{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/86865"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/86865","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Bicharacter Construction of Quantum Vertex Algebras","abstract":"This thesis is a study of the axiomatics of quantum vertex algebras based on the bicharacter construction suggested by R. Borcherds in [Bor01]. One of the goals is to use the ideas of [Bor01] to incorporate the examples of quantum vertex operators in the literature ([Jin91], [Jin95], [FR92], [FR97]), in particular the quantum vertex operators describing classes of symmetric polynomials as considered by N. Jing ([Jin94b]). The bicharacter construction is a tool which hasn't been explored in this context. We develop this construction further and based on our theory we propose the notion of HD-quantum vertex algebra. We then prove that large classes of quantum vertex operators can be given in terms of a bicharacter construction, including the Jing quantum vertex operators, and thus can be incorporated in an HD-quantum vertex algebra structure. Unlike the case of classical vertex algebras it turns out that the HD-quantum vertex algebras are not necessarily complete with respect to operator product expansions. Therefore we define an object, which we call generalized vertex algebra with Hopf symmetry, which requires completeness with respect to operator product expansions as an axiom. We then prove that for particular cases of Hopf algebras we can project this generalized vertex algebra structure to the deformed chiral algebra defined by E. Frenkel and N. Reshetikhin in [FR97]. Some of the corollaries of the bicharacter construction enable us to give formulas for the braiding map and the operator product expansions for any quantum vertex operators in the deformed chiral algebra, thereby completing the description of the main example considered in [FR97].","abstract_html":"This thesis is a study of the axiomatics of quantum vertex algebras based on the bicharacter construction suggested by R. Borcherds in [Bor01]. One of the goals is to use the ideas of [Bor01] to incorporate the examples of quantum vertex operators in the literature ([Jin91], [Jin95], [FR92], [FR97]), in particular the quantum vertex operators describing classes of symmetric polynomials as considered by N. Jing ([Jin94b]). The bicharacter construction is a tool which hasn't been explored in this context. We develop this construction further and based on our theory we propose the notion of HD-quantum vertex algebra. We then prove that large classes of quantum vertex operators can be given in terms of a bicharacter construction, including the Jing quantum vertex operators, and thus can be incorporated in an HD-quantum vertex algebra structure. Unlike the case of classical vertex algebras it turns out that the HD-quantum vertex algebras are not necessarily complete with respect to operator product expansions. Therefore we define an object, which we call generalized vertex algebra with Hopf symmetry, which requires completeness with respect to operator product expansions as an axiom. We then prove that for particular cases of Hopf algebras we can project this generalized vertex algebra structure to the deformed chiral algebra defined by E. Frenkel and N. Reshetikhin in [FR97]. Some of the corollaries of the bicharacter construction enable us to give formulas for the braiding map and the operator product expansions for any quantum vertex operators in the deformed chiral algebra, thereby completing the description of the main example considered in [FR97].","abstract_has_math":false,"creators":["Anguelova, Iana I."],"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":"2015-09-28T15:19:55Z","date_published":"2015-09-28T15:19:55Z","updated_at":"2026-07-22T22:26:28Z","subjects":["Mathematics"],"languages":["eng"],"rights":[],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["(MiAaPQ)AAI3242782"],"render_values":[{"text":"(MiAaPQ)AAI3242782","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/86865","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":["Anguelova, Iana I."]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2015-09-28T15:19:55Z","10000-01-01","2006"]},{"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/2142/86865","(MiAaPQ)AAI3242782"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["This thesis is a study of the axiomatics of quantum vertex algebras based on the bicharacter construction suggested by R. Borcherds in [Bor01]. One of the goals is to use the ideas of [Bor01] to incorporate the examples of quantum vertex operators in the literature ([Jin91], [Jin95], [FR92], [FR97]), in particular the quantum vertex operators describing classes of symmetric polynomials as considered by N. Jing ([Jin94b]). The bicharacter construction is a tool which hasn't been explored in this context. We develop this construction further and based on our theory we propose the notion of HD-quantum vertex algebra. We then prove that large classes of quantum vertex operators can be given in terms of a bicharacter construction, including the Jing quantum vertex operators, and thus can be incorporated in an HD-quantum vertex algebra structure. Unlike the case of classical vertex algebras it turns out that the HD-quantum vertex algebras are not necessarily complete with respect to operator product expansions. Therefore we define an object, which we call generalized vertex algebra with Hopf symmetry, which requires completeness with respect to operator product expansions as an axiom. We then prove that for particular cases of Hopf algebras we can project this generalized vertex algebra structure to the deformed chiral algebra defined by E. Frenkel and N. Reshetikhin in [FR97]. Some of the corollaries of the bicharacter construction enable us to give formulas for the braiding map and the operator product expansions for any quantum vertex operators in the deformed chiral algebra, thereby completing the description of the main example considered in [FR97].","Made available in DSpace on 2015-09-28T15:19:55Z (GMT). No. of bitstreams: 2 license.txt: 4848 bytes, checksum: 96035ab3f5e1c23cc7138a224ce498bd (MD5) 3242782.pdf: 1745160 bytes, checksum: 55be3fe5970cc39fc7d8f28220e1ac09 (MD5) Previous issue date: 2006","Embargo set by: Seth Robbins for item 88146 Lift date: Forever Reason: Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","U of I Only","71 p.","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 2006."]},{"key":"dc:title","label":"Title","values":["Bicharacter Construction of Quantum Vertex Algebras"]}]}],"canonical_facts":{"dc:contributor":["Maarten Bergvelt"],"dc:creator":["Anguelova, Iana I."],"dc:date":["2015-09-28T15:19:55Z","10000-01-01","2006"],"dc:description":["This thesis is a study of the axiomatics of quantum vertex algebras based on the bicharacter construction suggested by R. Borcherds in [Bor01]. One of the goals is to use the ideas of [Bor01] to incorporate the examples of quantum vertex operators in the literature ([Jin91], [Jin95], [FR92], [FR97]), in particular the quantum vertex operators describing classes of symmetric polynomials as considered by N. Jing ([Jin94b]). The bicharacter construction is a tool which hasn't been explored in this context. We develop this construction further and based on our theory we propose the notion of HD-quantum vertex algebra. We then prove that large classes of quantum vertex operators can be given in terms of a bicharacter construction, including the Jing quantum vertex operators, and thus can be incorporated in an HD-quantum vertex algebra structure. Unlike the case of classical vertex algebras it turns out that the HD-quantum vertex algebras are not necessarily complete with respect to operator product expansions. Therefore we define an object, which we call generalized vertex algebra with Hopf symmetry, which requires completeness with respect to operator product expansions as an axiom. We then prove that for particular cases of Hopf algebras we can project this generalized vertex algebra structure to the deformed chiral algebra defined by E. Frenkel and N. Reshetikhin in [FR97]. Some of the corollaries of the bicharacter construction enable us to give formulas for the braiding map and the operator product expansions for any quantum vertex operators in the deformed chiral algebra, thereby completing the description of the main example considered in [FR97].","Made available in DSpace on 2015-09-28T15:19:55Z (GMT). No. of bitstreams: 2 license.txt: 4848 bytes, checksum: 96035ab3f5e1c23cc7138a224ce498bd (MD5) 3242782.pdf: 1745160 bytes, checksum: 55be3fe5970cc39fc7d8f28220e1ac09 (MD5) Previous issue date: 2006","Embargo set by: Seth Robbins for item 88146 Lift date: Forever Reason: Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","U of I Only","71 p.","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 2006."],"dc:identifier":["http://hdl.handle.net/2142/86865","(MiAaPQ)AAI3242782"],"dc:language":["eng"],"dc:subject":["Mathematics"],"dc:title":["Bicharacter Construction of Quantum Vertex Algebras"],"dc:type":["text"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:26:28Z"}