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Jacobi Structures on Affine Bundles

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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 \( A^{ + } = \cup _{{p \in M}} {\text{Aff}}{\left( {A_{p} ,\mathbb{R}} \right)} \) 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.

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Authors and Affiliations

  1. Mathematical Institute, Polish Academy of Sciences, Śniadeckich 8, 21, 00–956, Warsaw, Poland

    J. Grabowski

  2. Instituto de Matemáticas y Física Fundamental, Consejo Superior de Investigaciones Científicas, Serrano 123, 28006, Madrid, Spain

    D. Iglesias

  3. Departamento de Matemática Fundamental, Facultad de Matemáticas, Universidad de la Laguna, La Laguna, Tenerife, Canary Islands, Spain

    J. C. Marrero & E. Padrón

  4. Division of Mathematical Methods in Physics, University of Warsaw Hoża 74, 00–682, Warsaw, Poland

    P. Urbański

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  1. J. Grabowski
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  2. D. Iglesias
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  3. J. C. Marrero
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  4. E. Padrón
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  5. P. Urbański
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Correspondence to J. Grabowski.

Additional information

Research partially supported by the Polish Ministry of Scientific Research and Information Technology under the grant No. 2 P03A 036 25 and DGICYT grants BFM2000–0808 and BFM2003–01319. D. Iglesias wishes to thank the Spanish Ministry of Education and Culture for an FPU grant

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Grabowski, J., Iglesias, D., Marrero, J.C. et al. Jacobi Structures on Affine Bundles. Acta Math Sinica 23, 769–788 (2007). https://doi.org/10.1007/s10114-005-0716-0

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  • DOI: https://doi.org/10.1007/s10114-005-0716-0

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