About: Real form (Lie theory)

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In mathematics, the notion of a real form relates objects defined over the field of real and complex numbers. A real Lie algebra g0 is called a real form of a complex Lie algebra g if g is the complexification of g0: The notion of a real form can also be defined for complex Lie groups. Real forms of complex semisimple Lie groups and Lie algebras have been completely classified by Élie Cartan.

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rdf:type	Thing yago:WikicatLieAlgebras yago:WikicatLieGroups yago:Abstraction100002137 yago:Algebra106012726 yago:Cognition100023271 yago:Content105809192 yago:Discipline105996646 yago:Group100031264 yago:KnowledgeDomain105999266 yago:Mathematics106000644 yago:PsychologicalFeature100023100 yago:PureMathematics106003682 yago:Science105999797
rdfs:label	Reelle Form (de) Forma real (álgebra de Lie) (es) Real form (Lie theory) (en)
rdfs:comment	En matemáticas, la forma real a objetos definidos sobre el cuerpo de los números reales en relación con objetos similares definidos sobre los números complejos. Un álgebra de Lie real se llama una forma real de un álgebra de Lie compleja si es la de : La noción de forma real también se puede definir para grupos de Lie complejos. Las formas reales de grupos de Lie semisimples y álgebras de Lie fueron completamente clasificadas por Élie Cartan en 1894. (es) In mathematics, the notion of a real form relates objects defined over the field of real and complex numbers. A real Lie algebra g0 is called a real form of a complex Lie algebra g if g is the complexification of g0: The notion of a real form can also be defined for complex Lie groups. Real forms of complex semisimple Lie groups and Lie algebras have been completely classified by Élie Cartan. (en) Der Begriff reelle Form wird in der Mathematik verwendet, um über den reellen und komplexen Zahlen definierte Objekte, insbesondere algebraische Strukturen, miteinander in Beziehung zu setzen. Er wird vor allem in der Theorie der Lie-Algebren und Lie-Gruppen gebraucht. Eine reelle Lie-Algebra ist eine reelle Form einer komplexen Lie-Algebra , wenn die Komplexifizierung von ist, also . Eine reelle Form einer komplexen Lie-Gruppe ist eine Untergruppe, deren Lie-Algebra eine reelle Form der Lie-Algebra der komplexen Lie-Gruppe ist. (de)
rdfs:seeAlso	Compact Lie algebra Split Lie algebra
dcterms:subject	Lie algebras Lie groups
Wikipage page ID	22385144 (xsd:integer)
Wikipage revision ID	939053538 (xsd:integer)
Link from a Wikipage to another Wikipage	Dynkin diagram Lie group Sylvester's law of inertia Lie algebras Compact Lie group Complex number Complexification Complexification (Lie group) Mathematics Symmetric matrix Lie algebra Compact Lie algebra Complex Lie algebra Élie Cartan Linear algebraic group Algebraic geometry Lie groups Field (algebra) Riemannian symmetric space Killing form Special unitary group Negative definite Cartan's criterion Cartan decomposition Cartan subalgebra Real number Semisimple Lie group Satake diagram Special linear group Skew-Hermitian matrix Skew-symmetric matrix Semisimple Lie algebra Split Lie algebra Lie correspondence Classical Lie algebra
sameAs	Real form (Lie theory) Real form (Lie theory) Real form (Lie theory) Real form (Lie theory) Real form (Lie theory) Real form (Lie theory)
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has abstract	Der Begriff reelle Form wird in der Mathematik verwendet, um über den reellen und komplexen Zahlen definierte Objekte, insbesondere algebraische Strukturen, miteinander in Beziehung zu setzen. Er wird vor allem in der Theorie der Lie-Algebren und Lie-Gruppen gebraucht. Eine reelle Lie-Algebra ist eine reelle Form einer komplexen Lie-Algebra , wenn die Komplexifizierung von ist, also . Allgemein lässt sich analog eine reelle Form eines komplexen Vektorraumes durch die Bedingung definieren.Ein komplexer Vektorraum hat unendlich viele reelle Formen, zum Beispiel sind oder reelle Formen von . Eine reelle Form einer komplexen Lie-Gruppe ist eine Untergruppe, deren Lie-Algebra eine reelle Form der Lie-Algebra der komplexen Lie-Gruppe ist. (de) En matemáticas, la forma real a objetos definidos sobre el cuerpo de los números reales en relación con objetos similares definidos sobre los números complejos. Un álgebra de Lie real se llama una forma real de un álgebra de Lie compleja si es la de : La noción de forma real también se puede definir para grupos de Lie complejos. Las formas reales de grupos de Lie semisimples y álgebras de Lie fueron completamente clasificadas por Élie Cartan en 1894. (es) In mathematics, the notion of a real form relates objects defined over the field of real and complex numbers. A real Lie algebra g0 is called a real form of a complex Lie algebra g if g is the complexification of g0: The notion of a real form can also be defined for complex Lie groups. Real forms of complex semisimple Lie groups and Lie algebras have been completely classified by Élie Cartan. (en)
prov:wasDerivedFrom	wikipedia-en:Real_form_(Lie_theory)?oldid=939053538&ns=0
page length (characters) of wiki page	6015 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Real_form_(Lie_theory)
is Link from a Wikipage to another Wikipage of	Borel–Weil–Bott theorem Representation theory of the Lorentz group Dynkin diagram Compact real form Complexification (Lie group) Symplectic group Linear algebraic group List of Lie groups topics Simple Lie group Split-octonion Reductive group Satake diagram List of things named after Sophus Lie Semisimple Lie algebra Plancherel theorem for spherical functions Split real form Real form Real form of a complex Lie algebra Compact form Split form

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