# Drinfel'd double structures for Poincar\'e and Euclidean groups

## TL;DR

This paper classifies all three-dimensional Euclidean Poisson homogeneous spaces using Lie bialgebra structures, identifies the unique Drinfel'd double for the Euclidean group, and compares it with the Poincaré case, highlighting key differences.

## Contribution

It provides a complete classification of Euclidean Poisson homogeneous spaces and explicitly constructs the Drinfel'd double structure for the Euclidean group in 3D.

## Key findings

- Unique Drinfel'd double structure for Euclidean group identified
- All non-isomorphic 3D Euclidean Poisson homogeneous spaces constructed
- Differences between Lorentzian and Euclidean cases highlighted

## Abstract

All non-isomorphic three-dimensional Poisson homogeneous Euclidean spaces are constructed and analyzed, based on the classification of coboundary Lie bialgebra structures of the Euclidean group in 3-dimensions, and the only Drinfel'd double structure for this group is explicitly given. The similar construction for the Poincar\'e case is reviewed and the striking differences between the Lorentzian and Euclidean cases are underlined. Finally, the contraction scheme starting from Drinfel'd double structures of the $\mathfrak{so}(3,1)$ Lie algebra is presented.