Cartan–Killing form

E542117

Lie algebra invariant bilinear form invariant bilinear form mathematical object symmetric bilinear form

The Cartan–Killing form is a canonical symmetric bilinear form on a Lie algebra that plays a central role in classifying and studying the structure of Lie algebras and Lie groups.

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Statements (50)

Predicate	Object
instanceOf	Lie algebra invariant ⓘ bilinear form ⓘ invariant bilinear form ⓘ mathematical object ⓘ symmetric bilinear form ⓘ
alsoKnownAs	Killing form NERFINISHED ⓘ
appliesTo	finite-dimensional Lie algebras over fields of characteristic zero ⓘ
belongsToField	Lie theory ⓘ differential geometry ⓘ representation theory ⓘ
characterizes	semisimple Lie algebras via nondegeneracy ⓘ
definedOn	Lie algebra ⓘ
hasProperty	ad-invariant ⓘ bilinear ⓘ degenerate on solvable Lie algebras ⓘ invariant under adjoint representation ⓘ negative definite on compact semisimple Lie algebras ⓘ nondegenerate on semisimple Lie algebras ⓘ symmetric ⓘ trace form ⓘ
isCanonicalOn	semisimple Lie algebra ⓘ
isDefinedBy	K(x,y) = Trace(ad(x) ∘ ad(y)) ⓘ
isGivenBy	trace of composition of adjoint endomorphisms ⓘ
isInvariantUnder	adjoint action of Lie group ⓘ inner automorphisms of Lie algebra ⓘ
isProportionalTo	any other invariant symmetric bilinear form on a simple Lie algebra ⓘ
isUsedFor	classification of complex semisimple Lie algebras ⓘ classification of real semisimple Lie algebras ⓘ classification of semisimple Lie algebras ⓘ classification of simple Lie algebras ⓘ construction of Dynkin diagrams ⓘ decomposition of Lie algebras ⓘ definition of Cartan subalgebras ⓘ definition of Casimir operator ⓘ definition of dual Coxeter number ⓘ definition of invariant Riemannian metric on Lie group ⓘ definition of metric on Lie algebra ⓘ definition of root systems ⓘ detection of semisimplicity ⓘ study of Lie algebra structure ⓘ study of Lie group structure ⓘ
isUsedToDefine	Cartan matrix NERFINISHED ⓘ Weyl group reflections NERFINISHED ⓘ angles between roots ⓘ lengths of roots ⓘ orthogonality of roots ⓘ
namedAfter	Wilhelm Killing NERFINISHED ⓘ Élie Cartan NERFINISHED ⓘ
restrictsTo	nondegenerate form on derived algebra of semisimple Lie algebra ⓘ
vanishesOn	center of a Lie algebra ⓘ

Referenced by (2)

Full triples — surface form annotated when it differs from this entity's canonical label.

Cartan → notableFor → Cartan–Killing form ⓘ

subject surface form: Élie Cartan

Cartan → hasRelatedConcept → Cartan–Killing form ⓘ