{"id":"https://openalex.org/W7212180474","doi":"https://doi.org/10.48550/arxiv.2609.10373","title":"Amenability of Lie, Group and Hopf algebras","display_name":"Amenability of Lie, Group and Hopf algebras","publication_year":2026,"publication_date":"2026-09-09","ids":{"openalex":"https://openalex.org/W7212180474","doi":"https://doi.org/10.48550/arxiv.2609.10373"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.10373","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.10373","pdf_url":"https://arxiv.org/pdf/2609.10373","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/2609.10373","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5010881149","display_name":"Laurent Bartholdi","orcid":"https://orcid.org/0000-0002-1243-6384","observed_orcids":["https://orcid.org/0000-0002-1243-6384"]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Bartholdi, Laurent","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5010881149"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":1,"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/T11673","display_name":"Advanced Topics in Algebra","score":0.9878000020980835,"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"}},"topics":[{"id":"https://openalex.org/T11673","display_name":"Advanced Topics in Algebra","score":0.9878000020980835,"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/T10899","display_name":"Advanced Operator Algebra Research","score":0.009100000374019146,"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/T11703","display_name":"Rings, Modules, and Algebras","score":0.0006000000284984708,"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/amenability","display_name":"amenability","score":0.9750000238418579},{"id":"https://openalex.org/keywords/lie-algebras","display_name":"Lie algebras","score":0.9869999885559082},{"id":"https://openalex.org/keywords/hopf-algebras","display_name":"Hopf algebras","score":0.8659999966621399},{"id":"https://openalex.org/keywords/groups","display_name":"groups","score":0.625},{"id":"https://openalex.org/keywords/universal-enveloping-algebra","display_name":"universal enveloping algebra","score":0.9120000004768372},{"id":"https://openalex.org/keywords/cocommutative-hopf-algebras","display_name":"cocommutative Hopf algebras","score":0.8989999890327454},{"id":"https://openalex.org/keywords/group-rings","display_name":"group rings","score":0.8629999756813049},{"id":"https://openalex.org/keywords/cleft-extensions","display_name":"cleft extensions","score":0.703000009059906},{"id":"https://openalex.org/keywords/subexponential-growth","display_name":"subexponential growth","score":0.8669999837875366}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8011999726295471},{"id":"https://openalex.org/C29712632","wikidata":"https://www.wikidata.org/wiki/Q1627597","display_name":"Hopf algebra","level":3,"score":0.6952999830245972},{"id":"https://openalex.org/C2778249326","wikidata":"https://www.wikidata.org/wiki/Q1777803","display_name":"Coalgebra","level":3,"score":0.6079000234603882},{"id":"https://openalex.org/C155058155","wikidata":"https://www.wikidata.org/wiki/Q1673406","display_name":"Universal enveloping algebra","level":3,"score":0.572700023651123},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.5446000099182129},{"id":"https://openalex.org/C2781311116","wikidata":"https://www.wikidata.org/wiki/Q83306","display_name":"Group (periodic table)","level":2,"score":0.47760000824928284},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.4596000015735626},{"id":"https://openalex.org/C148647251","wikidata":"https://www.wikidata.org/wiki/Q2122223","display_name":"Quantum group","level":3,"score":0.4050999879837036},{"id":"https://openalex.org/C51568863","wikidata":"https://www.wikidata.org/wiki/Q664495","display_name":"Lie algebra","level":2,"score":0.4016000032424927},{"id":"https://openalex.org/C159423971","wikidata":"https://www.wikidata.org/wiki/Q177251","display_name":"Associative property","level":2,"score":0.39149999618530273},{"id":"https://openalex.org/C51292594","wikidata":"https://www.wikidata.org/wiki/Q744960","display_name":"Associative algebra","level":5,"score":0.35409998893737793},{"id":"https://openalex.org/C132170107","wikidata":"https://www.wikidata.org/wiki/Q834585","display_name":"Semilattice","level":3,"score":0.3506999909877777},{"id":"https://openalex.org/C90673727","wikidata":"https://www.wikidata.org/wiki/Q901718","display_name":"Product (mathematics)","level":2,"score":0.33480000495910645},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.32690000534057617},{"id":"https://openalex.org/C2780378348","wikidata":"https://www.wikidata.org/wiki/Q25351438","display_name":"Ring (chemistry)","level":2,"score":0.3206999897956848},{"id":"https://openalex.org/C2776379255","wikidata":"https://www.wikidata.org/wiki/Q3406705","display_name":"Crossed product","level":3,"score":0.3052999973297119},{"id":"https://openalex.org/C55192134","wikidata":"https://www.wikidata.org/wiki/Q7314221","display_name":"Representation theory of Hopf algebras","level":5,"score":0.29330000281333923},{"id":"https://openalex.org/C120047569","wikidata":"https://www.wikidata.org/wiki/Q92795049","display_name":"Group algebra","level":3,"score":0.2842999994754791},{"id":"https://openalex.org/C2779904274","wikidata":"https://www.wikidata.org/wiki/Q1945014","display_name":"Monoidal category","level":2,"score":0.2831000089645386},{"id":"https://openalex.org/C2778676360","wikidata":"https://www.wikidata.org/wiki/Q494141","display_name":"Bialgebra","level":4,"score":0.28290000557899475},{"id":"https://openalex.org/C31498916","wikidata":"https://www.wikidata.org/wiki/Q583034","display_name":"Locally compact space","level":2,"score":0.2700999975204468},{"id":"https://openalex.org/C188971069","wikidata":"https://www.wikidata.org/wiki/Q1636734","display_name":"Universal algebra","level":3,"score":0.2671999931335449},{"id":"https://openalex.org/C1680195","wikidata":"https://www.wikidata.org/wiki/Q2296021","display_name":"Tensor algebra","level":5,"score":0.266400009393692},{"id":"https://openalex.org/C518143113","wikidata":"https://www.wikidata.org/wiki/Q4391941","display_name":"Non-associative algebra","level":4,"score":0.266400009393692},{"id":"https://openalex.org/C130856480","wikidata":"https://www.wikidata.org/wiki/Q7269541","display_name":"Quasitriangular Hopf algebra","level":5,"score":0.26249998807907104}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.10373","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.10373","pdf_url":"https://arxiv.org/pdf/2609.10373","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.2609.10373","doi":"https://doi.org/10.48550/arxiv.2609.10373","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.10373","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:2609.10373","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.10373","pdf_url":"https://arxiv.org/pdf/2609.10373","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/G7148726868","display_name":null,"funder_award_id":"TMAG-2_216487/1","funder_id":"https://openalex.org/F4320320924","funder_display_name":"Schweizerischer Nationalfonds zur F\u00f6rderung der Wissenschaftlichen Forschung"}],"funders":[{"id":"https://openalex.org/F4320320924","display_name":"Schweizerischer Nationalfonds zur F\u00f6rderung der Wissenschaftlichen Forschung","ror":"https://ror.org/00yjd3n13"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7212180474.pdf","grobid_xml":"https://content.openalex.org/works/W7212180474.grobid-xml"},"referenced_works_count":16,"referenced_works":["https://openalex.org/W1543310605","https://openalex.org/W1638819197","https://openalex.org/W1969903778","https://openalex.org/W1973382897","https://openalex.org/W2016790029","https://openalex.org/W2025858089","https://openalex.org/W2031241956","https://openalex.org/W2033123440","https://openalex.org/W2048117558","https://openalex.org/W2055774213","https://openalex.org/W2056169755","https://openalex.org/W2061621717","https://openalex.org/W2612269455","https://openalex.org/W2896715421","https://openalex.org/W2963189495","https://openalex.org/W4310987381"],"related_works":["https://openalex.org/W7096487202","https://openalex.org/W7215796802","https://openalex.org/W2055774213","https://openalex.org/W1718193784","https://openalex.org/W1616440164","https://openalex.org/W796821736","https://openalex.org/W2239568789","https://openalex.org/W2953348419","https://openalex.org/W1787472029","https://openalex.org/W7100927788"],"abstract_inverted_index":{"We":[0,39,131,159],"propose":[1],"to":[2,71,177],"define":[3],"amenability":[4,73,175],"of":[5,12,16,26,74,79,154,162],"a":[6,23,27,41,97],"Lie":[7,62,164],"algebra":[8,30,83,152],"by":[9,57,129],"the":[10,54,75,80,94,178],"existence":[11],"almost-invariant":[13,36,45,58],"finite-dimensional":[14,37,46,59],"subcoalgebras":[15],"its":[17,105,120],"universal":[18,81],"enveloping":[19,82],"algebra.":[20,89],"More":[21],"generally,":[22],"module":[24,78],"coalgebra":[25],"cocommutative":[28],"Hopf":[29,135,151],"is":[31,69,99,108,157],"amenable":[32,100,117,134,163],"if":[33,101,104],"it":[34],"admits":[35],"subcoalgebras.":[38,60],"prove":[40,132],"coalgebraic":[42],"rounding":[43],"theorem:":[44],"subspaces":[47],"can":[48],"be":[49],"replaced,":[50],"without":[51],"loss":[52],"in":[53],"F\u00f8lner":[55],"constant,":[56],"For":[61,90],"algebras":[63,136,165],"this":[64,92],"implies":[65],"that":[66,96,133,149,174],"our":[67],"definition":[68],"equivalent":[70],"Elek's":[72],"left":[76],"regular":[77],"seen":[84],"merely":[85],"as":[86],"an":[87,127],"associative":[88],"groups":[91],"recovers":[93],"result":[95],"group":[98,106],"and":[102,145,148],"only":[103],"ring":[107],"algebraically":[109],"amenable.":[110,158,170],"It":[111],"furthermore":[112],"shows":[113],"that,":[114],"for":[115],"every":[116,150],"group,":[118],"all":[119],"nonzero":[121],"modules":[122],"are":[123,137,167],"amenable,":[124],"thus":[125],"proving":[126],"assertion":[128],"Gromov.":[130],"closed":[138],"under":[139],"taking":[140],"subalgebras,":[141],"quotients,":[142],"cleft":[143],"extensions,":[144],"directed":[146],"unions,":[147],"locally":[153],"subexponential":[155],"growth":[156],"give":[160],"examples":[161],"which":[166],"not":[168],"elementarily":[169],"Finally,":[171],"we":[172],"show":[173],"passes":[176],"associated":[179],"graded":[180],"Hopf-module":[181],"coalgebra.":[182]},"counts_by_year":[{"year":2026,"cited_by_count":1}],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-09-11T00:00:00"}