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 |
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 isomorphism
self-linksurface differs
ⓘ
surface form:
Harish-Chandra homomorphism
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 subalgebras
ⓘ
surface form:
Cartan subalgebra
Weyl group action ⓘ invariant theory ⓘ symmetric algebra ⓘ universal enveloping algebra ⓘ |
Referenced by (4)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject surface form:
Harish-Chandra
this entity surface form:
Harish-Chandra homomorphism