Temperley-Lieb algebra

Temperley-Lieb algebra

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.

Definition

The Temperley-Lieb algebra with parameter τ is generated by elements "e""n" for "n" = 1, ... , "N" - 1subject to the following relations:
*"e""n"2 = 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"

Further reading

*N. Temperley, E. Lieb, [http://www.jstor.org/search/BasicSearch?si=1&hp=25&Search=Search&Query=temperley+lieb+percolation "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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