Temperley–Lieb algebra

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 group, quantum groups and subfactors of von Neumann algebras.

Let ${\displaystyle R}$ be a commutative ring and fix ${\displaystyle \delta \in R}$. The Temperley–Lieb algebra ${\displaystyle TL_{n}(\delta )}$ is the ${\displaystyle R}$-algebra generated by the elements ${\displaystyle U_{1},U_{2},\ldots ,U_{n-1}}$, subject to the Jones relations:

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