Matrix product solution to the reflection equation associated with a coideal subalgebra of \(\varvec{U_q(A^{(1)}_{n-1})}\)

  • Atsuo KunibaEmail author
  • Masato Okado
  • Akihito Yoneyama


We present a new solution to the reflection equation associated with a coideal subalgebra of \(U_q(A^{(1)}_{n-1})\) in the symmetric tensor representations and their dual. Elements of the K matrix are expressed by a matrix product formula involving terminating q-hypergeometric series in q-boson generators. At \(q=0\), our result reproduces a known set-theoretical solution to the reflection equation connected to the crystal base theory.


Quantum groups Reflection equation Coideal subalgebra Onsager algebra Matrix product method 

Mathematics Subject Classification

Primary 17B37 Secondary 17B80 



The authors thank Vladimir Mangazeev, Zengo Tsuboi and Bart Vlaar for comments. A.K. is supported by Grants-in-Aid for Scientific Research No. 18H01141 from JSPS. M.O. is supported by Grants-in-Aid for Scientific Research No. 15K13429 and No. 16H03922 from JSPS.


  1. 1.
    Baseilhac, P., Belliard, S.: Generalized \(q\)-Onsager algebras and boundary affine Toda field theories. Lett. Math. Phys. 93, 213–228 (2010)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Batchelor, M.T., Fridkin, V., Kuniba, A., Zhou, Y.K.: Solutions of the reflection equation for face and vertex models associated with \(A^{(1)}_n\), \(B^{(1)}_n\), \(C^{(1)}_n\), \(D^{(1)}_n\) and \(A^{(2)}_n\). Phys. Lett. B 376, 266–274 (1996)ADSMathSciNetCrossRefGoogle Scholar
  3. 3.
    Baxter, R.J.: Exactly Solved Models in Statistical Mechanics. Dover, New York City (2007)zbMATHGoogle Scholar
  4. 4.
    Bosnjak, G., Mangazeev, V.V.: Construction of \(R\)-matrices for symmetric tensor representations related to \(U_q(\widehat{sl_n})\). J. Phys. A: Math. Theor. 49, 495204 (2016)CrossRefzbMATHGoogle Scholar
  5. 5.
    Cherednik, I.V.: Factorizing particles on a half-line and root systems. Theor. Math. Phys. 61, 35–44 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Delius, G.W., MacKay, N.J.: Affine Quantum Groups, Encyclopedia of Mathematical Physics. Elsevier, Amsterdam (2006)Google Scholar
  7. 7.
    Gandenberger, G. M.: New non-diagonal solutions to the \(a^{(1)}_n\) boundary Yang-Baxter equation. arXiv:hep-th/9911178 (1999)
  8. 8.
    Kashiwara, M.: On crystal bases of the \(q\)-analogue of universal enveloping algebras. Duke Math. J. 63, 465–516 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Kang, S.-J., Kashiwara, M., Misra, K.C., Miwa, T., Nakashima, T., Nakayashiki, A.: Affine crystals and vertex models. Int. J. Mod. Phys. A 7(suppl. 1A), 449–484 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Kolb, S.: Quantum symmetric Kac–Moody pairs. Adv. Math. 267, 395–469 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Kulish, P.P.: Yang–Baxter equation and reflection equations in integrable models. In: Low-Dimensional Models in Statistical Physics and Quantum Field Theory. Lecture Notes Physics, vol. 469, pp. 125–144. Schladming (1995)Google Scholar
  12. 12.
    Kuniba, A.: Tetrahedron equation and quantum \(R\) matrices for \(q\)-oscillator representations mixing particles and holes. SIGMA 14(067), 23 (2018)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Kuniba, A., Mangazeev, V.V., Maruyama, S., Okado, M.: Stochastic \(R\) matrix for \(U_q(A^{(1)}_n)\). Nucl. Phys. B 913, 248–277 (2016)ADSCrossRefzbMATHGoogle Scholar
  14. 14.
    Kuniba, A., Okado, M., Yamada, Y.: Box–ball system with reflecting end. J. Nonlinear Math. Phys. 12, 475–507 (2005)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Kuniba, A., Pasquier, V.: Matrix product solutions to the reflection equation from three dimensional integrability. J. Phys. A: Math. Theor. 51(255204), 26 (2018)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Malara, R., Lima-Santos, A.: On \(A^{(1)}_{n-1}\), \(B^{(1)}_{n}\), \(C^{(1)}_{n}\), \(D^{(1)}_{n}\), \(A^{(2)}_{2n}\), \(A^{(2)}_{2n-1}\) and \(D^{(2)}_{n+1}\) reflection \(K\)-matrices. J. Stat. Mech. 0609, P09013 (2006)Google Scholar
  17. 17.
    Nakayashiki, A., Yamada, Y.: Kostka polynomials and energy functions in solvable lattice models. Sel. Math., New Ser. 3, 547–599 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Nepomechie, R.I., Retore, A.L.: Surveying the quantum group symmetries of integrable open spin chains, arXiv:1802.04864 (2018)
  19. 19.
    Regelskis, V., Vlaar, B.: Reflection matrices, coideal subalgebras and generalized Satake diagrams of affine type. arXiv:1602.08471 (2016)
  20. 20.
    Sklyanin, E.K.: Boundary conditions for integrable quantum systems. J. Phys. A: Math. Gen. 21, 2375–2389 (1988)ADSMathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Institute of PhysicsUniversity of TokyoTokyoJapan
  2. 2.Department of MathematicsOsaka City UniversityOsakaJapan

Personalised recommendations