Jacobi Structures on Affine Bundles  

作　　者：J.GRABOWSKI D.IGLESIAS J.C.MARRERO E.PADRN P.URBA■SKI 

机构地区：[1]Mathematical InstitutePolish Academy of Sciences Sniadeckich 8P.O.Box 2100-956 WarsawPoland [2]Instituto de Matemáticas y Física FundamentalConsejo Superior de Investigaciones CientificasSerrano 12328006 MadridSpain [3]Departamentode Matemática FundamentalFacultad de MatemáticasUniversidad de la LagunaLa LagunaTenerifeCanary IslandsSpain [4]Division of Mathematical Methods in PhysicsUniversity of Warsaw Hoia 7400-682 WarsawPoland

出　　处：《Acta Mathematica Sinica,English Series》2007年第5期769-788,共20页数学学报（英文版）

基　　金：the Polish Ministry of Scientific Research and Information Technology under the grant No.2 P03A 036 25;DGICYT grants BFM2000-0808 and BFM2003-01319;D.Iglesias wishes to thank the Spanish Ministry of Education and Cuture for an FPU grant

摘　　要：Abstract We study affine Jacobi structures （brackets） on an affine bundle π ： A → M, 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^＋ = ∪p∈M Aff（Ap, R） 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.Abstract We study affine Jacobi structures （brackets） on an affine bundle π ： A → M, 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^＋ = ∪p∈M Aff（Ap, R） 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.

关 键 词：Vector and affine bundles Jacobi manifolds Lie algebroids 

分 类 号：O152.5[理学—数学]