Classification of $4$-dimensional complex Poisson algebras

TL;DR

This paper provides a comprehensive classification of 4-dimensional complex Poisson algebras by analyzing associated commutative associative and Lie algebra structures, offering a complete isomorphism classification.

Contribution

It presents the first complete classification of 4-dimensional complex Poisson algebras, integrating associative and Lie algebra classifications.

Findings

Complete list of 4-dimensional complex Poisson algebras

Isomorphism classes identified for associative structures

Lie algebra structures classified and incorporated

Abstract

The present paper is devoted to the complete classification of $4$-dimensional complex Poisson algebras, taking into account a classification, up to isomorphism, of the complex commutative associative algebras of dimension $4$, as well as by using a Lie algebra classification, up to isomorphism.