---
tag: 2PCPEX85PX
kind: definition
name: Temperley–Lieb algebra
msc: [16S99, 57K14, 82B20]
defines: TZNR514JQN
uses: [3CTW1P4HBP, 0310REJJY8, 0YWZJEABB2, 5J2P2AGY4N]
sources:
  - url: https://en.wikipedia.org/wiki/Temperley–Lieb_algebra
---

## Definition

Let $R$ be a [[commutative ring|0310REJJY8]] with identity, let $\delta \in R$ and let $n \ge 1$.
The *Temperley–Lieb algebra* $TL_n(\delta)$ is the [[associative algebra|3CTW1P4HBP]] over $R$ generated by elements $e_1, \dots, e_{n-1}$, subject to the relations:

1. $e_i^2 = \delta e_i$ for $1 \le i \le n - 1$.
2. $e_i e_{i+1} e_i = e_i$ and $e_{i+1} e_i e_{i+1} = e_{i+1}$ for $1 \le i \le n - 2$.
3. $e_i e_j = e_j e_i$ for $\abs{i - j} > 1$.

## Remarks

The algebra $TL_n(\delta)$ is a free $R$-module of rank equal to the [[Catalan number|0YWZJEABB2]] $C_n = \frac{1}{n+1}\binom{2n}{n}$, with a basis given by non-crossing planar pairings of $2n$ points.
Example: $TL_3(\delta)$ has rank $C_3 = 5$, with basis $1, e_1, e_2, e_1 e_2, e_2 e_1$.
The check: every longer word reduces to one of these, for instance $e_1 e_2 e_1 = e_1$ and $e_1 e_2 e_2 = \delta e_1 e_2$.
Non-example: in the group algebra $R[S_3]$, the transpositions $s_1 = (1\,2)$ and $s_2 = (2\,3)$ satisfy $s_1 s_2 s_1 = (1\,3) \ne s_1$, so they do not satisfy relation 2.
For suitable $\delta$, $TL_n(\delta)$ is a quotient of the [[Hecke algebra|5J2P2AGY4N]] of the symmetric group $S_n$.