{"id":"https://openalex.org/W7220669013","doi":"https://doi.org/10.48550/arxiv.2610.04001","title":"Finite-dimensional Lie subalgebras of the universal enveloping algebra of $\\mathfrak{sl}(2,\\mathbb{C})$","display_name":"Finite-dimensional Lie subalgebras of the universal enveloping algebra of $\\mathfrak{sl}(2,\\mathbb{C})$","publication_year":2026,"publication_date":"2026-10-02","ids":{"openalex":"https://openalex.org/W7220669013","doi":"https://doi.org/10.48550/arxiv.2610.04001"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2610.04001","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.04001","pdf_url":"https://arxiv.org/pdf/2610.04001","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":"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/2610.04001","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5148855990","display_name":"Tim Heib","orcid":"https://orcid.org/0009-0000-7463-0978","observed_orcids":["https://orcid.org/0009-0000-7463-0978"]},"institutions":[{"id":"https://openalex.org/I171892758","display_name":"Forschungszentrum J\u00fclich","ror":"https://ror.org/02nv7yv05","country_code":"DE","type":"facility","lineage":["https://openalex.org/I1305996414","https://openalex.org/I171892758"]},{"id":"https://openalex.org/I91712215","display_name":"Saarland University","ror":"https://ror.org/01jdpyv68","country_code":"DE","type":"education","lineage":["https://openalex.org/I91712215"]}],"countries":["DE"],"is_corresponding":true,"raw_author_name":"Heib, Tim","raw_affiliation_strings":["Institute for Quantum Computing Analytics (PGI-12) , Forschungszentrum J\u00fclich , 52425 J\u00fclich , Germany","Theoretical Physics , Universit\u00e4t des Saarlandes , 66123 Saarbr\u00fccken , Germany"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Institute for Quantum Computing Analytics (PGI-12) , Forschungszentrum J\u00fclich , 52425 J\u00fclich , Germany","institution_ids":["https://openalex.org/I171892758"]},{"raw_affiliation_string":"Theoretical Physics , Universit\u00e4t des Saarlandes , 66123 Saarbr\u00fccken , Germany","institution_ids":["https://openalex.org/I91712215"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5148855990"],"corresponding_institution_ids":["https://openalex.org/I171892758","https://openalex.org/I91712215"],"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":0.9990000128746033,"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":0.9990000128746033,"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":0.0003000000142492354,"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/T10287","display_name":"Algebraic structures and combinatorial models","score":0.00019999999494757503,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/universal-enveloping-algebra","display_name":"universal enveloping algebra","score":0.906000018119812},{"id":"https://openalex.org/keywords/sl-2-c","display_name":"SL(2,C)","score":0.8410000205039978},{"id":"https://openalex.org/keywords/abelian-subalgebras","display_name":"abelian subalgebras","score":0.7770000100135803},{"id":"https://openalex.org/keywords/semisimple-lie-algebras","display_name":"semisimple Lie algebras","score":0.7749999761581421},{"id":"https://openalex.org/keywords/weyl-algebra","display_name":"Weyl algebra","score":0.9549999833106995}],"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/C67996461","wikidata":"https://www.wikidata.org/wiki/Q629933","display_name":"Subalgebra","level":3,"score":0.6389999985694885},{"id":"https://openalex.org/C73648015","wikidata":"https://www.wikidata.org/wiki/Q6543817","display_name":"Lie conformal algebra","level":3,"score":0.6035000085830688},{"id":"https://openalex.org/C155058155","wikidata":"https://www.wikidata.org/wiki/Q1673406","display_name":"Universal enveloping algebra","level":3,"score":0.5648999810218811},{"id":"https://openalex.org/C144091092","wikidata":"https://www.wikidata.org/wiki/Q3611543","display_name":"Graded Lie algebra","level":3,"score":0.5611000061035156},{"id":"https://openalex.org/C65096084","wikidata":"https://www.wikidata.org/wiki/Q2912114","display_name":"Cartan subalgebra","level":5,"score":0.542900025844574},{"id":"https://openalex.org/C58910974","wikidata":"https://www.wikidata.org/wiki/Q1017106","display_name":"Adjoint representation","level":3,"score":0.5163000226020813},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.5080999732017517},{"id":"https://openalex.org/C51568863","wikidata":"https://www.wikidata.org/wiki/Q664495","display_name":"Lie algebra","level":2,"score":0.4991999864578247},{"id":"https://openalex.org/C5475112","wikidata":"https://www.wikidata.org/wiki/Q632521","display_name":"Affine Lie algebra","level":4,"score":0.46369999647140503},{"id":"https://openalex.org/C78804095","wikidata":"https://www.wikidata.org/wiki/Q1133024","display_name":"Lie superalgebra","level":5,"score":0.44780001044273376},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.444599986076355},{"id":"https://openalex.org/C203946495","wikidata":"https://www.wikidata.org/wiki/Q1017106","display_name":"Adjoint representation of a Lie algebra","level":4,"score":0.43689998984336853},{"id":"https://openalex.org/C22365015","wikidata":"https://www.wikidata.org/wiki/Q1003162","display_name":"Simple Lie group","level":3,"score":0.4219000041484833},{"id":"https://openalex.org/C136170076","wikidata":"https://www.wikidata.org/wiki/Q181296","display_name":"Abelian group","level":2,"score":0.40959998965263367},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.40369999408721924},{"id":"https://openalex.org/C2778052762","wikidata":"https://www.wikidata.org/wiki/Q2366896","display_name":"Semisimple Lie algebra","level":5,"score":0.40070000290870667},{"id":"https://openalex.org/C145620117","wikidata":"https://www.wikidata.org/wiki/Q2989763","display_name":"Commutator","level":4,"score":0.3862000107765198},{"id":"https://openalex.org/C99634282","wikidata":"https://www.wikidata.org/wiki/Q634557","display_name":"Killing form","level":5,"score":0.37070000171661377},{"id":"https://openalex.org/C186489112","wikidata":"https://www.wikidata.org/wiki/Q763735","display_name":"Representation of a Lie group","level":3,"score":0.3580999970436096},{"id":"https://openalex.org/C5106717","wikidata":"https://www.wikidata.org/wiki/Q3427416","display_name":"Fundamental representation","level":4,"score":0.33239999413490295},{"id":"https://openalex.org/C50555996","wikidata":"https://www.wikidata.org/wiki/Q840023","display_name":"Nilpotent","level":2,"score":0.32919999957084656},{"id":"https://openalex.org/C101862200","wikidata":"https://www.wikidata.org/wiki/Q1047475","display_name":"Casimir element","level":5,"score":0.3091000020503998},{"id":"https://openalex.org/C195592935","wikidata":"https://www.wikidata.org/wiki/Q4544947","display_name":"(g,K)-module","level":5,"score":0.2953000068664551},{"id":"https://openalex.org/C174696410","wikidata":"https://www.wikidata.org/wiki/Q7301156","display_name":"Real form","level":5,"score":0.27799999713897705}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2610.04001","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.04001","pdf_url":"https://arxiv.org/pdf/2610.04001","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":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"doi:10.48550/arxiv.2610.04001","doi":"https://doi.org/10.48550/arxiv.2610.04001","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2610.04001","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:2610.04001","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.04001","pdf_url":"https://arxiv.org/pdf/2610.04001","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":"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/W7220669013.pdf","grobid_xml":"https://content.openalex.org/works/W7220669013.grobid-xml"},"referenced_works_count":3,"referenced_works":["https://openalex.org/W4416158482","https://openalex.org/W7125510785","https://openalex.org/W7213352517"],"related_works":["https://openalex.org/W7098132990","https://openalex.org/W6979298705","https://openalex.org/W2069611199","https://openalex.org/W7132939519","https://openalex.org/W3043347683","https://openalex.org/W2069624706","https://openalex.org/W4392539547","https://openalex.org/W2949077561","https://openalex.org/W7222987478","https://openalex.org/W2964236468"],"abstract_inverted_index":{"In":[0],"this":[1],"work":[2],"we":[3,29,110],"provide":[4],"the":[5,16,21,59,70,103,112,124,127],"complete":[6],"classification":[7,114],"of":[8,15,20,62,69,72,102],"all":[9],"non-zero":[10],"finite-dimensional":[11],"complex":[12],"Lie":[13,24,35,42,56,74],"subalgebras":[14],"universal":[17],"enveloping":[18],"algebra":[19,25],"special":[22],"linear":[23],"$\\mathfrak{sl}(2,\\mathbb{C})$.":[26],"That":[27],"is,":[28],"show":[30],"that":[31,89],"every":[32,40],"such":[33],"nilpotent":[34],"subalgebra":[36,43,57],"is":[37,44,58],"abelian":[38,64],"and":[39,66,120,129],"non-solvable":[41],"reductive,":[45],"with":[46,96],"semisimple":[47],"component":[48],"isomorphic":[49],"to":[50,123],"$\\mathfrak{sl}(2,\\mathbb{C})$":[51],"itself.":[52],"Every":[53],"non-nilpotent":[54],"solvable":[55,73],"direct":[60],"sum":[61],"an":[63],"center":[65],"a":[67,79,97,116],"member":[68],"family":[71],"algebras":[75],"$\\mathfrak{r}(\\boldsymbol{d},\\boldsymbol{m})$,":[76],"parameterized":[77],"by":[78],"vector":[80,100],"$\\boldsymbol{d}$":[81],"whose":[82],"entries":[83],"are":[84],"strictly":[85],"increasing":[86],"positive":[87,98],"integers":[88],"have":[90],"greatest":[91],"common":[92],"divisor":[93],"one":[94],"together":[95],"integer":[99],"$\\boldsymbol{m}$":[101],"same":[104],"length":[105],"recording":[106],"their":[107],"multiplicities.":[108],"Moreover,":[109],"discuss":[111],"corresponding":[113],"in":[115,126],"real":[117],"skew-hermitian":[118],"setting":[119],"its":[121],"relation":[122],"realizations":[125],"single-":[128],"two-mode":[130],"Weyl":[131],"algebras.":[132]},"counts_by_year":[],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-10-07T00:00:00"}