Acta Mathematica Sinica, English Series

, Volume 23, Issue 5, pp 769–788

Jacobi Structures on Affine Bundles

  • J. Grabowski
  • D. Iglesias
  • J. C. Marrero
  • E. Padrón
  • P. Urbański
ORIGINAL ARTICLES

DOI: 10.1007/s10114-005-0716-0

Cite this article as:
Grabowski, J., Iglesias, D., Marrero, J.C. et al. Acta Math Sinica (2007) 23: 769. doi:10.1007/s10114-005-0716-0
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Abstract

We study affine Jacobi structures (brackets) on an affine bundle π : AM, i.e. Jacobi brackets that close on affine functions. We prove that if the rank of A is non–zero, there is a one–toone correspondence between affine Jacobi structures on A and Lie algebroid structures on the vector bundle \( A^{ + } = \cup _{{p \in M}} {\text{Aff}}{\left( {A_{p} ,\mathbb{R}} \right)} \) of affine functionals. In the case rank A = 0, it is shown that there is a one–to–one correspondence between affine Jacobi structures on A and local Lie algebras on A+. Some examples and applications, also for the linear case, are discussed. For a special type of affine Jacobi structures which are canonically exhibited (strongly–affine or affine–homogeneous Jacobi structures) over a real vector space of finite dimension, we describe the leaves of its characteristic foliation as the orbits of an affine representation. These affine Jacobi structures can be viewed as an analog of the Kostant–Arnold–Liouville linear Poisson structure on the dual space of a real finite–dimensional Lie algebra.

Keywords

Vector and affine bundlesJacobi manifoldsLie algebroids

MR (2000) Subject Classification

53D1753D0581S10

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • J. Grabowski
    • 1
  • D. Iglesias
    • 2
  • J. C. Marrero
    • 3
  • E. Padrón
    • 3
  • P. Urbański
    • 4
  1. 1.Mathematical InstitutePolish Academy of SciencesWarsawPoland
  2. 2.Instituto de Matemáticas y Física FundamentalConsejo Superior de Investigaciones CientíficasMadridSpain
  3. 3.Departamento de Matemática FundamentalFacultad de Matemáticas, Universidad de la LagunaLa Laguna, TenerifeCanary Islands, Spain
  4. 4.Division of Mathematical Methods in PhysicsUniversity of Warsaw Hoża 74WarsawPoland