TemperleyLiebAlgebra

$TL_n(\delta)$ : the PLANAR perfect matchings — the diagrams you can draw in a rectangle without crossings. Dimension $C(n)$ , the Catalan numbers.

TemperleyLiebAlgebra(n)the Temperley–Lieb algebra on

n

strands

SageMathTemperleyLiebAlgebra(n, enumeratio_delta, enumeratio_ring)OscarEnumeratioDiagramAlgebra(:temperley_lieb, n)

Typed signatures

Type	Package	Applies
`(integer) -> diagram_algebra`	`diagram`	always

Domain: Diagram algebras

Details

Dimension $C(n)$ = 1, 2, 5, 14, 42 — the Catalan numbers, and the reason non-crossing matchings and Dyck paths are the same objects
Generated by the cup-caps $e_i$ , with $e_i^2 = \delta e_i$ , $e_ie_{i\pm1}e_i = e_i$ , and $e_ie_j = e_je_i$ when $|i-j| \ge 2$
Those relations are what the stacking rule produces — they are not imposed separately
The home of the Jones polynomial, via the Markov trace on $TL_n$

Examples

Basic #

$C(4)$

\mathrm{AlgebraDimension}(\mathrm{TemperleyLiebAlgebra}(4))

Out

14

the cup-cap and the identity — all of $TL_2$

\mathrm{Basis}(\mathrm{TemperleyLiebAlgebra}(2))

Out

\bigl\lbrack\mathrm{Diagram}(\bigl\lbrack\bigl\lbrack1, 2\bigr\rbrack, \bigl\lbrack-1, -2\bigr\rbrack\bigr\rbrack), \mathrm{Diagram}(\bigl\lbrack\bigl\lbrack1, -1\bigr\rbrack, \bigl\lbrack2, -2\bigr\rbrack\bigr\rbrack)\bigr\rbrack

Properties #

the cup-cap is planar, so it belongs

\mathrm{Diagram}(\bigl\lbrack\bigl\lbrack1, 2\bigr\rbrack, \bigl\lbrack-1, -2\bigr\rbrack, \bigl\lbrack3, -3\bigr\rbrack\bigr\rbrack)∈\mathrm{TemperleyLiebAlgebra}(3)

Out

\mathrm{True}

$e_1^2 = \delta e_1$

\mathrm{NonCommutativeMultiply}(\mathrm{Diagram}(\bigl\lbrack\bigl\lbrack1, 2\bigr\rbrack, \bigl\lbrack-1, -2\bigr\rbrack\bigr\rbrack), \mathrm{Diagram}(\bigl\lbrack\bigl\lbrack1, 2\bigr\rbrack, \bigl\lbrack-1, -2\bigr\rbrack\bigr\rbrack))

Out

δ\mathrm{Diagram}(\bigl\lbrack\bigl\lbrack1, 2\bigr\rbrack, \bigl\lbrack-1, -2\bigr\rbrack\bigr\rbrack)

About

Examples	4
Checked against	SageMath 4 agreeOscar 1 agree
Crosswalk	Wikipedia WikidataSageMathOscar

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