$\theta$-almost twisted Poisson cohomology

TL;DR

This paper introduces $ heta$-almost twisted Poisson structures on manifolds, develops their cohomology theory, and provides explicit examples, expanding the understanding of twisted Poisson geometry with a closed 1-form component.

Contribution

It defines $ heta$-almost twisted Poisson structures, constructs their associated cohomology, and characterizes these structures on low-dimensional manifolds.

Findings

Characterization of $ heta$-almost twisted Poisson structures on low-dimensional manifolds.

Construction of the Lie-Rinehart algebra and cochain complex for these structures.

Explicit example of the cohomology on $R^5$.

Abstract

We introduce the notion of a $\theta$-almost twisted Poisson structure on manifolds, which involves incorporating a closed $1$-form $\theta$ into twisted Poisson structures under specific conditions. We provide a characterization of this structure on low-dimensional manifolds and construct the Lie-Rinehart algebra on the module of $1$-forms on manifolds equipped with this structure. This construction leads to a cochain complex and its associated cohomology, which we refer to as $\theta$-almost twisted Poisson cohomology. An example illustrating this cohomology is also presented on $\mathbb{R}^5$.