Casimir operator
E581262
The Casimir operator is a distinguished central element in the universal enveloping algebra of a Lie algebra that acts as a scalar on each irreducible representation and is used to classify and label those representations.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Casimir operator canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T6282522 — 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: Casimir operator Context triple: [Lie algebra representation, relatedConcept, Casimir operator]
-
A.
spin Casimir operator
The spin Casimir operator is a Lorentz-invariant operator associated with the Poincaré group that characterizes the intrinsic angular momentum (spin) of elementary particles in relativistic quantum theory.
-
B.
Wigner–Eckart theorem
The Wigner–Eckart theorem is a fundamental result in quantum mechanics that factorizes matrix elements of tensor operators into a reduced matrix element and a purely geometric part given by Clebsch–Gordan coefficients, greatly simplifying angular momentum calculations.
-
C.
Clebsch–Gordan coefficients
Clebsch–Gordan coefficients are numerical factors in quantum mechanics and representation theory that describe how to combine two angular momenta (or group representations) into a single resultant one.
-
D.
Dirac operator
The Dirac operator is a fundamental first-order differential operator on spinor fields that generalizes the classical Dirac equation and plays a central role in geometry, topology, and quantum field theory.
-
E.
Racah algebra
Racah algebra is a mathematical structure in representation theory and quantum mechanics that encodes the symmetries and coupling properties of angular momenta, particularly through Racah coefficients and related special functions.
- 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: Casimir operator Target entity description: The Casimir operator is a distinguished central element in the universal enveloping algebra of a Lie algebra that acts as a scalar on each irreducible representation and is used to classify and label those representations.
-
A.
spin Casimir operator
The spin Casimir operator is a Lorentz-invariant operator associated with the Poincaré group that characterizes the intrinsic angular momentum (spin) of elementary particles in relativistic quantum theory.
-
B.
Wigner–Eckart theorem
The Wigner–Eckart theorem is a fundamental result in quantum mechanics that factorizes matrix elements of tensor operators into a reduced matrix element and a purely geometric part given by Clebsch–Gordan coefficients, greatly simplifying angular momentum calculations.
-
C.
Clebsch–Gordan coefficients
Clebsch–Gordan coefficients are numerical factors in quantum mechanics and representation theory that describe how to combine two angular momenta (or group representations) into a single resultant one.
-
D.
Dirac operator
The Dirac operator is a fundamental first-order differential operator on spinor fields that generalizes the classical Dirac equation and plays a central role in geometry, topology, and quantum field theory.
-
E.
Racah algebra
Racah algebra is a mathematical structure in representation theory and quantum mechanics that encodes the symmetries and coupling properties of angular momenta, particularly through Racah coefficients and related special functions.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
central element
ⓘ
mathematical concept ⓘ operator ⓘ |
| actsAs | scalar on each irreducible representation ⓘ |
| appearsIn |
classification of unitary representations of semisimple Lie groups
ⓘ
conformal field theory ⓘ gauge theories ⓘ harmonic analysis on Lie groups ⓘ |
| appliesTo |
Lie algebras
ⓘ
Lie groups NERFINISHED ⓘ |
| codomain | vector space of a representation ⓘ |
| constructedFrom |
basis of the Lie algebra
ⓘ
dual basis with respect to the Killing form ⓘ |
| constructedUsing | Killing form ⓘ |
| definedIn | universal enveloping algebra of a Lie algebra ⓘ |
| domain | vector space of a representation ⓘ |
| eigenvalueDependsOn |
highest weight of the representation
ⓘ
quadratic form on the weight space ⓘ |
| example |
quadratic Casimir of su(3) in quark model
ⓘ
total angular momentum operator J^2 in su(2) representations ⓘ |
| field |
Lie theory
ⓘ
mathematics ⓘ representation theory ⓘ theoretical physics ⓘ |
| hasFormula | C = \sum_i X_i X^i for basis {X_i} and dual basis {X^i} ⓘ |
| hasOrder | typically quadratic in Lie algebra generators ⓘ |
| hasProperty |
central in the universal enveloping algebra
ⓘ
commutes with all elements of the Lie algebra representation ⓘ for semisimple Lie algebras, quadratic Casimir is essentially unique up to scale ⓘ its eigenvalues label irreducible representations ⓘ lies in the center of the universal enveloping algebra ⓘ takes constant value on each irreducible representation ⓘ |
| hasType | invariant operator ⓘ |
| hasVariant |
higher-order Casimir operators
ⓘ
quadratic Casimir operator ⓘ |
| namedAfter | Hendrik Casimir NERFINISHED ⓘ |
| relatedTo |
Casimir invariants
NERFINISHED
ⓘ
Harish-Chandra isomorphism NERFINISHED ⓘ Killing form ⓘ enveloping algebra center ⓘ |
| specialCaseOf | central elements of universal enveloping algebras ⓘ |
| usedFor |
classifying irreducible representations
ⓘ
labelling irreducible representations ⓘ |
| usedIn |
angular momentum theory
ⓘ
classification of particles by spin and mass ⓘ particle physics ⓘ quantum mechanics ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: Casimir operator Description of subject: The Casimir operator is a distinguished central element in the universal enveloping algebra of a Lie algebra that acts as a scalar on each irreducible representation and is used to classify and label those representations.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.