Poisson Diffeomorphism Groups

Abstract

We construct explicitly a class of coboundary Poisson-Lie structures on the group of formal diffeomorphisms of ${\Bbb R}^n$. Equivalently, these give rise to a class of coboundary triangular Lie bialgebra structures on the Lie algebra $W_n$ of formal vector fields on ${\Bbb R}^n$. We conjecture that this class accounts for all such coboundary structures. The natural action of the constructed Poisson-Lie diffeomorphism groups induces large classes of compatible Poisson structures on ${\Bbb R}^n$, thus making it a Poisson homogeneous space. Moreover, the left-right action of the Poisson-Lie groups $FDiff({\Bbb R}^m)\times FDiff({\Bbb R}^n)$ induces classes of compatible Poisson structures on the space $J^{\infty}({\Bbb R}^m,{\Bbb R}^n)$ of infinite jets of smooth maps ${\Bbb R}^m\to {\Bbb R}^n$, which makes it also a Poisson homogeneous space for this action. Initial steps towards classification of these structures are taken.