Splitting theorems for Poisson and related structures

TL;DR

This paper introduces a new approach to splitting theorems in Poisson geometry and related structures, generalizing classical results and extending them to equivariant and novel contexts.

Contribution

It develops a novel method for proving splitting theorems, leading to new generalizations and extensions in Poisson, Lie algebroid, Dirac, and generalized complex structures.

Findings

Generalized splitting theorems for Poisson and related structures

Extensions to equivariant settings

New formulations in broader contexts

Abstract

According to the Weinstein splitting theorem, any Poisson manifold is locally, near any given point, a product of a symplectic manifold with another Poisson manifold whose Poisson structure vanishes at the point. Similar splitting results are known e.g. for Lie algebroids, Dirac structures and generalized complex structures. In this paper, we develop a novel approach towards these results that leads to various generalizations, including their equivariant versions as well as their formulations in new contexts.