Poisson structures on the Poincare group

An introduction to inhomogeneous Poisson groups is given. Poisson inhomogeneous $O(p,q)$ are shown to be coboundary, the generalized classical Yang-Baxter equation having only one-dimensional right hand side. Normal forms of the classical $r$-matrices for the Poincar\'{e} group (inhomogeneous $O(1,3)$) are calculated.