Poincaré–Birkhoff–Witt theorem (original) (raw)
En mathématiques, et plus particulièrement en algèbre générale, dans la théorie des algèbres de Lie, le théorème de Poincaré-Birkhoff-Witt est un théorème fondamental qui permet de décrire précisément la structure de l'algèbre enveloppante d'une algèbre de Lie. Ce théorème est le résultat des travaux de Henri Poincaré en 1900, Garrett Birkhoff en 1937 et Ernst Witt en 1937. Il est parfois appelé en abrégé « théorème PBW ».
| Property | Value |
|---|---|
| dbo:abstract | En mathématiques, et plus particulièrement en algèbre générale, dans la théorie des algèbres de Lie, le théorème de Poincaré-Birkhoff-Witt est un théorème fondamental qui permet de décrire précisément la structure de l'algèbre enveloppante d'une algèbre de Lie. Ce théorème est le résultat des travaux de Henri Poincaré en 1900, Garrett Birkhoff en 1937 et Ernst Witt en 1937. Il est parfois appelé en abrégé « théorème PBW ». (fr) In mathematics, more specifically in the theory of Lie algebras, the Poincaré–Birkhoff–Witt theorem (or PBW theorem) is a result giving an explicit description of the universal enveloping algebra of a Lie algebra. It is named after Henri Poincaré, Garrett Birkhoff, and Ernst Witt. The terms PBW type theorem and PBW theorem may also refer to various analogues of the original theorem, comparing a filtered algebra to its associated graded algebra, in particular in the area of quantum groups. (en) Nella teoria delle algebre di Lie, il teorema Poincaré–Birkhoff–Witt è un risultato fondamentale che caratterizza l'algebra universale inviluppante di ogni algebra di Lie. Ricordiamo che ogni spazio vettoriale V su un campo ha una base, ossia un insieme S tale che ogni elemento di V si possa scrivere in un unico modo come combinazione lineare (finita) di elementi di S. Nella formulazione del teorema di Poincaré–Birkhoff–Witt si considera una base di Hamel, ossia una base totalmente ordinata da una relazione che chiameremo ≤. Se L è un'algebra di Lie su un campo K, allora dalla definizione esiste una K-mappa lineare canonica h da L verso l'algebra universale inviluppante U(L). Tale algebra è una K-algebra associativa unitaria. (it) Теорема Пуанкаре — Біркгофа — Вітта — твердження у математиці, що описує структуру універсальних обгортуючих алгебр і є одним із фундаментальних результатів теорії алгебр Лі і їх представлень. (uk) Теорема Пуанкаре — Биркгофа — Витта — утверждение, описывающее универсальную обёртывающую алгебру для заданной алгебры Ли над полем с базисом в векторном пространстве : элементы и образуют базис в линейном пространстве . В частности, отображение является вложением в , то есть ядро отображения равно . (ru) |
| dbo:wikiPageExternalLink | https://books.google.com/books%3Fid=QCcW1h835pwC https://books.google.com/books%3Fid=3yFYAAAAYAAJ https://books.google.com/books%3Fid=E-pUAAAAYAAJ https://books.google.com/books%3Fid=J8EGCAAAQBAJ http://www.numdam.org/article/RHM_1999__5_2_249_0.pdf http://www.numdam.org/item/ASNSP_1958_3_12_1-2_1_0/ http://gdz.sub.uni-goettingen.de/dms/load/img/%3FPPN=PPN243919689_0177&DMDID=dmdlog17 |
| dbo:wikiPageID | 762696 (xsd:integer) |
| dbo:wikiPageLength | 14273 (xsd:nonNegativeInteger) |
| dbo:wikiPageRevisionID | 1085094594 (xsd:integer) |
| dbo:wikiPageWikiLink | dbr:Samuel_Eilenberg dbr:Basis_(linear_algebra) dbr:Vector_space dbr:Dedekind_domain dbr:Quantum_groups dbc:Lie_algebras dbr:Mathematics dbr:General_linear_group dbr:Garrett_Birkhoff dbr:Lie_algebra dbr:Structure_constants dbc:Theorems_about_algebras dbr:Total_order dbr:Linear_combination dbr:Alfredo_Capelli dbr:Ernst_Witt dbr:Field_(mathematics) dbr:Nicolas_Bourbaki dbr:Universal_enveloping_algebra dbr:Henri_Cartan dbr:Henri_Poincaré dbr:Armand_Borel dbr:Abelian_group dbr:Symmetric_algebra dbr:Coalgebra dbr:Torsion-free_abelian_group dbr:Filtered_algebra dbr:Tensor_algebra dbr:Linear_transformation dbr:Injective dbr:Anthony_Knapp |
| dbp:first | T.S. (en) |
| dbp:id | B/b016540 (en) |
| dbp:last | Fofanova (en) |
| dbp:title | Birkhoff–Witt theorem (en) |
| dbp:wikiPageUsesTemplate | dbt:Springer dbt:Authority_control dbt:Cite_book dbt:Cite_journal dbt:For dbt:Reflist dbt:Short_description dbt:Harvid |
| dct:subject | dbc:Lie_algebras dbc:Theorems_about_algebras |
| gold:hypernym | dbr:Result |
| rdf:type | owl:Thing yago:WikicatLieAlgebras yago:WikicatTheorems yago:WikicatTheoremsInAbstractAlgebra yago:WikicatTheoremsInAlgebra yago:Abstraction100002137 yago:Algebra106012726 yago:Cognition100023271 yago:Communication100033020 yago:Content105809192 yago:Discipline105996646 yago:KnowledgeDomain105999266 yago:Mathematics106000644 yago:Message106598915 yago:Proposition106750804 yago:PsychologicalFeature100023100 yago:PureMathematics106003682 yago:Science105999797 yago:Statement106722453 yago:Theorem106752293 |
| rdfs:comment | En mathématiques, et plus particulièrement en algèbre générale, dans la théorie des algèbres de Lie, le théorème de Poincaré-Birkhoff-Witt est un théorème fondamental qui permet de décrire précisément la structure de l'algèbre enveloppante d'une algèbre de Lie. Ce théorème est le résultat des travaux de Henri Poincaré en 1900, Garrett Birkhoff en 1937 et Ernst Witt en 1937. Il est parfois appelé en abrégé « théorème PBW ». (fr) In mathematics, more specifically in the theory of Lie algebras, the Poincaré–Birkhoff–Witt theorem (or PBW theorem) is a result giving an explicit description of the universal enveloping algebra of a Lie algebra. It is named after Henri Poincaré, Garrett Birkhoff, and Ernst Witt. The terms PBW type theorem and PBW theorem may also refer to various analogues of the original theorem, comparing a filtered algebra to its associated graded algebra, in particular in the area of quantum groups. (en) Теорема Пуанкаре — Біркгофа — Вітта — твердження у математиці, що описує структуру універсальних обгортуючих алгебр і є одним із фундаментальних результатів теорії алгебр Лі і їх представлень. (uk) Теорема Пуанкаре — Биркгофа — Витта — утверждение, описывающее универсальную обёртывающую алгебру для заданной алгебры Ли над полем с базисом в векторном пространстве : элементы и образуют базис в линейном пространстве . В частности, отображение является вложением в , то есть ядро отображения равно . (ru) Nella teoria delle algebre di Lie, il teorema Poincaré–Birkhoff–Witt è un risultato fondamentale che caratterizza l'algebra universale inviluppante di ogni algebra di Lie. Ricordiamo che ogni spazio vettoriale V su un campo ha una base, ossia un insieme S tale che ogni elemento di V si possa scrivere in un unico modo come combinazione lineare (finita) di elementi di S. Nella formulazione del teorema di Poincaré–Birkhoff–Witt si considera una base di Hamel, ossia una base totalmente ordinata da una relazione che chiameremo ≤. (it) |
| rdfs:label | Théorème de Poincaré-Birkhoff-Witt (fr) Teorema di Poincaré-Birkhoff-Witt (it) Poincaré–Birkhoff–Witt theorem (en) Теорема Пуанкаре — Биркгофа — Витта (ru) Теорема Пуанкаре — Біркгофа — Вітта (uk) |
| owl:sameAs | freebase:Poincaré–Birkhoff–Witt theorem http://d-nb.info/gnd/4792948-0 wikidata:Poincaré–Birkhoff–Witt theorem dbpedia-fr:Poincaré–Birkhoff–Witt theorem dbpedia-it:Poincaré–Birkhoff–Witt theorem dbpedia-ka:Poincaré–Birkhoff–Witt theorem dbpedia-ru:Poincaré–Birkhoff–Witt theorem dbpedia-uk:Poincaré–Birkhoff–Witt theorem https://global.dbpedia.org/id/29ueb |
| prov:wasDerivedFrom | wikipedia-en:Poincaré–Birkhoff–Witt_theorem?oldid=1085094594&ns=0 |
| foaf:isPrimaryTopicOf | wikipedia-en:Poincaré–Birkhoff–Witt_theorem |
| is dbo:knownFor of | dbr:Ernst_Witt dbr:Henri_Poincaré |
| is dbo:wikiPageDisambiguates of | dbr:Birkhoff |
| is dbo:wikiPageRedirects of | dbr:Poincaré-Birkhoff-Witt_theorem dbr:Poincare–Birkhoff–Witt_theorem dbr:PBW_theorem dbr:Poincare-Birkhoff-Witt_theorem dbr:Birkhoff-Witt_theorem |
| is dbo:wikiPageWikiLink of | dbr:List_of_things_named_after_Henri_Poincaré dbr:Duflo_isomorphism dbr:Lie_algebra_representation dbr:Lie_superalgebra dbr:List_of_scientific_laws_named_after_people dbr:Generalized_Verma_module dbr:Nichols_algebra dbr:Garrett_Birkhoff dbr:Geometric_algebra dbr:Braided_vector_space dbr:Bergman's_diamond_lemma dbr:Hall_word dbr:Standard_basis dbr:Poincaré-Birkhoff-Witt_theorem dbr:Zonal_spherical_function dbr:Domain_(ring_theory) dbr:Heisenberg_group dbr:Representation_theorem dbr:Ernst_Witt dbr:Birkhoff dbr:Universal_enveloping_algebra dbr:Henri_Poincaré dbr:Associated_graded_ring dbr:Free_Lie_algebra dbr:Verma_module dbr:List_of_theorems dbr:List_of_things_named_after_Ernst_Witt dbr:Lyndon_word dbr:Quantum_group dbr:Poincare–Birkhoff–Witt_theorem dbr:Poincaré_theorem dbr:PBW_theorem dbr:Poincare-Birkhoff-Witt_theorem dbr:Birkhoff-Witt_theorem |
| is dbp:knownFor of | dbr:Ernst_Witt dbr:Henri_Poincaré |
| is foaf:primaryTopic of | wikipedia-en:Poincaré–Birkhoff–Witt_theorem |