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Representations of the quantum doubles of finite group algebras and spectral parameter dependent solutions of the Yang–Baxter equation

J. Math. Phys. 47, 103511 (2006); doi:10.1063/1.2359575

Published 30 October 2006

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K. A. Dancer, P. S. Isac, and J. Links
Centre for Mathematical Physics, School of Physical Sciences, The University of Queensland, Brisbane 4072, Australia
Quantum doubles of finite group algebras form a class of quasitriangular Hopf algebras that algebraically solve the Yang–Baxter equation. Each representation of the quantum double then gives a matrix solution of the Yang–Baxter equation. Such solutions do not depend on a spectral parameter, and to date there has been little investigation into extending these solutions such that they do depend on a spectral parameter. Here we first explicitly construct the matrix elements of the generators for all irreducible representations of quantum doubles of the dihedral groups Dn. These results may be used to determine constant solutions of the Yang–Baxter equation. We then discuss Baxterization ansätze to obtain solutions of the Yang–Baxter equation with a spectral parameter and give several examples, including a new 21-vertex model. We also describe this approach in terms of minimal-dimensional representations of the quantum doubles of the alternating group A4 and the symmetric group S4. ©2006 American Institute of Physics
History: Received 14 December 2005; accepted 12 September 2006; published 30 October 2006
Permalink: http://link.aip.org/link/?JMAPAQ/47/103511/1
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REFERENCES (53)
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For access to fully linked references, you need to log in. For access to fully linked references, you need to Log in.
  1. R. J. Baxter, Exactly Solved Models in Statistical Mechanics (Academic, London, 1982).
  2. M. Jimbo, in Yang–Baxter Equations in Integrable Systems (World Scientific, Singapore, 1990).
  3. M. Jimbo M and T. Miwa, Algebraic Analysis of Solvable Lattice Models (American Mathematical Society, Providence, RI, 1995).
  4. E. K. Sklyanin, "Quantum inverse scattering method. Selected topics," in Quantum Group and Quantum Integrable Systems, Nankai Lectures Math. Phys. (World Scientific, Singapore, 1992), pp. 63–97.
  5. J. Links, H.-Q. Zhou, R. H. McKenzie, and M. D. Gould, J. Phys. A 36, R63 (2003).
  6. C. N. Yang and M.-L. Ge, in Braid Group, Knot Theory and Statistical Mechanics (World Scientific, Singapore, 1989).
  7. V. G. Drinfeld, "Quantum groups," in Proceedings of the International Congress of Mathematicians, edited by A. M. Gleason (American Mathematical Society, Providence, RI, 1986), pp. 798–820.
  8. M. Jimbo, Lett. Math. Phys. 10, 63 (1985).
  9. M. Jimbo, Lett. Math. Phys. 11, 247 (1986).
  10. Y. Umeno, M. Shiroishi, and M. Wadati, J. Phys. Soc. Jpn. 67, 2242 (1998).
  11. H.-Q. Zhou, Phys. Lett. A 221, 104 (1996).
  12. R. J. Baxter, J. H. H. Perk, and H. Au-Yang, Phys. Lett. A 128, 138 (1988).
  13. V. V. Bazhanov and Yu G. Stroganov, Theor. Math. Phys. 62, 253 (1985).
  14. V. V. Bazhanov and R. M. Kashaev, Commun. Math. Phys. 136, 607 (1991).
  15. V. V. Bazhanov, R. M. Kashaev, V. V. Mangazeev, and Yu G. Stroganov, Commun. Math. Phys. 138, 393 (1991).
  16. A. J. Bracken, M. D. Gould, Y.-Z. Zhang, and G. W. Delius, J. Phys. A 27, 6551 (1994).
  17. M. C. Takizawa and J. Links, "Ladder operators for integrable one-dimensional lattice models," in Group 24: Physical and Mathematical Aspects of Symmetries, edited by J.-P. Gazeau, R. Kerner, J.-P. Antoine, S. Métens, and J.-Y. Thibon, Institute of Physics Conference Series, 2003, Vol. 173, pp. 417–420.
  18. R. Dijkgraaf, V. Pasquier, and P. Roche, Nucl. Phys. B 18B, 60 (1990).
  19. M. D. Gould, Bull. Aust. Math. Soc. 48, 275 (1993).
  20. I. Tsohantjis and M. D. Gould, Bull. Aust. Math. Soc. 49, 177 (1994).
  21. M. de Wild Propitius and F. A. Bais, "Discrete gauge theories," in Particles and Fields, CRM Series in Mathematical Physics, edited by G. Semenoff and L. Vinet (Springer-Verlag, New York, 1998), p. 353.
  22. M. D. F. de Wild Propitius, "Topological interactions in broken gauge theories," Ph.D. thesis, hep-th/9511195.
  23. X. G. Wen and Q. Niu, Phys. Rev. B 41, 9377 (1990).
  24. A. Yu Kitaev, Ann. Phys. 303, 2 (2003).
  25. C. Mochon, Phys. Rev. A 67, 022315 (2003).
  26. C. Mochon, Phys. Rev. A 69, 032306 (2004).
  27. E. Dennis, A. Kitaev, A. Landahl, and J. Preskill, J. Math. Phys. 43, 4452 (2002).
  28. N. E. Bonesteel, L. Hormozi, G. Zikos, and S. H. Simon, Phys. Rev. Lett. 95, 140503 (2005).
  29. V. E. Korepin and I. Roditi, "Charge and spin separation in the 1D Hubbard model," cond-mat/0012266.
  30. P. W. Anderson, Science 235, 1196 (1987).
  31. Y. Zhang, L. H. Kauffman, and M.-L. Ge, Quantum Inf. Process. 4, 159 (2005).
  32. J. Slingerland (private communication).
  33. V. F. R. Jones, Int. J. Mod. Phys. B 4, 701 (1990).
  34. M. T. Batchelor and A. Kuniba, J. Phys. A 24, 2599 (1991).
  35. Y. Cheng, M.-L. Ge, and K. Xue, Commun. Math. Phys. 136, 195 (1991).
  36. R. B. Zhang, M. D. Gould, and A. J. Bracken, Nucl. Phys. B 354, 625 (1991).
  37. G. W. Delius, M. D. Gould, and Y.-Z. Zhang, Nucl. Phys. B 432, 377 (1994).
  38. U. Grimm, J. Phys. A 27, 5897 (1994).
  39. G. W. Delius, M. D. Gould, and Y.-Z. Zhang, Int. J. Mod. Phys. A 11, 3415 (1996).
  40. S. Boukraa and J.-M. Maillard, J. Stat. Phys. 102, 641 (2001).
  41. W. Galleas and M. Martins, "New R matrices from representations of braid-monoid algebras based on superalgebras," Nucl. Phys. B to appear. nlin.SI/0509014.
  42. C. W. Curtis and I. Reiner, Representation Theory of Finite Groups and Associative Algebras (Wiley, New York, 1962).
  43. A. J. Bracken, M. D. Gould, Y.-Z. Zhang, and G. W. Delius, Int. J. Mod. Phys. B 8, 3679 (1994).
  44. B. Golzer, J. Phys. A 22, L25 (1989).
  45. A. B. Zamolodchikov and V. A. Fateev, Sov. J. Nucl. Phys. 32, 298 (1980).
  46. A. G. Izergin and V. E. Korepin, Commun. Math. Phys. 79, 303 (1981).
  47. T. Kennedy, J. Phys. A 25, 2809 (1992).
  48. M. T. Batchelor and C. M. Yung, J. Phys. A 27, 5033 (1994).
  49. F. C. Alcaraz and R. Z. Bariev, J. Phys. A 34, L467 (2001).
  50. L. A. Takhtajan and L. D. Faddeev, Russ. Math. Surveys 34, 11 (1979).
  51. N. Reshetikhin, Lett. Math. Phys. 20, 331 (1990).
  52. D. S. McAnally, Lett. Math. Phys. 33, 249 (1995).
  53. D. S. McAnally and A. J. Bracken, Bull. Aust. Math. Soc. 55, 425 (1997).

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