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∞ of formal diffeomorphisms of the real line R1 which leave the origin fixed, as well as the extended group of diffeomorphisms G0∞⊃G∞ whose action on R1 does not necessarily fix the origin. A complete local classification of all Poisson–Lie structures on the groups G∞ and G0∞ is given. This includes a classification of all Lie-bialgebra structures on the Lie algebra G∞ of G∞, which we prove to be all of the coboundary type, and a classification of all Lie-bialgebra strucutures on the Lie algebra G0∞ (the Witt algebra) of G0∞ which also turned out to be all of the coboundary type. A large class of Poisson structures on the space Vλ of λ-densities on the real line is found such that Vλ becomes a homogeneous Poisson space under the action of the Poisson–Lie group G∞. We construct a series of quantum semigroups whose quasiclassical limits are finite-dimensional Poisson–Lie quotient groups of G∞ and G0∞.