{"id":"https://openalex.org/W7215799713","doi":"https://doi.org/10.48550/arxiv.2610.01208","title":"The crystal groupoid of a compact semisimple Lie group and its flag varieties","display_name":"The crystal groupoid of a compact semisimple Lie group and its flag varieties","publication_year":2026,"publication_date":"2026-10-01","ids":{"openalex":"https://openalex.org/W7215799713","doi":"https://doi.org/10.48550/arxiv.2610.01208"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2610.01208","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.01208","pdf_url":"https://arxiv.org/pdf/2610.01208","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2610.01208","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5091380066","display_name":"Aidan Sims","orcid":"https://orcid.org/0000-0002-1965-6451","observed_orcids":["https://orcid.org/0000-0002-1965-6451"]},"institutions":[{"id":"https://openalex.org/I1294671590","display_name":"Centre National de la Recherche Scientifique","ror":"https://ror.org/02feahw73","country_code":"FR","type":"government","lineage":["https://openalex.org/I1294671590"]},{"id":"https://openalex.org/I3017839442","display_name":"Institut \u00c9lie Cartan de Lorraine","ror":"https://ror.org/04rvw8791","country_code":"FR","type":"facility","lineage":["https://openalex.org/I1294671590","https://openalex.org/I1326498283","https://openalex.org/I3017839442","https://openalex.org/I4210141950","https://openalex.org/I90183372"]},{"id":"https://openalex.org/I31746571","display_name":"UNSW Sydney","ror":"https://ror.org/03r8z3t63","country_code":"AU","type":"education","lineage":["https://openalex.org/I31746571"]},{"id":"https://openalex.org/I90183372","display_name":"Universit\u00e9 de Lorraine","ror":"https://ror.org/04vfs2w97","country_code":"FR","type":"education","lineage":["https://openalex.org/I90183372"]}],"countries":["AU","FR"],"is_corresponding":false,"raw_author_name":"Sims, Aidan","raw_affiliation_strings":["School of Mathematics and Statistics , UNSW Sydney , NSW 2052 , Australia","Universit\u00e9 de Lorraine , CNRS , IECL , F-57000 Metz , France"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Mathematics and Statistics , UNSW Sydney , NSW 2052 , Australia","institution_ids":["https://openalex.org/I31746571"]},{"raw_affiliation_string":"Universit\u00e9 de Lorraine , CNRS , IECL , F-57000 Metz , France","institution_ids":["https://openalex.org/I1294671590","https://openalex.org/I3017839442","https://openalex.org/I90183372"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5066528318","display_name":"Robert Yuncken","orcid":"https://orcid.org/0000-0002-2747-3276","observed_orcids":["https://orcid.org/0000-0002-2747-3276"]},"institutions":[{"id":"https://openalex.org/I31746571","display_name":"UNSW Sydney","ror":"https://ror.org/03r8z3t63","country_code":"AU","type":"education","lineage":["https://openalex.org/I31746571"]}],"countries":["AU"],"is_corresponding":false,"raw_author_name":"Yuncken, Robert","raw_affiliation_strings":["School of Mathematics and Statistics , UNSW Sydney , NSW 2052 , Australia"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Mathematics and Statistics , UNSW Sydney , NSW 2052 , Australia","institution_ids":["https://openalex.org/I31746571"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":4,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10899","display_name":"Advanced Operator Algebra Research","score":0.8970000147819519,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10899","display_name":"Advanced Operator Algebra Research","score":0.8970000147819519,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10896","display_name":"Homotopy and Cohomology in Algebraic Topology","score":0.029899999499320984,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11680","display_name":"Advanced Algebra and Geometry","score":0.01209999993443489,"subfield":{"id":"https://openalex.org/subfields/2602","display_name":"Algebra and Number Theory"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/compact-semisimple-lie-groups","display_name":"compact semisimple Lie groups","score":0.9670000076293945},{"id":"https://openalex.org/keywords/subshifts-of-finite-type","display_name":"subshifts of finite type","score":0.8989999890327454},{"id":"https://openalex.org/keywords/c-algebras","display_name":"C*-algebras","score":0.8650000095367432},{"id":"https://openalex.org/keywords/c-envelope","display_name":"C*-envelope","score":0.9409999847412109},{"id":"https://openalex.org/keywords/weyl-group","display_name":"Weyl group","score":0.8320000171661377},{"id":"https://openalex.org/keywords/bruhat-order","display_name":"Bruhat order","score":1.0},{"id":"https://openalex.org/keywords/flag-varieties","display_name":"flag varieties","score":0.902999997138977},{"id":"https://openalex.org/keywords/poisson-homogeneous-spaces","display_name":"Poisson homogeneous spaces","score":0.9169999957084656},{"id":"https://openalex.org/keywords/orbit-spaces","display_name":"orbit spaces","score":0.5659999847412109}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7910000085830688},{"id":"https://openalex.org/C24424167","wikidata":"https://www.wikidata.org/wiki/Q180907","display_name":"Bijection","level":2,"score":0.7495999932289124},{"id":"https://openalex.org/C75328951","wikidata":"https://www.wikidata.org/wiki/Q17008126","display_name":"Double groupoid","level":2,"score":0.6532999873161316},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.5662000179290771},{"id":"https://openalex.org/C2776730729","wikidata":"https://www.wikidata.org/wiki/Q1392130","display_name":"Flag (linear algebra)","level":3,"score":0.5580000281333923},{"id":"https://openalex.org/C196644772","wikidata":"https://www.wikidata.org/wiki/Q3884964","display_name":"Orbit (dynamics)","level":2,"score":0.4893999993801117},{"id":"https://openalex.org/C2781311116","wikidata":"https://www.wikidata.org/wiki/Q83306","display_name":"Group (periodic table)","level":2,"score":0.48170000314712524},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.4700999855995178},{"id":"https://openalex.org/C63548660","wikidata":"https://www.wikidata.org/wiki/Q858223","display_name":"Limit point","level":2,"score":0.46720001101493835},{"id":"https://openalex.org/C151201525","wikidata":"https://www.wikidata.org/wiki/Q177239","display_name":"Limit (mathematics)","level":2,"score":0.4523000121116638},{"id":"https://openalex.org/C51568863","wikidata":"https://www.wikidata.org/wiki/Q664495","display_name":"Lie algebra","level":2,"score":0.43290001153945923},{"id":"https://openalex.org/C2778052762","wikidata":"https://www.wikidata.org/wiki/Q2366896","display_name":"Semisimple Lie algebra","level":5,"score":0.4251999855041504},{"id":"https://openalex.org/C182306322","wikidata":"https://www.wikidata.org/wiki/Q1779371","display_name":"Order (exchange)","level":2,"score":0.41659998893737793},{"id":"https://openalex.org/C2778572836","wikidata":"https://www.wikidata.org/wiki/Q380933","display_name":"Space (punctuation)","level":2,"score":0.40610000491142273},{"id":"https://openalex.org/C2781285689","wikidata":"https://www.wikidata.org/wiki/Q21921428","display_name":"Crystal (programming language)","level":2,"score":0.4009999930858612},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.38929998874664307},{"id":"https://openalex.org/C207467116","wikidata":"https://www.wikidata.org/wiki/Q4385666","display_name":"Inverse","level":2,"score":0.384799987077713},{"id":"https://openalex.org/C130133044","wikidata":"https://www.wikidata.org/wiki/Q1887083","display_name":"Compact group","level":3,"score":0.3714999854564667},{"id":"https://openalex.org/C53936288","wikidata":"https://www.wikidata.org/wiki/Q2634828","display_name":"Inverse limit","level":2,"score":0.3506999909877777},{"id":"https://openalex.org/C2777299769","wikidata":"https://www.wikidata.org/wiki/Q3707858","display_name":"Type (biology)","level":2,"score":0.3393000066280365},{"id":"https://openalex.org/C40233952","wikidata":"https://www.wikidata.org/wiki/Q768074","display_name":"Weyl group","level":2,"score":0.3305000066757202},{"id":"https://openalex.org/C187915474","wikidata":"https://www.wikidata.org/wiki/Q622679","display_name":"Lie group","level":2,"score":0.3301999866962433},{"id":"https://openalex.org/C96469262","wikidata":"https://www.wikidata.org/wiki/Q1324364","display_name":"Homogeneous space","level":2,"score":0.3109000027179718},{"id":"https://openalex.org/C136197465","wikidata":"https://www.wikidata.org/wiki/Q1729295","display_name":"Variety (cybernetics)","level":2,"score":0.30630001425743103},{"id":"https://openalex.org/C119238805","wikidata":"https://www.wikidata.org/wiki/Q229102","display_name":"Surjective function","level":2,"score":0.3021000027656555},{"id":"https://openalex.org/C75174853","wikidata":"https://www.wikidata.org/wiki/Q190629","display_name":"Centralizer and normalizer","level":2,"score":0.28859999775886536},{"id":"https://openalex.org/C2777404646","wikidata":"https://www.wikidata.org/wiki/Q1057968","display_name":"Finite group","level":3,"score":0.2856000065803528},{"id":"https://openalex.org/C2779152781","wikidata":"https://www.wikidata.org/wiki/Q1069378","display_name":"Weyl algebra","level":3,"score":0.2777999937534332},{"id":"https://openalex.org/C200288055","wikidata":"https://www.wikidata.org/wiki/Q2621792","display_name":"Element (criminal law)","level":2,"score":0.2757999897003174},{"id":"https://openalex.org/C144091092","wikidata":"https://www.wikidata.org/wiki/Q3611543","display_name":"Graded Lie algebra","level":3,"score":0.2705000042915344},{"id":"https://openalex.org/C4042151","wikidata":"https://www.wikidata.org/wiki/Q215111","display_name":"Homomorphism","level":2,"score":0.26809999346733093},{"id":"https://openalex.org/C22365015","wikidata":"https://www.wikidata.org/wiki/Q1003162","display_name":"Simple Lie group","level":3,"score":0.26159998774528503},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.26089999079704285},{"id":"https://openalex.org/C197273675","wikidata":"https://www.wikidata.org/wiki/Q13220368","display_name":"Representation theory","level":2,"score":0.2524000108242035},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.2508000135421753}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2610.01208","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.01208","pdf_url":"https://arxiv.org/pdf/2610.01208","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"doi:10.48550/arxiv.2610.01208","doi":"https://doi.org/10.48550/arxiv.2610.01208","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2610.01208","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2610.01208","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.01208","pdf_url":"https://arxiv.org/pdf/2610.01208","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"study_designs":[],"awards":[{"id":"https://openalex.org/G6297983914","display_name":"Operator algebras and representation theory of Lie groups and groupoids","funder_award_id":"ANR-23-CE40-0016","funder_id":"https://openalex.org/F4320320883","funder_display_name":"Agence Nationale de la Recherche"},{"id":"https://openalex.org/G6356721191","display_name":"Crossroads in the theory of compact quantum groups","funder_award_id":"ANR-25-CE40-5010","funder_id":"https://openalex.org/F4320320883","funder_display_name":"Agence Nationale de la Recherche"},{"id":"https://openalex.org/G7080048889","display_name":null,"funder_award_id":"ANR-23","funder_id":"https://openalex.org/F4320320883","funder_display_name":"Agence Nationale de la Recherche"}],"funders":[{"id":"https://openalex.org/F4320309530","display_name":"Stanley Medical Research Institute","ror":"https://ror.org/01pj5nn22"},{"id":"https://openalex.org/F4320320883","display_name":"Agence Nationale de la Recherche","ror":"https://ror.org/00rbzpz17"},{"id":"https://openalex.org/F4320320969","display_name":"University of Wollongong","ror":"https://ror.org/00jtmb277"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7215799713.pdf","grobid_xml":"https://content.openalex.org/works/W7215799713.grobid-xml"},"referenced_works_count":19,"referenced_works":["https://openalex.org/W1633336122","https://openalex.org/W1809432865","https://openalex.org/W1971977200","https://openalex.org/W1973807996","https://openalex.org/W1984884211","https://openalex.org/W1988032111","https://openalex.org/W1989107173","https://openalex.org/W1997529948","https://openalex.org/W2039041861","https://openalex.org/W2042357641","https://openalex.org/W2068504889","https://openalex.org/W2162427658","https://openalex.org/W2165121148","https://openalex.org/W4232723477","https://openalex.org/W4386119854","https://openalex.org/W4386687603","https://openalex.org/W4403503246","https://openalex.org/W4407745591","https://openalex.org/W7128768974"],"related_works":["https://openalex.org/W2963528330","https://openalex.org/W2989757198","https://openalex.org/W7094326676","https://openalex.org/W4386119854","https://openalex.org/W2999042083","https://openalex.org/W2508178437","https://openalex.org/W2039501996","https://openalex.org/W7140213088","https://openalex.org/W4389984156","https://openalex.org/W1659180456"],"abstract_inverted_index":{"We":[0,34,67,97],"introduce":[1],"the":[2,14,44,48,59,64,70,74,81,90,94,113],"crystal":[3,45,75,101],"groupoid":[4,16,76,102],"of":[5,17,21,24,31,47,52,61,73,84],"a":[6,18,22,100],"compact":[7],"semisimple":[8],"Lie":[9],"group":[10,83],"$K$;":[11],"it":[12],"is":[13,77],"Deaconu-Renault":[15],"higher-rank":[19],"analogue":[20],"subshift":[23],"finite":[25],"type":[26],"on":[27,93],"an":[28],"inverse":[29],"limit":[30],"Kashiwara":[32],"crystals.":[33],"show":[35,68],"that":[36,69],"its":[37,41,55],"Steinberg":[38],"algebra":[39,51],"and":[40,54,63,106,111],"$C^*$-algebra":[42],"realise":[43],"limits":[46],"polynomial":[49],"function":[50],"$K$":[53],"$C^*$-envelope,":[56],"respectively,":[57],"in":[58,78],"sense":[60],"Matassa":[62],"second":[65],"author.":[66],"orbit":[71],"space":[72],"bijection":[79],"with":[80,86],"Weyl":[82,95],"$K$,":[85],"topology":[87],"determined":[88],"by":[89],"Bruhat":[91],"order":[92],"group.":[96],"also":[98],"define":[99],"for":[103],"flag":[104],"manifolds":[105],"related":[107],"Poisson":[108],"homogeneous":[109],"spaces,":[110],"prove":[112],"analogous":[114],"structural":[115],"results.":[116]},"counts_by_year":[],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-10-03T00:00:00"}