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

SIGMA 11 (2015), 040, 14 pages      arXiv:1505.03619
Contribution to the Special Issue on Exact Solvability and Symmetry Avatars in honour of Luc Vinet

From Twisted Quantum Loop Algebras to Twisted Yangians

Patrick Conner a and Nicolas Guay b
a) Science Department, Red Deer College, Red Deer, Alberta T4N 5H5, Canada
b) Department of Mathematical and Statistical Sciences, University of Alberta, CAB 632, Edmonton, Alberta T6G 2G1, Canada

Received January 30, 2015, in final form May 01, 2015; Published online May 14, 2015

We prove how the Yangian of $\mathfrak{gl}_N$ in its RTT presentation and Olshanski's twisted Yangians for the orthogonal and symplectic Lie algebras can be obtained by a degeneration process from the corresponding quantum loop algebra and some of its twisted analogues.

Key words: twisted Yangians; twisted quantum loop algebras; degeneration; RTT-presentation.

pdf (422 kb)   tex (22 kb)


  1. Arnaudon D., Molev A., Ragoucy E., On the $R$-matrix realization of Yangians and their representations, Ann. Henri Poincaré 7 (2006), 1269-1325, math.QA/0511481.
  2. Bernard D., Hidden Yangians in $2$D massive current algebras, Comm. Math. Phys. 137 (1991), 191-208.
  3. Bernard D., LeClair A., Quantum group symmetries and nonlocal currents in $2$D QFT, Comm. Math. Phys. 142 (1991), 99-138.
  4. Bernard D., Maassarani Z., Mathieu P., Logarithmic Yangians in WZW models, Modern Phys. Lett. A 12 (1997), 535-544, hep-th/9612217.
  5. Brundan J., Kleshchev A., Parabolic presentations of the Yangian $Y({\mathfrak{gl}}_n)$, Comm. Math. Phys. 254 (2005), 191-220, math.QA/0407011.
  6. Chari V., Pressley A., Yangians, integrable quantum systems and Dorey's rule, Comm. Math. Phys. 181 (1996), 265-302, hep-th/9505085.
  7. Chen H., Guay N., Twisted affine Lie superalgebra of type $Q$ and quantization of its enveloping superalgebra, Math. Z. 272 (2012), 317-347.
  8. Chen H., Guay N., Ma X., Twisted Yangians, twisted quantum loop algebras and affine Hecke algebras of type $BC$, Trans. Amer. Math. Soc. 366 (2014), 2517-2574.
  9. Cherednik I.V., A new interpretation of Gelfand-Tzetlin bases, Duke Math. J. 54 (1987), 563-577.
  10. De La Rosa Gomez A., MacKay N.J., Twisted Yangian symmetry of the open Hubbard model, J. Phys. A: Math. Theor. 47 (2014), 305203, 9 pages, arXiv:1404.2095.
  11. 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 (2001), 335-344, hep-th/0109115.
  12. Ding J.T., Frenkel I.B., Isomorphism of two realizations of quantum affine algebra $U_q(\widehat{\mathfrak{gl}(n)})$, Comm. Math. Phys. 156 (1993), 277-300.
  13. Drinfeld V.G., Quantum groups, in Proceedings of the International Congress of Mathematicians, Vols. 1, 2 (Berkeley, Calif., 1986), Amer. Math. Soc., Providence, RI, 1987, 798-820.
  14. Drinfeld V.G., A new realization of Yangians and of quantum affine algebras, Soviet Math. Dokl. 36 (1988), 212-216.
  15. Gautam S., Toledano Laredo V., Yangians, quantum loop algebras and abelian difference equations, arXiv:1310.7318.
  16. Gautam S., Toledano Laredo V., Meromorphic Kazhdan-Lusztig equivalence for Yangians and quantum loop algebras, arXiv:1403.5251.
  17. Gautam S., Toledano Laredo V., Yangians and quantum loop algebras, Selecta Math. (N.S.) 19 (2013), 271-336, arXiv:1012.3687.
  18. Guay N., Ma X., From quantum loop algebras to Yangians, J. Lond. Math. Soc. 86 (2012), 683-700.
  19. Guay N., Regelskis V., Twisted Yangians for symmetric pairs of types $B$, $C$, $D$, arXiv:1407.5247.
  20. Haldane F.D.M., Ha Z.N.C., Talstra J.C., Bernard D., Pasquier V., Yangian symmetry of integrable quantum chains with long-range interactions and a new description of states in conformal field theory, Phys. Rev. Lett. 69 (1992), 2021-2025.
  21. Kolb S., Quantum symmetric Kac-Moody pairs, Adv. Math. 267 (2014), 395-469, arXiv:1207.6036.
  22. MacKay N., Regelskis V., Achiral boundaries and the twisted Yangian of the D5-brane, J. High Energy Phys. 2011 (2011), no. 8, 019, 22 pages, arXiv:1105.4128.
  23. MacKay N., Regelskis V., Reflection algebra, Yangian symmetry and bound-states in AdS/CFT, J. High Energy Phys. 2012 (2012), no. 1, 134, 31 pages, arXiv:1101.6062.
  24. MacKay N.J., Rational $K$-matrices and representations of twisted Yangians, J. Phys. A: Math. Gen. 35 (2002), 7865-7876, math.QA/0205155.
  25. MacKay N.J., Introduction to Yangian symmetry in integrable field theory, Internat. J. Modern Phys. A 20 (2005), 7189-7217, hep-th/0409183.
  26. Mintchev M., Ragoucy E., Sorba P., Zaugg Ph., Yangian symmetry in the nonlinear Schrödinger hierarchy, J. Phys. A: Math. Gen. 32 (1999), 5885-5900, hep-th/9905105.
  27. Molev A., Yangians and classical Lie algebras, Mathematical Surveys and Monographs, Vol. 143, Amer. Math. Soc., Providence, RI, 2007.
  28. Molev A., Ragoucy E., Representations of reflection algebras, Rev. Math. Phys. 14 (2002), 317-342, math.QA/0107213.
  29. Molev A., Ragoucy E., Sorba P., Coideal subalgebras in quantum affine algebras, Rev. Math. Phys. 15 (2003), 789-822, math.QA/0208140.
  30. Nazarov M., Yangian of the queer Lie superalgebra, Comm. Math. Phys. 208 (1999), 195-223, math.QA/9902146.
  31. Olshanskii G.I., Twisted Yangians and infinite-dimensional classical Lie algebras, in Quantum Groups (Leningrad, 1990), Lecture Notes in Math., Vol. 1510, Springer, Berlin, 1992, 104-119.
  32. Reshetikhin N.Yu., Takhtadzhyan L.A., Faddeev L.D., Quantization of Lie groups and Lie algebras, Leningrad Math. J. 1 (1990), 193-225.

Previous article  Next article   Contents of Volume 11 (2015)