{"id":{"repo_id":"cambridge","oai_identifier":"oai:www.repository.cam.ac.uk:1810/291025"},"canonical_url":"https://search.dev.ndltd.org/etd/cambridge/oai:www.repository.cam.ac.uk:1810/291025","repository":{"repo_id":"cambridge","name":"Cambridge University","base_url":"https://api.repository.cam.ac.uk/server/oai/request"},"display":{"title":"Rigid Analytic Quantum Groups","abstract":"Following constructions in rigid analytic geometry, we introduce a theory of $p$-adic analytic quantum groups. We first define Fréchet completions $\\wideparen{U_q(\\mathfrak{g})}$ and $\\wideparen{\\mathcal{O}_q(G)}$ of the quantized enveloping algebra of a semisimple Lie algebra $\\mathfrak{g}$ and the quantized coordinate ring of the corresponding semisimple algebraic group $G$ respectively. We consider these to be quantum analogues of the Arens-Michael envelope of the enveloping algebra $U(\\mathfrak{g})$ and of the algebra of rigid analytic functions on the rigid analytification of $G$ respectively. We show that these algebras are topological Hopf algebras and, by adapting techniques extracted from work of Ardakov-Wadsley, Schmidt and Emerton in $p$-adic representation theory, we also show that they are Fréchet-Stein algebras and use this to investigate an analogue of category $\\mathcal{O}$ for $\\wideparen{U_q(\\mathfrak{g})}$. We then introduce a $p$-adic analytic analogue of Backelin and Kremnizer's construction of the quantum flag variety of a semisimple algebraic group, using a Banach completion of $\\wideparen{\\mathcal{O}_q(G)}$. Our main result is a Beilinson-Bernstein localisation theorem in this context. We define a category of $\\lambda$-twisted $D$-modules on this analytic quantum flag variety. This category has a distinguished object $\\widehat{\\D_q^\\lambda}$ which plays the role of the sheaf of $\\lambda$-twisted differential operators. We show that when $\\lambda$ is regular and dominant, the global section functor gives an equivalence of categories between the coherent $\\lambda$-twisted $D$-modules and the finitely presented modules over the global sections of $\\widehat{\\mathcal{D}_q^\\lambda}$. The construction of this analytic quantum flag variety involves working with Banach comodules over the Banach completion $\\OqBhat$ of the quantum coordinate algebra of the Borel. Along the way, we also show that Banach comodules over $\\OqBhat$ can be naturally identified with what we call topologically integrable modules over the Banach completion of Lusztig's integral form of the quantum Borel.","abstract_html":"Following constructions in rigid analytic geometry, we introduce a theory of $p$-adic analytic quantum groups. We first define Fréchet completions <span class=\"etd-inline-math\">\\wideparen{U<sub>q</sub>(\\mathfrak{g})}</span> and <span class=\"etd-inline-math\">\\wideparen{\\mathcal{O}<sub>q</sub>(G)}</span> of the quantized enveloping algebra of a semisimple Lie algebra $\\mathfrak{g}$ and the quantized coordinate ring of the corresponding semisimple algebraic group $G$ respectively. We consider these to be quantum analogues of the Arens-Michael envelope of the enveloping algebra $U(\\mathfrak{g})$ and of the algebra of rigid analytic functions on the rigid analytification of $G$ respectively. We show that these algebras are topological Hopf algebras and, by adapting techniques extracted from work of Ardakov-Wadsley, Schmidt and Emerton in $p$-adic representation theory, we also show that they are Fréchet-Stein algebras and use this to investigate an analogue of category $\\mathcal{O}$ for <span class=\"etd-inline-math\">\\wideparen{U<sub>q</sub>(\\mathfrak{g})}</span>. We then introduce a $p$-adic analytic analogue of Backelin and Kremnizer&#x27;s construction of the quantum flag variety of a semisimple algebraic group, using a Banach completion of <span class=\"etd-inline-math\">\\wideparen{\\mathcal{O}<sub>q</sub>(G)}</span>. Our main result is a Beilinson-Bernstein localisation theorem in this context. We define a category of $\\lambda$-twisted $D$-modules on this analytic quantum flag variety. This category has a distinguished object <span class=\"etd-inline-math\">\\widehat{\\D<sub>q</sub><sup>\\</sup>lambda}</span> which plays the role of the sheaf of $\\lambda$-twisted differential operators. We show that when $\\lambda$ is regular and dominant, the global section functor gives an equivalence of categories between the coherent $\\lambda$-twisted $D$-modules and the finitely presented modules over the global sections of <span class=\"etd-inline-math\">\\widehat{\\mathcal{D}<sub>q</sub><sup>\\</sup>lambda}</span>. The construction of this analytic quantum flag variety involves working with Banach comodules over the Banach completion $\\OqBhat$ of the quantum coordinate algebra of the Borel. Along the way, we also show that Banach comodules over $\\OqBhat$ can be naturally identified with what we call topologically integrable modules over the Banach completion of Lusztig&#x27;s integral form of the quantum Borel.","abstract_has_math":true,"creators":["Dupré, Nicolas"],"institution":"University of Cambridge","degree_name":"Doctor of Philosophy (PhD)","degree_level":"Doctoral","degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":["Wadsley, Simon James"],"committee_chairs":[],"committee_members":[],"year":2019,"date_issued":"2019-04-27","date_published":"2019-04-27","updated_at":"2026-07-22T22:24:17Z","subjects":["Quantum groups","D-modules","p-adic representation theory"],"languages":["en"],"rights":[],"rights_urls":["https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/b266428f-8711-4386-b9ca-6b4ff4a3f4c5/download","https://creativecommons.org/licenses/by-nc-sa/4.0/"],"identifier_entries":[]},"links":{"outbound_url":"https://doi.org/10.17863/CAM.38204","outbound_label":"DOI","outbound_source":"dc:identifier.doi"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Wadsley, Simon James"]},{"key":"dc:creator","label":"Author","values":["Dupré, Nicolas"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.issued","label":"Date","values":["2019-04-27"]},{"key":"dc:publisher.institution","label":"Dc Publisher Institution","values":["University of Cambridge"]},{"key":"dc:relation.isreferencedby.uri","label":"Dc Relation Isreferencedby URI","values":["https://www.repository.cam.ac.uk/handle/1810/291025"]},{"key":"dc:type","label":"Dc Type","values":["Thesis"]},{"key":"dc:type.qualificationlevel","label":"Dc Type Qualificationlevel","values":["Doctoral"]},{"key":"dc:type.qualificationname","label":"Dc Type Qualificationname","values":["Doctor of Philosophy (PhD)"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Quantum groups","D-modules","p-adic representation theory"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/b266428f-8711-4386-b9ca-6b4ff4a3f4c5/download","https://creativecommons.org/licenses/by-nc-sa/4.0/"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.doi","label":"DOI","values":["10.17863/CAM.38204"]},{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/48e09fef-b4fb-4e23-acb0-5af4b40eefbb/download"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["Following constructions in rigid analytic geometry, we introduce a theory of $p$-adic analytic quantum groups. We first define Fréchet completions $\\wideparen{U_q(\\mathfrak{g})}$ and $\\wideparen{\\mathcal{O}_q(G)}$ of the quantized enveloping algebra of a semisimple Lie algebra $\\mathfrak{g}$ and the quantized coordinate ring of the corresponding semisimple algebraic group $G$ respectively. We consider these to be quantum analogues of the Arens-Michael envelope of the enveloping algebra $U(\\mathfrak{g})$ and of the algebra of rigid analytic functions on the rigid analytification of $G$ respectively. We show that these algebras are topological Hopf algebras and, by adapting techniques extracted from work of Ardakov-Wadsley, Schmidt and Emerton in $p$-adic representation theory, we also show that they are Fréchet-Stein algebras and use this to investigate an analogue of category $\\mathcal{O}$ for $\\wideparen{U_q(\\mathfrak{g})}$. We then introduce a $p$-adic analytic analogue of Backelin and Kremnizer's construction of the quantum flag variety of a semisimple algebraic group, using a Banach completion of $\\wideparen{\\mathcal{O}_q(G)}$. Our main result is a Beilinson-Bernstein localisation theorem in this context. We define a category of $\\lambda$-twisted $D$-modules on this analytic quantum flag variety. This category has a distinguished object $\\widehat{\\D_q^\\lambda}$ which plays the role of the sheaf of $\\lambda$-twisted differential operators. We show that when $\\lambda$ is regular and dominant, the global section functor gives an equivalence of categories between the coherent $\\lambda$-twisted $D$-modules and the finitely presented modules over the global sections of $\\widehat{\\mathcal{D}_q^\\lambda}$. The construction of this analytic quantum flag variety involves working with Banach comodules over the Banach completion $\\OqBhat$ of the quantum coordinate algebra of the Borel. Along the way, we also show that Banach comodules over $\\OqBhat$ can be naturally identified with what we call topologically integrable modules over the Banach completion of Lusztig's integral form of the quantum Borel."]},{"key":"dc:format.checksum.md5","label":"Dc Format Checksum Md5","values":["6e34f3456e5d5dd2ae0f3c865c32a9d2","87eda9de84448d1f82354d60eee3eb5f"]},{"key":"dc:title","label":"Title","values":["Rigid Analytic Quantum Groups"]}]}],"canonical_facts":{"dc:contributor.advisor":["Wadsley, Simon James"],"dc:creator":["Dupré, Nicolas"],"dc:date.issued":["2019-04-27"],"dc:description.abstract":["Following constructions in rigid analytic geometry, we introduce a theory of $p$-adic analytic quantum groups. We first define Fréchet completions $\\wideparen{U_q(\\mathfrak{g})}$ and $\\wideparen{\\mathcal{O}_q(G)}$ of the quantized enveloping algebra of a semisimple Lie algebra $\\mathfrak{g}$ and the quantized coordinate ring of the corresponding semisimple algebraic group $G$ respectively. We consider these to be quantum analogues of the Arens-Michael envelope of the enveloping algebra $U(\\mathfrak{g})$ and of the algebra of rigid analytic functions on the rigid analytification of $G$ respectively. We show that these algebras are topological Hopf algebras and, by adapting techniques extracted from work of Ardakov-Wadsley, Schmidt and Emerton in $p$-adic representation theory, we also show that they are Fréchet-Stein algebras and use this to investigate an analogue of category $\\mathcal{O}$ for $\\wideparen{U_q(\\mathfrak{g})}$. We then introduce a $p$-adic analytic analogue of Backelin and Kremnizer's construction of the quantum flag variety of a semisimple algebraic group, using a Banach completion of $\\wideparen{\\mathcal{O}_q(G)}$. Our main result is a Beilinson-Bernstein localisation theorem in this context. We define a category of $\\lambda$-twisted $D$-modules on this analytic quantum flag variety. This category has a distinguished object $\\widehat{\\D_q^\\lambda}$ which plays the role of the sheaf of $\\lambda$-twisted differential operators. We show that when $\\lambda$ is regular and dominant, the global section functor gives an equivalence of categories between the coherent $\\lambda$-twisted $D$-modules and the finitely presented modules over the global sections of $\\widehat{\\mathcal{D}_q^\\lambda}$. The construction of this analytic quantum flag variety involves working with Banach comodules over the Banach completion $\\OqBhat$ of the quantum coordinate algebra of the Borel. Along the way, we also show that Banach comodules over $\\OqBhat$ can be naturally identified with what we call topologically integrable modules over the Banach completion of Lusztig's integral form of the quantum Borel."],"dc:format.checksum.md5":["6e34f3456e5d5dd2ae0f3c865c32a9d2","87eda9de84448d1f82354d60eee3eb5f"],"dc:identifier.doi":["10.17863/CAM.38204"],"dc:identifier.uri":["https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/48e09fef-b4fb-4e23-acb0-5af4b40eefbb/download"],"dc:language":["en"],"dc:publisher.institution":["University of Cambridge"],"dc:relation.isreferencedby.uri":["https://www.repository.cam.ac.uk/handle/1810/291025"],"dc:rights":["https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/b266428f-8711-4386-b9ca-6b4ff4a3f4c5/download","https://creativecommons.org/licenses/by-nc-sa/4.0/"],"dc:subject":["Quantum groups","D-modules","p-adic representation theory"],"dc:title":["Rigid Analytic Quantum Groups"],"dc:type":["Thesis"],"dc:type.qualificationlevel":["Doctoral"],"dc:type.qualificationname":["Doctor of Philosophy (PhD)"]},"updated_at":"2026-07-22T22:24:17Z"}