Double Poisson extensions

TL;DR

This paper introduces Poisson double extensions as a Poisson algebra analogue of double Ore extensions, showing their role in deformation quantization and exploring their properties and relationships with other Poisson structures.

Contribution

It defines Poisson double extensions, connects them to double Ore extensions via deformation quantization, and studies their modular derivations and structural relationships.

Findings

Poisson double extensions are deformation quantizations of certain double Ore extensions.

The paper characterizes modular derivations of Poisson double extensions.

Examples illustrate the theoretical concepts and relationships.

Abstract

A double Ore extension was introduced by James Zhang and Jun Zhang [26] to study a class of Artin-Schelter regular algebras. Here we give a definition of Poisson double extension which may be considered as an analogue of double Ore extension and show that algebras in a class of double Ore extensions are deformation quantizations of Poisson double extensions. We also investigate the modular derivations of Poisson double extensions and the relationship between Poisson double extensions and iterated Poisson polynomial extensions. Results are illustrated by examples.