2002/12/04 by Janusz Grabowski, J. Grabowski, Grabowski, J. +12
Mathematics · Medicine · #53D05 #53D17 #81S10 #Advanced Algebra and Geometry #Differential Geometry (math.DG) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Ophthalmology and Eye Disorders #Symplectic Geometry (math.SG) #math.DG #math.SG #msc:53D05 #msc:53D17 #msc:81S10
paper · pdf · doi:10.48550/arxiv.math/0212052
26 pages; minor changes, one reference added. The final version to appear in Acta Math. Sinica, English Series
openalex publication_date 2002/12/04 · arxiv created 2006/03/13 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study affine Jacobi structures on an affine bundle π:A→ M, i.e. Jacobi brackets that close on affine functions. We prove that there is a one-to-one correspondence between affine Jacobi structures on A and Lie algebroid structures on the vector bundle A+=\bigcupp∈ MAff(Ap,\R) of affine functionals. 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.