Harish-Chandra isomorphism
E250729
The Harish-Chandra isomorphism is a fundamental result in representation theory that identifies the center of the universal enveloping algebra of a semisimple Lie algebra with the algebra of Weyl group–invariant polynomials on a Cartan subalgebra.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Harish-Chandra isomorphism canonical | 3 |
| Harish-Chandra homomorphism | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T2267714 — 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: Harish-Chandra isomorphism Context triple: [Harish-Chandra, notableWork, Harish-Chandra isomorphism]
-
A.
Weil representation
The Weil representation is a fundamental projective unitary representation of symplectic groups (or their metaplectic covers) on spaces of functions, central to number theory, automorphic forms, and the theory of theta functions.
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B.
Weyl character formula
The Weyl character formula is a fundamental result in representation theory that gives an explicit expression for the characters of irreducible finite-dimensional representations of semisimple Lie algebras and Lie groups.
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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.
Harish-Chandra
Harish-Chandra was a pioneering mathematician and physicist best known for his fundamental contributions to representation theory and harmonic analysis on Lie groups.
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E.
Conway–Norton collaboration
The Conway–Norton collaboration was a joint mathematical effort, led by John Conway and Simon Norton, that played a key role in developing the theory of monstrous moonshine and the construction of the Monster group.
- 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: Harish-Chandra isomorphism Target entity description: The Harish-Chandra isomorphism is a fundamental result in representation theory that identifies the center of the universal enveloping algebra of a semisimple Lie algebra with the algebra of Weyl group–invariant polynomials on a Cartan subalgebra.
-
A.
Weil representation
The Weil representation is a fundamental projective unitary representation of symplectic groups (or their metaplectic covers) on spaces of functions, central to number theory, automorphic forms, and the theory of theta functions.
-
B.
Weyl character formula
The Weyl character formula is a fundamental result in representation theory that gives an explicit expression for the characters of irreducible finite-dimensional representations of semisimple Lie algebras and Lie groups.
-
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.
Harish-Chandra
Harish-Chandra was a pioneering mathematician and physicist best known for his fundamental contributions to representation theory and harmonic analysis on Lie groups.
-
E.
Conway–Norton collaboration
The Conway–Norton collaboration was a joint mathematical effort, led by John Conway and Simon Norton, that played a key role in developing the theory of monstrous moonshine and the construction of the Monster group.
- F. None of above. chosen
Statements (37)
| Predicate | Object |
|---|---|
| instanceOf |
isomorphism
ⓘ
mathematical theorem ⓘ |
| appliesTo | semisimple Lie algebra ⓘ |
| assumes |
choice of Cartan subalgebra
ⓘ
semisimplicity of the Lie algebra ⓘ |
| characterizes | center of the universal enveloping algebra of a semisimple Lie algebra ⓘ |
| codomain | Weyl group–invariant polynomials on a Cartan subalgebra ⓘ |
| context | complex semisimple Lie algebra ⓘ |
| domain | center of the universal enveloping algebra ⓘ |
| expressedAs | Z(U(g)) ≅ S(h)^W ⓘ |
| field |
Lie theory
ⓘ
representation theory ⓘ |
| givesIsomorphismBetween | center of U(g) and S(h)^W ⓘ |
| hasConsequence |
description of central characters of representations
ⓘ
parametrization of infinitesimal characters ⓘ |
| holdsFor | complex semisimple Lie algebra g with Cartan subalgebra h and Weyl group W ⓘ |
| involves |
Harish-Chandra homomorphism
ⓘ
linked to:
Harish-Chandra isomorphism
Harish-Chandra projection ⓘ |
| isFundamentalResultIn |
representation theory of semisimple Lie algebras
ⓘ
structure theory of semisimple Lie algebras ⓘ |
| namedAfter | Harish-Chandra ⓘ |
| objectOfStudy |
W-invariant elements of S(h)
ⓘ
center Z(U(g)) of the universal enveloping algebra U(g) ⓘ |
| relates |
Cartan subalgebra
ⓘ
Weyl group ⓘ universal enveloping algebra ⓘ |
| typeOf | algebra isomorphism ⓘ |
| usedIn |
classification of irreducible representations of semisimple Lie algebras
ⓘ
harmonic analysis on Lie groups ⓘ representation theory of real reductive Lie groups ⓘ study of primitive ideals in enveloping algebras ⓘ theory of highest weight modules ⓘ |
| usesConcept |
Cartan subalgebra
ⓘ
linked to:
Cartan subalgebras
Weyl group action ⓘ invariant theory ⓘ symmetric algebra ⓘ universal enveloping 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: Harish-Chandra isomorphism Description of subject: The Harish-Chandra isomorphism is a fundamental result in representation theory that identifies the center of the universal enveloping algebra of a semisimple Lie algebra with the algebra of Weyl group–invariant polynomials on a Cartan subalgebra.
Referenced by (4)
Full triples — surface form annotated when it differs from this entity's canonical label.