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.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Cartan–Killing form canonical | 2 |
How this entity was disambiguated
This entity first appeared as the object of triple T5705372 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Cartan–Killing form Context triple: [Élie Cartan, notableFor, Cartan–Killing form]
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A.
Kähler form
A Kähler form is a closed, positive-definite (1,1)-form that defines the compatible symplectic and Hermitian structure on a Kähler manifold.
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B.
Cartan decomposition
Cartan decomposition is a fundamental structural result in Lie theory that expresses a Lie algebra or Lie group as a direct sum or product of subspaces or subgroups with specific symmetry properties, widely used in differential geometry and representation theory.
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C.
Cartan
Cartan is a French surname most famously associated with mathematician Élie Cartan and his influential family of mathematicians.
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D.
Cartan structure equations
Cartan structure equations are fundamental differential geometric relations that express curvature and torsion in terms of connection 1-forms on a manifold.
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E.
Lie bracket
The Lie bracket is a bilinear, antisymmetric operation on a Lie algebra that measures the noncommutativity of its elements and encodes its infinitesimal structure.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Cartan–Killing form Target entity description: 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.
-
A.
Kähler form
A Kähler form is a closed, positive-definite (1,1)-form that defines the compatible symplectic and Hermitian structure on a Kähler manifold.
-
B.
Cartan decomposition
Cartan decomposition is a fundamental structural result in Lie theory that expresses a Lie algebra or Lie group as a direct sum or product of subspaces or subgroups with specific symmetry properties, widely used in differential geometry and representation theory.
-
C.
Cartan
Cartan is a French surname most famously associated with mathematician Élie Cartan and his influential family of mathematicians.
-
D.
Cartan structure equations
Cartan structure equations are fundamental differential geometric relations that express curvature and torsion in terms of connection 1-forms on a manifold.
-
E.
Lie bracket
The Lie bracket is a bilinear, antisymmetric operation on a Lie algebra that measures the noncommutativity of its elements and encodes its infinitesimal structure.
- F. None of above. chosen
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 ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Cartan–Killing form Description of subject: 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.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject surface form:
Élie Cartan