ABSTRACT
This chapter deals with Jacobi/Poisson algebras, the algebraic counterparts of Jacobi/Poisson manifolds. We introduce representations of a Jacobi algebra A and Frobenius Jacobi algebras as symmetric objects in the category. A characterization theorem for Frobenius Jacobi algebras is given in terms of integrals on Jacobi algebras. For a vector space V a non-abelian cohomological type object is constructed: it classifies all Jacobi algebras containing A as a subalgebra of codimension equal to the dimension of V. Representations of A are used in order to give the decomposition of the classifying object as a coproduct over all Jacobi A-module structures on V. The bicrossed product of two Poisson algebras recently introduced by Ni and Bai [189] appears as a special case of our construction. A new type of deformation of a given Poisson algebra Q is introduced and a cohomological type object is explicitly constructed as a classifying set for the bicrossed descent problem for extensions of Poisson algebras. Several examples and applications are provided.
