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.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Gel'fand–Kirillov conjecture canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T11411710 — 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: Gel'fand–Kirillov conjecture Context triple: [Gelfand–Kirillov dimension, relatedTo, Gel'fand–Kirillov conjecture]
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A.
Gelfand–Kirillov dimension
The Gelfand–Kirillov dimension is an invariant in noncommutative algebra that measures the growth rate of algebras and modules, serving as an analogue of Krull dimension for noncommutative settings.
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B.
Connes embedding problem
The Connes embedding problem is a central open question in operator algebras and quantum theory that asks whether every separable II₁ factor can be approximated in a specific way by finite-dimensional matrix algebras.
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C.
Hilbert’s fourteenth problem
Hilbert’s fourteenth problem is one of David Hilbert’s famous list of 23 problems, concerning the finite generation of certain algebras of invariants in algebraic geometry and invariant theory.
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D.
The Poincaré-Birkhoff-Witt theorem in ring theory
"The Poincaré-Birkhoff-Witt theorem in ring theory" is a mathematical work, attributed here to N. G. de Bruijn, that studies and applies the Poincaré–Birkhoff–Witt theorem in the context of associative and Lie-theoretic ring structures.
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E.
Zassenhaus conjecture
The Zassenhaus conjecture is a prominent open problem in group theory concerning the structure of units in integral group rings and their relation to the underlying finite 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: Gel'fand–Kirillov conjecture Target entity description: 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.
-
A.
Gelfand–Kirillov dimension
The Gelfand–Kirillov dimension is an invariant in noncommutative algebra that measures the growth rate of algebras and modules, serving as an analogue of Krull dimension for noncommutative settings.
-
B.
Connes embedding problem
The Connes embedding problem is a central open question in operator algebras and quantum theory that asks whether every separable II₁ factor can be approximated in a specific way by finite-dimensional matrix algebras.
-
C.
Hilbert’s fourteenth problem
Hilbert’s fourteenth problem is one of David Hilbert’s famous list of 23 problems, concerning the finite generation of certain algebras of invariants in algebraic geometry and invariant theory.
-
D.
The Poincaré-Birkhoff-Witt theorem in ring theory
"The Poincaré-Birkhoff-Witt theorem in ring theory" is a mathematical work, attributed here to N. G. de Bruijn, that studies and applies the Poincaré–Birkhoff–Witt theorem in the context of associative and Lie-theoretic ring structures.
-
E.
Zassenhaus conjecture
The Zassenhaus conjecture is a prominent open problem in group theory concerning the structure of units in integral group rings and their relation to the underlying finite group.
- F. None of above. chosen
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 ⓘ |
How these facts were elicited
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Input
Subject: Gel'fand–Kirillov conjecture Description of subject: 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.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.