Abstract
We determine the structure of two variations on the Temperley-Lieb algebra, both used for dealing with special kinds of boundary conditions in statistical mechanics models. The first is a new algebra, the ‘blob’ algebra. We determine both the generic and all the exceptional structures for this two parameter algebra. The second is the periodic Temperley-Lieb algebra. The generic structure and part of the exceptional structure of this algebra have already been studied. We complete the analysis using results from the study of the blob algebra.
Similar content being viewed by others
References
Albertini, G., Dasmahapatra, S. and McCoy, B. M.,Internat. J. Modern Phys. A 7 Suppl. 1 A, 1 (1993); Alcaraz, F., Grimm, U and Rittenberg, V.,Nuclear Phys. B 316, 735 (1989).
Baxter, R. J.,Exactly Solved Models in Statistical Mechanics, Academic Press, New York, 1982.
Pasquier, V. and Saleur, H.,Nuclear Phys. 330, 523 (1990).
Levy, D., The structure of the affine Hecke, quotient underlying the translation invariantXXZ chain, Tel-Aviv preprint TAUP 1986–92.
Levy, D.,Phys. Rev. Lett. 67, 1971 (1991).
Martin, P. P. and Saleur, H.,Comm. Math. Phys. 158, 155 (1993).
Belavin, A., Polyakov, A. and Zamolodchikov, Al. B.,Nuclear Phys. B 241, 333 (1984).
Temperley, H. N. V. and Lieb, E.,Proc. Roy Soc. London Ser. A 322 25 (1971).
Penrose, R., in (ed), T. Bastin,Quantum Theory and Beyond, Cambridge Univ Press, Cambridge, 1971.
Westbury, B. W., PhD Thesis Manchester University, 1990 and unpublished 1991 notes ‘On the Temperley-Lieb algebras’.
James, G. D. and Kerber, A.,The Representation Theory of the Symmetric Group, Addison-Wesley, Reading, Mass, 1981.
Green, J. A.,Polynomial Representations of Gl n, Lecture Notes in Math. 830, Springer-Verlag, Berlin, 1980.
Cline, E., Parshal, B. and Scott, L.,J. Reine Angew. Math,391, 85 (1988).
Martin, P. P., Non-planar statistical mechanics-the partition algebra construction, Yale preprint YCTP-P34-92, to appear inJ.K.T.R.
Martin, P. P.,Potts Models and Related Problems in Statistical Mechanics, World Scientific, Singapore, 1991.
Cohn, P. M.,Algebra, vol. 2, Wiley, London, 1989.
Pasquier, V.,Nuclear Phys. B 285, 162 (1987);J. Phys., Am 20, L1229 (1987).
Andrews, G. E., Baxter, R. J., and Forrester, P. J.,J. Stat. Phys. 35, 193 (1984).
Di Francesco, P., Saleur, H. and Zuber, J. B.,Nuclear Phys. 285, 454 (1987).