Jacobi and Poisson algebras

Abstract

Jacobi/Poisson algebras are 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 $\mathcal{J}{\mathcal{H}}^2(V,A)$ is constructed: it classifies all Jacobi algebras containing $A$ as a sub algebra of codimension equal to $\dim (V)$. Representations of $A$ are used in order to give the decomposition of $\mathcal{J}{\mathcal{H}}^2(V,A)$ as a coproduct over all Jacobi $A$-module structures on $V$. The bicrossed product $P\bowtie Q$ of two Poisson algebras recently introduced by Ni and Bai appears as a special case of our construction. A new type of deformations of a given Poisson algebra $Q$ is introduced and a cohomological type object $\mathcal{H}{\mathcal{A}}^2(P,Q|(◃,▹,\leftharpoonup ,\rightharpoonup ))$ is explicitly constructed as a classifying set for the bicrossed descent problem for extensions of Poisson algebras. Several examples and applications are provided.