Poisson-Lie Structures on Infinite-Dimensional Jet Groups and Quantum Groups Related to Them

Abstract

We study the problem of classifying all Poisson-Lie structures on the group $G_{\infty}$ of formal diffeomorphisms of the real line $\zR^{1}$ which leave the origin fixed, as well as the extended group of diffeomorphisms $G_{0\infty}\supset G_{\infty}$ whose action on $\zR^{1}$ does not necessarily fix the origin. A complete local classification of all Poisson-Lie structures on the groups $G_{\infty}$ and $G_{0\infty}$ is given. This includes a classification of all Lie-bialgebra structures on the Lie algebra $\Cal G_{\infty}$ of $G_{\infty}$, which we prove to be all of coboundary type, and a classification of all Lie-bialgebra strucutures on the Lie algebra $\Cal G_{0\infty}$ (the Witt algebra) of $G_{0\infty}$ which also turned out to be all of coboundary type. A large class of Poisson structures on the space $V_{\lambda}$ of $\lambda$-densities on the real line is found such that $V_{\lambda}$ becomes a homogeneous Poisson space under the action of the Poisson-Lie group $G_{\infty}$. We construct a series of quantum semigroups whose quasiclassical limits are finite-dimensional Poisson-Lie factor groups of $G_{\infty}$ and $G_{0\infty}$.