Abstract
We establish an explicit algebra isomorphism between the quantum reflection algebra for the \({U_q(\widehat{sl_2}) R}\)-matrix and a new type of current algebra. These two algebras are shown to be two realizations of a special case of tridiagonal algebras (q-Onsager).
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Au-Yang H., McCoy B.M., Perk J.H.H., Tang S.: Solvable models in statistical mechanics and Riemann surfaces of genus greater than one. In: Kashiwara, M., Kawai, T. (eds) Algebraic Analysis, vol. 1, pp. 29–40. Academic Press, San Diego (1988)
Baseilhac P.: Deformed Dolan-Grady relations in quantum integrable models. Nucl. Phys. B 709, 491–521 (2005) arXiv:hep-th/0404149
Baseilhac P.: An integrable structure related with tridiagonal algebras. Nucl. Phys. B 705, 605–619 (2005) arXiv:math-ph/0408025
Baseilhac P.: A family of tridiagonal pairs and related symmetric functions. J. Phys. A 39, 11773–11791 (2006) arXiv:math-ph/0604035v3
Baseilhac P., Koizumi K.: A new (in)finite dimensional algebra for quantum integrable models. Nucl. Phys. B 720, 325–347 (2005) arXiv:math-ph/0503036
Baxter R.: Exactly solved models in statistical mechanics. Academic Press, New York (1982)
Beck J.: Braid group action and quantum affine algebras. Commun. Math. Phys. 165, 555–568 (1994)
Chari V., Pressley A.: A Guide to Quantum Groups. Cambridge University Press, Cambridge (1994)
Cherednik I.V.: Factorizing particles on the half-line and root systems. Teor. Mat. Fiz. 61, 35–44 (1984)
Damiani I.: A basis of type Poincaré–Birkhoff–Witt for the quantum algebra of \({\widehat{sl_2}}\). J. Algebra 161, 291–310 (1993)
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/9911018
Date E., Roan S.S.: The algebraic structure of the Onsager algebra. Czech. J. Phys. 50, 37–44 (2000) cond-mat/0002418
Davies B.: Onsager’s algebra and superintegrability. J. Phys. A 23, 2245–2261 (1990)
Davies B.: Onsager’s algebra and the Dolan-Grady condition in the non-self-dual case. J. Math. Phys. 32, 2945–2950 (1991)
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/0208043
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/0112023
Delius, G.W., MacKay, N.J., Short, B.J.: Boundary remnant of Yangian symmetry and the structure of rational reflection matrices. Phys. Lett. B 522 335–344 (2001); Erratum-ibid. B 524 (2002) 401. arXiv:hep-th/0109115v2
Ding J., Frenkel I.B.: Isomorphism of two realizations of quantum affine algebra \({U_q(\widehat{gl(n)})}\). Commun. Math. Phys. 156, 277–300 (1993)
Dolan L., Grady M.: Conserved charges from self-duality. Phys. Rev. D 25, 1587–1604 (1982)
Drinfeld V.G.: Hopf algebras and the quantum Yang–Baxter equation. Sov. Math. Doklady 32, 254–258 (1985)
Drinfeld V.G.: A new realization of Yangians and quantum affine algebras. Sov. Math. Doklady 36, 212–216 (1988)
Faddeev L.D.: Integrable models in (1+1)-dimensional quantum field theory. In: Zuber, J.-B., Stora, R. (eds) Recent Advances in Field Theory and Statistical Mechanics, Les Houches 1982, pp. 561–608. North-Holland, Amsterdam (1984)
Faddeev, L.D., Reshetikhin, N.Yu., Takhtajan, L.A.: Quantization of Lie groups and Lie algebras, Yang–Baxter equation and quantum integrable systems. Advanced Series in Mathematical Physics, vol. 10, pp. 299–309. World Scientific, Singapore (1989)
Ito, T., Tanabe, K., Terwilliger, P.: Some algebra related to P- and Q-polynomial association schemes, Codes and association schemes (Piscataway, NJ, 1999), pp. 167–192, DIMACS Ser. Discrete Math. Theoret. Comput. Sci., vol. 56. American Mathematical Society, Providence (2001). arXiv:math/0406556v1
Jimbo M.: A q-difference analogue of U(g) and the Yang–Baxter equation. Lett. Math. Phys. 10, 63–69 (1985)
Jimbo M.: A q-analog of U(gl(N + 1)), Hecke algebra and the Yang–Baxter equation. Lett. Math. Phys. 11, 247–252 (1986)
Jing, N.: On Drinfeld realization of quantum affine algebras. In: Ferrar, J., Harada, K. (eds.) Proceedings of Conf. on Lie Alg. at Ohio State Univ., May 1996; in Monster and Lie Algebras, pp. 195–206. OSU Math. Res. Inst. Publ. 7. de Gruyter, Berlin (1998)
Kulish P.P., Sklyanin E.K.: Solutions of the Yang–Baxter equation. J. Soviet. Math. 19, 1596–1620 (1982)
Kulish, P.P., Sklyanin, E.K.: Quantum spectral transform method. In: Hietarinta, J., Montonen, C. (eds.) Recent Developments in Integrable Quantum Field Theories, Tvarminne (1981). Lecture Notes in Physics, vol. 151, pp. 61–119. Springer, Berlin (1981)
Kulish P.P., Reshetikhin N.Yu., Sklyanin E.K.: Yang–Baxter equations and representation theory: I. Lett. Math. Phys. 5, 393–403 (1981)
Lusztig G.: Quantum deformations of certain simple modules over enveloping algebras. Adv. Math. 70, 237–249 (1988)
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/9709078v1
Nepomechie R.I.: Boundary quantum group generators of type A. Lett. Math. Phys. 62, 83–89 (2002) arXiv:hep-th/0204181
Onsager L.: Crystal statistics. I. A two-dimensional model with an order–disorder transition. Phys. Rev. 65, 117–149 (1944)
Perk, J.H.H.: Star-triangle equations, quantum Lax operators, and higher genus curves. In: Proceedings 1987 Summer Research Institute on Theta functions, Proc. Symp. Pure. Math., vol. 49, part 1, pp. 341–354. American Mathematical Society, Providence (1989)
Reshetikhin N.Yu., Semenov-Tian-Shansky M.A.: Central extensions of quantum current groups. Lett. Math. Phys. 19, 133–142 (1990)
Sklyanin E.K.: Boundary conditions for integrable quantum systems. J. Phys. A 21, 2375–2389 (1988)
Terwilliger P.: The subconstituent algebra of an association scheme. III. J. Algebraic Combin. 2, 177–210 (1993)
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/0307016
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Baseilhac, P., Shigechi, K. A New Current Algebra and the Reflection Equation. Lett Math Phys 92, 47–65 (2010). https://doi.org/10.1007/s11005-010-0380-x
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DOI: https://doi.org/10.1007/s11005-010-0380-x