Poisson Manifolds, Lie Algebroids, Modular Classes: a Survey

2007/10/31 by Yvette Kosmann–Schwarzbach, Yvette Kosmann-Schwarzbach · 1 citation
Mathematics · Physics and Astronomy · #Advanced Topics in Algebra #Algebra over a field #Algebraic structures and combinatorial models #Artificial intelligence #Class (philosophy) #Computer science #Homotopy and Cohomology in Algebraic Topology #Mathematics #Modular design #Modular form #Poisson distribution #Poisson manifold #Pure mathematics #Statistics #math-ph #math.MP #math.SG #msc:15A66 #msc:17-02 #msc:17B70 #msc:53-02 #msc:53C15 #msc:58-02 #msc:58A10

paper · pdf · doi:10.3842/sigma.2008.005

published as SIGMA 4 (2008), 005, 30 pages · 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/

arxiv created 2008/01/16 · openalex publication_date 2008/01/16 · arxiv updated 2012/12/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

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.

Citations

• Introduction to Foliations and Lie Groupoids
• Lie Groupoids and Lie Algebroids in Differential Geometry
• General Theory of Lie Groupoids and Lie Algebroids
• Boundary Solutions of the Classical Yang-Baxter Equation
• On the modular classes of Poisson-Nijenhuis manifolds
• Symplectic reduction, BRS cohomology, and infinite-dimensional Clifford algebras
• The Cohomology Structure of an Associative Ring
• The modular automorphism group of a Poisson manifold
• Lie algebroids and Poisson-Nijenhuis structures
• Transverse measures, the modular class and a cohomology pairing for Lie algebroids
• Symplectic Geometry and Analytical Mechanics
• Quasi-Poisson Manifolds
• Poisson Sigma Model on the Sphere
• Holomorphic Poisson Manifolds and Holomorphic Lie Algebroids
• Differentiable and algebroid cohomology, Van Est isomorphisms, and characteristic classes
• Lie algebroid morphisms, Poisson sigma models, and off-shell closed gauge symmetries
• Characteristic classes associated to Q-bundles
• Poisson harmonic forms, Kostant harmonic forms, and the S¹-equivariant cohomology of K/T
• Integrating Lie algebroids via stacks
• Generalized Calabi-Yau Manifolds

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