Temperley–Lieb algebra

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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.

[edit] Definition

Let $R$ be a commutative ring and fix $\delta \in R$ . The Temperley-Lieb algebra $T n (δ)$ is the $R$ -algebra generated by the elements $U_1, U_2, \ldots, U_{n-1}$ , subject to the following relations:

$U_i^2 = \delta U_i$ for all $1 \leq i \leq n-1$
$U i U i + 1 U i = U i$ for all $1 \leq i \leq n-2$
$U i U i - 1 U i = U i$ for all $2 \leq i \leq n-1$
$U i U j = U j U i$ for all $1 \leq i,j \leq n-1$ such that $|i-j| \neq 1$

[edit] Further reading

R.J. Baxter, Exactly solved models in statistical mechanics Academic Press Inc. (1982)
N. Temperley, E. Lieb, Relations between the percolation and colouring problem and other graph-theoretical problems associated with regular planar lattices: some exact results for the percolation problem. Proceedings of the Royal Society Series A 322 (1971), 251-280.

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Temperley–Lieb algebra

From Wikipedia, the free encyclopedia

[edit] Definition

[edit] Further reading

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