Spin Chains as Modules over the Affine Temperley–Lieb Algebra

The affine Temperley–Lieb algebra a T L N ( β ) is an infinite-dimensional algebra over ℂ parametrized by a number β∈ℂ and an integer N∈ℕ . It naturally acts on (ℂ^2)^⊗ N to produce a family of representations labeled by an additional parameter z∈ℂ^× . The structure of these representations, which were first introduced by Pasquier and Saleur (Nucl. Phys., 330 , 523 1990 ) in their study of spin chains, is here made explicit. They share their composition factors with the cellular a T L N ( β )-modules of Graham and Lehrer (Enseign. Math., 44 , 173 1998 ), but differ from the latter by the direction of about half of the arrows of their Loewy diagrams. The proof of this statement uses a morphism introduced by Morin-Duchesne and Saint-Aubin (J. Phys. A, 46 , 285207 2013 ) as well as new maps that intertwine various a T L N ( β )-actions on the periodic chain and generalize applications studied by Deguchi et al. (J. Stat. Phys., 102 , 701 2001 ) and after by Morin-Duchesne and Saint-Aubin (J. Phys. A, 46 , 494013 2013 ).