Poisson–Lie Diffeomorphism Groups

We explicitly construct a class of coboundary Poisson–Lie structures on the group of formal diffeomorphisms of ℝn. Equivalently, these give rise to a class of coboundary triangular Lie bialgebra structures on the Lie algebra Wn of formal vector fields on ℝn. We conjecture that this class accounts for all such coboundary structures. The natural action of the constructed Poisson–Lie diffeomorphism groups gives rise to large classes of compatible Poisson structures on ℝn, thus making it a Poisson space. Moreover, the canonical action of the Poisson–Lie groups FDiff(ℝm) × FDiffℝn) gives rise to classes of compatible Poisson structures on the space J∞(ℝm,ℝn) of infinite jets of smooth maps ℝm → ℝn, which makes it also a Poisson space for this action. Poisson modules of generalized densities are also constructed. Initial steps towards a classification of these structures are taken.