{"id":{"repo_id":"anu","oai_identifier":"oai:openresearch-repository.anu.edu.au:1885/270268"},"canonical_url":"https://search.dev.ndltd.org/etd/anu/oai:openresearch-repository.anu.edu.au:1885/270268","repository":{"repo_id":"anu","name":"Australian National University","base_url":"https://openresearch-repository.anu.edu.au/server/oai/request"},"display":{"title":"From Homological Algebra To Topology via Type B Zigzag Algebra and Heisenberg Algebra","abstract":"We construct a faithful categorical action of the type $B$ braid group on the bounded homotopy category of finitely generated projective modules over a finite dimensional algebra which we call the type $B$ zigzag algebra. This categorical action is closely related to the action of the type $B$ braid group on curves on the disc. Thus, our exposition can be seen as a type $B$ analogue of the work of Khovanov-Seidel. Moreover, we relate our topological (respectively categorical) action of the type $B$ Artin braid group to their topological (respectively categorical) action of the type $A$ Artin braid group. Then, we prove Rouquier's conjecture \\cite[Conjecture 9.8]{Rouq} on the faithfulness of Type $B$ $2$-braid group on Soergel category following the strategy used by Jensen's master with the diagrammatic tools from Elias-Williamson. In the final part of the thesis, we produce a graded Fock vector in the Laurent ring $\\Z[t,t^{-1}]$ for a crossingless matching using Heisenberg algebra. We conjecture that the span of such vectors forms a Temperley-Lieb representation, and hence, a new presentation of Jones polynomial can be obtained.","abstract_html":"We construct a faithful categorical action of the type $B$ braid group on the bounded homotopy category of finitely generated projective modules over a finite dimensional algebra which we call the type $B$ zigzag algebra. This categorical action is closely related to the action of the type $B$ braid group on curves on the disc. Thus, our exposition can be seen as a type $B$ analogue of the work of Khovanov-Seidel. Moreover, we relate our topological (respectively categorical) action of the type $B$ Artin braid group to their topological (respectively categorical) action of the type $A$ Artin braid group. Then, we prove Rouquier&#x27;s conjecture \\cite[Conjecture 9.8]{Rouq} on the faithfulness of Type $B$ $2$-braid group on Soergel category following the strategy used by Jensen&#x27;s master with the diagrammatic tools from Elias-Williamson. In the final part of the thesis, we produce a graded Fock vector in the Laurent ring <span class=\"etd-inline-math\">\\Z[t,t<sup>-1</sup>]</span> for a crossingless matching using Heisenberg algebra. We conjecture that the span of such vectors forms a Temperley-Lieb representation, and hence, a new presentation of Jones polynomial can be obtained.","abstract_has_math":true,"creators":["Nge, Kie Seng"],"institution":null,"degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2022,"date_issued":"2022","date_published":"2022","updated_at":"2026-07-24T00:55:07Z","subjects":[],"languages":["en_AU"],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/1885/270268","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:creator","label":"Author","values":["Nge, Kie Seng"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2022-08-08T04:47:19Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2022-08-08T04:47:19Z"]},{"key":"dc:date.issued","label":"Date","values":["2022"]},{"key":"dc:type","label":"Dc Type","values":["Thesis (PhD)"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language.iso","label":"Language (ISO)","values":["en_AU"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["http://hdl.handle.net/1885/270268"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["We construct a faithful categorical action of the type $B$ braid group on the bounded homotopy category of finitely generated projective modules over a finite dimensional algebra which we call the type $B$ zigzag algebra. This categorical action is closely related to the action of the type $B$ braid group on curves on the disc. Thus, our exposition can be seen as a type $B$ analogue of the work of Khovanov-Seidel. Moreover, we relate our topological (respectively categorical) action of the type $B$ Artin braid group to their topological (respectively categorical) action of the type $A$ Artin braid group. Then, we prove Rouquier's conjecture \\cite[Conjecture 9.8]{Rouq} on the faithfulness of Type $B$ $2$-braid group on Soergel category following the strategy used by Jensen's master with the diagrammatic tools from Elias-Williamson. In the final part of the thesis, we produce a graded Fock vector in the Laurent ring $\\Z[t,t^{-1}]$ for a crossingless matching using Heisenberg algebra. We conjecture that the span of such vectors forms a Temperley-Lieb representation, and hence, a new presentation of Jones polynomial can be obtained."]},{"key":"dc:title","label":"Title","values":["From Homological Algebra To Topology via Type B Zigzag Algebra and Heisenberg Algebra"]}]}],"canonical_facts":{"dc:creator":["Nge, Kie Seng"],"dc:date.accessioned":["2022-08-08T04:47:19Z"],"dc:date.available":["2022-08-08T04:47:19Z"],"dc:date.issued":["2022"],"dc:description.abstract":["We construct a faithful categorical action of the type $B$ braid group on the bounded homotopy category of finitely generated projective modules over a finite dimensional algebra which we call the type $B$ zigzag algebra. This categorical action is closely related to the action of the type $B$ braid group on curves on the disc. Thus, our exposition can be seen as a type $B$ analogue of the work of Khovanov-Seidel. Moreover, we relate our topological (respectively categorical) action of the type $B$ Artin braid group to their topological (respectively categorical) action of the type $A$ Artin braid group. Then, we prove Rouquier's conjecture \\cite[Conjecture 9.8]{Rouq} on the faithfulness of Type $B$ $2$-braid group on Soergel category following the strategy used by Jensen's master with the diagrammatic tools from Elias-Williamson. In the final part of the thesis, we produce a graded Fock vector in the Laurent ring $\\Z[t,t^{-1}]$ for a crossingless matching using Heisenberg algebra. We conjecture that the span of such vectors forms a Temperley-Lieb representation, and hence, a new presentation of Jones polynomial can be obtained."],"dc:identifier.uri":["http://hdl.handle.net/1885/270268"],"dc:language.iso":["en_AU"],"dc:title":["From Homological Algebra To Topology via Type B Zigzag Algebra and Heisenberg Algebra"],"dc:type":["Thesis (PhD)"]},"updated_at":"2026-07-24T00:55:07Z"}