Poisson 2-groups

TL;DR

This paper establishes a one-to-one correspondence between certain classes of Poisson and quasi-Poisson Lie 2-groups and their algebraic counterparts, Lie 2-bialgebras and quasi-Lie 2-bialgebras, using a universal lifting theorem.

Contribution

It extends classical results to a 2-categorical setting, providing a new framework for understanding Poisson Lie 2-groups and their algebraic structures.

Findings

Established correspondence between connected, simply-connected Poisson Lie 2-groups and Lie 2-bialgebras.

Extended the correspondence to quasi-Poisson 2-groups and quasi-Lie 2-bialgebras.

Developed a universal lifting theorem relating multiplicative polyvector fields and polydifferentials.

Abstract

We prove a 2-categorical analogue of a classical result of Drinfeld: there is a one-to-one correspondence between connected, simply-connected Poisson Lie 2-groups and Lie 2-bialgebras. In fact, we also prove that there is a one-to-one correspondence between connected, simply connected quasi-Poisson 2-groups and quasi-Lie 2-bialgebras. Our approach relies on a "universal lifting theorem" for Lie 2-groups: an isomorphism between the graded Lie algebras of multiplicative polyvector fields on the Lie 2-group on one hand and of polydifferentials on the corresponding Lie 2-algebra on the other hand.