Hopf algebra (concept named after him)
E679321
A Hopf algebra is an abstract algebraic structure that unifies and generalizes groups, rings, and vector spaces, playing a central role in areas such as algebraic topology, quantum groups, and category theory.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Hopf algebra (concept named after him) canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T7648310 — 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.
Target entity: Hopf algebra (concept named after him) Context triple: [Heinz Hopf, notableWork, Hopf algebra (concept named after him)]
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A.
Griess algebra
The Griess algebra is a 196,884-dimensional commutative nonassociative algebra over the real numbers whose automorphism group is the Monster, providing a concrete algebraic realization of this largest sporadic simple group.
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B.
Rota–Baxter algebra
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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C.
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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D.
Eilenberg–Zilber theorem
The Eilenberg–Zilber theorem is a fundamental result in algebraic topology that establishes a chain homotopy equivalence between the singular chain complex of a product space and the tensor product of the singular chain complexes of the factors.
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E.
Atiyah–Segal axioms
The Atiyah–Segal axioms are a set of mathematical conditions that rigorously define topological quantum field theories as functorial assignments from geometric data to algebraic structures.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
Target entity: Hopf algebra (concept named after him) Target entity description: A Hopf algebra is an abstract algebraic structure that unifies and generalizes groups, rings, and vector spaces, playing a central role in areas such as algebraic topology, quantum groups, and category theory.
-
A.
Griess algebra
The Griess algebra is a 196,884-dimensional commutative nonassociative algebra over the real numbers whose automorphism group is the Monster, providing a concrete algebraic realization of this largest sporadic simple group.
-
B.
Rota–Baxter algebra
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.
-
C.
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.
-
D.
Eilenberg–Zilber theorem
The Eilenberg–Zilber theorem is a fundamental result in algebraic topology that establishes a chain homotopy equivalence between the singular chain complex of a product space and the tensor product of the singular chain complexes of the factors.
-
E.
Atiyah–Segal axioms
The Atiyah–Segal axioms are a set of mathematical conditions that rigorously define topological quantum field theories as functorial assignments from geometric data to algebraic structures.
- F. None of above. chosen
Statements (54)
| Predicate | Object |
|---|---|
| instanceOf |
algebraic structure
ⓘ
mathematical concept ⓘ |
| appearsIn |
algebraic K-theory
NERFINISHED
ⓘ
stable homotopy theory ⓘ |
| condition |
antipode is convolution inverse of identity
ⓘ
comultiplication is an algebra homomorphism ⓘ counit is an algebra homomorphism ⓘ |
| definedOver |
commutative ring
ⓘ
field ⓘ |
| example |
Connes–Kreimer Hopf algebra
NERFINISHED
ⓘ
Hopf algebra of symmetric functions ⓘ Sweedler’s 4-dimensional Hopf algebra NERFINISHED ⓘ coordinate ring of an algebraic group ⓘ group algebra of a group ⓘ quantum enveloping algebra U_q(g) ⓘ universal enveloping algebra of a Lie algebra ⓘ |
| field |
abstract algebra
ⓘ
algebraic topology ⓘ category theory NERFINISHED ⓘ noncommutative geometry ⓘ quantum algebra ⓘ representation theory ⓘ |
| generalizes |
Lie algebra (via universal enveloping algebra)
NERFINISHED
ⓘ
bialgebra ⓘ coalgebra ⓘ group ⓘ group algebra ⓘ ring ⓘ |
| hasPart |
algebra structure
ⓘ
antipode ⓘ coalgebra structure ⓘ comultiplication ⓘ counit ⓘ multiplication ⓘ unit ⓘ |
| namedAfter | Heinz Hopf NERFINISHED ⓘ |
| property |
algebra and coalgebra structures are compatible
ⓘ
antipode is anticomultiplicative ⓘ antipode is antimultiplicative ⓘ comultiplication is coassociative ⓘ counit satisfies counit axioms ⓘ simultaneously an algebra and a coalgebra ⓘ |
| relatedConcept |
Tannaka–Krein duality
NERFINISHED
ⓘ
bialgebra ⓘ monoidal category ⓘ quantum group ⓘ |
| usedIn |
Hopf–Galois theory
NERFINISHED
ⓘ
algebraic topology ⓘ combinatorics ⓘ deformation quantization ⓘ homotopy theory ⓘ knot invariants ⓘ quantum groups ⓘ topological quantum field theory ⓘ |
How these facts were elicited
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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.
Subject: Hopf algebra (concept named after him) Description of subject: A Hopf algebra is an abstract algebraic structure that unifies and generalizes groups, rings, and vector spaces, playing a central role in areas such as algebraic topology, quantum groups, and category theory.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.