Cartan subalgebras

E125774

mathematical concept subalgebra of a Lie algebra

Cartan subalgebras are maximal abelian subalgebras of a Lie algebra consisting of semisimple elements, fundamental for classifying and understanding the structure of Lie algebras.

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All labels observed (2)

Label	Occurrences
Cartan subalgebra	5
Cartan subalgebras canonical	1

Statements (49)

Predicate	Object
instanceOf	mathematical concept ⓘ subalgebra of a Lie algebra ⓘ
appearsIn	Cartan–Weyl theory ⓘ classification of real forms of complex semisimple Lie algebras ⓘ structure theory of semisimple Lie algebras ⓘ
characterizedBy	being a maximal toral subalgebra in a reductive Lie algebra ⓘ being nilpotent and self-normalizing in an arbitrary finite-dimensional Lie algebra ⓘ consisting of elements simultaneously diagonalizable in all finite-dimensional representations for semisimple Lie algebras ⓘ
conjugacyProperty	all Cartan subalgebras of a finite-dimensional complex semisimple Lie algebra are conjugate ⓘ
context	complex semisimple Lie algebras ⓘ finite-dimensional Lie algebras over fields of characteristic zero ⓘ real semisimple Lie algebras ⓘ
definedIn	Lie algebra ⓘ
dimensionProperty	dimension equals rank of the Lie algebra for semisimple Lie algebras ⓘ
example	diagonal matrices in the Lie algebra of all complex n×n matrices ⓘ diagonal traceless matrices in sl(n,ℂ) ⓘ maximal toral subalgebras of compact Lie algebras ⓘ
existenceProperty	every finite-dimensional Lie algebra over an algebraically closed field of characteristic zero has a Cartan subalgebra ⓘ
field	Lie theory ⓘ algebra ⓘ representation theory ⓘ
generalizationOf	maximal tori in Lie groups ⓘ
hasInvariant	rank of the Lie algebra ⓘ
hasOperation	induction to Levi subalgebras ⓘ intersection with ideals and Levi factors ⓘ
namedAfter	Élie Cartan ⓘ
property	consist of ad-diagonalizable elements over algebraically closed fields of characteristic zero ⓘ equal to their own normalizer ⓘ maximal abelian subalgebra consisting of semisimple elements ⓘ nilpotent subalgebra in general Lie algebras ⓘ self-normalizing subalgebra ⓘ
relatedTo	Borel subalgebras ⓘ Cartan decomposition ⓘ Cartan decomposition ⓘ surface form: Cartan involution Coxeter–Dynkin diagrams ⓘ surface form: Cartan matrix Coxeter–Dynkin diagrams ⓘ surface form: Dynkin diagrams Killing form ⓘ Weyl group ⓘ surface form: Weyl groups maximal tori in Lie groups ⓘ root systems ⓘ
studiedIn	Lie algebra monographs ⓘ advanced algebra textbooks ⓘ
usedFor	Harish-Chandra character formula ⓘ surface form: Harish-Chandra theory of representations classification of complex semisimple Lie groups ⓘ classification of finite-dimensional semisimple Lie algebras ⓘ construction of Dynkin diagrams ⓘ definition of root systems ⓘ root space decomposition of Lie algebras ⓘ weight space decompositions in representation theory ⓘ