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Modular classes of skew algebroid relations

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Abstract

Skew algebroid is a natural generalization of the concept of Lie algebroid. In this paper, for a skew algebroid E, its modular class mod(E) is defined in the classical as well as in the supergeometric formulation. It is proved that there is a homogeneous nowhere-vanishing 1-density on E * which is invariant with respect to all Hamiltonian vector fields if and only if E is modular, i.e., mod(E) = 0. Further, the relative modular class of a subalgebroid is introduced and studied together with its application to holonomy, as well as the modular class of a skew algebroid relation. These notions provide, in particular, a unified approach to the concepts of a modular class of a Lie algebroid morphism and of a Poisson map.

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Authors and Affiliations

  1. Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, P.O. Box 21, 00-956, Warszawa, Poland

    Janusz Grabowski

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  1. Janusz Grabowski
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Correspondence to Janusz Grabowski.

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Research supported by the Polish Ministry of Science and Higher Education under the grant N N201 416839.

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Grabowski, J. Modular classes of skew algebroid relations. Transformation Groups 17, 989–1010 (2012). https://doi.org/10.1007/s00031-012-9197-2

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