{"authors":[{"id":null,"fullName":"Carotenuto, Alessandro","name":"Alessandro","surname":"Carotenuto","rank":1,"pid":null},{"id":null,"fullName":"Buachalla, Réamonn Ó","name":"Réamonn Ó","surname":"Buachalla","rank":2,"pid":null},{"id":null,"fullName":"Razzaq, Junaid","name":"Junaid","surname":"Razzaq","rank":3,"pid":null}],"openAccessColor":null,"publiclyFunded":false,"eoscIfGuidelines":null,"type":"publication","language":{"code":"und","label":"Undetermined"},"countries":null,"subjects":[{"subject":{"scheme":"keyword","value":"Mathematics - Differential Geometry"},"provenance":null},{"subject":{"scheme":"keyword","value":"Differential Geometry (math.DG)"},"provenance":null},{"subject":{"scheme":"keyword","value":"Mathematics - Quantum Algebra"},"provenance":null},{"subject":{"scheme":"keyword","value":"FOS: Mathematics"},"provenance":null},{"subject":{"scheme":"keyword","value":"Quantum Algebra (math.QA)"},"provenance":null},{"subject":{"scheme":"FOS","value":"0101 mathematics"},"provenance":null},{"subject":{"scheme":"FOS","value":"01 natural sciences"},"provenance":null}],"mainTitle":"Noncommutative Complex Structures for the Full Quantum Flag Manifold of Quantum SU(3)","subTitle":null,"descriptions":["In recent work, Lusztig's positive root vectors (with respect to a distinguished choice of reduced decomposition of the longest element of the Weyl group) were shown to give a quantum tangent space for every $A$-series Drinfeld--Jimbo full quantum flag manifold $\\mathcal{O}_q(\\mathrm{F}_n)$. Moreover, the associated differential calculus $Ω^{(0,\\bullet)}_q(\\mathrm{F}_n)$ was shown to have classical dimension, giving a direct $q$-deformation of the classical anti-holomorphic Dolbeault complex of $\\mathrm{F}_n$. Here we examine in detail the rank two case, namely the full quantum flag manifold of $\\mathcal{O}_q(\\mathrm{SU}_3)$. In particular, we examine the $*$-differential calculus associated to $Ω^{(0,\\bullet)}_q(\\mathrm{F}_3)$ and its non-commutative complex geometry. We find that the number of almost-complex structures reduces from $8$ (that is $2$ to the power of the number of positive roots of $\\frak{sl}_3$) to $4$ (that is $2$ to the power of the number of simple roots of $\\frak{sl}_3$). Moreover, we show that each of these almost-complex structures is integrable, which is to say, each of them is a complex structure. Finally, we observe that, due to non-centrality of all the non-degenerate coinvariant $2$-forms, none of these complex structures admits a left $\\mathcal{O}_q(\\mathrm{SU}_3)$-covariant noncommutative Kähler structure."],"publicationDate":"2024-01-01","publisher":null,"embargoEndDate":"2024-11-01","sources":null,"formats":null,"contributors":null,"coverages":null,"bestAccessRight":{"code":"c_abf2","label":"OPEN","scheme":"http://vocabularies.coar-repositories.org/documentation/access_rights/"},"container":null,"documentationUrls":null,"codeRepositoryUrl":null,"programmingLanguage":null,"contactPeople":null,"contactGroups":null,"tools":null,"size":null,"version":null,"geoLocations":null,"id":"doi_dedup___::16abab8435f8dc9a039c0fd17cab7c8d","originalIds":["50|datacite____::16abab8435f8dc9a039c0fd17cab7c8d","10.48550/arxiv.2411.07767","oai:arXiv.org:2411.07767","50|od________18::7874d6d6ac0e6413ba99e5c5d4096962"],"pids":[{"scheme":"doi","value":"10.48550/arxiv.2411.07767"},{"scheme":"arXiv","value":"2411.07767"}],"dateOfCollection":null,"lastUpdateTimeStamp":null,"indicators":{"citationImpact":{"citationCount":0.0,"influence":2.1746283E-9,"popularity":1.944044E-9,"impulse":0.0,"citationClass":"C5","influenceClass":"C5","impulseClass":"C5","popularityClass":"C5"}},"projects":[{"id":"corda_____he::623cdd9c735a3d8acfb2c679d5d5099b","code":"101119552","acronym":"CaLiForNIA","title":"Cartan and differential geometry, Lie theory, quantum groups and non commutative geometry For novel and Innovative Applications to quantum algorithms and geometric deep learning","funder":"European Commission","pids":[{"scheme":"doi","value":"10.3030/101119552"}]}],"organizations":null,"communities":null,"collectedFrom":[{"key":"openaire____::9e3be59865b2c1c335d32dae2fe7b254","value":"Datacite"},{"key":"opendoar____::6f4922f45568161a8cdf4ad2299f6d23","value":"arXiv.org e-Print Archive"}],"instances":[{"pids":[{"scheme":"doi","value":"10.48550/arxiv.2411.07767"}],"license":"CC BY","type":"Article","urls":["https://dx.doi.org/10.48550/arxiv.2411.07767"],"publicationDate":"2024-01-01","refereed":"nonPeerReviewed","hostedBy":{"key":"openaire____::55045bd2a65019fd8e6741a755395c8c","value":"Unknown Repository"},"collectedFrom":{"key":"openaire____::9e3be59865b2c1c335d32dae2fe7b254","value":"Datacite"}},{"pids":[{"scheme":"arXiv","value":"2411.07767"}],"accessRight":{"code":"c_abf2","label":"OPEN","scheme":"http://vocabularies.coar-repositories.org/documentation/access_rights/","openAccessRoute":null},"type":"Preprint","urls":["http://arxiv.org/abs/2411.07767"],"publicationDate":"2024-11-12","refereed":"nonPeerReviewed","hostedBy":{"key":"opendoar____::6f4922f45568161a8cdf4ad2299f6d23","value":"arXiv.org e-Print Archive"},"collectedFrom":{"key":"opendoar____::6f4922f45568161a8cdf4ad2299f6d23","value":"arXiv.org e-Print Archive"}}],"links":[{"header":{"relationType":"resultProject","relationClass":"isProducedBy","relatedIdentifier":"corda_____he::623cdd9c735a3d8acfb2c679d5d5099b","relatedRecordType":"project","relationProvenance":"iis","trust":"0.897"},"collectedfrom":[{"dsId":"openaire____::3f264f93cf3b0cfc4ede188a6300455c","dsName":"CORDA - COmmon Research DAta Warehouse - Horizon Europe"}],"projectTitle":"Cartan and differential geometry, Lie theory, quantum groups and non commutative geometry For novel and Innovative Applications to quantum algorithms and geometric deep learning","code":"101119552","funding":{"funder":{"id":"ec__________::EC","shortname":"EC","name":"European Commission","jurisdiction":{"code":"EU","label":"European Union"},"pid":null},"level0":{"id":"ec__________::EC::HE","description":"Horizon Europe Framework Programme","name":"HE"},"level1":{"id":"ec__________::EC::HE::HORIZON-TMA-MSCA-DN","description":"HORIZON TMA MSCA Doctoral Networks","name":"HORIZON-TMA-MSCA-DN"},"level2":{"id":null,"description":null,"name":null}},"startDate":"2024-01-01","endDate":"2027-12-31"}],"otherTitles":null,"green":true,"inDiamondJournal":false,"isGreen":true,"isInDiamondJournal":false}