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

SIGMA 3 (2007), 009, 19 pages      math-ph/0606040
Contribution to the Vadim Kuznetsov Memorial Issue

Asymmetric Twin Representation: the Transfer Matrix Symmetry

Anastasia Doikou
INFN Section of Bologna, Physics Department, University of Bologna, Via Irnerio 46, 40126 Bologna, Italy

Received August 02, 2006, in final form December 26, 2006; Published online January 09, 2007

The symmetry of the Hamiltonian describing the asymmetric twin model was partially studied in earlier works, and our aim here is to generalize these results for the open transfer matrix. In this spirit we first prove, that the so called boundary quantum algebra provides a symmetry for any generic - independent of the choice of model - open transfer matrix with a trivial left boundary. In addition it is shown that the boundary quantum algebra is the centralizer of the B type Hecke algebra. We then focus on the asymmetric twin representation of the boundary Temperley-Lieb algebra. More precisely, by exploiting exchange relations dictated by the reflection equation we show that the transfer matrix with trivial boundary conditions enjoys the recognized Uq(sl2) Ui(sl2) symmetry. When a non-diagonal boundary is implemented the symmetry as expected is reduced, however again certain familiar boundary non-local charges turn out to commute with the corresponding transfer matrix.

Key words: quantum integrability; boundary symmetries; quantum algebras; Hecke algebras.

pdf (343 kb)   ps (209 kb)   tex (24 kb)


  1. Baxter R.J., Partition function of the eight vertex lattice model, Ann. Phys. 70 (1972), 193-228.
  2. Baxter R.J., Exactly solved models in statistical mechanics, Academic Press, 1982.
  3. Korepin V.E., The mass spectrum and the S-matrix of the massive Thirring model in the repulsive case, Comm. Math. Phys. 76 (1980), 165-176.
  4. Korepin V.E., Izergin G., Bogoliubov N.M., Quantum inverse scattering method, correlation functions and algebraic Bethe ansatz, Cambridge University Press, 1993.
  5. Cherednik I.V., Factorizing particles on a half line and root systems, Theoret. and Math. Phys. 61 (1984), 977-983.
  6. Sklyanin E.K., Boundary conditions for integrable quantum systems, J. Phys. A: Math. Gen. 21 (1988), 2375-2389.
  7. Takhtajan L.A., Introduction to quantum groups and intergable massive models of quantum field theory, Nankai Lectures on Mathematical Physics, Editors M.-L. Ge and B.-H. Zhao, World Scientific, 1990, 69-197.
  8. Jimbo M., A q analog of U(gl(N+1)) Hecke algebra and the Yang-Baxter equation, Lett. Math. Phys. 11 (1986), 247-252.
  9. Drinfeld V.G., Hopf algebras and the quantum Yang-Baxter equation, Sov. Math. Dokl. 32 (1985), 254-258.
  10. Mezincescu L., Nepomechie R.I., Fractional-spin integrals of motion for the boundary sine-Gordon model at the free fermion point, Internat. J. Modern Phys. A 13 (1998), 2747-2764, hep-th/9709078.
  11. Molev A.I., Ragoucy E., Representations of reflection algebras, Rev. Math. Phys. 14 (2002), 317-342, math.QA/0107213.
  12. Delius G., Mackay N., Quantum group symmetry in sine-Gordon and affine Toda field theories on the half-line, Comm. Math. Phys. 233 (2003), 173-190, hep-th/0112023.
  13. Doikou A., Boundary non-local charges from the open spin chain, J. Stat. Mech. Theory Exp. (2004), P12005, 14 pages, math-ph/0402067.
  14. Doikou A., From affine Hecke algebras to boundary symmetries, Nuclear Phys. B 725 (2005), 493-530, math-ph/0409060.
  15. Levy D., Martin P.P., Hecke algebra solutions to the reflection equation, J. Phys. A: Math. Gen. 27 (1994), L521-L526.
  16. Martin P.P., Woodcock D., Levy D., A diagrammatic approach to Hecke algebras of the reflection equation, J. Phys. A: Math. Gen. 33 (2000), 1265-1296.
  17. Doikou A., Martin P.P., Hecke algebraic approach to the reflection equation for spin chains, J. Phys. A: Math. Gen. 36 (2003), 2203-2226, hep-th/0206076.
  18. Martin P.P., Woodcock D., Generalized blob algebras and alcove geometry, J. Comput. Math. 6 (2003), 249-296, math.RT/0205263.
  19. Doikou A., Martin P.P., On quantum group symmetry and Bethe ansatz for the asymmetric twin spin chain with integrable boundary, J. Stat. Mech. Theory Exp. (2006), P06004, 43 pages, hep-th/0503019.
  20. Martin P.P., Saleur H., The blob algebra and the periodic Temperley-Lieb algebra, Lett. Math. Phys. 30 (1994), 189-206, hep-th/9302094.
  21. de Vega H.J., Gonzalez-Ruiz A., Boundary K-matrices for the XYZ, XXZ and XXX spin chains, J. Phys. A: Math. Gen. 27 (1994), 6129-6137, hep-th/9306089.
  22. Ghoshal S., Zamolodchikov A.B., Boundary S-matrix and boundary state in two-dimensional integrable quantum field theory, Internat. J. Modern Phys. A 9 (1994), 3841-3886, hep-th/9306002.
  23. Faddeev L.D., Takhtajan L.A., What is the spin of a spin wave?, Phys. Lett. A 85 (1981), 375-377.
  24. Mezincescu L., Nepomechie R.I., Quantum algebra structure of exactly soluble quantum spin chains, Modern Phys. Lett. A 6 (1991), 2497-2508.
  25. Mezincescu L., Nepomechie R.I., Addendum to "Integrability of open spin chains with quantum algebra symmetry", Internat. J. Modern Phys. A 7 (1992), 5657-5660, hep-th/9206047.
  26. Doikou A., Nepomechie R.I., Duality and quantum-algebra symmetry of the AN-1(1) open spin chain with diagonal boundary fields, Nuclear Phys. B 530 (1998), 641-664, hep-th/9807065.
  27. Saponov P.A., The Weyl approach to the representation theory of reflection equation algebra, J. Phys. A: Math. Gen. 37 (2004), 5021-5046, math.QA/0307024.
  28. Kulish P.P., Sklyanin E.K., The general U(q)(sl(2)) invariant XXZ integrable quantum spin chain, J. Phys. A: Math. Gen. 24 (1991), L435-L439.
  29. Pasquier V., Saleur H., Common structures between finite systems and conformal field theories through quantum groups, Nuclear Phys. B 330 (1990), 523-556.
  30. Jimbo M., Quantum R matrix for the generalized toda system, Comm. Math. Phys. 102 (1986), 537-547.
  31. Deguchi T., Fabricius K., McCoy B., The sl2 loop algebra symmetry of the six-vertex model at roots of unity, J. Statist. Phys. 102 (2001), 701-736, cond-mat/9912141.
  32. Doikou A., The open XXZ and associated models at q root of unity, J. Stat. Mech. Theory Exp. (2006), P09010, 32 pages, hep-th/0603112.
  33. Baseilhac P., The q-deformed analogue of the Onsager algebra: beyond the Bethe ansatz approach, Nuclear Phys. B 754 (2006), 309-328, math-ph/0604036.
  34. Baseilhac P., A family of tridiagonal pairs and related symmetric functions, J. Phys. A: Math. Gen. 39 (2006), 11773-11791, math-ph/0604035.
  35. Nichols A., Rittenberg V., de Gier J., One-boundary Temperley-Lieb algebras in the XXZ and loop models, J. Stat. Mech. Theory Exp. (2005), P05003, 32 pages, cond-mat/0411512.
  36. de Gier J., Nichols A., Pyatov P., Rittenberg V., Magic in the spectra of the XXZ quantum chain with boundaries at D = 0 and D = -1/2, Nuclear Phys. B 729 (2005), 387-418, hep-th/0505062.

Previous article   Next article   Contents of Volume 3 (2007)