"Transitive Lie algebroids of rank 1 and locally conformal symplectic structures"

(joint work with R. Kadobianski, V. Kushnirevitch and R.Wolak, Journal of Geometry and Physiscs, in print)
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The aim of the paper is the study of transitive Lie algebroids with the trivial 1-rank adjoint bundle of isotropy Lie algebras g@M×R. We show that a locally conformal symplectic (l.c.s.) structure defines such a Lie algebroid, so our algebroids are a natural generalisation of l.c.s. structures. We prove that such a Lie algebroid has the Poincare duality property for the Lie algebroid cohomology (TUIO-property) if and only if the top dimensional cohomology space is non trivial. Moreover, if an algebroid is defined by a l.c.s. structure, then this algebroid is a TUIO-Lie algebroid if and only if the associated l.c.s. structure is a globally conformal symplectic structure.