Poisson–Lie Diffeomorphism Groups

Abstract

We explicitly construct a class of coboundary Poisson–Lie structures on the group of formal diffeomorphisms of ℝ n . Equivalently, these give rise to a class of coboundary triangular Lie bialgebra structures on the Lie algebra W n of formal vector fields on ℝ n . 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 ℝ n , thus making it a Poisson space. Moreover, the canonical action of the Poisson–Lie groups FDiff(ℝ m ) × FDiffℝ n ) gives rise to classes of compatible Poisson structures on the space J ∞(ℝ m ,ℝ n ) of infinite jets of smooth maps ℝ m → ℝ n , 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.