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Poisson-lie structures on infinite-dimensional jet groups and their quantizationStoyanov, Ognyan S. 06 June 2008 (has links)
We study the problem of classifying all Poisson-Lie structures on the group Gy of local diffeomorphisms of the real line R¹ which leave the origin fixed, as well as the extended group of diffeomorphisms G₀<sub>∞</sub> ⊃ G<sub>∞</sub> whose action on R¹ does not necessarily fix the origin.
A complete classification of all Poisson-Lie structures on the group G<sub>∞</sub> is given. All Poisson-Lie structures of coboundary type on the group G₀<sub>∞</sub> are classified. This includes a classification of all Lie-bialgebra structures on the Lie algebra G<sub>∞</sub> of G<sub>∞</sub>, which we prove to be all of coboundary type, and a classification of all Lie-bialgebra structures of coboundary type on the Lie algebra Go<sub>∞</sub> of Go<sub>∞</sub> which is the Witt algebra.
A large class of Poisson structures on the space V<sub>λ</sub> of λ-densities on the real line is found such that V<sub>λ</sub> becomes a homogeneous Poisson space under the action of the Poisson-Lie group G<sub>∞</sub>.
We construct a series of finite-dimensional quantum groups whose quasiclassical limits are finite-dimensional Poisson-Lie factor groups of G<sub>∞</sub> and G₀<sub>∞</sub>. / Ph. D.
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