Drinfel'd double structures for Poincaré and Euclidean groups

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 Poincaré 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.

12 pages. Based on the contribution presented at "The 32nd International Colloquium on Group Theoretical Methods in Physics" (Group32), July 9-13, 2018