Degenerations of Poisson algebras

Abstract

We construct a method to obtain the algebraic classification of Poisson algebras defined on a commutative associative alge bra, and we apply it to obtain the classification of the 3-dimensional Poisson algebras. In addition, we study the geometric classification, the graph of degenerations and the closures of the orbits of the variety of 3-dimensional Poisson algebras. Finally, we also study the algebraic classification of the Poisson algebras defined on a commutative associative null-filiform or filiform algebra and, to enrich this classification, we study the degenerations between these particular Poisson algebras.