Steinberg relations
E911353
Steinberg relations are algebraic identities in Milnor K-theory that impose the condition that symbols of pairs of field elements summing to one vanish, playing a central role in defining the structure of these K-groups.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Steinberg relations canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T11219353 — 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: Steinberg relations Context triple: [Milnor K-theory, usesConcept, Steinberg relations]
-
A.
Dolan–Grady relations
The Dolan–Grady relations are algebraic commutation relations between two operators that generate the Onsager algebra and play a key role in the study of exactly solvable models in statistical mechanics.
-
B.
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.
-
C.
Schur–Weyl duality
Schur–Weyl duality is a fundamental result in representation theory that links representations of the symmetric group and the general linear group via their commuting actions on tensor powers of a vector space.
-
D.
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.
-
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: Steinberg relations Target entity description: Steinberg relations are algebraic identities in Milnor K-theory that impose the condition that symbols of pairs of field elements summing to one vanish, playing a central role in defining the structure of these K-groups.
-
A.
Dolan–Grady relations
The Dolan–Grady relations are algebraic commutation relations between two operators that generate the Onsager algebra and play a key role in the study of exactly solvable models in statistical mechanics.
-
B.
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.
-
C.
Schur–Weyl duality
Schur–Weyl duality is a fundamental result in representation theory that links representations of the symmetric group and the general linear group via their commuting actions on tensor powers of a vector space.
-
D.
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.
-
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 |
algebraic relation
ⓘ
defining relation in Milnor K-theory ⓘ |
| appearsIn |
definition of K_2^{M}(F) as (F^× ⊗ F^×)/R where R is generated by {a,1-a}
ⓘ
higher Milnor K-groups via tensor products modulo Steinberg relations ⓘ |
| appliesTo |
K_2^{M}(F)
ⓘ
K_n^{M}(F) ⓘ Milnor K-groups ⓘ
linked to:
Milnor K-theory
|
| category |
relation in graded ring
ⓘ
universal relation for symbols on F^× ⓘ |
| condition |
symbols of pairs of field elements summing to one vanish
ⓘ
{a,1-a,a_3, … ,a_n}=0 in K_n^{M}(F) for a,1-a in F^× ⓘ {a,1-a}=0 in K_2^{M}(F) for a,1-a in F^× ⓘ |
| context |
algebraic geometry
ⓘ
algebraic number theory ⓘ motivic cohomology ⓘ |
| definedOver | field F ⓘ |
| field |
Milnor K-theory
ⓘ
algebraic K-theory ⓘ
linked to:
K-theory
|
| generalizationOf | classical relation in K_2 of a field ⓘ |
| implies |
relations among cross-ratios in K_2^{M}(F)
ⓘ
{a,-a}=0 in K_2^{M}(F) for a in F^× with characteristic not 2 ⓘ |
| inspired | terminology for Steinberg groups in algebraic K-theory ⓘ |
| involves |
Milnor symbols
ⓘ
multiplicative group F^× ⓘ symbols {a,b} in K_2^{M}(F) ⓘ symbols {a_1, … ,a_n} in K_n^{M}(F) ⓘ |
| logicalForm |
for all a in F^× with 1-a in F^×, {a,1-a}=0
ⓘ
for all a_1,a_2 in F^× with a_1+a_2=1, {a_1,a_2}=0 ⓘ |
| namedAfter | Robert Steinberg ⓘ |
| property |
bilinear in each argument of the Milnor symbol
ⓘ
compatible with graded-commutativity of Milnor K-theory ⓘ functorial with respect to field homomorphisms ⓘ |
| relatedConcept |
Bloch group
ⓘ
Matsumoto theorem ⓘ Milnor K-theory ⓘ Quillen K-theory ⓘ universal central extension of Chevalley groups ⓘ |
| role |
define the structure of Milnor K-theory
ⓘ
identify trivial symbols in Milnor K-theory ⓘ impose relations among generators of Milnor K-groups ⓘ |
| usedFor |
defining generators-and-relations description of Milnor K-groups
ⓘ
presentation of K_n^{M}(F) as quotient of tensor powers of F^× ⓘ |
| usedIn |
computations of K_2 of global fields
ⓘ
construction of symbols in Galois cohomology ⓘ regulator maps from K-theory to cohomology ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: Steinberg relations Description of subject: Steinberg relations are algebraic identities in Milnor K-theory that impose the condition that symbols of pairs of field elements summing to one vanish, playing a central role in defining the structure of these K-groups.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.