Degenerations of Poisson algebras

TL;DR

This paper develops a method for classifying 3-dimensional Poisson algebras both algebraically and geometrically, analyzing degenerations and orbit closures, and extends the classification to special types of associative algebras.

Contribution

It introduces a new algebraic classification method for Poisson algebras and applies it to classify 3-dimensional cases and specific associative algebra structures.

Findings

Complete classification of 3D Poisson algebras

Description of the degeneration graph and orbit closures

Classification of Poisson algebras on null-filiform and filiform algebras

Abstract

We construct a method to obtain the algebraic classification of Poisson algebras defined on a commutative associative algebra, 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.