Associative rings and algebras Definition

Temperley–Lieb algebra

Builds on Associative algebra Commutative ring Catalan numbers Hecke algebra

Definition

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

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

Remarks

The algebra $TL_n(\delta)$ is a free $R$ -module of rank equal to the Catalan number $C_n = \frac{1}{n+1}\binom{2n}{n}$ , 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 of the symmetric group $S_n$ .

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Tag	`2PCPEX85PX`
URI	`m2://2PCPEX85PX`
Kind	definition
Writing	W0 stub
MSC	16S99, 57K14, 82B20
Defines	Temperley–Lieb algebra
Uses	Associative algebra, Commutative ring, Catalan numbers, Hecke algebra

Temperley–Lieb algebra

Definition

Remarks

Reading path

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