Star calculus on Jacobi manifolds.
A linear isomorphism from the cotangent bundle onto the tangent bundle of a manifold, together with the standard Schouten-Nijenhuis bracket on the space of multivector fields, allows to define a nondegenerate Gerstenhaber bracket on the space of differential forms. Two examples of this construction are well known: the first corresponds to the isomorphism produced by a Riemannian metric, the second to the isomorphism coming from a symplectic structure. In both cases, the resulting Gerstenhaber bracket turnes out to be generated by an appropriate codifferential operator. In this paper, the author considers two examples of Jacobi manifolds (contact manifolds and locally conformal symplectic manifolds) which induce an isomorphism of cotangent and tangent bundle. In both cases, the author finds an explicit description of the corresponding Gerstenhaber bracket on the space of differential forms. Moreover, he shows that for each of these brackets there is an appropriate generating operator. The generating operators can be chosen to be of square zero, thus making the Gerstenhaber algebra structure on the space of forms an exact one. Finally, the author introduces the star operators for the contact and the locally conformal symplectic manifolds in such a way, that in both cases the generating operators of the Gerstenhaber brackets are obtained by conjugation of the de Rham differential by these star operators.
scientific article; zbMATH DE number 1780869
scientific article