Skip to main content
Log in

The R-matrix factorization, Q-operator, and variable separation in the case of the XXX spin chain with the SL(2, C) symmetry group

  • Published:
Theoretical and Mathematical Physics Aims and scope Submit manuscript

Abstract

We show a connection between the R-matrix factorization, the Baxter Q-operator, and separation of variables in the example of an integrable spin chain with the SL(2, ℂ) symmetry group.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. E. K. Sklyanin, L. A. Takhtadzhyan, and L. D. Faddeev, Theor. Math. Phys., 40, 688–706 (1979).

    Article  MathSciNet  Google Scholar 

  2. L. A. Takhtadzhyan and L. D. Faddeev, Russ. Math. Surveys, 34, 11–68 (1979).

    Article  ADS  MathSciNet  Google Scholar 

  3. P. P. Kulish and E. K. Sklyanin, “Quantum spectral transform method recent developments,” in: Integrable Quantum Field Theories (Lect. Notes Phys., Vol. 151, J. Hietarinta and C. Montonen, eds.), Springer, Berlin (1982), pp. 61–119.

    Chapter  Google Scholar 

  4. E. K. Sklyanin, “Quantum inverse scattering method: Selected topics,” in: Quantum Group and Quantum Integrable Systems (Nankai Lect. Math. Phys., M.-L. Ge, ed.), World Scientific, River Edge, N. J. (1992), p. 63–97; arXiv:hep-th/9211111v1 (1992).

    Google Scholar 

  5. L. D. Faddeev, “How the algebraic Bethe ansatz works for integrable models,” in: Symétries quantiques = Quantum Symmetries: Les Houches, Session LXIV, 1 Août-8 Septembre 1995 (A. Connes, K. Gawedzki, and J. Zinn-Justin, eds.), North Holland, Amsterdam (1998), pp. 149–219.

    Google Scholar 

  6. R. J. Baxter, Exactly Solved Models in Statistical Mechanics, Acad. Press, London (1982).

    MATH  Google Scholar 

  7. V. Pasquier and M. Gaudin, J. Phys. A, 25, 5243–5252 (1992).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  8. E. K. Sklyanin, Prog. Theoret. Phys. Suppl., 118, 35–60 (1995); arXiv:solv-int/9504001v1 (1995).

    Article  ADS  MathSciNet  Google Scholar 

  9. V. V. Bazhanov, S. L. Lukyanov, and A. B. Zamolodchikov, Commun. Math. Phys., 177, 381–398 (1996); arXiv:hep-th/9412229v1 (1994); 190, 247–278 (1997); arXiv:hep-th/9604044v2 (1996); 200, 297–324 (1999); arXiv:hep-th/9805008v2 (1998).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  10. S. M. Khoroshkin and V. N. Tolstoy, Commun. Math. Phys., 141, 599–617 (1991); Lett. Math. Phys., 24, 231–244 (1992); S. M. Khoroshkin, A. A. Stolin, and V. N. Tolstoy, Modern Phys. Lett. A, 10, 1375–1392 (1995); arXiv:hep-th/9404038v1 (1994).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  11. S. É. Derkachev, J. Math. Sci. (New York), 151, 2880–2893 (2008); arXiv:math.qa/0507252v2 (2005).

    Article  MathSciNet  Google Scholar 

  12. S. É. Derkachov and A. N. Manashov, J. Phys. A, 39, 4147–4159 (2006); arXiv:nlin.si/0512047v2 (2005).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  13. S. É. Derkachev and A. N. Manashov, St. Petersburg Math. J., 21, 513–577 (2010).

    Article  MATH  MathSciNet  Google Scholar 

  14. L. N. Lipatov, JETP Letters, 59, 596–599 (1994); Nucl. Phys. B, 548, 328–362 (1999); arXiv:hep-ph/9812336v3 (1998).

    ADS  Google Scholar 

  15. L. D. Faddeev and G. P. Korchemsky, Phys. Lett. B, 342, 311–322 (1995); arXiv:hep-th/9404173v1 (1994).

    Article  ADS  Google Scholar 

  16. D. R. Karakhanyan and R. Kirschner, Phys. Atomic Nuclei, 65, 1501–1512 (2002); arXiv:hep-th/9902147v1 (1999); Fortschr. Phys., 48, 139–142 (2000); arXiv:hep-th/9902031v1 (1999).

    Article  ADS  Google Scholar 

  17. S. É. Derkachov, G. P. Korchemsky, and A. N. Manashov, Nucl. Phys. B, 617, 375–440 (2001); arXiv:hep-th/0107193v2 (2001).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  18. I. M. Gel’fand, M. I. Graev, and N. Ya. Vilenkin, Generalized Functions [in Russian], Vol. 5, Integral Geometry and Questions of Representation Theory Related to It, Fizmatlit, Moscow (1962); English transl.: Vol. 5, Integral Geometry and Representation Theory, Acad. Press, New York (1966).

    Google Scholar 

  19. P. P. Kulish, N. Yu. Reshetikhin, and E. K. Sklyanin, Lett. Math. Phys., 5, 393–403 (1981).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  20. A. P. Isaev, Nucl. Phys. B, 662, 461–475 (2003); arXiv:hep-th/0303056v3 (2003).

    Article  MATH  ADS  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to S. É. Derkachev.

Additional information

__________

Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 169, No. 2, pp. 204–217, November, 2011.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Derkachev, S.É. The R-matrix factorization, Q-operator, and variable separation in the case of the XXX spin chain with the SL(2, C) symmetry group. Theor Math Phys 169, 1539–1550 (2011). https://doi.org/10.1007/s11232-011-0131-x

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11232-011-0131-x

Keywords

Navigation