum map for the action of the group of volume-preserving diffeomorphisms on the multisymplectic 3-dimensional Euclidean space is proposed. This map can be naturally related to hydrodynamics. For instance, it transgresses to the standard hydrodynamical co-momentum map of Arnol'd, Marsden and Weinstein and others. A slight generalization of this construction to a special class of Riemannian manifolds is also provided. The explicitly constructed homotopy comomentum map can be also related to knot theory by virtue of the aforementioned hydrodynamical interpretation. Namely, it allows for a reinterpretation of (higher-order) linking numbers in terms of multisymplectic conserved quantities. As an aside, it also paves the road for a semiclassical interpretation of the HOMFLYPT polynomial in the language of geometric quantization.