1995/06/08 by Ognyan Stoyanov, Stoyanov, Ognyan
Mathematics · #FOS: Mathematics #Quantum Algebra (math.QA) #math.QA #q-alg
paper · pdf · doi:10.48550/arxiv.q-alg/9506008
79 pages, AmSTeX file
arxiv created 1995/06/08 · arxiv updated 2009/11/30
We study the problem of classifying all Poisson-Lie structures on the group G∞ of formal diffeomorphisms of the real line \zR1 which leave the origin fixed, as well as the extended group of diffeomorphisms G0∞⊃ G∞ whose action on \zR1 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 \Cal G∞ of G∞, which we prove to be all of coboundary type, and a classification of all Lie-bialgebra strucutures on the Lie algebra \Cal G0∞ (the Witt algebra) of G0∞ which also turned out to be all of 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 factor groups of G∞ and G0∞.