Temperley–Lieb algebra
E911208
algebra
associative algebra
cellular algebra
diagram algebra
diagrammatic algebra
finite-dimensional algebra
The Temperley–Lieb algebra is a diagrammatic algebra arising in statistical mechanics and knot theory, central to the study of exactly solvable models and link invariants.
Statements (52)
| Predicate | Object |
|---|---|
| instanceOf |
algebra
ⓘ
associative algebra ⓘ cellular algebra ⓘ diagram algebra ⓘ diagrammatic algebra ⓘ finite-dimensional algebra ⓘ |
| appearsIn |
conformal field theory
NERFINISHED
ⓘ
integrable systems ⓘ |
| arisesIn |
Potts model
NERFINISHED
ⓘ
exactly solvable models ⓘ ice-type models ⓘ lattice models in statistical mechanics ⓘ |
| centralTo |
Jones polynomial
NERFINISHED
ⓘ
planar algebras ⓘ study of link invariants ⓘ subfactor theory ⓘ |
| definedOver |
commutative ring
ⓘ
field ⓘ |
| dependsOn | integer n ⓘ |
| dimensionFormula | Catalan number C_n ⓘ |
| fieldOfStudy |
knot theory
ⓘ
low-dimensional topology ⓘ mathematics ⓘ quantum algebra ⓘ representation theory ⓘ statistical mechanics ⓘ |
| hasBasis |
non-crossing pairings
ⓘ
planar diagrams ⓘ |
| hasMultiplicationDefinedBy | concatenation of diagrams ⓘ |
| hasNotation | TL_n(q) NERFINISHED ⓘ |
| hasParameter | loop parameter q ⓘ |
| hasProperty |
non-semisimple at roots of unity
ⓘ
semisimple for generic q ⓘ |
| hasRelation |
e_i e_j = e_j e_i for |i-j|>1
ⓘ
e_i e_{i\\pm1} e_i = e_i ⓘ e_i^2 = \\delta e_i ⓘ |
| introducedBy |
Elliott Lieb
NERFINISHED
ⓘ
Neville Temperley NERFINISHED ⓘ |
| quotientOf | Hecke algebra of type A NERFINISHED ⓘ |
| relatedTo |
Catalan numbers
ⓘ
Hecke algebra of type A NERFINISHED ⓘ Jones–Wenzl idempotents NERFINISHED ⓘ Kauffman bracket NERFINISHED ⓘ braid group NERFINISHED ⓘ planar non-crossing partitions ⓘ subfactors of type II_1 ⓘ |
| usedFor |
construction of Jones polynomial
ⓘ
exact diagonalization of lattice models ⓘ representation theory of quantum groups ⓘ study of quantum spin chains ⓘ transfer matrices in statistical mechanics ⓘ |
| yearOfIntroduction | 1971 ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.