Lie groups of Poisson diffeomorphisms

TL;DR

This paper develops a method using Poisson groupoids to establish Lie group structures on subgroups of Poisson diffeomorphisms, demonstrating regular infinite-dimensional Lie groups for various classes of Poisson manifolds.

Contribution

It introduces a novel approach leveraging Poisson groupoids to endow Poisson diffeomorphism groups with Lie group structures, including for complex classes like log-symplectic and scattering-symplectic manifolds.

Findings

Poisson diffeomorphism groups are regular infinite-dimensional Lie groups for several classes of Poisson manifolds.

The approach applies to normal-crossing log-symplectic, elliptic symplectic, scattering-symplectic, and cosymplectic manifolds.

Establishes Lie group structures on subgroups of Poisson diffeomorphisms using Poisson groupoids.

Abstract

By considering suitable Poisson groupoids, we develop an approach to obtain Lie group structures on (subgroups of) the Poisson diffeomorphism groups of various classes of Poisson manifolds. As applications, we show that the Poisson diffeomorphism groups of (normal-crossing) log-symplectic, elliptic symplectic, scattering-symplectic and cosymplectic manifolds are regular infinite-dimensional Lie groups.