Weyl algebra
E117658
Noetherian ring
Weyl algebra
algebra over a field
associative algebra
domain
filtered algebra
noncommutative algebra
simple ring
The Weyl algebra is a fundamental noncommutative algebra generated by position and momentum operators satisfying canonical commutation relations, central in quantum mechanics and representation theory.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Weyl algebra canonical | 1 |
| first Weyl algebra A1(k) | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T990136 — 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: Weyl algebra Context triple: [Hermann Weyl, knownFor, Weyl algebra]
-
A.
Weierstrass preparation theorem
The Weierstrass preparation theorem is a fundamental result in complex analysis and analytic geometry that locally expresses analytic functions near a zero as a product of a polynomial and a unit, enabling a power-series analogue of factorization.
-
B.
Hilbert’s syzygy theorem
Hilbert’s syzygy theorem is a fundamental result in commutative algebra that describes the finite length and structure of free resolutions of modules over polynomial rings.
-
C.
Hilbert’s Nullstellensatz
Hilbert’s Nullstellensatz is a foundational theorem in algebraic geometry that establishes a deep correspondence between ideals in polynomial rings and algebraic sets, linking algebra and geometry.
-
D.
Wick’s theorem
Wick’s theorem is a fundamental result in quantum field theory that expresses time-ordered products of field operators as sums of normal-ordered products with all possible contractions, forming the basis for deriving Feynman rules and diagrammatic expansions.
-
E.
Hilbert basis theorem
The Hilbert basis theorem is a fundamental result in commutative algebra stating that if a ring is Noetherian then any polynomial ring over it is also Noetherian, ensuring that ideals in such rings are finitely generated.
- 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: Weyl algebra Target entity description: The Weyl algebra is a fundamental noncommutative algebra generated by position and momentum operators satisfying canonical commutation relations, central in quantum mechanics and representation theory.
-
A.
Weierstrass preparation theorem
The Weierstrass preparation theorem is a fundamental result in complex analysis and analytic geometry that locally expresses analytic functions near a zero as a product of a polynomial and a unit, enabling a power-series analogue of factorization.
-
B.
Hilbert’s syzygy theorem
Hilbert’s syzygy theorem is a fundamental result in commutative algebra that describes the finite length and structure of free resolutions of modules over polynomial rings.
-
C.
Hilbert’s Nullstellensatz
Hilbert’s Nullstellensatz is a foundational theorem in algebraic geometry that establishes a deep correspondence between ideals in polynomial rings and algebraic sets, linking algebra and geometry.
-
D.
Wick’s theorem
Wick’s theorem is a fundamental result in quantum field theory that expresses time-ordered products of field operators as sums of normal-ordered products with all possible contractions, forming the basis for deriving Feynman rules and diagrammatic expansions.
-
E.
Hilbert basis theorem
The Hilbert basis theorem is a fundamental result in commutative algebra stating that if a ring is Noetherian then any polynomial ring over it is also Noetherian, ensuring that ideals in such rings are finitely generated.
- F. None of above. chosen
Statements (50)
| Predicate | Object |
|---|---|
| instanceOf |
Noetherian ring
ⓘ
Weyl algebra ⓘ algebra over a field ⓘ associative algebra ⓘ domain ⓘ filtered algebra ⓘ noncommutative algebra ⓘ simple ring ⓘ |
| appearsIn |
algebraic geometry
ⓘ
microlocal analysis ⓘ |
| associatedGradedIs | polynomial ring in 2n variables ⓘ |
| generalization | nth Weyl algebra An(k) ⓘ |
| hasAlternativeDescription |
algebra generated by x and d/dx with [d/dx,x]=1
ⓘ
algebra of polynomial differential operators ⓘ |
| hasApplicationIn | mathematical physics ⓘ |
| hasBaseField | field of characteristic zero ⓘ |
| hasCanonicalRepresentation |
Schrödinger representation on L2(Rn)
ⓘ
action on polynomial ring by differential operators ⓘ |
| hasCenter | base field ⓘ |
| hasCommutationRelations | canonical commutation relations ⓘ |
| hasDimension | countably infinite as vector space over base field ⓘ |
| hasFiltration | order of differential operators ⓘ |
| hasGenerator |
momentum operator p
ⓘ
position operator x ⓘ |
| hasGenerators | xi, ∂i for i=1,…,n ⓘ |
| hasModuleCategory | category of D-modules on affine space ⓘ |
| hasPresentation | k⟨x,∂⟩/(∂x - x∂ - 1) ⓘ |
| hasStandardExample |
Weyl algebra
self-linksurface differs
ⓘ
surface form:
first Weyl algebra A1(k)
|
| isCentralIn |
D-module theory
ⓘ
algebraic analysis ⓘ quantum mechanics ⓘ representation theory ⓘ |
| isGradedBy | order filtration degree ⓘ |
| isNoetherian | true ⓘ |
| isPIAlgebra | false ⓘ |
| isQuantizationOf | polynomial algebra on symplectic vector space ⓘ |
| isSimple | true ⓘ |
| namedAfter | Hermann Weyl ⓘ |
| obtainedBy | universal enveloping algebra of Heisenberg Lie algebra modulo central relation ⓘ |
| relatedTo | Heisenberg Lie algebra ⓘ |
| satisfiesProperty |
Auslander regular
ⓘ
homologically smooth ⓘ |
| satisfiesRelation |
[p,x] = 1
ⓘ
[xi,xj] = 0 ⓘ [∂i,xj] = δij ⓘ [∂i,∂j] = 0 ⓘ p x - x p = 1 ⓘ |
| usedIn |
Heisenberg operator formulation of quantum mechanics
ⓘ
surface form:
Heisenberg representation
Schrödinger representation ⓘ canonical quantization ⓘ |
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: Weyl algebra Description of subject: The Weyl algebra is a fundamental noncommutative algebra generated by position and momentum operators satisfying canonical commutation relations, central in quantum mechanics and representation theory.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
first Weyl algebra A1(k)