Gelfand–Tsetlin algebra
E929568
The Gelfand–Tsetlin algebra is a commutative subalgebra of the universal enveloping algebra of a Lie algebra that acts diagonally in the Gelfand–Tsetlin basis and plays a central role in the explicit description of representations.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Gelfand–Tsetlin algebra canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T11411663 — 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: Gelfand–Tsetlin algebra Context triple: [Gelfand–Tsetlin basis, relatedTo, Gelfand–Tsetlin algebra]
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A.
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.
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B.
Gelfand–Tsetlin graph
The Gelfand–Tsetlin graph is a combinatorial structure whose vertices encode interlacing patterns corresponding to representations of unitary groups, organizing the branching of these representations in a graded, graph-theoretic form.
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C.
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.
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D.
Kazhdan–Lusztig theory
Kazhdan–Lusztig theory is a framework in representation theory and algebraic geometry that studies Hecke algebras and their bases via Kazhdan–Lusztig polynomials, with deep connections to the representation theory of Lie algebras and geometry of Schubert varieties.
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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: Gelfand–Tsetlin algebra Target entity description: The Gelfand–Tsetlin algebra is a commutative subalgebra of the universal enveloping algebra of a Lie algebra that acts diagonally in the Gelfand–Tsetlin basis and plays a central role in the explicit description of representations.
-
A.
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.
-
B.
Gelfand–Tsetlin graph
The Gelfand–Tsetlin graph is a combinatorial structure whose vertices encode interlacing patterns corresponding to representations of unitary groups, organizing the branching of these representations in a graded, graph-theoretic form.
-
C.
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.
-
D.
Kazhdan–Lusztig theory
Kazhdan–Lusztig theory is a framework in representation theory and algebraic geometry that studies Hecke algebras and their bases via Kazhdan–Lusztig polynomials, with deep connections to the representation theory of Lie algebras and geometry of Schubert varieties.
-
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 (45)
| Predicate | Object |
|---|---|
| instanceOf |
commutative algebra
ⓘ
representation-theoretic object ⓘ subalgebra ⓘ |
| actsDiagonallyOn | Gelfand–Tsetlin basis NERFINISHED ⓘ |
| actsOn |
finite-dimensional representations of gl_n
ⓘ
highest weight modules ⓘ |
| associatedWith |
spectra of commuting operators
ⓘ
weight decomposition of representations ⓘ |
| centralIn | description of Gelfand–Tsetlin modules ⓘ |
| containedIn |
U(g) for a Lie algebra g
ⓘ
universal enveloping algebra of a Lie algebra ⓘ |
| definedFor |
chains of Lie algebras
ⓘ
gl_1 ⊂ gl_2 ⊂ … ⊂ gl_n ⓘ |
| fieldOfStudy |
Lie theory
NERFINISHED
ⓘ
integrable systems ⓘ mathematics ⓘ noncommutative algebra ⓘ representation theory ⓘ |
| generatedBy | centers of U(gl_k) for k=1,…,n ⓘ |
| hasApplicationIn |
algebraic combinatorics
ⓘ
harmonic analysis on Lie groups ⓘ quantum integrable models ⓘ |
| hasProperty |
commutative
ⓘ
diagonalizable on Gelfand–Tsetlin basis ⓘ |
| hasRole |
diagonalizing algebra for Gelfand–Tsetlin basis
ⓘ
maximal commutative subalgebra in U(gl_n) ⓘ |
| isSubalgebraOf |
U(gl_n)
ⓘ
U(sl_n) ⓘ universal enveloping algebra ⓘ |
| namedAfter |
Israel Gelfand
NERFINISHED
ⓘ
Mikhail Tsetlin NERFINISHED ⓘ |
| relatedTo |
Gelfand–Tsetlin basis
NERFINISHED
ⓘ
Gelfand–Tsetlin integrable system NERFINISHED ⓘ Gelfand–Tsetlin modules NERFINISHED ⓘ Gelfand–Tsetlin patterns ⓘ branching rules for representations ⓘ highest weight representations ⓘ integrable systems ⓘ |
| studiedIn |
noncommutative algebra
ⓘ
representation theory of classical Lie algebras ⓘ |
| usedIn |
construction of Gelfand–Tsetlin bases
ⓘ
explicit description of representations ⓘ representation theory of gl_n ⓘ representation theory of sl_n ⓘ spectral decomposition of representations ⓘ |
How these facts were elicited
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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: Gelfand–Tsetlin algebra Description of subject: The Gelfand–Tsetlin algebra is a commutative subalgebra of the universal enveloping algebra of a Lie algebra that acts diagonally in the Gelfand–Tsetlin basis and plays a central role in the explicit description of representations.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.