Spin chains as modules over the affine Temperley-Lieb algebra. (English) Zbl 1537.16024
Summary: The affine Temperley-Lieb algebra \(\mathsf{aTL}_N ( \beta )\) is an infinite-dimensional algebra over \(\mathbb{C}\) parametrized by a number \(\beta \in \mathbb{C}\) and an integer \(N\in \mathbb{N} \). It naturally acts on \((\mathbb{C}^2)^{\otimes N}\) to produce a family of representations labeled by an additional parameter \(z\in \mathbb{C}^{\times } \). The structure of these representations, which were first introduced by V. Pasquier and H. Saleur [“Common structures between finite systems and conformal field theories through quantum groups”, Nucl. Phys., B 330, No. 2–3, 523–553 (1990; doi:10.1016/0550-3213(90)90122-T)] in their study of spin chains, is here made explicit. They share their composition factors with the cellular \(\mathsf{aTL}_N ( \beta )\)-modules of J. J. Graham and G. I. Lehrer [Enseign. Math. (2) 44, No. 3–4, 173–218 (1998; Zbl 0964.20002)], 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 A. Morin-Duchesne and Y. Saint-Aubin [J. Phys. A, Math. Theor. 46, No. 28, Article ID 285207, 34 p. (2013; Zbl 1285.82018)] as well as new maps that intertwine various \(\mathsf{aTL}_N ( \beta )\)-actions on the periodic chain and generalize applications studied by T. Deguchi et al. [J. Stat. Phys. 102, No. 3–4, 701–736 (2001; Zbl 0990.82008)] and after by A. Morin-Duchesne and Y. Saint-Aubin [J. Phys. A, Math. Theor. 46, No. 49, Article ID 494013, 46 p. (2013; Zbl 1283.81112)].
MSC:
| 16G99 | Representation theory of associative rings and algebras |
| 17B37 | Quantum groups (quantized enveloping algebras) and related deformations |
| 20G42 | Quantum groups (quantized function algebras) and their representations |
| 82B20 | Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics |
Keywords:
affine Temperley-Lieb algebra; Temperley-Lieb algebra; quantum groups; \(\mathrm{Uqsl}2\); Feigin-Fuchs module; Schur-Weyl duality; indecomposable projective; XXZ chain; periodic XXZ chainReferences:
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