{"id":{"repo_id":"vt","oai_identifier":"oai:vtechworks.lib.vt.edu:10919/38421"},"canonical_url":"https://search.dev.ndltd.org/etd/vt/oai:vtechworks.lib.vt.edu:10919/38421","repository":{"repo_id":"vt","name":"Virginia Tech","base_url":"https://vtechworks.lib.vt.edu/oai/request"},"display":{"title":"Poisson-lie structures on infinite-dimensional jet groups and their quantization","abstract":"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>.","abstract_html":"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&lt;sub&gt;∞&lt;/sub&gt; ⊃ G&lt;sub&gt;∞&lt;/sub&gt; whose action on R¹ does not necessarily fix the origin. A complete classification of all Poisson-Lie structures on the group G&lt;sub&gt;∞&lt;/sub&gt; is given. All Poisson-Lie structures of coboundary type on the group G₀&lt;sub&gt;∞&lt;/sub&gt; are classified. This includes a classification of all Lie-bialgebra structures on the Lie algebra G&lt;sub&gt;∞&lt;/sub&gt; of G&lt;sub&gt;∞&lt;/sub&gt;, 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&lt;sub&gt;∞&lt;/sub&gt; of Go&lt;sub&gt;∞&lt;/sub&gt; which is the Witt algebra. A large class of Poisson structures on the space V&lt;sub&gt;λ&lt;/sub&gt; of λ-densities on the real line is found such that V&lt;sub&gt;λ&lt;/sub&gt; becomes a homogeneous Poisson space under the action of the Poisson-Lie group G&lt;sub&gt;∞&lt;/sub&gt;. We construct a series of finite-dimensional quantum groups whose quasiclassical limits are finite-dimensional Poisson-Lie factor groups of G&lt;sub&gt;∞&lt;/sub&gt; and G₀&lt;sub&gt;∞&lt;/sub&gt;.","abstract_has_math":false,"creators":["Stoyanov, Ognyan S."],"institution":"Virginia Tech","degree_name":"Ph. D.","degree_level":"doctoral","degree_discipline":"Mathematical Physics","degree_department":"Mathematical Physics","school":null,"contributors":[],"advisors":[],"committee_chairs":["Kupershmidt, Boris","Slawny, Joseph"],"committee_members":["Zweifel, Paul F.","Greenberg, William","Haskell, Peter","Klaus, Martin","Bowden, Robert L."],"year":1993,"date_issued":"1993","date_published":"1993","updated_at":"2026-07-22T22:19:24Z","subjects":[],"languages":["en"],"rights":["In Copyright"],"rights_urls":["http://rightsstatements.org/vocab/InC/1.0/"],"identifier_entries":[{"key":"dc:identifier.other","label":"Dc Identifier Other","values":["etd-06062008-170612"],"render_values":[{"text":"etd-06062008-170612","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/10919/38421","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.committeechair","label":"Committee Chair","values":["Kupershmidt, Boris","Slawny, Joseph"]},{"key":"dc:contributor.committeemember","label":"Committee Member","values":["Zweifel, Paul F.","Greenberg, William","Haskell, Peter","Klaus, Martin","Bowden, Robert L."]},{"key":"dc:contributor.department","label":"Department","values":["Mathematical Physics"]},{"key":"dc:creator","label":"Author","values":["Stoyanov, Ognyan S."]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2014-03-14T21:14:19Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2014-03-14T21:14:19Z","2008-06-06"]},{"key":"dc:date.issued","label":"Date","values":["1993"]},{"key":"dc:publisher","label":"Institution","values":["Virginia Tech"]},{"key":"dc:type","label":"Dc Type","values":["Dissertation"]},{"key":"dc:type.dcmitype","label":"Dc Type Dcmitype","values":["Text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematical Physics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["doctoral"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph. 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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>."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["Ph. D."]},{"key":"dc:format.medium","label":"Dc Format Medium","values":["BTD"]},{"key":"dc:format.mimetype","label":"Dc Format Mimetype","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Poisson-lie structures on infinite-dimensional jet groups and their quantization"]}]}],"canonical_facts":{"dc:contributor.committeechair":["Kupershmidt, Boris","Slawny, Joseph"],"dc:contributor.committeemember":["Zweifel, Paul F.","Greenberg, William","Haskell, Peter","Klaus, Martin","Bowden, Robert L."],"dc:contributor.department":["Mathematical Physics"],"dc:creator":["Stoyanov, Ognyan S."],"dc:date.accessioned":["2014-03-14T21:14:19Z"],"dc:date.available":["2014-03-14T21:14:19Z","2008-06-06"],"dc:date.issued":["1993"],"dc:description.abstract":["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>."],"dc:description.degree":["Ph. D."],"dc:format.medium":["BTD"],"dc:format.mimetype":["application/pdf"],"dc:identifier.other":["etd-06062008-170612"],"dc:identifier.uri":["http://hdl.handle.net/10919/38421"],"dc:language.iso":["en"],"dc:publisher":["Virginia Tech"],"dc:rights":["In Copyright"],"dc:rights.uri":["http://rightsstatements.org/vocab/InC/1.0/"],"dc:title":["Poisson-lie structures on infinite-dimensional jet groups and their quantization"],"dc:type":["Dissertation"],"dc:type.dcmitype":["Text"],"thesis:degree_discipline":["Mathematical Physics"],"thesis:degree_level":["doctoral"],"thesis:degree_name":["Ph. D."],"thesis:institution_name":["Virginia Polytechnic Institute and State University"]},"updated_at":"2026-07-22T22:19:24Z"}