In statistical mechanics, the Temperley–Lieb algebra is an algebra from which are built certain transfer matrices, invented by Neville Temperley and Elliott Lieb. It is also related to integrable models, knot theory and the braid groups, quantum groups and subfactors of von Neumann algebras.
Structure
Generators and relations Let R {\displaystyle R} be a commutative ring and fix δ ∈ R {\displaystyle \delta \in R} . The Temperley–Lieb algebra T L n ( δ ) {\displaystyle TL_{n}(\delta )} is the unital associative R {\displaystyle R} -algebra generated by the elements e 1 , e 2 , … , e n − 1 {\displaystyle e_{1},e_{2},\ldots ,e_{n-1}} , subject to the Jones relations:
e i 2 = δ e i {\displaystyle e_{i}^{2}=\delta e_{i}} for all 1 ≤ i ≤ n − 1 {\displaystyle 1\leq i\leq n-1}
e i e i + 1 e i = e i {\displaystyle e_{i}e_{i+1}e_{i}=e_{i}} for all 1 ≤ i ≤ n − 2 {\displaystyle 1\leq i\leq n-2}
e i e i − 1 e i = e i {\displaystyle e_{i}e_{i-1}e_{i}=e_{i}} for all 2 ≤ i ≤ n − 1 {\displaystyle 2\leq i\leq n-1}
e i e j = e j e i {\displaystyle e_{i}e_{j}=e_{j}e_{i}} for all 1 ≤ i , j ≤ n − 1 {\displaystyle 1\leq i,j\leq n-1} such that | i − j | ≠ 1 {\displaystyle |i-j|\neq 1}
Using these relations, any product of generators e i {\displaystyle e_{i}} can be brought to Jones' normal form:
E = ( e i 1 e i 1 − 1 ⋯ e j 1 ) ( e i 2 e i 2 − 1 ⋯ e j 2 ) ⋯ ( e i r e i r − 1 ⋯ e j r ) {\displaystyle E={\big (}e_{i_{1}}e_{i_{1}-1}\cdots e_{j_{1}}{\big )}{\big (}e_{i_{2}}e_{i_{2}-1}\cdots e_{j_{2}}{\big )}\cdots {\big (}e_{i_{r}}e_{i_{r}-1}\cdots e_{j_{r}}{\big )}}
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