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

SIGMA 6 (2010), 029, 16 pages      arXiv:0910.5144
Contribution to the Proceedings of the XVIIIth International Colloquium on Integrable Systems and Quantum Symmetries

Jordan-Schwinger Representations and Factorised Yang-Baxter Operators

David Karakhanyan a and Roland Kirschner b
a) Yerevan Physics Institute, Br. Alikhanian Str. 2, 375036 Yerevan, Armenia
b) Institut für Theoretische Physik, Universität Leipzig, PF 100 920, D-04009 Leipzig, Germany

Received October 28, 2009, in final form March 30, 2010; Published online April 07, 2010

The construction elements of the factorised form of the Yang-Baxter R operator acting on generic representations of q-deformed sl(n+1) are studied. We rely on the iterative construction of such representations by the restricted class of Jordan-Schwinger representations. The latter are formulated explicitly. On this basis the parameter exchange and intertwining operators are derived.

Key words: Yang-Baxter equation; factorisation method.

pdf (272 kb)   ps (187 kb)   tex (21 kb)


  1. Biedenharn L.C., Lohe M.A., An extension of the Borel-Weil construction to the quantum group Uq(n), Comm. Math. Phys. 146 (1992) 483-504.
    Biedenharn L.C., Lohe M.A., Quantum group symmetry and q-tensor algebras, World Scientific Publishing Co., Inc., River Edge, NJ, 1995.
  2. Gelfand I.M., Naimark M.A., Unitary representations of the classical groups, Trudy Math. Inst. Steklov., Vol. 36, Izdat. Nauk SSSR, Moscow - Leningrad, 1950 (German transl.: Akademie Verlag, Berlin, 1957).
  3. Borel A., Weil A., Representations lineaires et espaces homogenes Kählerians des groupes de Lie compactes, Sem. Bourbaki, May 1954 (expose J.-P. Serre).
  4. Derkachov S.E., Karakhanyan D.R., Kirschner R., Valinevich P., Iterative construction of Uq(sl(n+1)) representations and Lax matrix factorisation, Lett. Math. Phys. 85 (2008), 221-234, arXiv:0805.4724.
  5. Isaev A.P., Quantum groups and Yang-Baxter equations, Sov. J. Part. Nucl. 26 (1995), 501-525 (see also the extended version: Preprint, Bonn, 2004, MPI 2004-132).
  6. Khoroshkin S.M., Tolstoy V.N., Universal R-matrix for quantized (super)algebras, Comm. Math. Phys. 141 (1991), 599-617.
  7. Date E., Jimbo M., Miki K., Miwa T., Generalized chiral Potts model and minimal cyclic representations of Uq(^gl(n,C)), Comm. Math. Phys. 137 (1991), 133-147.
  8. Bazhanov V.V., Kashaev R.M., Mangazeev V.V., Stroganov Yu.G., (ZN×)n–1 generalization of the chiral Potts model, Comm. Math. Phys. 138 (1991), 393-408.
  9. Delius G.W., Gould M.D., Zhang Y.Z., On the construction of trigonometric solutions of the Yang-Baxter equation, Nuclear Phys. B 432 (1994), 377-403, hep-th/9405030.
  10. Bazhanov V.V., Stroganov Yu.G., Chiral Potts model as a descendant of the six-vertex model, J. Statist. Phys. 59 (1990), 799-817.
  11. Hasegawa K., Yamada Y., Algebraic derivation of the broken ZN symmetric model, Phys. Lett. A 146 (1990), 387-396.
  12. Tarasov V.O., Cyclic monodromy matrices for the R-matrix of the six-vertex model and the chiral Potts model with fixed spin boundary conditions, in Infinite Analysis, Part A, B (Kyoto, 1991), Adv. Ser. Math. Phys., Vol. 16, World Sci. Publ., River Edge, NJ, 1992, 963-975.
  13. Belavin A.A., Odesskii A.V., Usmanov R.A., New relations in the algebra of the Baxter Q-operators, Theoret. and Math. Phys. 130 (2002), 323-350, hep-th/0110126.
  14. Kashaev R.M., Mangazeev V.V., Nakanishi T., Yang-Baxter equation for the sl(n) chiral Potts model, Nuclear Phys. B 362 (1991), 563-582.
  15. Tarasov V., Cyclic monodromy matrices for sl(n) trigonometric R-matrices, Comm. Math. Phys. 158 (1993), 459-483, hep-th/9211105.
  16. Lipatov L.N., High energy asymptotics of multi-color QCD and exactly solvable lattice models, JETP Lett. 59 (1994), 596-599, hep-th/9311037.
  17. Braun V.M., Derkachov S.E., Manashov A.N., Integrability of three-particle evolution equations in QCD, Phys. Rev. Lett. 81 (1998), 2020-2023, hep-ph/9805225.
  18. Karakhanyan D., Kirschner R., Mirumyan M., Universal R operator with deformed conformal symmetry, Nuclear Phys. B 636 (2002), 529-548, nlin.SI/0111032.
  19. Derkachov S.E., Factorization of R-matrix and Baxter's Q-operator, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 335 (2007), 144-166 (English transl.: J. Math. Sci. (N.Y.) 151 (2008), 2880-2893), math.QA/0507252.
  20. Derkachov S., Karakhanyan D., Kirschner R., Baxter Q-operators of the XXZ chain and R-matrix factorization, Nuclear Phys. B 738 (2006), 368-390, hep-th/0511024.
  21. Derkachov S., Karakhanyan D., Kirschner R., Yang-Baxter R-operators and parameter permutations, Nuclear Phys. B 785 (2007), 263-285, hep-th/0703076.
  22. Derkachov S.E., Manashov A.N., R-matrix and Baxter Q-operators for the noncompact SL(N,C) invariant spin chain, SIGMA 2 (2006), 084, 20 pages, nlin.SI/0612003.
  23. Valinevich P.A., Derkachov S.E., Karakhanyan D., Kirschner R., Factorization of the R-matrix for the quantum algebra slq(3), Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 335 (2007), 88-92 (English transl.: J. Math. Sci. (N.Y.) 151 (2008), 2848-2858).
  24. Jordan P., Der Zusammenhang der symmetrischen und linearen Gruppen und das Mehrkörperproblem, Z. Phys. 94 (1935), 531-535.
  25. Schwinger J., Quantum mechanics of angular momentum. A collection of reprints and original papers, Editors L.C. Biedenharn and H. van Dam, Academic Press, New York - London 1965.
  26. Jimbo M., A q-difference analog of U(g) and the Yang-Baxter equation, Lett. Math. Phys. 10 (1985) 63-69.
    Jimbo M., A q-analogue of U(gl(N+1)), Hecke algebra, and the Yang-Baxter equation, Lett. Math. Phys. 11 (1986), 247-252.
  27. Faddeev L.D., Kashaev R.M., Quantum dilogarithm, Modern Phys. Lett. A 9 (1994), 427-434, hep-th/9310070.
    Kirillov A.N., Dilogarithm identities, Progr. Theor. Phys. Suppl. (1995), no. 118, 61-142.

Previous article   Next article   Contents of Volume 6 (2010)