Linearization of Poisson groupoids

TL;DR

This paper investigates the conditions under which Poisson structures of Poisson groupoids can be linearized around the unit, extending existing theorems and introducing new examples of Lie algebroids and LA-groupoids.

Contribution

It extends the Lagrangian neighbourhood theorem to cosymplectic Lie algebroids and proves linearizability of dual integrations of triangular bialgebroids, introducing new structures in the process.

Findings

Dual integrations of triangular bialgebroids are always linearizable.

Linearizability of a triangular Lie bialgebroid depends on the $r$-matrix being of cosymplectic type.

Product Poisson groupoid linearizability requires the Poisson structure on the unit space to be regular.

Abstract

Motivated by a search for Lie group structures on groups of Poisson diffeomorphisms [24], we investigate linearizability of Poisson structures of Poisson groupoids around the unit section. After extending the Lagrangian neighbourhood theorem to the setting of cosymplectic Lie algebroids, we establish that dual integrations of triangular bialgebroids are always linearizable. Additionally, we show that the (non-dual) integration of a triangular Lie bialgebroid is linearizable whenever the $r$-matrix is of so-called cosymplectic type. The proof relies on the integration of a triangular Lie bialgebroid to a symplectic LA-groupoid, and in the process we define interesting new examples of double Lie algebroids and LA-groupoids. We also show that the product Poisson groupoid can only be linearizable when the Poisson structure on the unit space is regular.