Temperley-Lieb algebra

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In statistical mechanics, the Temperley-Lieb algebra is an algebra generated by certain transfer matrices, invented by Temperley and Lieb in about 1971. It is also related to knot theory, the braid group, and subfactors of von Neumann algebras.

[edit] Definition

The Temperley-Lieb algebra is generated by elements e_n for n≥1 subject to the following relations:

• e_n²=e_n

• e_me_n=e_ne_m if |m-n|>1.

• e_n+1e_ne_n+1= τe_n+1

• e_ne_n+1e_n= τe_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 precolation problem. Proceedings of the Royal Society Series A 322 (1971), 251-280.