Poisson-Lie Group structures on semidirect products

TL;DR

This paper investigates Poisson-Lie structures on semidirect product groups derived from matched pairs of Lie groups, especially those from Iwasawa decompositions, revealing coboundary Lie bialgebras.

Contribution

It demonstrates how dual bundles of Lie algebroids from matched pairs form Poisson-Lie groups and connects this to Iwasawa decompositions with coboundary Lie bialgebras.

Findings

Dual bundles form Poisson-Lie groups with semidirect product structures.

Matched pairs from Iwasawa decompositions lead to coboundary Lie bialgebras.

The work extends understanding of Poisson-Lie structures in Lie group theory.

Abstract

We look at the Poisson structure on the total space of the dual bundle to the Lie algebroid arising from a matched pair of Lie groups. This dual bundle, with the natural semidirect product group structure, becomes a Poisson-Lie group as suggested by a recent work of Stachura. Moreover, when we start from matched pairs given by the Iwasawa decomposition of simple Lie groups, the associated Lie bialgebra is coboundary.