Drinfeld associators
E884936
algebraic structure
associator
object in braided monoidal category theory
structure in quantum group theory
Drinfeld associators are algebraic structures arising in the study of quantum groups and braided monoidal categories that encode solutions to the Knizhnik–Zamolodchikov equations and play a central role in deformation theory and low-dimensional topology.
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
algebraic structure
ⓘ
associator ⓘ object in braided monoidal category theory ⓘ structure in quantum group theory ⓘ |
| actOn | completed free Lie algebra on two generators ⓘ |
| appearIn | Drinfeld’s work on quasi-Hopf algebras and KZ equations ⓘ |
| ariseIn |
braided monoidal categories
ⓘ
quantum groups ⓘ |
| classifiedBy | Grothendieck–Teichmüller group (up to gauge equivalence) NERFINISHED ⓘ |
| context | monoidal categories with non-strict associativity ⓘ |
| definedOver | fields of characteristic zero ⓘ |
| encode |
coherence data for associativity in braided monoidal categories
ⓘ
solutions to the Knizhnik–Zamolodchikov equations ⓘ |
| formalObjectType | group-like element in the completed universal enveloping algebra of a free Lie algebra ⓘ |
| haveProperty | non-uniqueness; form a torsor under the Grothendieck–Teichmüller group ⓘ |
| haveVariant |
Drinfeld associator over Q
NERFINISHED
ⓘ
p-adic Drinfeld associator NERFINISHED ⓘ rational Drinfeld associator ⓘ |
| introducedBy | Vladimir Drinfeld NERFINISHED ⓘ |
| introducedInContextOf | quasi-triangular quasi-Hopf algebras ⓘ |
| invariantUnder | gauge transformations up to equivalence ⓘ |
| liveIn | completed tensor algebra on two non-commuting variables ⓘ |
| namedAfter | Vladimir Drinfeld NERFINISHED ⓘ |
| oftenConsideredOver | the field of complex numbers ⓘ |
| playRoleIn |
Grothendieck–Teichmüller theory
NERFINISHED
ⓘ
deformation theory ⓘ finite-type (Vassiliev) invariants ⓘ knot invariants ⓘ low-dimensional topology ⓘ theory of quasi-Hopf algebras ⓘ |
| relatedTo |
Grothendieck’s program on the fundamental group of P1 minus three points
ⓘ
Knizhnik–Zamolodchikov connection NERFINISHED ⓘ formality of the little disks operad ⓘ monodromy of the KZ equations ⓘ multiple zeta values (for specific associators) ⓘ |
| satisfy |
group-like condition in a completed tensor algebra
ⓘ
hexagon equations ⓘ normalization conditions (e.g., trivial constant term) ⓘ pentagon equation ⓘ |
| usedIn |
construction of quantum invariants of 3-manifolds
ⓘ
deformation quantization ⓘ theory of braided tensor categories ⓘ |
| usedToConstruct |
braidings in tensor categories
ⓘ
quasi-Hopf algebra structures ⓘ universal Vassiliev knot invariant ⓘ |
| usedToDefine |
Drinfeld associator invariants of links
ⓘ
Grothendieck–Teichmüller group NERFINISHED ⓘ |
| usedToRelate |
braid groups and quantum groups
ⓘ
knot theory and perturbative quantum field theory ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.