Poisson double structures

TL;DR

This paper introduces Poisson double algebroids and double Lie bialgebroids, exploring their Lie theory and relation to Poisson double groupoids, and revisits Lie 2-bialgebras through these structures.

Contribution

It defines and develops the theory of Poisson double algebroids and double Lie bialgebroids, linking them to Poisson double groupoids and Lie 2-bialgebras.

Findings

Established the Lie theory for Poisson double algebroids.

Demonstrated the differentiation and integration relations between these structures.

Revisited Lie 2-bialgebras using Poisson double structures.

Abstract

We introduce Poisson double algebroids, and the equivalent concept of double Lie bialgebroid, which arise as second-order infinitesimal counterparts of Poisson double groupoids. We develop their underlying Lie theory, showing how these objects are related by differentiation and integration. We use these results to revisit Lie 2-bialgebras by means of Poisson double structures.