{"id":"https://openalex.org/W7214751638","doi":"https://doi.org/10.48550/arxiv.2609.32371","title":"A Note on the Schouten--Nijenhuis Bracket of Lie--Rinehart Algebras","display_name":"A Note on the Schouten--Nijenhuis Bracket of Lie--Rinehart Algebras","publication_year":2026,"publication_date":"2026-09-26","ids":{"openalex":"https://openalex.org/W7214751638","doi":"https://doi.org/10.48550/arxiv.2609.32371"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.32371","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.32371","pdf_url":"https://arxiv.org/pdf/2609.32371","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","indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2609.32371","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5004603697","display_name":"Servais Cyr Gatse","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Nsiloulou, Servais Cyr Gats\u00e9 \\and Basile Guy Richard Bossoto \\and Irmely Gladesh Mabanza","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5004603697"],"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/T11673","display_name":"Advanced Topics in Algebra","score":0.9922000169754028,"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":0.9922000169754028,"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/T10896","display_name":"Homotopy and Cohomology in Algebraic Topology","score":0.007300000172108412,"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/T13080","display_name":"Advanced Differential Geometry Research","score":9.999999747378752e-05,"subfield":{"id":"https://openalex.org/subfields/3103","display_name":"Astronomy and Astrophysics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/schouten-nijenhuis-bracket","display_name":"Schouten-Nijenhuis bracket","score":0.9769999980926514},{"id":"https://openalex.org/keywords/lie-rinehart-algebras","display_name":"Lie-Rinehart algebras","score":0.9980000257492065},{"id":"https://openalex.org/keywords/gerstenhaber-bracket","display_name":"Gerstenhaber bracket","score":0.9150000214576721},{"id":"https://openalex.org/keywords/exterior-algebra","display_name":"exterior algebra","score":0.9319999814033508},{"id":"https://openalex.org/keywords/inner-derivations","display_name":"inner derivations","score":0.6850000023841858}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.833899974822998},{"id":"https://openalex.org/C2776809875","wikidata":"https://www.wikidata.org/wiki/Q1375963","display_name":"Converse","level":2,"score":0.6414999961853027},{"id":"https://openalex.org/C133769200","wikidata":"https://www.wikidata.org/wiki/Q174565","display_name":"Bracket","level":2,"score":0.5918999910354614},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.5604000091552734},{"id":"https://openalex.org/C51292594","wikidata":"https://www.wikidata.org/wiki/Q744960","display_name":"Associative algebra","level":5,"score":0.5566999912261963},{"id":"https://openalex.org/C160873635","wikidata":"https://www.wikidata.org/wiki/Q17148807","display_name":"Unital","level":3,"score":0.554099977016449},{"id":"https://openalex.org/C159423971","wikidata":"https://www.wikidata.org/wiki/Q177251","display_name":"Associative property","level":2,"score":0.5271000266075134},{"id":"https://openalex.org/C2778029271","wikidata":"https://www.wikidata.org/wiki/Q5421931","display_name":"Extension (predicate logic)","level":2,"score":0.5235999822616577},{"id":"https://openalex.org/C2778355321","wikidata":"https://www.wikidata.org/wiki/Q17079427","display_name":"Identity (music)","level":2,"score":0.5131000280380249},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.49219998717308044},{"id":"https://openalex.org/C183778304","wikidata":"https://www.wikidata.org/wiki/Q165474","display_name":"Commutative property","level":2,"score":0.48500001430511475},{"id":"https://openalex.org/C206159579","wikidata":"https://www.wikidata.org/wiki/Q1151539","display_name":"Jacobi identity","level":3,"score":0.47510001063346863},{"id":"https://openalex.org/C9652623","wikidata":"https://www.wikidata.org/wiki/Q190109","display_name":"Field (mathematics)","level":2,"score":0.46700000762939453},{"id":"https://openalex.org/C14394260","wikidata":"https://www.wikidata.org/wiki/Q4723980","display_name":"Algebra representation","level":3,"score":0.4235000014305115},{"id":"https://openalex.org/C28719098","wikidata":"https://www.wikidata.org/wiki/Q44946","display_name":"Point (geometry)","level":2,"score":0.40720000863075256},{"id":"https://openalex.org/C147073270","wikidata":"https://www.wikidata.org/wiki/Q567083","display_name":"Annihilator","level":3,"score":0.3905999958515167},{"id":"https://openalex.org/C133558598","wikidata":"https://www.wikidata.org/wiki/Q2916283","display_name":"Differential graded algebra","level":5,"score":0.3903000056743622},{"id":"https://openalex.org/C115179944","wikidata":"https://www.wikidata.org/wiki/Q371133","display_name":"Incidence algebra","level":5,"score":0.329800009727478},{"id":"https://openalex.org/C176954179","wikidata":"https://www.wikidata.org/wiki/Q1052674","display_name":"Symmetric algebra","level":5,"score":0.3221000134944916},{"id":"https://openalex.org/C161491579","wikidata":"https://www.wikidata.org/wiki/Q858656","display_name":"Commutative ring","level":3,"score":0.3206999897956848},{"id":"https://openalex.org/C100899422","wikidata":"https://www.wikidata.org/wiki/Q5058347","display_name":"Cellular algebra","level":4,"score":0.2824999988079071},{"id":"https://openalex.org/C169171071","wikidata":"https://www.wikidata.org/wiki/Q5449241","display_name":"Filtered algebra","level":3,"score":0.2815000116825104},{"id":"https://openalex.org/C138354692","wikidata":"https://www.wikidata.org/wiki/Q1231309","display_name":"Division algebra","level":4,"score":0.27790001034736633},{"id":"https://openalex.org/C2777299769","wikidata":"https://www.wikidata.org/wiki/Q3707858","display_name":"Type (biology)","level":2,"score":0.2759000062942505},{"id":"https://openalex.org/C139080492","wikidata":"https://www.wikidata.org/wiki/Q7756337","display_name":"Differential algebra","level":2,"score":0.27379998564720154},{"id":"https://openalex.org/C167239746","wikidata":"https://www.wikidata.org/wiki/Q15534632","display_name":"Graded ring","level":3,"score":0.2727000117301941},{"id":"https://openalex.org/C188971069","wikidata":"https://www.wikidata.org/wiki/Q1636734","display_name":"Universal algebra","level":3,"score":0.2718000113964081},{"id":"https://openalex.org/C37253","wikidata":"https://www.wikidata.org/wiki/Q5156502","display_name":"Complete intersection","level":2,"score":0.26269999146461487},{"id":"https://openalex.org/C100376341","wikidata":"https://www.wikidata.org/wiki/Q649977","display_name":"Jordan algebra","level":4,"score":0.25999999046325684}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.32371","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.32371","pdf_url":"https://arxiv.org/pdf/2609.32371","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.2609.32371","doi":"https://doi.org/10.48550/arxiv.2609.32371","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.32371","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:2609.32371","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.32371","pdf_url":"https://arxiv.org/pdf/2609.32371","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":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7214751638.pdf","grobid_xml":"https://content.openalex.org/works/W7214751638.grobid-xml"},"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W1809367921","https://openalex.org/W1634182604","https://openalex.org/W7099071833","https://openalex.org/W4402955323","https://openalex.org/W4407008240","https://openalex.org/W4386251086","https://openalex.org/W3047994032","https://openalex.org/W7096650865","https://openalex.org/W1843313164","https://openalex.org/W4415030227"],"abstract_inverted_index":{"Let":[0],"$A$":[1],"be":[2,18],"a":[3,9,19,111],"commutative":[4],"associative":[5],"unital":[6],"algebra":[7,21,33,58],"over":[8,22,55],"field":[10],"$K$":[11],"of":[12,39,41,51,114],"characteristic":[13],"zero,":[14],"and":[15,89,103],"let":[16],"$(G,\u03c1)$":[17],"Lie--Rinehart":[20,61,128],"$A$.":[23],"We":[24],"revisit":[25],"the":[26,31,37,49,52,56,60,68,73,78,86,90,98,115],"Gerstenhaber":[27,118],"structure":[28],"carried":[29],"by":[30],"exterior":[32,80],"$\\bigwedge_A":[34,121],"G$":[35,122],"from":[36],"point":[38],"view":[40],"graded":[42,84,91,100],"derivations.":[43],"Particular":[44],"attention":[45],"is":[46,63],"paid":[47],"to":[48,65,75,77],"well-definedness":[50],"Schouten--Nijenhuis":[53],"extension":[54],"coefficient":[57],"$A$:":[59],"identity":[62,93],"shown":[64],"provide":[66],"exactly":[67],"correction":[69],"terms":[70],"needed":[71],"for":[72],"bracket":[74],"descend":[76],"$A$-balanced":[79],"algebra.":[81],"After":[82],"establishing":[83],"antisymmetry,":[85],"Leibniz":[87],"rule":[88],"Jacobi":[92],"without":[94],"circularity,":[95],"we":[96,109],"study":[97],"inner":[99],"derivations":[101],"$\\adSN(P)=[P,\\cdot]_{SN}$":[102],"prove":[104],"\\[":[105],"[\\adSN(P),\\adSN(Q)]=\\adSN([P,Q]_{SN}).":[106],"\\]":[107],"Finally,":[108],"give":[110],"complete":[112],"proof":[113],"converse":[116],"construction:":[117],"brackets":[119],"on":[120,130],"are":[123],"in":[124],"one-to-one":[125],"correspondence":[126],"with":[127],"structures":[129],"$G$.":[131]},"counts_by_year":[],"updated_date":"2026-10-28T10:33:24.094632","created_date":"2026-09-30T00:00:00"}