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 Temperley and Lieb in about 1971. It is also related to integrable models, knot theory and the braid group, and subfactors of von Neumann algebras.

[edit] Definition

The Temperley-Lieb algebra with parameter τ is generated by elements e_n for n = 1, ... , N - 1 subject to the following relations:

• e_n² = τ e_n

• e_m e_n = e_n e_m if |m-n|>1.

• e_n+1 e_n e_n+1 = e_n+1

• e_n e_n+1 e_n = e_n

[edit] Further reading

• 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.