Letters in Mathematical Physics

, Volume 93, Issue 3, pp 213–228 | Cite as

Generalized q-Onsager Algebras and Boundary Affine Toda Field Theories

  • Pascal Baseilhac
  • Samuel Belliard


Generalizations of the q-Onsager algebra are introduced and studied. In one of the simplest case and q = 1, the algebra reduces to the one proposed by Uglov–Ivanov. In the general case and q ≠ 1, an explicit algebra homomorphism associated with coideal subalgebras of quantum affine Lie algebras (simply and non-simply laced) is exhibited. Boundary (soliton non-preserving) integrable quantum Toda field theories are then considered in light of these results. For the first time, all defining relations for the underlying non-Abelian symmetry algebra are explicitly obtained. As a consequence, based on purely algebraic arguments all integrable (fixed or dynamical) boundary conditions are classified.

Mathematics Subject Classification (2000)

81R50 81R10 81U15 81T40 


q-Onsager algebra quantum group symmetry boundary affine Toda field theory 


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  1. 1.
    Ahn C., Kim Ch., Rim Ch.: Reflection amplitudes of boundary Toda theories and thermodynamic Bethe Ansatz. Nucl. Phys. B 628, 486–504 (2002) arXiv:hep-th/0110218v1MATHCrossRefMathSciNetADSGoogle Scholar
  2. 2.
    Alnajjar H., Curtin B.: A family of tridiagonal pairs related to the quantum affine algebra \({U_q(\widehat{sl2})}\). Electron. J. Linear Algebra 13, 1–9 (2005)MATHMathSciNetGoogle Scholar
  3. 3.
    Avan J., Doikou A.: Boundary Lax pairs for the \({A_{n}^{(1)}}\) Toda field theories. Nucl. Phys. B 821, 481–505 (2009) arXiv:0809.2734v3CrossRefMathSciNetADSGoogle Scholar
  4. 4.
    Baseilhac P.: Deformed Dolan-Grady relations in quantum integrable models. Nucl. Phys. B 709, 491–521 (2005) arXiv:hep-th/0404149MATHCrossRefMathSciNetADSGoogle Scholar
  5. 5.
    Baseilhac P.: An integrable structure related with tridiagonal algebras. Nucl. Phys. B 705, 605–619 (2005) arXiv:math-ph/0408025MATHCrossRefMathSciNetADSGoogle Scholar
  6. 6.
    Baseilhac P.: A family of tridiagonal pairs and related symmetric functions. J. Phys. A 39, 11773–11791 (2006) arXiv:math-ph/0604035v3MATHCrossRefMathSciNetADSGoogle Scholar
  7. 7.
    Baseilhac P., Delius G.W.: Coupling integrable field theories to mechanical systems at the boundary. J. Phys. A 34, 8259–8270 (2001) arXiv:hep-th/0106275MATHCrossRefMathSciNetADSGoogle Scholar
  8. 8.
    Baseilhac P., Koizumi K.: A new (in)finite dimensional algebra for quantum integrable models. Nucl. Phys. B 720, 325–347 (2005) arXiv:math-ph/0503036MATHCrossRefMathSciNetADSGoogle Scholar
  9. 9.
    Baseilhac P., Koizumi K.: A deformed analogue of Onsager’s symmetry in the XXZ open spin chain. J. Stat. Mech. 0510, P005 (2005) arXiv:hep-th/0507053Google Scholar
  10. 10.
    Baseilhac, P., Koizumi, K.: Exact spectrum of the XXZ open spin chain from the q-Onsager algebra representation theory. J. Stat. Mech. P09006 (2007). arXiv:hep-th/0703106Google Scholar
  11. 11.
    Baseilhac P., Koizumi K.: Sine-Gordon quantum field theory on the half-line with quantum boundary degrees of freedom. Nucl. Phys. B 649, 491–510 (2003) arXiv:hep-th/0208005MATHCrossRefMathSciNetADSGoogle Scholar
  12. 12.
    Baseilhac P., Shigechi K.: A new current algebra and the reflection equation. Lett. Math. Phys. 92, 47–65 (2010) arXiv:0906.1215MATHCrossRefMathSciNetADSGoogle Scholar
  13. 13.
    Baseilhac, P., Belliard, S., Shigechi, K.: in preparationGoogle Scholar
  14. 14.
    Bazhanov V.V., Hibberd A.N., Khoroshkin S.M.: Integrable structure of W 3 conformal field theory, quantum Boussinesq theory and boundary affine Toda theory. Nucl. Phys. B 622, 475–547 (2002) arXiv:hep-th/0105177v3MATHCrossRefMathSciNetADSGoogle Scholar
  15. 15.
    Bernard D., Leclair A.: Quantum group symmetries and nonlocal currents in 2-D QFT. Commun. Math. Phys. 142, 99–138 (1991)MATHCrossRefMathSciNetADSGoogle Scholar
  16. 16.
    Bowcock P., Corrigan E., Dorey P.E., Rietdijk R.H.: Classically integrable boundary conditions for affine Toda field theories. Nucl. Phys. B 445, 469–500 (1995) hep-th/9501098MATHCrossRefMathSciNetADSGoogle Scholar
  17. 17.
    Corrigan E., Dorey P.E., Rietdijk R.H., Sasaki R.: Affine Toda field theory on a half line, Phys. Lett. B 333, 83–91 (1994) arXiv:hep-th/9404108MathSciNetADSGoogle Scholar
  18. 18.
    Chari V., Pressley A.: A Guide to Quantum Groups. Cambridge University Press, Cambridge (1994)MATHGoogle Scholar
  19. 19.
    Date E., Roan S.S.: The structure of quotients of the Onsager algebra by closed ideals. J. Phys. A Math. Gen. 33, 3275–3296 (2000) math.QA/9911018MATHCrossRefMathSciNetADSGoogle Scholar
  20. 20.
    Date E., Roan S.S.: The algebraic structure of the Onsager algebra. Czech. J. Phys. 50, 37–44 (2000) cond-mat/0002418MATHCrossRefMathSciNetADSGoogle Scholar
  21. 21.
    Davies B.: Onsager’s algebra and superintegrability. J. Phys. A 23, 2245–2261 (1990)MATHCrossRefMathSciNetADSGoogle Scholar
  22. 22.
    Davies B.: Onsager’s algebra and the Dolan–Grady condition in the non-self-dual case. J. Math. Phys. 32, 2945–2950 (1991)MATHCrossRefMathSciNetADSGoogle Scholar
  23. 23.
    Delius G.W.: Soliton-preserving boundary condition in affine Toda field theories. Phys. Lett. B 444, 217 (1998) arXiv:hep-th/9809140v2CrossRefMathSciNetADSGoogle Scholar
  24. 24.
    Delius G.W., George A.: Quantum affine reflection algebras of type \({d_n^{(1)}}\) and reflection matrices. Lett. Math. Phys. 62, 211–217 (2002) arXiv:math/0208043MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Delius G.W., Gandenberger G.M.: Particle reflection amplitudes in \({a_n^{(1)}}\) Toda Field Theories. Nucl. Phys. B 554, 325–364 (1999) arXiv:hep-th/9904002MATHCrossRefMathSciNetADSGoogle Scholar
  26. 26.
    Delius G.W., MacKay N.J.: Quantum group symmetry in sine-Gordon and affine Toda field theories on the half-line. Commun. Math. Phys. 233, 173–190 (2003) arXiv:hep-th/0112023MATHCrossRefMathSciNetADSGoogle Scholar
  27. 27.
    Dolan L., Grady M.: Conserved charges from self-duality. Phys. Rev. D 25, 1587–1604 (1982)CrossRefMathSciNetADSGoogle Scholar
  28. 28.
    Doikou A.: \({a_n^{(1)}}\) affine Toda field theories with integrable boundary conditions revisited. JHEP 0805, 091 (2008) arXiv:0803.0943CrossRefMathSciNetADSGoogle Scholar
  29. 29.
    Doikou A.: From affine Hecke algebras to boundary symmetries. Nucl. Phys. B 725, 493–530 (2005) arXiv:math-ph/0409060MATHCrossRefMathSciNetADSGoogle Scholar
  30. 30.
    Fateev V.A., Onofri E.: Boundary One-point functions, scattering theory and vacuum solutions in integrable systems. Nucl. Phys. B 634, 546–570 (2002) arXiv:hep-th/0203131MATHCrossRefMathSciNetADSGoogle Scholar
  31. 31.
    Fring A., Köberle R.: Boundary Bound States in Affine Toda Field Theory. Int. J. Mod. Phys. A 10, 739–752 (1995) arXiv:hep-th/9404188MATHCrossRefADSGoogle Scholar
  32. 32.
    Fring A., Köberle R.: Affine Toda field theory in the presence of reflecting boundaries. Nucl. Phys. B 419, 647–664 (1994) arXiv:hep-th/9309142MATHCrossRefADSGoogle Scholar
  33. 33.
    Gandenberger G.M.: On \({a_2^{(1)}}\) reflection matrices and affine Toda theories. Nucl. Phys. B 542, 659–693 (1999) arXiv:hep-th/9806003MATHCrossRefMathSciNetADSGoogle Scholar
  34. 34.
    Gandenberger, G.M.: New non-diagonal solutions to the \({a_n^{(1)}}\) boundary Yang-Baxter equation. arXiv:hep-th/9911178Google Scholar
  35. 35.
    Gavrilik A.M., Iorgov N.Z.: q-deformed algebras U q(son) and their representations. Methods Funct. Anal. Topol. 3, 51–63 (1997)MATHMathSciNetGoogle Scholar
  36. 36.
    von Gehlen G., Rittenberg V.: Zn-symmetric quantum chains with an infinite set of conserved charges and Zn zero modes. Nucl. Phys. B 257(FS14), 351–370 (1985)CrossRefADSGoogle Scholar
  37. 37.
    Ghoshal S.: Bound state boundary S-matrix of the sine-Gordon Model. Int. J. Mod. Phys. A 9, 4801–4810 (1994) arXiv:hep-th/9310188MATHCrossRefMathSciNetADSGoogle Scholar
  38. 38.
    Ghoshal, S., Zamolodchikov, A.: Boundary S-matrix and boundary state in two-dimensional integrable quantum field theory. Int. J. Mod. Phys. A 9, 3841–3886 (1994) (Erratum-ibid. A 9, 4353 (1994), arXiv:hep-th/9306002)Google Scholar
  39. 39.
    Grünbaum F.A., Haine L.: The q-version of a theorem of Bochner. J. Comput. Appl. Math. 68, 103–114 (1996)MATHCrossRefMathSciNetGoogle Scholar
  40. 40.
    Ito T., Terwilliger P.: Tridiagonal pairs and the quantum affine algebra \({U_q(\widehat{sl2})}\). Ramanujan J. 13, 39–62 (2007) arXiv:math.QA/0310042MATHCrossRefMathSciNetGoogle Scholar
  41. 41.
    Ito, T., Terwilliger, P.: Tridiagonal pairs of q-Racah type. arXiv:0807.0271v1Google Scholar
  42. 42.
    Ito, T., Tanabe, K., Terwilliger, P.: Some algebra related to P- and Q-polynomial association schemes. Codes and association schemes (Piscataway, NJ, 1999). DIMACS Ser. Discrete Math. Theoret. Comput. Sci., vol. 56, pp. 167–192. American Mathematical Society, Providence (2001). arXiv:math/0406556v1Google Scholar
  43. 43.
    Jimbo M.: A q-difference analogue of U(g) and the Yang-Baxter equation. Lett. Math. Phys. 10, 63–69 (1985)MATHCrossRefMathSciNetADSGoogle Scholar
  44. 44.
    Jimbo M.: A q-analog of U(gl(N + 1)), Hecke algebra and the Yang–Baxter equation. Lett. Math. Phys. 11, 247–252 (1986)MATHCrossRefMathSciNetADSGoogle Scholar
  45. 45.
    Kac V.G.: Infinite dimensional Lie algebras. Birkhäuser, Boston (1983)MATHGoogle Scholar
  46. 46.
    Klimyk A.U.: The nonstandard q-deformation of enveloping algebra U(so n): results and problems. Czech. J. Phys. 51, 331–340 (2001)CrossRefMathSciNetADSGoogle Scholar
  47. 47.
    Klimyk, A.U.: Classification of irreducible representations of the q-deformed algebra \({U'_q(so_n)}\). arXiv:math/0110038v1Google Scholar
  48. 48.
    Letzter, G.: Coideal subalgebras and quantum symmetric pairs. MSRI volume 1999, Hopf Algebra Workshop. arXiv:math/0103228Google Scholar
  49. 49.
    Mezincescu L., Nepomechie R.I.: Fractional-spin integrals of motion for the boundary sine-Gordon model at the free fermion point. Int. J. Mod. Phys. A 13, 2747–2764 (1998) arXiv:hep-th/9709078MATHCrossRefMathSciNetADSGoogle Scholar
  50. 50.
    Molev A.I., Ragoucy E., Sorba P.: Coideal subalgebras in quantum affine algebras. Rev. Math. Phys. 15, 789–822 (2003) arXiv:math/0208140MATHCrossRefMathSciNetGoogle Scholar
  51. 51.
    Onsager L.: Crystal statistics. I. A two-dimensional model with an order-disorder transition. Phys. Rev. 65, 117–149 (1944)MATHCrossRefMathSciNetADSGoogle Scholar
  52. 52.
    Perk, J.H.H.: Star-triangle equations, quantum Lax operators, and higher genus curves. In: Proceedings 1987 Summer Research Institute on Theta functions. Proceedings of Symposium on Pure Mathematics, vol. 49, part 1, pp. 341–354. American Mathematical Society, Providence (1989)Google Scholar
  53. 53.
    Penati S., Refolli A., Zanon D.: Classical Versus quantum symmetries for Toda theories with a nontrivial boundary perturbation. Nucl. Phys. B 470, 396–418 (1996) arXiv:hep-th/9512174MATHCrossRefMathSciNetADSGoogle Scholar
  54. 54.
    Sklyanin E.K.: Boundary conditions for integrable quantum systems. J. Phys. A 21, 2375–2389 (1988)MATHCrossRefMathSciNetADSGoogle Scholar
  55. 55.
    Terwilliger P.: The subconstituent algebra of an association scheme. III. J. Algebr. Comb. 2, 177–210 (1993)MATHCrossRefMathSciNetGoogle Scholar
  56. 56.
    Terwilliger, P.: Two relations that generalize the q-Serre relations and the Dolan–Grady relations. In: Kirillov, A.N., Tsuchiya, A., Umemura, H. (eds.) Proceedings of the Nagoya 1999 International Workshop on Physics and Combinatorics, pp. 377–398. math.QA/0307016Google Scholar
  57. 57.
    Terwilliger P.: Two linear transformations each tridiagonal with respect to an eigenbasis of the other. Linear Algebra Appl. 330, 149–203 (2001) arXiv:math.RA/0406555MATHCrossRefMathSciNetGoogle Scholar
  58. 58.
    Uglov D., Ivanov L.: sl(N) Onsager’s algebra and integrability. J. Stat. Phys. 82, 87 (1996) arXiv:hep-th/9502068v1MATHCrossRefMathSciNetADSGoogle Scholar
  59. 59.
    Zhedanov A.S.: Hidden symmetry of Askey–Wilson polynomials. Teoret. Mat. Fiz. 89, 190–204 (1991)MATHMathSciNetGoogle Scholar

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© Springer 2010

Authors and Affiliations

  1. 1.Laboratoire de Mathématiques et Physique Théorique, CNRS/UMR 6083, Fédération Denis PoissonUniversité de ToursToursFrance
  2. 2.Istituto Nazionale di Fisica NucleareSezione di BolognaBolognaItaly

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