About: Cartan subalgebra

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In mathematics, a Cartan subalgebra, often abbreviated as CSA, is a nilpotent subalgebra of a Lie algebra that is self-normalising (if for all , then ). They were introduced by Élie Cartan in his doctoral thesis. It controls the representation theory of a semi-simple Lie algebra over a field of characteristic . Kac–Moody algebras and generalized Kac–Moody algebras also have subalgebras that play the same role as the Cartan subalgebras of semisimple Lie algebras (over a field of characteristic zero).

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rdf:type	Thing yago:WikicatLieAlgebras yago:Abstraction100002137 yago:Algebra106012726 yago:Cognition100023271 yago:Content105809192 yago:Discipline105996646 yago:KnowledgeDomain105999266 yago:Mathematics106000644 yago:PsychologicalFeature100023100 yago:PureMathematics106003682 yago:Science105999797
rdfs:label	Cartan subalgebra (en) Cartan-Unteralgebra (de) カルタン部分環 (ja) 카르탕 부분 대수 (ko) Подалгебра Картана (ru) Підалгебра Картана (uk) 嘉当子代数 (zh)
rdfs:comment	In der Mathematik, speziell in der Theorie der Lie-Algebren, werden Cartan-Unteralgebren unter anderem in der Klassifikation der halbeinfachen Lie-Algebren und in der Theorie der symmetrischen Räume verwendet. Der Rang einer Lie-Algebra (oder der zugehörigen Lie-Gruppe) ist definiert als die Dimension der Cartan-Unteralgebra. Ein Beispiel einer Cartan-Unteralgebra ist die Algebra der Diagonalmatrizen. (de) 数学において，カルタン部分環（カルタンぶぶんかん，英: Cartan subalgebra，しばしば CSA と略される）とは，リー環の冪零部分環であって，なもの（すべてのに対してであるならば，であるもの）のことである．エリ・カルタンによって彼の博士論文において導入された． (ja) 리 대수 이론에서, 카르탕 부분 대수(Cartan部分代數, 영어: Cartan subalgebra)는 리 대수의 최대 아벨 부분 대수의 일종이다. (ko) Подалгебра Картана — подалгебра Ли , равная своему нормализатору: * для некоторого (нильпотентность), * (самонормализованность). Понятие имеет большое значение для классификации полупростых алгебр Ли и в теории симметричных пространств. Названа в честь французского математика Эли Картана. Эквивалентное определение: нильпотентная подалгебра является подалгеброй Картана, если она равна своей нуль-компоненте Фиттинга, то есть множеству: где — присоединённое представление группы Ли. (ru) 在数学中，嘉当子代数（Cartan subalgebra，缩写为 CSA），是一个李代数的自正规化（如果对所有，那么）、幂零子代数，通常用表示。 (zh) В математиці, зокрема теорії алгебр Лі, підалгебрами Картана називаються певні нільпотентні підалгебри, які зокрема мають велике значення для класифікації напівпростих алгебр Лі і в теорії симетричних просторів. Названі на честь французького математика Елі Картана. (uk) In mathematics, a Cartan subalgebra, often abbreviated as CSA, is a nilpotent subalgebra of a Lie algebra that is self-normalising (if for all , then ). They were introduced by Élie Cartan in his doctoral thesis. It controls the representation theory of a semi-simple Lie algebra over a field of characteristic . Kac–Moody algebras and generalized Kac–Moody algebras also have subalgebras that play the same role as the Cartan subalgebras of semisimple Lie algebras (over a field of characteristic zero). (en)
rdfs:seeAlso	Semisimple Lie algebra
dcterms:subject	Lie algebras
Wikipage page ID	1336000 (xsd:integer)
Wikipage revision ID	1117754196 (xsd:integer)
Link from a Wikipage to another Wikipage	David Vogan Algebraically closed field Lie algebra representation Lie group Lie algebras Mathematics Generalized Kac–Moody algebra Lie algebra Élie Cartan Harish-Chandra isomorphism Identity component Automorphism Toral Lie algebra Linear Lie algebra Square matrix Field (mathematics) Nilpotent Lie algebra Graduate Texts in Mathematics Isomorphism Kac–Moody algebra Regular element of a Lie algebra Adjoint representation of a Lie group Abelian Lie algebra Maximal torus Diagonalizable matrix Dover Publications Root system Toral subalgebra Semisimple Lie algebra Semisimple operator Representation theory of semisimple Lie algebras Weyl group Split Lie algebra Subalgebra Splitting Cartan subalgebra Springer-Verlag Centralizer Fundamental Weyl chamber Rank of a Lie algebra Adjoint endomorphism Self-normalising
Link from a Wikipage to an external page	https://archive.org/details/introductiontoli00jame https://web.archive.org/web/20191029165616/http:/pi.math.cornell.edu/~dmehrle/notes/partiii/liealg_partiii_notes.pdf https://web.archive.org/web/20201006224222/https:/www.maths.ed.ac.uk/~igordon/LA1.pdf
sameAs	Cartan subalgebra Cartan subalgebra Cartan subalgebra Cartan subalgebra Cartan subalgebra Cartan subalgebra Cartan subalgebra Cartan subalgebra Cartan subalgebra Cartan subalgebra Cartan subalgebra
dbp:wikiPageUsesTemplate	dbt:Footnotes dbt:Citation dbt:Cite_book dbt:Clarify dbt:Fact dbt:For dbt:Main dbt:See_also dbt:Short_description dbt:Expand-section dbt:Eom dbt:Lie_groups
authorlink	Vladimir L. Popov (en)
first	V.L. (en)
last	Popov (en)

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