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  "body": "\n## Definition\n\nLet $R$ be a [[commutative ring|0310REJJY8]] with identity, let $\\delta \\in R$ and let $n \\ge 1$.\nThe *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:\n\n1. $e_i^2 = \\delta e_i$ for $1 \\le i \\le n - 1$.\n2. $e_i e_{i+1} e_i = e_i$ and $e_{i+1} e_i e_{i+1} = e_{i+1}$ for $1 \\le i \\le n - 2$.\n3. $e_i e_j = e_j e_i$ for $\\abs{i - j} \u003e 1$.\n\n## Remarks\n\nThe 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.\nExample: $TL_3(\\delta)$ has rank $C_3 = 5$, with basis $1, e_1, e_2, e_1 e_2, e_2 e_1$.\nThe check: every longer word reduces to one of these, for instance $e_1 e_2 e_1 = e_1$ and $e_1 e_2 e_2 = \\delta e_1 e_2$.\nNon-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.\nFor suitable $\\delta$, $TL_n(\\delta)$ is a quotient of the [[Hecke algebra|5J2P2AGY4N]] of the symmetric group $S_n$.\n",
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