Jacobi and Poisson algebras

TL;DR

This paper develops a comprehensive algebraic framework for Jacobi and Poisson algebras, introducing new classification, representation, and deformation theories, including cohomological tools and bicrossed product constructions, with various examples and applications.

Contribution

It introduces a new cohomological classification for Jacobi and Poisson algebra extensions, including Frobenius structures, and generalizes bicrossed product constructions for Poisson algebras.

Findings

Characterization of Frobenius Jacobi algebras via integrals.

Construction of a cohomological object classifying Jacobi algebra extensions.

Identification of bicrossed products as a special case of the new framework.

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 subalgebra of codimension equal to ${\rm dim} (V)$. 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 $\mathcal{H}\mathcal{A}^{2} \bigl(P,\, Q ~|~ (\triangleleft, \, \triangleright, \, \leftharpoonup, \, \rightharpoonup)\bigl)$ is explicitly constructed as a classifying set for the bicrossed descent problem for extensions of Poisson algebras. Several examples and applications are provided.