Rota–Baxter algebra

E421312

algebraic structure associative algebra with operator

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.

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All labels observed (5)

Label	Occurrences
Baxter algebra	2
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]

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
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	Baxter algebra ⓘ linked to: Rota–Baxter algebra
hasAlternativeSpelling	Rota-Baxter algebra ⓘ
hasConcept	Rota–Baxter ideal ⓘ Rota–Baxter module ⓘ linked to: Rota–Baxter algebra 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 Hopf algebra ⓘ linked to: Rota–Baxter algebra Rota–Baxter bialgebra ⓘ Rota–Baxter coalgebra ⓘ linked to: Rota–Baxter algebra 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 ⓘ 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