Poisson–Lie group structures on semidirect products

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 the matched pairs given by the Iwasawa decomposition of simple Lie groups, we find that the associated Lie bialgebra is coboundary.