Rota–Baxter algebra
E421312
A Rota–Baxter algebra is an associative algebra equipped with a linear operator satisfying a specific integration-like identity that generalizes the properties of integral and summation operators in algebraic form.
All labels observed (5)
| Label | Occurrences |
|---|---|
| Baxter algebra | 1 |
| Rota–Baxter Hopf algebra | 1 |
| Rota–Baxter algebra canonical | 1 |
| Rota–Baxter coalgebra | 1 |
| Rota–Baxter module | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T4207190 — 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: Rota–Baxter algebra Context triple: [Gian-Carlo Rota, knownFor, Rota–Baxter algebra]
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A.
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.
-
B.
Weyl algebra
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.
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C.
Yang–Baxter equation
The Yang–Baxter equation is a fundamental consistency condition in mathematical physics and integrable systems that underlies exactly solvable models, quantum groups, and braid group representations.
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D.
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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E.
Onsager algebra
The Onsager algebra is an infinite-dimensional Lie algebra introduced in the study of exactly solvable models in statistical mechanics, particularly the two-dimensional Ising model.
- 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: Rota–Baxter algebra Target entity description: A Rota–Baxter algebra is an associative algebra equipped with a linear operator satisfying a specific integration-like identity that generalizes the properties of integral and summation operators in algebraic form.
-
A.
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.
-
B.
Weyl algebra
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.
-
C.
Yang–Baxter equation
The Yang–Baxter equation is a fundamental consistency condition in mathematical physics and integrable systems that underlies exactly solvable models, quantum groups, and braid group representations.
-
D.
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.
-
E.
Onsager algebra
The Onsager algebra is an infinite-dimensional Lie algebra introduced in the study of exactly solvable models in statistical mechanics, particularly the two-dimensional Ising model.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
algebraic structure
ⓘ
associative algebra with operator ⓘ |
| appearedIn | probability theory ⓘ |
| definedOver | associative algebra ⓘ |
| equippedWith | linear operator ⓘ |
| formalizes |
discrete summation by parts–type identity
ⓘ
integration by parts–type identity ⓘ |
| generalizes |
integration operator
ⓘ
summation operator ⓘ |
| hasAlternativeName |
Rota–Baxter algebra
ⓘ
surface form:
Baxter algebra
|
| hasAlternativeSpelling | Rota-Baxter algebra ⓘ |
| hasConcept |
Rota–Baxter ideal
ⓘ
Rota–Baxter algebra self-linksurface differs ⓘ
surface form:
Rota–Baxter module
free Rota–Baxter algebra ⓘ morphism of Rota–Baxter algebras ⓘ |
| hasExample |
Connes–Kreimer Hopf algebra with renormalization operator
ⓘ
algebra of Laurent series with projection operator ⓘ algebra of continuous functions with definite integral operator ⓘ algebra of functions with integral operator ⓘ algebra of polynomials with Rota–Baxter operator given by integration from 0 ⓘ algebra of sequences with summation operator ⓘ |
| hasGeneralization |
Rota–Baxter algebra
self-linksurface differs
ⓘ
surface form:
Rota–Baxter Hopf algebra
Rota–Baxter bialgebra ⓘ Rota–Baxter algebra self-linksurface differs ⓘ
surface form:
Rota–Baxter coalgebra
Rota–Baxter operator on nonassociative algebras ⓘ |
| hasParameter | weight ⓘ |
| hasSpecialCase |
Rota–Baxter algebra of weight one
ⓘ
weight zero Rota–Baxter algebra ⓘ |
| hasStructure |
associative multiplication
ⓘ
linear endomorphism ⓘ |
| isNamedAfter |
Gian-Carlo Rota
ⓘ
Glen Baxter ⓘ |
| relatedTo |
dendriform algebra
ⓘ
pre-Lie algebra ⓘ quasi-shuffle algebra ⓘ shuffle algebra ⓘ |
| researchedBy |
Frédéric Patras
NERFINISHED
ⓘ
Kurusch Ebrahimi-Fard ⓘ Li Guo ⓘ |
| satisfies | Rota–Baxter identity ⓘ |
| studiedIn | noncommutative geometry ⓘ |
| usedIn |
Hopf algebra theory
ⓘ
combinatorics ⓘ multiple zeta values ⓘ number theory ⓘ operad theory ⓘ probability theory ⓘ renormalization in quantum field theory ⓘ |
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: Rota–Baxter algebra Description of subject: A Rota–Baxter algebra is an associative algebra equipped with a linear operator satisfying a specific integration-like identity that generalizes the properties of integral and summation operators in algebraic form.
Referenced by (5)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Baxter algebra
this entity surface form:
Rota–Baxter module
this entity surface form:
Rota–Baxter coalgebra
this entity surface form:
Rota–Baxter Hopf algebra