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Modular classes of Lie algebroid morphisms. (English) Zbl 1167.53068

This paper is a study of modular classes of Lie algebroid morphisms. The modular class of a Poisson manifold was introduced by Koszul. In 1997, A. Weinstein [J. Geom. Phys. 23, No. 3–4, 379–394 (1997; Zbl 0902.58013)] showed that this class is the obstruction to the existence of a smooth Poisson trace. The modular class was then generalized for Lie algebroids and studied by several authors. In this paper, the authors concentrate on the notion of relative modular class. Using this notion, they study the behavior of the modular class of a Lie algebroid under general Lie algebroid morphisms. Generalized Lie algebroid morphisms and the Morita equivalence of Lie algebroids are also defined. It is proved in particular that the modular classes of isomorphic generalized morphisms are equal. Finally, many questions concerning the modular classes of Lie algebroids and their morphisms are discussed.

MSC:

53D17 Poisson manifolds; Poisson groupoids and algebroids
53D50 Geometric quantization
53D55 Deformation quantization, star products

Citations:

Zbl 0902.58013
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References:

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