Gel'fand–Kirillov conjecture
E924201
The Gel'fand–Kirillov conjecture is a statement in noncommutative algebra proposing that certain universal enveloping algebras of Lie algebras are birationally equivalent to Weyl algebras, linking their structure to that of algebras of differential operators.
Statements (45)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical conjecture
ⓘ
statement in noncommutative algebra ⓘ |
| aimsTo | classify enveloping algebras up to birational equivalence ⓘ |
| assumes | existence of a skew field of fractions for the enveloping algebra ⓘ |
| compares | skew field of fractions of an enveloping algebra with skew field of fractions of a Weyl algebra ⓘ |
| concerns |
Weyl algebras
NERFINISHED
ⓘ
algebras of differential operators ⓘ universal enveloping algebras of Lie algebras ⓘ |
| connectedTo |
classification of primitive ideals via associated varieties
ⓘ
study of symplectic leaves in representation theory ⓘ |
| context |
Weyl algebra as algebra of polynomial differential operators
ⓘ
universal enveloping algebra of a Lie algebra as a Noetherian domain ⓘ |
| field |
Lie theory
NERFINISHED
ⓘ
noncommutative algebra ⓘ representation theory ⓘ ring theory ⓘ |
| formulatedInContextOf | enveloping algebra of a finite-dimensional Lie algebra over a field of characteristic zero ⓘ |
| hasVariant |
Gel'fand–Kirillov conjecture for Poisson algebras
NERFINISHED
ⓘ
quantum Gel'fand–Kirillov conjecture ⓘ |
| influenced |
development of noncommutative birational geometry
ⓘ
study of Noetherian domains in ring theory ⓘ |
| involvesConcept |
Gel'fand–Kirillov dimension
NERFINISHED
ⓘ
Ore localization ⓘ division ring of fractions ⓘ |
| knownToFailFor |
some Lie algebras of Cartan type
ⓘ
some simple Lie algebras ⓘ |
| knownToHoldFor |
finite-dimensional nilpotent Lie algebras
ⓘ
finite-dimensional solvable Lie algebras of certain types ⓘ |
| motivation | to understand the structure of enveloping algebras via differential-operator-like models ⓘ |
| namedAfter |
Alexandre Kirillov
NERFINISHED
ⓘ
Israel Gel'fand NERFINISHED ⓘ |
| originallyFormulatedFor | finite-dimensional complex Lie algebras ⓘ |
| proposes | that certain universal enveloping algebras are birationally equivalent to Weyl algebras ⓘ |
| relatedTo |
Dixmier's work on enveloping algebras
ⓘ
algebraic geometry of noncommutative algebras ⓘ theory of primitive ideals in enveloping algebras ⓘ |
| relates | structure of universal enveloping algebras to structure of algebras of differential operators ⓘ |
| status |
false in general
ⓘ
proved for some classes of Lie algebras ⓘ |
| topic |
Weyl skew field
NERFINISHED
ⓘ
birational equivalence of algebras ⓘ enveloping algebra of a Lie algebra ⓘ skew fields of fractions ⓘ |
| typeOfEquivalence | birational equivalence of noncommutative algebras GENERATED ⓘ |
| usedIn | understanding representations of Lie algebras through differential operators ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.