{"id":{"repo_id":"toronto-retro","oai_identifier":"oai:utoronto.scholaris.ca:1807/32806"},"canonical_url":"https://search.dev.ndltd.org/etd/toronto-retro/oai:utoronto.scholaris.ca:1807/32806","repository":{"repo_id":"toronto-retro","name":"University of Toronto","base_url":"https://utoronto.scholaris.ca/server/oai/request"},"display":{"title":"The Pure Virtual Braid Group is Quadratic","abstract":"If an augmented algebra K over Q is filtered by powers of its augmentation ideal I, the associated graded algebra grK need not in general be quadratic: although it is generated in degree 1, its relations may not be generated by homogeneous relations of degree 2. In this thesis we give a sufficient criterion (called the PVH Criterion) for grK to be quadratic. When K is the group algebra of a group G, quadraticity is known to be equivalent to the existence of a (not necessarily homomorphic) universal finite type invariant for G. Thus the PVH Criterion also implies the existence of a universal finite type invariant for the group G. We apply the PVH Criterion to the group algebra of the pure virtual braid group (also known as the quasi-triangular group), and show that the corresponding associated graded algebra is quadratic, and hence that these groups have a universal finite type invariant.","abstract_html":"If an augmented algebra K over Q is filtered by powers of its augmentation ideal I, the associated graded algebra grK need not in general be quadratic: although it is generated in degree 1, its relations may not be generated by homogeneous relations of degree 2. In this thesis we give a sufficient criterion (called the PVH Criterion) for grK to be quadratic. When K is the group algebra of a group G, quadraticity is known to be equivalent to the existence of a (not necessarily homomorphic) universal finite type invariant for G. Thus the PVH Criterion also implies the existence of a universal finite type invariant for the group G. We apply the PVH Criterion to the group algebra of the pure virtual braid group (also known as the quasi-triangular group), and show that the corresponding associated graded algebra is quadratic, and hence that these groups have a universal finite type invariant.","abstract_has_math":false,"creators":["Lee, Peter"],"institution":null,"degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":"Mathematics","school":null,"contributors":[],"advisors":["Bar-Natan, Dror"],"committee_chairs":[],"committee_members":[],"year":2012,"date_issued":"2012-08-31","date_published":"2012-08-31","updated_at":"2026-07-27T21:28:22Z","subjects":["Pure Virtual Braids","Quadraticity","Graded 1-Formal","Universal Finite Type Invariant"],"languages":["en_ca"],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/1807/32806","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Bar-Natan, Dror"]},{"key":"dc:contributor.department","label":"Department","values":["Mathematics"]},{"key":"dc:creator","label":"Author","values":["Lee, Peter"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2012-06"]},{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2012-08-31T14:10:04Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["NO_RESTRICTION","2012-08-31T14:10:04Z"]},{"key":"dc:date.issued","label":"Date","values":["2012-08-31"]},{"key":"dc:type","label":"Dc Type","values":["Thesis"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Pure Virtual Braids","Quadraticity","Graded 1-Formal","Universal Finite Type Invariant"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language.iso","label":"Language (ISO)","values":["en_ca"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["http://hdl.handle.net/1807/32806"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["If an augmented algebra K over Q is filtered by powers of its augmentation ideal I, the associated graded algebra grK need not in general be quadratic: although it is generated in degree 1, its relations may not be generated by homogeneous relations of degree 2. In this thesis we give a sufficient criterion (called the PVH Criterion) for grK to be quadratic. When K is the group algebra of a group G, quadraticity is known to be equivalent to the existence of a (not necessarily homomorphic) universal finite type invariant for G. Thus the PVH Criterion also implies the existence of a universal finite type invariant for the group G. We apply the PVH Criterion to the group algebra of the pure virtual braid group (also known as the quasi-triangular group), and show that the corresponding associated graded algebra is quadratic, and hence that these groups have a universal finite type invariant."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["PhD"]},{"key":"dc:title","label":"Title","values":["The Pure Virtual Braid Group is Quadratic"]}]}],"canonical_facts":{"dc:contributor.advisor":["Bar-Natan, Dror"],"dc:contributor.department":["Mathematics"],"dc:creator":["Lee, Peter"],"dc:date":["2012-06"],"dc:date.accessioned":["2012-08-31T14:10:04Z"],"dc:date.available":["NO_RESTRICTION","2012-08-31T14:10:04Z"],"dc:date.issued":["2012-08-31"],"dc:description.abstract":["If an augmented algebra K over Q is filtered by powers of its augmentation ideal I, the associated graded algebra grK need not in general be quadratic: although it is generated in degree 1, its relations may not be generated by homogeneous relations of degree 2. In this thesis we give a sufficient criterion (called the PVH Criterion) for grK to be quadratic. When K is the group algebra of a group G, quadraticity is known to be equivalent to the existence of a (not necessarily homomorphic) universal finite type invariant for G. Thus the PVH Criterion also implies the existence of a universal finite type invariant for the group G. We apply the PVH Criterion to the group algebra of the pure virtual braid group (also known as the quasi-triangular group), and show that the corresponding associated graded algebra is quadratic, and hence that these groups have a universal finite type invariant."],"dc:description.degree":["PhD"],"dc:identifier.uri":["http://hdl.handle.net/1807/32806"],"dc:language.iso":["en_ca"],"dc:subject":["Pure Virtual Braids","Quadraticity","Graded 1-Formal","Universal Finite Type Invariant"],"dc:title":["The Pure Virtual Braid Group is Quadratic"],"dc:type":["Thesis"]},"updated_at":"2026-07-27T21:28:22Z"}