{"id":{"repo_id":"mit","oai_identifier":"oai:dspace.mit.edu:1721.1/67790"},"canonical_url":"https://search.dev.ndltd.org/etd/mit/oai:dspace.mit.edu:1721.1/67790","repository":{"repo_id":"mit","name":"MIT","base_url":"https://dspace.mit.edu/oai/request"},"display":{"title":"Quantized multiplicative quiver varieties and actions of higher genus braid groups","abstract":"In this thesis, a new class of algebras called quantized multiplicative quiver varieties A (Q), is constructed, depending upon a quiver Q, its dimension vector d, and a certain \"moment map\" parameter . The algebras Ad(Q) are obtained via quantum Hamiltonian reduction of another algebra D,(Matd(Q)) relative to a quantum moment map pq, both of which are also constructed herein. The algebras Dq(Matd(Q)) and A (Q) bear relations to many constructions in representation theory, some of which are spelled out herein, and many more whose precise formulation remains conjectural. When Q consists of a single vertex of dimension N with a single loop, the algebra Dq(MatA(Q)) is isomorphic to the algebra of quantum differential operators on G = GLN. In this case, for any n E Z>o, we construct a functor from the category of Dq-modules to representations of the type A double affine Hecke algebra of rank n. This functor is an instance of a more general construction which may be applied to any quasi-triangular Hopf algebra H, and yields representations of the elliptic braid group of rank n.","abstract_html":"In this thesis, a new class of algebras called quantized multiplicative quiver varieties A (Q), is constructed, depending upon a quiver Q, its dimension vector d, and a certain "moment map" parameter . The algebras Ad(Q) are obtained via quantum Hamiltonian reduction of another algebra D,(Matd(Q)) relative to a quantum moment map pq, both of which are also constructed herein. The algebras Dq(Matd(Q)) and A (Q) bear relations to many constructions in representation theory, some of which are spelled out herein, and many more whose precise formulation remains conjectural. When Q consists of a single vertex of dimension N with a single loop, the algebra Dq(MatA(Q)) is isomorphic to the algebra of quantum differential operators on G = GLN. In this case, for any n E Z>o, we construct a functor from the category of Dq-modules to representations of the type A double affine Hecke algebra of rank n. This functor is an instance of a more general construction which may be applied to any quasi-triangular Hopf algebra H, and yields representations of the elliptic braid group of rank n.","abstract_has_math":false,"creators":["Jordan, David Andrew"],"institution":"Massachusetts Institute of Technology","degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":"Massachusetts Institute of Technology. Dept. of Mathematics.","school":null,"contributors":[],"advisors":["Pavel Etingof."],"committee_chairs":[],"committee_members":[],"year":2011,"date_issued":"2011","date_published":"2011","updated_at":"2026-07-22T22:22:28Z","subjects":["Mathematics."],"languages":["eng"],"rights":["M.I.T. theses are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. See provided URL for inquiries about permission."],"rights_urls":["http://dspace.mit.edu/handle/1721.1/7582"],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/1721.1/67790","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Pavel Etingof."]},{"key":"dc:contributor.department","label":"Department","values":["Massachusetts Institute of Technology. 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They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. See provided URL for inquiries about permission."]},{"key":"dc:rights.uri","label":"Rights URI","values":["http://dspace.mit.edu/handle/1721.1/7582"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["http://hdl.handle.net/1721.1/67790"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2011.","Cataloged from PDF version of thesis.","Includes bibliographical references (p. 109-112)."]},{"key":"dc:description.abstract","label":"Abstract","values":["In this thesis, a new class of algebras called quantized multiplicative quiver varieties A (Q), is constructed, depending upon a quiver Q, its dimension vector d, and a certain \"moment map\" parameter . The algebras Ad(Q) are obtained via quantum Hamiltonian reduction of another algebra D,(Matd(Q)) relative to a quantum moment map pq, both of which are also constructed herein. The algebras Dq(Matd(Q)) and A (Q) bear relations to many constructions in representation theory, some of which are spelled out herein, and many more whose precise formulation remains conjectural. When Q consists of a single vertex of dimension N with a single loop, the algebra Dq(MatA(Q)) is isomorphic to the algebra of quantum differential operators on G = GLN. In this case, for any n E Z>o, we construct a functor from the category of Dq-modules to representations of the type A double affine Hecke algebra of rank n. This functor is an instance of a more general construction which may be applied to any quasi-triangular Hopf algebra H, and yields representations of the elliptic braid group of rank n."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["Ph.D."]},{"key":"dc:title","label":"Title","values":["Quantized multiplicative quiver varieties and actions of higher genus braid groups"]}]}],"canonical_facts":{"dc:contributor.advisor":["Pavel Etingof."],"dc:contributor.department":["Massachusetts Institute of Technology. Dept. of Mathematics."],"dc:contributor.other":["Massachusetts Institute of Technology. Dept. of Mathematics."],"dc:creator":["Jordan, David Andrew"],"dc:date.accessioned":["2011-12-19T18:51:45Z"],"dc:date.available":["2011-12-19T18:51:45Z"],"dc:date.issued":["2011"],"dc:description":["Thesis (Ph. 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In this case, for any n E Z>o, we construct a functor from the category of Dq-modules to representations of the type A double affine Hecke algebra of rank n. This functor is an instance of a more general construction which may be applied to any quasi-triangular Hopf algebra H, and yields representations of the elliptic braid group of rank n."],"dc:description.degree":["Ph.D."],"dc:identifier.uri":["http://hdl.handle.net/1721.1/67790"],"dc:language.iso":["eng"],"dc:publisher":["Massachusetts Institute of Technology"],"dc:rights":["M.I.T. theses are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. See provided URL for inquiries about permission."],"dc:rights.uri":["http://dspace.mit.edu/handle/1721.1/7582"],"dc:subject":["Mathematics."],"dc:title":["Quantized multiplicative quiver varieties and actions of higher genus braid groups"],"dc:type":["Thesis"]},"updated_at":"2026-07-22T22:22:28Z"}