Geodesic
Universal integrability objects
Teoretičeskaâ i matematičeskaâ fizika, Tome 174 (2013) no. 1, pp. 25-45 Cet article a éte moissonné depuis la source Math-Net.Ru

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We discuss the main points of the quantum group approach in the theory of quantum integrable systems and illustrate them for the case of the quantum group $U_q(\mathcal L(\mathfrak{sl}_2))$. We give a complete set of the functional relations correcting inexactitudes in the previous considerations. We especially attend to the interrelation of the representations used to construct the universal transfer operators and $Q$-operators.
Keywords: integrable system, quantum group, representation, functional relation. 
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H. Boos; F. Gohmann; A. Klümper; Kh. Nirov; A. V. Razumov. Universal integrability objects. Teoretičeskaâ i matematičeskaâ fizika, Tome 174 (2013) no. 1, pp. 25-45. http://geodesic.mathdoc.fr/item/TMF_2013_174_1_a1/

[1] V. G. Drinfel'd, “Quantum groups”, Proceedings of the International Congress of Mathematicians (Berkeley, California, August 3–11, 1986), v. 1, ed. A. M. Gleason, AMS, Providence, RI, 1988, 798–820 | MR

[2] M. Jimbo, Lett. Math. Phys., 10:1 (1985), 63–69 | DOI | MR | Zbl

[3] V. V. Bazhanov, S. L. Lukyanov, A. B. Zamolodchikov, Commun. Math. Phys., 177:2 (1996), 381–398, arXiv: hep-th/9412229 | DOI | MR | Zbl

[4] V. V. Bazhanov, S. L. Lukyanov, A. B. Zamolodchikov, Commun. Math. Phys., 190:2 (1997), 247–278, arXiv: hep-th/9604044 | DOI | MR | Zbl

[5] V. V. Bazhanov, S. L. Lukyanov, A. B. Zamolodchikov, Commun. Math. Phys., 200:2 (1999), 297–324, arXiv: hep-th/9805008 | DOI | MR | Zbl

[6] A. Antonov, B. Feigin, Phys. Lett. B, 392:1–2 (1997), 115–122, arXiv: hep-th/9603105 | MR

[7] V. V. Bazhanov, Z. Tsuboi, Nucl. Phys. B, 805:3 (2008), 451–516, arXiv: 0805.4274 | DOI | MR | Zbl

[8] H. Boos, F. Göhmann, A. Klümper, Kh. S. Nirov, A. V. Razumov, Universal ${R}$-matrix and functional relations, arXiv: 1205.1631

[9] V. Kats, Beskonechnomernye algebry Li, Mir, M., 1993 | MR | Zbl

[10] M. Jimbo, T. Miwa, “Algebraic Analysis of Solvable Lattice Models”, CBMS Regional Conference Series in Mathematics, 85, AMS, Providence, RI, 1985 | MR

[11] V. Chari, A. Pressley, A Guide to Quantum Groups, Cambridge Univ. Press, Cambridge, 1994 | MR | Zbl

[12] P. I. Etingof, I. B. Frenkel, A. A. Kirillov (Jr.), Lectures on Representation Theory and Knizhnik–Zamolodchikov Equations, Mathematical Surveys and Monographs, 58, AMS, Providence, RI, 1998 | DOI | MR | Zbl

[13] M. Rosso, Commun. Math. Phys., 124:2 (1989), 307–318 | DOI | MR | Zbl

[14] A. N. Kirillov, N. Reshetikhin, Commun. Math. Phys., 134:2 (1990), 421–431 | DOI | MR | Zbl

[15] S. Z. Levenderovskiĭ, Ya. S. Soibelman, J. Geom. Phys., 7:2 (1990), 241–254 | DOI | MR

[16] V. N. Tolstoi, S. M. Khoroshkin, Funkts. analiz i ego pril., 26:1 (1992), 85–88 | MR | Zbl

[17] S. M. Khoroshkin, V. N. Tolstoy, Lett. Math. Phys., 24:3 (1992), 231–244 | DOI | MR | Zbl

[18] S. Levenderovskii, Y. Soibelman, V. Stukopin, Lett. Math. Phys., 27:4 (1993), 253–264 | DOI | MR

[19] Y.-Z. Zhang, M. D. Gould, Lett. Math. Phys., 31:2 (1994), 101–110, arXiv: hep-th/9307007 | DOI | MR | Zbl

[20] A. J. Bracken, M. D. Gould, Y.-Z. Zhang, G. W. Delius, Internat. J. Modern Phys. B, 8 (1994), 3679–3691, arXiv: hep-th/9310183 | DOI | MR

[21] A. J. Bracken, M. D. Gould, Y.-Z. Zhang, Bull. Austral. Math. Soc., 51:2 (1995), 177–194 | DOI | MR | Zbl

[22] H. Boos, F. Göhmann, A. Klümper, Kh. S. Nirov, A. V. Razumov, J. Phys. A, 43:41 (2010), 415208, 35 pp., arXiv: 1004.5342 | DOI | MR | Zbl

[23] H. Boos, F. Göhmann, A. Klümper, Kh. S. Nirov, A. V. Razumov, J. Phys. A, 44:35 (2011), 355202, 25 pp., arXiv: 1104.5696 | DOI | MR | Zbl

[24] V. V. Bazhanov, A. N. Hibberd, S. M. Khoroshkin, Nucl. Phys. B, 622:3 (2002), 475–574, arXiv: hep-th/0105177 | DOI | MR

[25] D. Hernandez, M. Jimbo, Compos. Math., 148:5 (2012), 1593–1623, arXiv: 1104.1891 | DOI | MR | Zbl

[26] A. Klimyk, K. Schmüdgen, Quantum Groups and Their Representations, Springer, Berlin, 1997 | MR | Zbl

[27] M. Jimbo, Lett. Math. Phys., 11:3 (1986), 247–252 | DOI | MR | Zbl

[28] V. V. Bazhanov, T. Łukowski, C. Meneghelli, M. Staudacher, J. Stat. Mech., 2010 (2010), P11002, 37 pp., arXiv: 1005.3261 | DOI

[29] M. Rossi, R. Weston, J. Phys. A, 35:47 (2002), 10015–10032, arXiv: math-ph/0207004 | DOI | MR | Zbl

[30] A. V. Antonov, TMF, 113:3 (1997), 384–396, arXiv: hep-th/9607031 | DOI | DOI | MR | Zbl

[31] H. Boos, M. Jimbo, T. Miwa, F. Smirnov, Y. Takeyama, Commun. Math. Phys., 286:3 (2009), 875–932, arXiv: 0801.1176 | DOI | MR | Zbl

[32] S. É. Derkachov, A. N. Manashov, J. Phys. A, 39:16 (2006), 4147–4159, arXiv: nlin/0512047 | DOI | MR | Zbl