Lectures on Poisson Geometry

  title={Lectures on Poisson Geometry},
  author={Marius Crainic and Rui Loja Fernandes and Ioan Marcut},
  journal={Graduate Studies in Mathematics},

Submanifolds in Koszul–Vinberg Geometry

A Koszul–Vinberg manifold is a manifold M endowed with a pair (∇,h)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb}

Poly-Jacobi Manifolds: the Dimensioned Approach to Jacobi Geometry

The standard formulation of Jacobi manifolds in terms of differential operators on line bundles, although effective at capturing most of the relevant geometric features, lacks a clear algebraic

Unimodularity and invariant volume forms for Hamiltonian dynamics on Poisson–Lie groups

In this paper, we discuss several relations between the existence of invariant volume forms for Hamiltonian systems on Poisson–Lie groups and the unimodularity of the Poisson–Lie structure. In

Symplectic Groupoids for Poisson Integrators

A BSTRACT . We use local symplectic Lie groupoids to construct Poisson integrators for generic Poisson structures. More precisely, recursively obtained solutions of a Hamilton-Jacobi-like equation


  • 2022

Basic notions of Poisson and symplectic geometry in local coordinates, with applications to Hamiltonian systems

This work contains a brief and elementary exposition of the foundations of Poisson and symplectic geometries, with an emphasis on applications for Hamiltonian systems with second-class constraints.

Measurand Spaces and Dimensioned Hamiltonian Mechanics

In this paper we introduce a generalization of Hamiltonian mechanics that replaces configuration spaces, conventionally regarded simply as smooth manifolds, with line bundles over smooth manifolds.

SU(2) Lie-Poisson algebra and its descendants

In this paper, a novel discrete algebra is presented which follows by combining the SU(2) Lie-Poisson bracket with the discrete Frenet equation. Physically, the construction describes a discrete

Triangular structures on flat Lie algebras

In this work we study a large class of exact Lie bialgebras arising from noncommutative deformations of Poisson-Lie groups endowed with a left invariant Riemannian metric. We call these structures

Target space diffeomorphisms in Poisson sigma models and asymptotic symmetries in 2D dilaton gravities

The dilaton gravity models in two dimensions, including the Jackiw–Teitelboim model and its deformations, are particular cases of Poisson sigma models. Target space diffeomorphisms map one Poisson



On Hausdorff integrations of Lie algebroids

In this note we present Hausdorff versions for Lie Integration Theorems 1 and 2, and apply them to study Hausdorff symplectic groupoids arising from Poisson manifolds. To prepare for these results we

Bott–Samelson atlases, total positivity, and Poisson structures on some homogeneous spaces

Let G be a connected and simply connected complex semisimple Lie group. For a collection of homogeneous G-spaces G/Q, we construct a finite atlas ABS(G/Q)\documentclass[12pt]{minimal}

Quantum Groups

Here is an introduction to the theory of quantum groups with emphasis on the spectacular connections with knot theory and Drinfeld's recent fundamental contributions. It presents the quantum groups

Géométrie du moment

On definit le cadre naturel dans lequel se generalise l'etude du moment d'une action hamiltonienne. On donne une nouvelle demonstration des theoremes de convexite du moment

Intégration Symplectique des Variétés de Poisson Régulières

Asymplectic integration of a Poisson manifold (M, Λ) is a symplectic groupoid (Γ,η) whichrealizes the given Poisson manifold, i.e. such that the space of units Γ0 with the induced Poisson structure

Normal forms for smooth Poisson structures

On classe, localement, les tenseurs de Poisson C ∞ de rang variable, soumis a une condition de non-degenerescence