Askey–Wilson algebra
E653520
The Askey–Wilson algebra is a quadratic algebra arising in the theory of orthogonal polynomials and quantum groups, closely linked to the Askey–Wilson polynomials and related integrable models.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Askey–Wilson algebra canonical | 3 |
| Leonard pairs | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T7287606 — 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: Askey–Wilson algebra Context triple: [Onsager algebra, connectedTo, Askey–Wilson algebra]
-
A.
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.
-
B.
Schur–Weyl duality
Schur–Weyl duality is a fundamental result in representation theory that links representations of the symmetric group and the general linear group via their commuting actions on tensor powers of a vector space.
-
C.
Yang–Baxter equation
The Yang–Baxter equation is a fundamental consistency condition in mathematical physics and integrable systems that underlies exactly solvable models, quantum groups, and braid group representations.
-
D.
Rogers–Ramanujan-type identities
Rogers–Ramanujan-type identities are a class of deep q-series and partition identities generalizing the classical Rogers–Ramanujan identities, with rich connections to combinatorics, number theory, and modular forms.
-
E.
Gelfand–Tsetlin basis
The Gelfand–Tsetlin basis is a canonical, combinatorially defined basis for representations of certain Lie algebras and groups, particularly used in the representation theory of GL(n) and related structures.
- 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: Askey–Wilson algebra Target entity description: The Askey–Wilson algebra is a quadratic algebra arising in the theory of orthogonal polynomials and quantum groups, closely linked to the Askey–Wilson polynomials and related integrable models.
-
A.
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.
-
B.
Schur–Weyl duality
Schur–Weyl duality is a fundamental result in representation theory that links representations of the symmetric group and the general linear group via their commuting actions on tensor powers of a vector space.
-
C.
Yang–Baxter equation
The Yang–Baxter equation is a fundamental consistency condition in mathematical physics and integrable systems that underlies exactly solvable models, quantum groups, and braid group representations.
-
D.
Rogers–Ramanujan-type identities
Rogers–Ramanujan-type identities are a class of deep q-series and partition identities generalizing the classical Rogers–Ramanujan identities, with rich connections to combinatorics, number theory, and modular forms.
-
E.
Gelfand–Tsetlin basis
The Gelfand–Tsetlin basis is a canonical, combinatorially defined basis for representations of certain Lie algebras and groups, particularly used in the representation theory of GL(n) and related structures.
- F. None of above. chosen
Statements (45)
| Predicate | Object |
|---|---|
| instanceOf |
associative algebra
ⓘ
quadratic algebra ⓘ |
| appearsIn |
algebraic combinatorics
ⓘ
theory of bispectral problems ⓘ |
| arisesIn |
exactly solvable models
ⓘ
integrable lattice models ⓘ representation theory of quantum groups ⓘ theory of orthogonal polynomials ⓘ |
| definingRelations | quadratic relations among generators A, B, C ⓘ |
| field | mathematics ⓘ |
| generalizes | algebras associated with classical orthogonal polynomials ⓘ |
| hasCenter | central elements depending on parameters ⓘ |
| hasGenerator |
A
ⓘ
B ⓘ C ⓘ |
| hasRealization |
operators acting on polynomial spaces
ⓘ
q-difference operators ⓘ |
| hasRepresentation |
finite-dimensional modules
ⓘ
infinite-dimensional modules ⓘ |
| namedAfter |
James A. Wilson
ⓘ
Richard Askey ⓘ |
| parameterDependsOn |
q
ⓘ
structure constants ⓘ |
| relatedConcept |
Askey scheme
ⓘ
q-Askey scheme ⓘ |
| relatedTo |
Askey–Wilson polynomials
ⓘ
Leonard pairs ⓘ
linked to:
Askey–Wilson algebra
U_q(sl_2) ⓘ
linked to:
Drinfeld–Jimbo quantum groups
double affine Hecke algebras ⓘ
linked to:
Hecke algebra
q-Racah polynomials ⓘ
linked to:
Racah polynomials
q-orthogonal polynomials ⓘ quantum groups ⓘ quantum integrable models ⓘ tridiagonal pairs ⓘ |
| structure | defined by quadratic commutation relations ⓘ |
| subfield |
algebra
ⓘ
integrable systems ⓘ orthogonal polynomials ⓘ quantum groups ⓘ |
| symmetryAlgebraOf |
Askey–Wilson polynomials
ⓘ
certain q-hypergeometric operators ⓘ |
| usedFor |
algebraic description of Askey–Wilson polynomials
ⓘ
classification of tridiagonal pairs ⓘ construction of Leonard pairs ⓘ spectral analysis of q-difference operators ⓘ |
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: Askey–Wilson algebra Description of subject: The Askey–Wilson algebra is a quadratic algebra arising in the theory of orthogonal polynomials and quantum groups, closely linked to the Askey–Wilson polynomials and related integrable models.
Referenced by (4)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Askey–Wilson algebra