Poisson-Lie groupoids and the contraction procedure

TL;DR

This paper extends the contraction procedure from Lie algebras to Lie bialgebroids, providing explicit integration and exploring the resulting structures, thereby broadening the understanding of Lie groupoid contractions.

Contribution

It introduces a novel extension of the contraction method to Lie bialgebroids, including explicit integration of the dual structures.

Findings

Constructed a bundle of central extensions with Lie bialgebroid structures.

Explicitly integrated the dual Lie bialgebroid.

Connected contraction procedures with Lie groupoid theory.

Abstract

On the level of Lie algebras, the contraction procedure is a method to create a new Lie algebra from a given Lie algebra by rescaling generators and letting the scaling parameter tend to zero. One of the most well-known examples is the contraction from su(2) to e(2), the Lie algebra of upper-triangular matrices with zero trace and purely imaginary diagonal. In this paper, we will consider an extension of this contraction by taking also into consideration the natural bialgebra structures on these Lie algebras. This will give a bundle of central extensions of the above Lie algebras with a Lie bialgebroid structure having transversal component. We consider as well the dual Lie bialgebroid, which is in a sense easier to understand, and whose integration can be explicitly presented.