Abstract
We review some recent achievements in studies of symmetry properties of systems solved by the quantum inverse scattering method and connected with reflection equations. Special attention is given to the twist procedure, which, in particular, relates three constant R-matrices corresponding to the Lie superalgebra \(\mathfrak{g}\mathfrak{l}(1|1)\). Bibliography: 39 titles.
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REFERENCES
L. D. Faddeev, "How algebraic Bethe Ansatz works for integrable models," in: Les Houches Session LXIV(1995), pp. 149–219; Quantum symmetries, eds. A. Connes, K. Gawedzki, and J. Zinn-Justin, Elsevie Science (1998).
V. G. Drinfeld, "Quantum Groups," in: Proceedings of the International Congress of Mathematicians, Vol. 1, Academic Press, New York (1986), p. 798.
L. D. Faddeev, N. Yu. Reshetikhin, and L. A. Takhtajan, Algebra Analiz, 1, 1781989).
P. P. Kulish and E. K. Sklyanin, Lect. Notes Phys., 151,69(1982).
M. Jimbo (ed.), Yang-Baxter Equation in Integrable Systems, Singapore (1990).
M. Gaudin, La fonction d'onde de Bethe, Masson, Paris (1983).
F. A. Smirnov, Form Factors in Completely Integrable Models of Quantum Field Theory, Singapore (1992).
V. E. Korepin, N. M. Bogoliubov, and A. G. Izergin, Quantum Inverse Scattering Method and Correlation Functions, Cambridge University Press (1993).
M. Jimbo and T. Miwa, "Algebraic analysis of solvable lattice models," Preprint CBMS/85N (1994).
A. Montorsi and M. Rasetti, Phys. Rev. Lett., 72, 17301994); Zap. Nauch. Semin. POMI, 251,69(1998).
D. Arnaudon, "Algebraic approach to q-deformed supersymmetric variants of the Hubbard model with pair hopping," Preprint ENSLAPP-A-650/97 (1997)
P. P. Kulish, "Yang-Baxter equation and reflection equations in integrable models," in: Proceedings of the International Universitatswochen f¨ ur Kern-und Teilchenphysik, Schladming, eds. H. Grosse and L. Pittner, Springer-Verlag (1996), p. 125.
J.-M. Maillet and J. Sanchez de Santos, "Drinfeld twists and algebraic Bethe Ansatz," q-alg/9612012 (1996).
O. Babelon, D. Bernard, and E. Billey, Phys. Lett.B., 375,89(1966); M. Jimbo, H. Konno, S. Odake, and J. Shiraishi, "Quasi-Hopf twistors for elliptic quantum groups," q-alg/9712029 (1997); D. Arnaudon, E. Bu_enoire, E. Ragoucy, and Ph. Roche, Lett. Math. Phys., 44, 201(1998).
E. K. Sklyanin, "Quantum inverse scattering method: selected topics," in: Quantum Group and Quantum Integrable Systems, ed. M.-L. Ge, Singapore (1992); E. K. Sklyanin, Comm. Math. Phys., 142, 123(1992).
V. B. Kuznetsov and E. K. Sklyanin, J. Phys., A31, 22411998).
R. Kashaev, I. Korepanov, and S. Sergeev, TMP, 117, 3701998).
A. P. Isaev and P. P. Kulish, Mod. Phys. Lett., A12, 4271997).
P. P. Kulish and R. Sasaki, Prog. Theor. Phys., 89, 7411993).
P. P. Kulish and E. K. Sklyanin, J. Phys., A24, L435 (1991).
L. Mezincescu, R. I. Nepomechie, and V. Rittenberg, Phys. Lett., A147,70(1990); L. Mezincescu and R. I. Nepomechie, J. Phys., A24, L17 (1991); L. Mezincescu and R. I. Nepomechie, Int. J. Mod. Phys., A6, 5231(1991); A7, 5657(1992).
M. Karowski and A. Zapletal, Nucl. Phys., B419, 5671994).
M.-L. Ge and Q. Zhao, Phys. Lett., A175, 1991993).
W. G. Drinfeld, Dokl. Akad. Nauk SSSR, 273, 5311983); W. G. Drinfeld, Algebra Analiz, 1, 1419(dy1990).
M. Gerstenhaber, A. Giaquinto, and S. Schack, in: Quantum Groups. Proc. EIMI1990, Springer-Verlag (1992), p. 9; O. V. Ogievetsky, in: Proceedings of the Winter School on Geometry and Physics (1993), p. 185.
P. P. Kulish, V. D. Lyakhovsky, and A. I. Mudrov, J. Math. Phys., 40, 45691999).
P. P. Kulish, V. D. Lyakhovskii, and M. A. del Olmo, J. Phys., A32, 86711999).
E. Celeghini and P. P. Kulish, J. Phys., A31, L79 (1998); P. P. Kulish, "Super-jordanian deformation of the orthosymplectic Lie superalgebras," Preprint DFF 315/6/98 (1998).
J. Hietarinta, J. Math. Phys., 34, 17251993); H. Ewen and O. Ogievetsky, "Classification of GL(3) quantum matrix groups," Preprint MPI-PhT/94–93 (1993); M. Gerstenhaber and A. Giaquinto, Lett. Math. Phys., 44, 131(1998).
P. P. Kulish and A. Stolin, Czech. Jour. Phys., 47, 1231997); 47, 1207(1997).
A. Antonov, K. Hasegawa, and A. Zabrodin, Nucl. Phys., B503, 7471997).
B. Enriquez and G. Felder, "A construction of Hopf algebra cocycles for the double Yangian DY (SL2)," q-alg/9703012 (1997).
P. P. Kulish and N. Yu. Reshetikhin, J. Phys., A16, L591 (1983).
E. Ogievetskii, N. Yu. Reshetikhin, and P. B. Wiegmann, Nucl. Phys., B280,45(1987).
G. Felder and A. Varchenko, "Algebraic Bethe ansatz for the elliptic quantum group Eτη(sl2)," q-alg/9605024 (1996).
F. C. Alcaraz, M. Droz, M. Henkel, and V. Rittenberg, Ann. Phys., 230, 2501994).
H. Hinrichsen and V. Rittenberg, Phys. Lett., B275, 3501992).
P. P. Kulish and A. I. Mudrov, Lett. Math. Phys., 47, 1371999).
E. V. Damaskinskii, P. P. Kulish, and Chaichian, Teor. Mat. Fiz., 116, 1011998).
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Damaskinskii, E.V., Kulish, P.P. Symmetries Connected with Yang-Baxter and Reflection Equations. Journal of Mathematical Sciences 115, 1986–1993 (2003). https://doi.org/10.1023/A:1022699729324
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DOI: https://doi.org/10.1023/A:1022699729324