Jacobi and Poisson algebras

DOI10.4171/JNCG/224zbMATH Open1378.17034arXiv1406.3529MaRDI QIDQ5963572

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Publication date: 22 February 2016

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

Full work available at URL: https://arxiv.org/abs/1406.3529

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