Racah algebra
E443146
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.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Racah algebra canonical | 1 |
| Wigner 6-j symbols | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T4461379 — 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: Racah algebra Context triple: [Wigner–Eckart theorem, relatedTo, Racah algebra]
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A.
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.
-
B.
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.
-
C.
The Classical Groups: Their Invariants and Representations
The Classical Groups: Their Invariants and Representations is a foundational mathematical monograph by Hermann Weyl that systematically develops the theory of classical Lie groups, their invariants, and their representation theory.
-
D.
Onsager algebra
The Onsager algebra is an infinite-dimensional Lie algebra introduced in the study of exactly solvable models in statistical mechanics, particularly the two-dimensional Ising model.
-
E.
Methods of Representation Theory
Methods of Representation Theory is a foundational multi-volume work in mathematics that systematically develops the theory of group and algebra representations, coauthored by Israel Gelfand and collaborators.
- 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: Racah algebra Target entity description: 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.
-
A.
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.
-
B.
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.
-
C.
The Classical Groups: Their Invariants and Representations
The Classical Groups: Their Invariants and Representations is a foundational mathematical monograph by Hermann Weyl that systematically develops the theory of classical Lie groups, their invariants, and their representation theory.
-
D.
Onsager algebra
The Onsager algebra is an infinite-dimensional Lie algebra introduced in the study of exactly solvable models in statistical mechanics, particularly the two-dimensional Ising model.
-
E.
Methods of Representation Theory
Methods of Representation Theory is a foundational multi-volume work in mathematics that systematically develops the theory of group and algebra representations, coauthored by Israel Gelfand and collaborators.
- F. None of above. chosen
Statements (43)
| Predicate | Object |
|---|---|
| instanceOf |
algebra
ⓘ
mathematical structure ⓘ noncommutative algebra ⓘ |
| appearsIn |
representation theory of rotation group SO(3)
ⓘ
representation theory of su(2) ⓘ |
| connectedTo |
Clebsch–Gordan coefficients
NERFINISHED
ⓘ
Lie algebra representations ⓘ Wigner 3-j symbols NERFINISHED ⓘ Wigner 9-j symbols NERFINISHED ⓘ quantum group symmetries ⓘ |
| encodes |
coupling properties of angular momenta
ⓘ
symmetries of angular momentum coupling ⓘ |
| field |
mathematical physics
ⓘ
mathematics ⓘ quantum mechanics ⓘ representation theory ⓘ |
| frameworkFor |
algebraic treatment of 6-j symbols
ⓘ
recoupling theory ⓘ |
| generalizationOf | angular momentum coupling algebra ⓘ |
| governs | recoupling transformations of angular momenta ⓘ |
| hasApplication |
calculation of coupling coefficients
ⓘ
calculation of transition amplitudes ⓘ spectral analysis of quantum systems ⓘ |
| hasProperty |
associative
ⓘ
noncommutative ⓘ |
| hasRole |
algebraic framework for Racah coefficients
ⓘ
symmetry algebra of certain special functions ⓘ |
| namedAfter | Giulio Racah NERFINISHED ⓘ |
| relatedTo |
Racah coefficients
ⓘ
Racah polynomials NERFINISHED ⓘ Wigner 6-j symbols NERFINISHED ⓘ orthogonal polynomials ⓘ special functions ⓘ |
| studiedIn |
algebraic methods in quantum mechanics
ⓘ
theory of exactly solvable models ⓘ |
| symmetryOf |
Racah polynomials
NERFINISHED
ⓘ
certain hypergeometric-type difference equations ⓘ |
| usedIn |
addition of angular momenta
ⓘ
atomic physics ⓘ molecular physics ⓘ nuclear physics ⓘ quantum angular momentum theory ⓘ spectroscopic coupling schemes ⓘ |
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: Racah algebra Description of subject: 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.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Wigner 6-j symbols