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Transitive Lie algebroids of rank 1 and locally conformal symplectic structures. (English) Zbl 1034.53084

The subject of this paper is Poincaré duality in the framework of Lie algebroid cohomology. This was first studied by S. Evens, J.-H. Lu and A. Weinstein [Q. J. Math., Oxf. II. Ser. 50, 417–436 (1999; Zbl 0968.58014)] (in the geometric setting) and by J. Huebschmann [J. Reine Angew. Math. 510, 103–159 (1999; Zbl 1034.53083)] (in the algebraic setting). Unfortunately, the authors fail to acknowledge these works, and they develop an unnecessarily complicated set up, along the lines of one of the authors previous work [J. Geom. Phys. 46, 151–158 (2003; Zbl 1048.58014)]. Moreover, their methods only apply in the transitive case.
The main result of the paper states that Poincaré duality holds for a transitive Lie algebroid, with trivial, rank one, adjoint bundle, over a compact, oriented, connected manifold if and only if the top Lie algebroid cohomology is non-zero.

MSC:

53D17 Poisson manifolds; Poisson groupoids and algebroids
58H05 Pseudogroups and differentiable groupoids
22A22 Topological groupoids (including differentiable and Lie groupoids)
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