Poisson-Lie structures on infinite-dimensional jet groups and quantum groups related to them

We study the problem of classifying all Poisson-Lie structures on the group G(infinity) of formal diffeomorphisms of the real line R-1 which leave the origin fixed, as well as the extended group of diffeomorphisms G(0 infinity)superset of G(infinity) whose action on R-1 does not necessarily fix the origin. A complete local classification of all Poisson-Lie structures on the groups G(infinity) and G(0 infinity) is given. This includes a classification of all Lie-bialgebra structures on the Lie algebra G(infinity) of G(infinity), which we prove to be all of the coboundary type, and a classification of all Lie-bialgebra structures on the Lie algebra G(0 infinity) (the Witt algebra) of G(0 infinity) which also turned out to be all of the coboundary type. A large class of Poisson structures on the space V-lambda becomes a homogeneous Poisson space under the action of the Poisson-Lie group G(infinity). We construct a series of quantum semigroups whose quasiclassical limits are finite-dimensional Poisson-Lie quotient groups of G(infinity) and G(0 infinity). (C) 1999 American Institute of Physics. [S0022- 2488(99)00201-7].