Algebroid Desingularizable Poisson Structures

TL;DR

This paper introduces algebroid desingularizable Poisson manifolds, a new class generalizing several known structures, and explores their properties and limitations through Lie algebra examples.

Contribution

It defines a new class of Poisson manifolds induced by symplectic Lie algebroids with almost-injective anchors and analyzes their existence in various Lie algebra contexts.

Findings

Duals of certain Lie algebras do not admit these structures.

Provides examples of nilpotent Lie algebras with and without desingularizable Poisson structures.

Abstract

We introduce algebroid desingularizable Poisson manifolds, a class of Poisson manifolds induced by symplectic Lie algebroids with almost-injective anchors, generalizing structures including log-symplectic, $b^m$-symplectic, $E$-symplectic manifolds, and hypersurface algebroids. We show that the dual of real, finite-dimensional, non-abelian, reductive Lie algebras never admit such algebroids. We finish by giving two infinite families of $2$-step nilpotent Lie algebras, one of which is desingularizable, and one of which is not.