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Yang-Baxter algebras, integrable theories and Betre Ansatz

  • II. Representation of Special Quantum Groups
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Quantum Groups

Part of the book series: Lecture Notes in Physics ((LNP,volume 370))

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Abstract

The Yang-Baxter algebras (YBA) are introduced in a general framework stressing their power to exactly solve the lattice models associated to them. The algebraic Bethe Ansatz is developed as an eigenvector construction based on the YBA. The six-vertex model solution is given explicitly. The generalization of YB algebras to face language is considered. The algebraic BA for the SOS model of Andrews, Baxter and Forrester is described using these face YB algebras.

It is explained how these lattice models yield both solvable massive QFT and conformal models in appropiated scaling (continuous) limits within the lattice light-cone approach. This approach permit to define and solve rigorously massive OFT as an appropiate continuum limit of gapless vertex models.

The deep links between the YBA and Lie algebras are analyzed including the quantum groups that underly the trigonometric/hyperbolic YBA. Braid and quantum groups are derived from trigonometric/hyperbolic YBA in the limit of infinite spectral parameter.

To conclude, some recent developments in the domain of integrable theories are summarized.

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References

  1. R.J. Baxter, Exactly solved models in Statistical Mechanics, Academic Press (1982).

    Google Scholar 

  2. L.D. Faddeev, Soviet Sci. Review C1, 107 (1980) and Les Houches lectures, North Holland (1982) and references contained therein.

    Google Scholar 

  3. P.P. Kulish and E.K. Sklyanin, in Tvärmine lectures, Springer Lectures in Physics, Vol. 151 (1981).

    Google Scholar 

  4. H.J. de Vega, Int. J. Mod. Phys., 4, 2371–2463 (1989) and in Advanced Studies in Pure Mathematics, vol.19, Kinokuniya-Academic,1989.

    Google Scholar 

  5. H.J. de Vega, Nucl. Phys. B240, 495 (1984).

    Google Scholar 

  6. R. J. Baxter, Studies in Appl. Math. L51, (1971).

    Google Scholar 

  7. H. J. de Vega and E. Lopes, Phys. Lett. B 186, 180 (1987).

    Google Scholar 

  8. see for example: L. D. Faddeev and L. A. Takhtadzhyan, ‘Hamiltonian approach to Soliton Theory', Nauka 1986.

    Google Scholar 

  9. R.J. Baxter, Phil. Trans. Roy. Soc. 289, 315 (1978).

    Google Scholar 

  10. M. Löscher, Nucl. Phys. B117, 475 (1976).

    Google Scholar 

  11. J.B. Mc Guire, J.M.P. 5, 622 (1964).

    Google Scholar 

  12. A.B. Zamolodchikov and AI.B. Zamolodchikov,Ann. Phys. 120, 253 (1979).

    Google Scholar 

  13. H.B. Thacker, Revs. Mods. Phys. 53, 253 (1981).

    Google Scholar 

  14. M. Lüscher, Nucl. Phys. B135, 1 (1978). See also: H.J. de Vega, H. Eichenherr and J.M. Maillet, Nucl. Phys. B240, 377 (1984), Phys. Lett. B132, 337 (1983) and Comm. Math. Phys. 92, 507 (1984).

    Google Scholar 

  15. The Analytic S-Matrix. R. J. Eden, P. Landshoff, D. Olive and J.C.Polkinghorne, Cambridge Univ. Press (1966).

    Google Scholar 

  16. A.B. Zamolodchikov, Comm. Math. Phys. 69, 165 (1979).

    Google Scholar 

  17. R. Shankar, Phys. Rev. Lett. 47, 1177 (1981).

    Google Scholar 

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

    Google Scholar 

  19. O. Babelon,H. J. de Vega and C. M. Viallet, Nucl. Phys. B200, 266 (1982).

    Google Scholar 

  20. L.A. Takhtadzhyan, Journal of Soviet Math. 2470 (1983).

    Google Scholar 

  21. P.P. Kulish and N. Yu. Reshetikhin, J. Phys. A16, L591 (1983).

    Google Scholar 

  22. N.Y. Reshetikhin, Theor. Math. Phys. 63, 555 (1986).

    Google Scholar 

  23. H.J. de Vega and M. Karowski, Nucl. Phys. B280, 225 (1987).

    Google Scholar 

  24. Zhou Yu-Kui, Yan Mu-Lin and Hou Bo-Yu, J. Phys. A 21, L929 (1988).

    Google Scholar 

  25. C. Destri and H. J. de Vega, LPTHE Paris preprint 89-20, to appear in Mod. Phys. Lett.

    Google Scholar 

  26. H. Au-Yang, B. M. McCoy, J.H.H. Perk, S. Tang and M. L. Yan, Phys. Lett. A 123, 219 (1987).

    Google Scholar 

  27. B. M. McCoy, J.H.H. Perk, S. Tang and C. H. Sah, Phys. Lett. A 125, 9 (1987).

    Google Scholar 

  28. R. J. Baxter, J. H. H. Perk and H. Au-Yang, Phys. Lett. A 128, 138 (1988).

    Google Scholar 

  29. V. V. Bazhanov and Yu. G. Stroganov, Canberra preprint, 1989.

    Google Scholar 

  30. Y. Akutsu, T. Deguchi and M. Wadati, J. Phys. Soc. Japan, 56, 3039 (1987).

    Google Scholar 

  31. F. Woynarovich, J. Phys. A15, 2985 (1982).

    Google Scholar 

  32. C. Destri and J. H. Lowenstein, Nucl. Phys. B205, 369 (1982).

    Google Scholar 

  33. L. D. Faddeev and L.A. Takhtadzhyan, Zap. Nauch. Sem. LOMI, 109, 134 (1982).

    Google Scholar 

  34. O. Babelon, H.J. de Vega and C.M. Viallet, Nucl. Phys. B220, 13 (1983) and 282 (1983).

    Google Scholar 

  35. L. V. Avdeev and B. D. Dörfel, Theor. Math. Phys. 71, 272 (1987).

    Google Scholar 

  36. F. C. Alcaraz, M. N. Barber and M.T. Batchelor,Phys. Rev. Lett. 58, 771 (1987) and Ann. Phys. 182, 280 (1988).

    Google Scholar 

  37. F. C. Alcaraz and M. J. Martins, J. Phys. A 22, 1829 (1989) and A 21, 4397 (1988).

    Google Scholar 

  38. H.J. de Vega and F. Woynarovich, Nucl. Phys. B251, 439 (1985).

    Google Scholar 

  39. H.O. Martin and H.J. de Vega, Phys. Rev. B32, 5959 (1985).

    Google Scholar 

  40. M.T. Batchelor et al. J. Phys. A 20,5677 (1987)

    Google Scholar 

  41. C. J. Hamer and M. T. Batchelor J. Phys. A 21, L173 (1988).

    Google Scholar 

  42. F. Woynarovich and H.P. Eckle, J. Phys. A 20, L443 (1987).

    Google Scholar 

  43. H.J. de Vega, J. Phys. A 20, 6023 (1987) and J. Phys. A 21, L1089 (1988).

    Google Scholar 

  44. H.P. Eckle and F. Woynarovich, J. Phys. A 20, L97 (1987).

    Google Scholar 

  45. F. Woynarovich, Phys. Rev. Lett. 59, 259 (1987).

    Google Scholar 

  46. H. J. de Vega and F. Woynarovich, LPTHE Paris preprint 89-32.

    Google Scholar 

  47. R. J. Baxter, Ann. Phys. 76, 1, 25 and 48 (1973).

    Google Scholar 

  48. C. Destri, H.J. de Vega and H.J. Giacomini, J. Stat. Phys. 56, 291 (1989).

    Google Scholar 

  49. J.H.H. Perk and F. Y. Wu, J. Stat. Phys. 42, 727 (1986).

    Google Scholar 

  50. G.E. Andrews, R. J. Baxter and P. J. Forrester, J. Stat. Phys. 35, 193 (1984).

    Google Scholar 

  51. M. Jimbo, T. Miwa and M. Okado, Comm. Math. Phys. 116, 507 (1988) and preprint RIMS-579 (1987).

    Google Scholar 

  52. E. Date, M. Jimbo, A. Kuniba T. Miwa and M. Okado,Nucl. Phys. B 290, 231 (1987).

    Google Scholar 

  53. A. L. Owczarek and R. J. Baxter, J. Stat. Phys. 49, 1093 (1987).

    Google Scholar 

  54. V. Pasquier, J. Phys. A 20, L217 and L221 (1987) and Nucl. Phys. B 285, 162 (1987).

    Google Scholar 

  55. Y. Akutsu, T. Deguchi and M. Wadati, J. Phys. Soc. of Japan, 57, 1173 (1988).

    Google Scholar 

  56. H. J. de Vega, LPTHE preprint 89/23, to be published in Int. J. Mod. Phys.

    Google Scholar 

  57. H. J. de Vega and H. J. Giacomini, J. Phys. A 22, 2759 (1989).

    Google Scholar 

  58. C. Destri and H.J. de Vega, Nucl. Phys. 8290, 363 (1987); Phys. Lett. B201, 261 (1988) and J. Phys. A22, 1329 (1989).

    Google Scholar 

  59. T.T. Truong and M. D.Schotte, Nucl. Phys. B220, 77 (1983) and B230, 1 (1984).

    Google Scholar 

  60. M.F. Weiss and M.D. Schotte, Nucl. Phys. B225, 247 (1983).

    Google Scholar 

  61. T.T. Truong in Non-linear Eqs. in Classical and QFT Springer Lectures in Physics, Vol. 226 (1985), Ed. N. Sá nchez.

    Google Scholar 

  62. L.V. Avdeev and M.V. Chizov, Phys. Lett. B184,363 (1987).

    Google Scholar 

  63. O. Babelon, H.J. de Vega and C.M. Viallet, Nucl. Phys. B190, 542 (1981).

    Google Scholar 

  64. H. Freudenthal and H. de Vries, Linear Lie Groups, New-York, Academic Press.

    Google Scholar 

  65. H. M. Babujan, Nucl. Phys. B215, 317 (1983).

    Google Scholar 

  66. P. P. Kulish and N. Yu. Reshetikhin, Zap. Nauk Sem. LOMI 101, 101 (1981).

    Google Scholar 

  67. L. A. Takhtadzhyan, Phys. Lett. A 87, 479 (1982).

    Google Scholar 

  68. V. E. Korepin, Theor. Math. Phys. 41, N°2 (1979).

    Google Scholar 

  69. N. Andrei and J. H. Lowenstein, Phys. Lett.B 91, 401 (1980).

    Google Scholar 

  70. E. Ogievetski, N. Y. Reshetikhin and P. B. Wiegmann, Nucl. Phys. B280, 45 (1987).

    Google Scholar 

  71. L. D. Faddeev and N. Y. Reshetikhin, Ann. Phys. 167,227 (1986).

    Google Scholar 

  72. P. P. Kulish, N. Y. Reshetikhin and E. K. Sklianin, L.M.P. 5, 393 (1981).

    Google Scholar 

  73. C. Destri and H. J. de Vega, Phys. Lett. B201, 245 (1988).

    Google Scholar 

  74. A. N. Kirillov and N. Y. Reshetikhin in Proceedings of the Paris-Meudon Colloquium. H. J. de Vega and N. Sanchez, Editors. World Scientific 1987.

    Google Scholar 

  75. R. J. Baxter, J. Stat. Phys. 28,1 (1982).

    Google Scholar 

  76. E. Artin, Ann. of Math. 48, 101 (1947).

    Google Scholar 

  77. J. S. Birman, Braids et al., Princeton Univ. Press, 1974.

    Google Scholar 

  78. T. Deguchi, M. Wadati and Y. Akutsu, Tokyo Univ. preprint, 1988.

    Google Scholar 

  79. M. Wadati and Y. Akutsu, C. M. P. 117, 243 (1988).

    Google Scholar 

  80. V. G. Turaev, LOMI preprint E-3-87, Leningrad.

    Google Scholar 

  81. N. Y. Reshetikhin, LOMI preprints E-4-87 and E-17-87, Leningrad.

    Google Scholar 

  82. V. Jones (unpublished).

    Google Scholar 

  83. A. Tsuchiya and Y. Kanie, LMP 13, 303 (1987) and in Adv. Studies Pure Math., 16, 297 (1988), Kinokuniya, Tokyo, 1988.

    Google Scholar 

  84. J. Fröhlich, Cargèse Lectures, 1987, Plenum New York.

    Google Scholar 

  85. K.H. Rehren, CMP 116, 675 (1988).

    Google Scholar 

  86. K. H. Rehren and B. Schroer, Nucl.Phys. B312, 715 (1989).

    Google Scholar 

  87. H. M. Babujian and A. M. Tsevelik, Nucl. Phys. B265, 24 (1986).

    Google Scholar 

  88. M. Jimbo, L.M.P.11,247 (1986) and 10, 63 (1985).

    Google Scholar 

  89. V. G. Drinfeld Dokl. Akad. Nauk, 283, 1060 (1985) and 296, 13 (1987).

    Google Scholar 

  90. E. Date, M. Jimbo, T. Miwa and M. Okado, RIMS preprint 590,(1987) and references therein.

    Google Scholar 

  91. M. Jimbo in Springer Notes in Physics, vol. 246, H. J. de Vega and N. Sanchez, editors and Comm. Math. Phys. 102, 537 (1986).

    Google Scholar 

  92. N. Yu. Reshetikhin, Theor. Math. Phys. 63, 197 and 347 (1985).

    Google Scholar 

  93. N. Yu. Reshetikhin and P. B. Wiegman, Phys. Lett. B 168, 360 (1986).

    Google Scholar 

  94. E. I. Ogievetski and P. B. Wiegman, Phys. Lett. B 189, 125 (1987).

    Google Scholar 

  95. C. Destri and H. J. de Vega, Phys. Lett. B223, 365 (1989).

    Google Scholar 

  96. H. W. Blöte, J. L. Cardy and M.P. Nightingale, Phys. Rev. Lett. 56, 742 (1986).

    Google Scholar 

  97. I. Affleck, Phys. Rev. Lett. 56, 746 (1986).

    Google Scholar 

  98. H. J. de Vega and T. M. J. Simoes, Phys. Lett. B 217, 142 (1989).

    Google Scholar 

  99. B. Klaiber, Boulder Lectures, Vol. XA, 141 (1968).

    Google Scholar 

  100. J. A. Swieca, Fortschr. Phys. 25, 303 (1977).

    Google Scholar 

  101. A. B. Zamolodchikov, Comm. Math. Phys. 69, 165 (1979).

    Google Scholar 

  102. A. B. Zamolodchikov in Advanced Studies in Pure Mathematics, vol.19,Kinokuniya-Academic,1989 and Int. J. Mod. Phys. 16, 4235 (1989).

    Google Scholar 

  103. C. Destri and H. J. de Vega, LPTHE Paris preprint 89-33 (to appear in Phys. Lett. B). H. W. Braden, E. Corrigan, P. E. Dorey and R. Sasaki, Durham preprint.

    Google Scholar 

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  1. Laboratoire de Physique Théorique et Hautes Energies, Paris

    H. J. De Vega

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H. -D. Doebner J. -D. Hennig

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De Vega, H.J. (1990). Yang-Baxter algebras, integrable theories and Betre Ansatz. In: Doebner, H.D., Hennig, J.D. (eds) Quantum Groups. Lecture Notes in Physics, vol 370. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-53503-9_46

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  • DOI: https://doi.org/10.1007/3-540-53503-9_46

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