{"id":"https://openalex.org/W7213268276","doi":"https://doi.org/10.48550/arxiv.2609.15658","title":"Singular automorphic products and BKM algebras","display_name":"Singular automorphic products and BKM algebras","publication_year":2026,"publication_date":"2026-09-14","ids":{"openalex":"https://openalex.org/W7213268276","doi":"https://doi.org/10.48550/arxiv.2609.15658"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.15658","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.15658","pdf_url":"https://arxiv.org/pdf/2609.15658","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/2609.15658","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5151768029","display_name":"Kaiwen Sun","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Sun, Kaiwen","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5007111544","display_name":"Haowu Wang","orcid":"https://orcid.org/0000-0003-2506-5110","observed_orcids":["https://orcid.org/0000-0003-2506-5110"]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Wang, Haowu","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5043541238","display_name":"Brandon Williams","orcid":"https://orcid.org/0000-0001-7599-6058","observed_orcids":["https://orcid.org/0000-0001-7599-6058"]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Williams, Brandon","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"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/T10287","display_name":"Algebraic structures and combinatorial models","score":0.9398999810218811,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"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/T10287","display_name":"Algebraic structures and combinatorial models","score":0.9398999810218811,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"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/T11428","display_name":"Advanced Mathematical Identities","score":0.0203000009059906,"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"}},{"id":"https://openalex.org/T11680","display_name":"Advanced Algebra and Geometry","score":0.011599999852478504,"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/affine-lie-superalgebras","display_name":"affine Lie superalgebras","score":0.8230000138282776},{"id":"https://openalex.org/keywords/vertex-operator-superalgebras","display_name":"vertex operator superalgebras","score":0.8880000114440918},{"id":"https://openalex.org/keywords/modularity","display_name":"modularity","score":0.675000011920929},{"id":"https://openalex.org/keywords/cusp-forms","display_name":"cusp forms","score":0.45899999141693115}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7680000066757202},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.6460000276565552},{"id":"https://openalex.org/C92757383","wikidata":"https://www.wikidata.org/wiki/Q382497","display_name":"Affine transformation","level":2,"score":0.5182999968528748},{"id":"https://openalex.org/C204575570","wikidata":"https://www.wikidata.org/wiki/Q207476","display_name":"Holomorphic function","level":2,"score":0.48579999804496765},{"id":"https://openalex.org/C80899671","wikidata":"https://www.wikidata.org/wiki/Q1304193","display_name":"Vertex (graph theory)","level":3,"score":0.46619999408721924},{"id":"https://openalex.org/C51568863","wikidata":"https://www.wikidata.org/wiki/Q664495","display_name":"Lie algebra","level":2,"score":0.4496999979019165},{"id":"https://openalex.org/C186863843","wikidata":"https://www.wikidata.org/wiki/Q3977654","display_name":"Superalgebra","level":3,"score":0.39419999718666077},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.38339999318122864},{"id":"https://openalex.org/C17020691","wikidata":"https://www.wikidata.org/wiki/Q139677","display_name":"Operator (biology)","level":5,"score":0.3772999942302704},{"id":"https://openalex.org/C90673727","wikidata":"https://www.wikidata.org/wiki/Q901718","display_name":"Product (mathematics)","level":2,"score":0.3693999946117401},{"id":"https://openalex.org/C98184364","wikidata":"https://www.wikidata.org/wiki/Q1780131","display_name":"Argument (complex analysis)","level":2,"score":0.3529999852180481},{"id":"https://openalex.org/C5475112","wikidata":"https://www.wikidata.org/wiki/Q632521","display_name":"Affine Lie algebra","level":4,"score":0.34380000829696655},{"id":"https://openalex.org/C73648015","wikidata":"https://www.wikidata.org/wiki/Q6543817","display_name":"Lie conformal algebra","level":3,"score":0.3073999881744385},{"id":"https://openalex.org/C132817261","wikidata":"https://www.wikidata.org/wiki/Q1134435","display_name":"Automorphic form","level":2,"score":0.2971000075340271},{"id":"https://openalex.org/C2777299769","wikidata":"https://www.wikidata.org/wiki/Q3707858","display_name":"Type (biology)","level":2,"score":0.2969000041484833},{"id":"https://openalex.org/C2778400075","wikidata":"https://www.wikidata.org/wiki/Q655275","display_name":"Cusp (singularity)","level":2,"score":0.26820001006126404},{"id":"https://openalex.org/C156772000","wikidata":"https://www.wikidata.org/wiki/Q864475","display_name":"Functor","level":2,"score":0.2574000060558319},{"id":"https://openalex.org/C78804095","wikidata":"https://www.wikidata.org/wiki/Q1133024","display_name":"Lie superalgebra","level":5,"score":0.2533000111579895}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.15658","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.15658","pdf_url":"https://arxiv.org/pdf/2609.15658","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.2609.15658","doi":"https://doi.org/10.48550/arxiv.2609.15658","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.15658","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":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2609.15658","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.15658","pdf_url":"https://arxiv.org/pdf/2609.15658","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/G5846448348","display_name":null,"funder_award_id":"TRR 326","funder_id":"https://openalex.org/F4320320879","funder_display_name":"Deutsche Forschungsgemeinschaft"},{"id":"https://openalex.org/G8483945739","display_name":"TRR 326: Geometrie und Arithmetik uniformisierter Strukturen (GAUS)","funder_award_id":"444845124","funder_id":"https://openalex.org/F4320320879","funder_display_name":"Deutsche Forschungsgemeinschaft"}],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"},{"id":"https://openalex.org/F4320320879","display_name":"Deutsche Forschungsgemeinschaft","ror":"https://ror.org/018mejw64"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7213268276.pdf","grobid_xml":"https://content.openalex.org/works/W7213268276.grobid-xml"},"referenced_works_count":4,"referenced_works":["https://openalex.org/W2746213619","https://openalex.org/W4317548395","https://openalex.org/W4318903520","https://openalex.org/W4389421871"],"related_works":["https://openalex.org/W4405887447","https://openalex.org/W1975132635","https://openalex.org/W4389470990","https://openalex.org/W2081122515","https://openalex.org/W2067873933","https://openalex.org/W2018734002","https://openalex.org/W2013378999","https://openalex.org/W1624744390","https://openalex.org/W2945899825","https://openalex.org/W2144769277"],"abstract_inverted_index":{"After":[0],"proving":[1],"the":[2,36,53,83,103,136],"moonshine":[3],"conjecture,":[4],"Borcherds":[5],"asked":[6],"in":[7],"1995":[8],"whether":[9],"there":[10,27],"exist":[11],"only":[12],"finitely":[13],"many":[14],"holomorphic":[15],"automorphic":[16],"products":[17,32],"of":[18,35,55,89,115,138],"singular":[19],"weight.":[20],"We":[21],"show":[22],"that,":[23],"up":[24],"to":[25,77,85,102,130],"conjugation,":[26],"are":[28,75],"exactly":[29,96],"27":[30,104],"such":[31],"on":[33],"lattices":[34],"form":[37],"$2U\\oplus":[38],"L$.":[39],"The":[40,71],"classification":[41,107],"proceeds":[42],"by":[43],"identifying":[44],"each":[45],"product":[46],"at":[47],"a":[48,56,61,122],"standard":[49],"$1$-dimensional":[50],"cusp":[51],"with":[52,111,133],"superdenominator":[54],"BKM":[57],"superalgebra":[58,69],"that":[59,81,100],"carries":[60],"graded":[62],"module":[63],"structure":[64],"over":[65],"an":[66,86],"affine":[67,127],"Lie":[68,128],"$\\hat{\\mathfrak{g}}$.":[70],"auxiliary":[72],"algebras":[73,98],"$\\hat{\\mathfrak{g}}$":[74,99],"shown":[76],"satisfy":[78],"strong":[79],"conditions":[80],"reduce":[82],"possibilities":[84],"initial":[87],"list":[88],"1372":[90],"candidates.":[91],"An":[92],"elimination":[93],"argument":[94],"leaves":[95],"85":[97],"correspond":[101],"products.":[105],"This":[106],"is":[108],"closely":[109],"connected":[110],"certain":[112],"distinguished":[113],"families":[114],"vertex":[116],"operator":[117],"superalgebras.":[118],"Our":[119],"results":[120],"provide":[121],"systematic":[123],"framework":[124],"for":[125],"extending":[126],"superalgebras":[129],"hyperbolic":[131],"ones":[132],"modularity,":[134],"generalizing":[135],"work":[137],"Feingold":[139],"and":[140],"Frenkel":[141],"(1983).":[142]},"counts_by_year":[],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-09-16T00:00:00"}