{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/87004"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/87004","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Geometry of Affine Actions","abstract":"We introduce a new factorization for elements of the loop group LSL2 = SL2( C ((lambda-1))). We then define a new decomposition for the flag manifold FL2 = LSL2/ B+ (= SL2&d14;/B&d14; + ), into isomorphic overlapping cells. This allows us to examine globally the left vector field action of the homogeneous Heisenberg subalgebra Hsl2 on FL2. On each cell of the quotient H -\\FL2, we find coordinates for the modified non-linear Schrodinger (mNLS) hierarchy, with Miura, maps to NLS coordinates on the Grassmannian. We also find transformation rules relating coordinates across cells. The right vector field action of n&d14;-⊂ sl2&d14; attaches to the coordinate ring of each cell an object e +/-2&phis;. We show that this gives a special line bundle structure on H-\\FL2. Finally, we define on each cell a Hamiltonian structure. This leads to a vertex algebra structure, and from the n&d14;- -action, a vertex algebra module structure, on each cell. We establish the compatibility of these structures across cells to show that H -\\FL2 is a vertex variety, i.e. a variety whose structure sheaf is a sheaf of vertex algebras.","abstract_html":"We introduce a new factorization for elements of the loop group LSL2 = SL2( C ((lambda-1))). We then define a new decomposition for the flag manifold FL2 = LSL2/ B+ (= SL2&d14;/B&d14; + ), into isomorphic overlapping cells. This allows us to examine globally the left vector field action of the homogeneous Heisenberg subalgebra Hsl2 on FL2. On each cell of the quotient H -\\FL2, we find coordinates for the modified non-linear Schrodinger (mNLS) hierarchy, with Miura, maps to NLS coordinates on the Grassmannian. We also find transformation rules relating coordinates across cells. The right vector field action of n&d14;-⊂ sl2&d14; attaches to the coordinate ring of each cell an object e +/-2&phis;. We show that this gives a special line bundle structure on H-\\FL2. Finally, we define on each cell a Hamiltonian structure. This leads to a vertex algebra structure, and from the n&d14;- -action, a vertex algebra module structure, on each cell. We establish the compatibility of these structures across cells to show that H -\\FL2 is a vertex variety, i.e. a variety whose structure sheaf is a sheaf of vertex algebras.","abstract_has_math":false,"creators":["Kilmurray, Donough"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Maarten Bergvelt"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2015,"date_issued":"2015-09-28T15:20:34Z","date_published":"2015-09-28T15:20:34Z","updated_at":"2026-07-22T22:26:28Z","subjects":["Mathematics"],"languages":["eng"],"rights":[],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["(MiAaPQ)AAI9990040"],"render_values":[{"text":"(MiAaPQ)AAI9990040","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/87004","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Maarten Bergvelt"]},{"key":"dc:creator","label":"Author","values":["Kilmurray, Donough"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2015-09-28T15:20:34Z","10000-01-01","2000"]},{"key":"dc:type","label":"Dc Type","values":["text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Mathematics"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["eng"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/87004","(MiAaPQ)AAI9990040"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["We introduce a new factorization for elements of the loop group LSL2 = SL2( C ((lambda-1))). We then define a new decomposition for the flag manifold FL2 = LSL2/ B+ (= SL2&d14;/B&d14; + ), into isomorphic overlapping cells. This allows us to examine globally the left vector field action of the homogeneous Heisenberg subalgebra Hsl2 on FL2. On each cell of the quotient H -\\FL2, we find coordinates for the modified non-linear Schrodinger (mNLS) hierarchy, with Miura, maps to NLS coordinates on the Grassmannian. We also find transformation rules relating coordinates across cells. The right vector field action of n&d14;-⊂ sl2&d14; attaches to the coordinate ring of each cell an object e +/-2&phis;. We show that this gives a special line bundle structure on H-\\FL2. Finally, we define on each cell a Hamiltonian structure. This leads to a vertex algebra structure, and from the n&d14;- -action, a vertex algebra module structure, on each cell. We establish the compatibility of these structures across cells to show that H -\\FL2 is a vertex variety, i.e. a variety whose structure sheaf is a sheaf of vertex algebras.","Made available in DSpace on 2015-09-28T15:20:34Z (GMT). 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We then define a new decomposition for the flag manifold FL2 = LSL2/ B+ (= SL2&d14;/B&d14; + ), into isomorphic overlapping cells. This allows us to examine globally the left vector field action of the homogeneous Heisenberg subalgebra Hsl2 on FL2. On each cell of the quotient H -\\FL2, we find coordinates for the modified non-linear Schrodinger (mNLS) hierarchy, with Miura, maps to NLS coordinates on the Grassmannian. We also find transformation rules relating coordinates across cells. The right vector field action of n&d14;-⊂ sl2&d14; attaches to the coordinate ring of each cell an object e +/-2&phis;. We show that this gives a special line bundle structure on H-\\FL2. Finally, we define on each cell a Hamiltonian structure. This leads to a vertex algebra structure, and from the n&d14;- -action, a vertex algebra module structure, on each cell. We establish the compatibility of these structures across cells to show that H -\\FL2 is a vertex variety, i.e. a variety whose structure sheaf is a sheaf of vertex algebras.","Made available in DSpace on 2015-09-28T15:20:34Z (GMT). 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