Poisson Manifolds, Lie Algebroids, Modular Classes: a Survey

Abstract

After a brief summary of the main properties of Poisson manifolds and Lie algebroids in general, we survey recent work on the modular classes of Poisson and twisted Poisson manifolds, of Lie algebroids with a Poisson or twisted Poisson structure, and of Poisson-Nijenhuis manifolds. A review of the spinor approach to the modular class concludes the paper.

Dedicated to the memory of Thomas Branson. This is a contribution to the Proceedings of the 2007 Midwest Geometry Conference in honor of Thomas P. Branson, published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA/

References in corpus (14)

• Holomorphic Poisson Manifolds and Holomorphic Lie Algebroids
• Lie algebroid morphisms, Poisson Sigma Models, and off-shell closed gauge symmetries
• Lectures on Generalized Complex Geometry and Supersymmetry
• Poisson sigma model on the sphere
• The modular class of a twisted Poisson structure
• Characteristic classes associated to Q-bundles
• Picard groups in Poisson geometry
• Deformation Quantization of Poisson Structures Associated to Lie Algebroids
• On the modular classes of Poisson-Nijenhuis manifolds
• Pure Spinors on Lie groups
• Jacobi-Nijenhuis algebroids and their modular classes
• Modular classes of regular twisted Poisson structures on Lie algebroids
• Modular class of even symplectic manifolds
• On (Co-)morphisms of Lie Pseudoalgebras and Groupoids

Cited by in corpus (27)

• Symplectic and Poisson geometry on b-manifolds
• Conformal Twists, Yang-Baxter $\sigma$-models & Holographic Noncommutativity
• Higher Poisson Brackets and Differential Forms
• Courant Algebroids. A Short History
• Gravity theory on Poisson manifold with $R$-flux
• The geometry of variations in Batalin-Vilkovisky formalism
• Contravariant Gravity on Poisson Manifolds and Einstein Gravity
• Particle-like Structure of Lie algebras
• On a Batalin--Vilkovisky operator generating higher Koszul brackets on differential forms
• Thick morphisms, higher Koszul brackets, and $L_{\infty }$-algebroids
• Modular Classes of Lie Groupoid Representations up to Homotopy
• Integrability of weak distributions on Banach manifolds
• The geometry of Lie algebroids and its applications to optimal control
• Higher Koszul brackets on the cotangent complex
• On Computational Poisson Geometry II: Numerical Methods
• On Computational Poisson Geometry I: Symbolic Foundations
• The modular class of a singular foliation
• Contravariant geometry and emergent gravity from noncommutative gauge theories
• Unimodularity Criteria for Poisson Structures on Foliated Manifolds
• The modular class of a Dirac map
• Modular classes of Q-manifolds: a review and some applications
• On coisotropic deformations of holomorphic submanifolds
• A Poisson bracket on the space of Poisson structures
• Morita Invariance of Intrinsic Characteristic Classes of Lie Algebroids
• Compatibility of a Jacobi structure and a Riemannian structure on a Lie algebroid
• Logic of differential calculus and the zoo of geometric strujctures
• Assembling Lie Algebras from Lieons