Transitive Lie algebroids of rank 1 and locally conformal symplectic structures

Abstract

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 Poincaré 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 an 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.