Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 6 (2010), 089, 19 pages      arXiv:1005.4338      https://doi.org/10.3842/SIGMA.2010.089
Contribution to the Proceedings of the International Workshop “Recent Advances in Quantum Integrable Systems”

Junction Type Representations of the Temperley-Lieb Algebra and Associated Symmetries

Anastasia Doikou a and Nikos Karaiskos a, b
a) Department of Engineering Sciences, University of Patras, GR-26500 Patras, Greece
b) Centre de Physique Théorique, Ecole Polytechnique, CNRS-UMR 7644, 91128 Palaiseau, France

Received September 06, 2010, in final form November 29, 2010; Published online December 07, 2010

Abstract
Inspired by earlier works on representations of the Temperley-Lieb algebra we introduce a novel family of representations of the algebra. This may be seen as a generalization of the so called asymmetric twin representation. The underlying symmetry algebra is also examined and it is shown that in addition to certain obvious exact quantum symmetries non trivial quantum algebraic realizations that exactly commute with the representation also exist. Non trivial representations of the boundary Temperley-Lieb algebra as well as the related residual symmetries are also discussed. The corresponding novel R and K matrices solutions of the Yang-Baxter and reflection equations are identified, the relevant quantum spin chain is also constructed and its exact symmetries are studied.

Key words: quantum integrability; Temperley-Lieb algebras; symmetries associated to integrable models.

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