Poisson Structures on Trivial Extension Algebras

TL;DR

This paper introduces a new class of Poisson structures on trivial extension algebras, linking them to contravariant derivatives and Lie algebroids, and explores their cohomological properties with relevant examples.

Contribution

It generalizes known Poisson structures on trivial extensions and establishes a correspondence with contravariant derivatives and Lie algebroids.

Findings

Established a one-to-one correspondence between Poisson structures and contravariant derivatives.

Analyzed properties of the first Poisson cohomology.

Provided examples from Poisson modules and submanifolds.

Abstract

We present a class of Poisson structures on trivial extension algebras which generalize some known structures induced by Poisson modules. We show that there exists a one-to-one correspondence between such a class of Poisson structures and some data involving (not necessarily flat) contravariant derivatives, and then we give a formulation of this result in terms of Lie algebroids. Some properties of the first Poisson cohomology are also presented. Examples coming from Poisson modules and Poisson submanifolds are given.