Jacobi and Poisson algebras

  • Ana L. Agore

    Vrije Universiteit Brussel, Brussels, Belgium
  • Gigel Militaru

    University of Bucharest, Romania

Abstract

Jacobi/Poisson algebras are algebraic counterparts of Jacobi/Poisson manifolds. We introduce representations of a Jacobi algebra 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 a non-abelian cohomological type object is constructed: it classifies all Jacobi algebras containing as a sub algebra of codimension equal to . Representations of are used in order to give the decomposition of as a coproduct over all Jacobi -module structures on . 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 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.

Cite this article

Ana L. Agore, Gigel Militaru, Jacobi and Poisson algebras. J. Noncommut. Geom. 9 (2015), no. 4, pp. 1295–1342

DOI 10.4171/JNCG/224