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.
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 ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.