The modular class of a Poisson map

We introduce the modular class of a Poisson map. We look at several examples and we use the modular classes of Poisson maps to study the behavior of the modular class of a Poisson manifold under different kinds of reduction. We also discuss their symplectic groupoid version, which lives in groupoid cohomology.Comment: Final version accepted for publication in Annales de l'Institut Fourier. Several changes made to the manuscript, based on referees' remark

A note on modular classes of Lie algebroids

We introduce the modular class of a twisted Jacobi structure on a Lie algebroid and establish a relation between this modular class and the modular class of the underlying Lie algebroid

A note on modular classes of Lie algebroids

We introduce the modular class of a twisted Jacobi structure on a Lie algebroid and establish a relation between this modular class and the modular class of the underlying Lie algebroid

Inversion of a mapping associated with the Aomoto-Forrester system

This article is devoted to the study of a general class of Hamiltonian systems which extends the Calogero systems with external quadratic potential associated to any root system. The interest for such a class comes from a previous article of Aomoto and Forrester. We consider first the one-degree of freedom case and compute the Birkhoff series defined near each of its stationary points. In general, the analysis of the system motivates finding some expression for the inverses of a rational map introduced by Aomoto and Forrester. We derive here some diagrammatic expansion series for these inverses

On Poisson quasi-Nijenhuis Lie algebroids

We propose a definition of Poisson quasi-Nijenhuis Lie algebroids as a natural generalization of Poisson quasi-Nijenhuis manifolds and show that any such Lie algebroid has an associated quasi-Lie bialgebroid. Therefore, also an associated Courant algebroid is obtained. We introduce the notion of a morphism of quasi-Lie bialgebroids and of the induced Courant algebroids morphism and provide some examples of Courant algebroid morphisms. Finally, we use paired operators to deform doubles of Lie and quasi-Lie bialgebroids and find an application to generalized complex geometry.Comment: 12 page

Síntese de calix[4]pirróis sulfonados e complexação com aniões

Mestrado em Química Orgânica e Produtos NaturaisO trabalho apresentado ao longo desta dissertação teve como objetivo a síntese de novos calix[4]pirróis contendo grupos sulfonamida de modo a torná-los mais seletivos na complexação com aniões. Esta dissertação está dividida em três capítulos essenciais. O primeiro apresenta uma breve introdução focando as características, métodos de síntese e funcionalização bem como as aplicações dos calix[4]pirróis. No capítulo que se segue encontra-se descrito o trabalho de síntese, purificação e caracterização de novos compostos, tais como dipirrometano, calix[4]pirróis e calix[4]firinas. A caracterização dos compostos obtidos foi realizada recorrendo a espectroscopia de ressonância magnética nuclear e espectrometria de massa. Nesta parte também são descritos e apresentados os estudos onde se verifica a capacidade dos calix[4]pirróis e dos dipirrometanos sintetizados atuarem como sensores de aniões. Esta deteção é feita recorrendo à leitura da absorvância das soluções na zona do visível e utilizando o método indireto. No último capítulo descreve-se, pormenorizadamente, os métodos de síntese e de purificação e caracterização estrutural dos compostos sintetizados.The work presented in this dissertation lays on the synthesis of new calix[4]pyrroles with sulfonamide groups aiming to obtain novel selective sensors for anion binding applications. This dissertation is the combination of three essential chapters. The first one introducing the subjects of characterization, synthetic routes and functionalization, as well as the applications of calix[4]pyrroles. The following chapter describes the synthesis implemented for the proposed work, purification and characterization of the new compounds, such as dipyrromethanes, calix[4]pyrroles and calix[4]phyrines. The compounds were characterized by nuclear magnetic resonance (1H NMR, 13C NMR, and bidimensional techniques when required) and mass spectrometry techniques. It is also presented the anion binding tests using compounds 31, 35a, 35b, 37a and 37b. The ability of such compounds to act as anion sensor was followed by UV-Vis spectroscopy using the indirect method. The last chapter describes the experimental procedures and the structural characterization of synthesized compounds

On Jacobi quasi-Nijenhuis algebroids and Courant-Jacobi algebroid morphisms

We propose a definition of Jacobi quasi-Nijenhuis algebroid and show that any such Jacobi algebroid has an associated quasi-Jacobi bialgebroid. Therefore, also an associated Courant-Jacobi algebroid is obtained. We introduce the notions of quasi-Jacobi bialgebroid morphism and Courant-Jacobi algebroid morphism providing also some examples of Courant-Jacobi algebroid morphisms.Comment: 14 pages, to appear in Journal of Geometry and Physic

Modular classes of Poisson-Nijenhuis Lie algebroids

The modular vector field of a Poisson-Nijenhuis Lie algebroid $A$ is defined and we prove that, in case of non-degeneracy, this vector field defines a hierarchy of bi-Hamiltonian $A$-vector fields. This hierarchy covers an integrable hierarchy on the base manifold, which may not have a Poisson-Nijenhuis structure.Comment: To appear in Letters in Mathematical Physic

Jacobi-Nijenhuis algebroids and their modular classes

Jacobi-Nijenhuis algebroids are defined as a natural generalization of Poisson-Nijenhuis algebroids, in the case where there exists a Nijenhuis operator on a Jacobi algebroid which is compatible with it. We study modular classes of Jacobi and Jacobi-Nijenhuis algebroids