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@article{AA_2009_21_4_a0, author = {S. E. Derkachev and A. N. Manashov}, title = {General solution of the {Yung--Baxter} equation with symmetry group $\mathrm{SL}(\mathrm n,\mathbb C)$}, journal = {Algebra i analiz}, pages = {1--94}, publisher = {mathdoc}, volume = {21}, number = {4}, year = {2009}, language = {ru}, url = {http://geodesic.mathdoc.fr/item/AA_2009_21_4_a0/} }
TY - JOUR AU - S. E. Derkachev AU - A. N. Manashov TI - General solution of the Yung--Baxter equation with symmetry group $\mathrm{SL}(\mathrm n,\mathbb C)$ JO - Algebra i analiz PY - 2009 SP - 1 EP - 94 VL - 21 IS - 4 PB - mathdoc UR - http://geodesic.mathdoc.fr/item/AA_2009_21_4_a0/ LA - ru ID - AA_2009_21_4_a0 ER -
S. E. Derkachev; A. N. Manashov. General solution of the Yung--Baxter equation with symmetry group $\mathrm{SL}(\mathrm n,\mathbb C)$. Algebra i analiz, Tome 21 (2009) no. 4, pp. 1-94. http://geodesic.mathdoc.fr/item/AA_2009_21_4_a0/
[1] Theoret. and Math. Phys., 40:2 (1979), 688–706 | DOI | MR
[2] Sklyanin E. K., Takhtadzhyan L. A., Faddeev L. D., Theoret. and Math. Phys., 40:2 (1979) | DOI
[3] Takhtadzhyan L. A., Faddeev L. D., “Kvantovyi metod obratnoi zadachi i $\mathbf{XYZ}$-model Geizenberga”, Uspekhi mat. nauk, 34:5 (1979), 13–63 | MR
[4] Kulish P. P., Sklyanin E. K., “Quantum spectral transform method. Recent developments”, Lecture Notes in Phys., 151, Springer, Berlin–New York, 1982, 61–119 | MR
[5] Sklyanin E. K., “Quantum inverse scattering method. Selected topics”, Quantum Group and Quantum Integrable Systems, Nankai Lectures Math. Phys., ed. Mo-Lin Ge, World Sci. Publ., River Edge, NJ, 1992, 63–97 ; arXiv: hep-th/9211111 | MR
[6] Faddeev L. D., “How the algebraic Bethe ansatz works for integrable models”, Quantum Symmetries/Symétries Qantiques, Les Houches, 1995, eds. A. Connes, K. Kawedzki, J. Zinn-Justin, North-Holland, Amsterdam, 1998, 149–219 ; arXiv: hep-th/9605187 | MR | Zbl
[7] Kulish P. P., Sklyanin E. K., “O resheniyakh uravneniya Yanga–Bakstera”, Zap. nauch. semin. LOMI, 95, 1980, 129–160 | MR
[8] Jimbo M., “Introduction to the Yang–Baxter equation”, Internat. J. Modern Phys. A, 4 (1989), 3759–3777 | DOI | MR | Zbl
[9] Jimbo M. (ed.), Yang–Baxter equation in integrable systems, Adv. Ser. Math. Phys., 10, World Sci. Publ. Co., Inc., Teaneck, NJ, 1989 | MR | Zbl
[10] Drinfeld V. G., “Algebry Khopfa i kvantovoe uravnenie Yanga–Bakstera”, Dokl. AN SSSR, 283:5 (1985), 1060–1064 | MR
[11] Drinfeld V. G., “Quantum groups”, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, CA, 1986), Providence, RI, 1987, Amer. Math. Soc., | MR
[12] Baxter R. J., Exactly solved models in statistical mechanics, Acad. Press, Inc., London, 1982 | MR | Zbl
[13] Sutherland B., “A general model for multicomponent quantum systems”, Phys. Rev. B, 12 (1975), 3795 | DOI
[14] Kulish P. P., Reshetikhin N. Yu., “O $GL_3$-invariantnykh resheniyakh uravneniya Yanga–Bakstera i assotsiirovannykh kvantovykh sistemakh”, Zap. nauch. semin. LOMI, 120, 1982, 92–121 | MR
[15] Kulish P. P., Reshetikhin N. Y., “Diagonalisation of $GL(N)$ invariant transfer matrices and quantum $N$-wave system (Lee model)”, J. Phys. A, 16 (1983), L591–L596 | DOI | MR | Zbl
[16] Sklyanin E. K., “The quantum Toda chain”, Nonlinear Equations in Classical and Quantum Field Theory (Meudon/Paris, 1983/1984), Lecture Notes in Phys., 226, Springer, Berlin, 1985, 196–233 ; “Functional Bethe ansatz”, Integrable and Superintegrable Systems, ed. B. A. Kupershmidt, World Sci. Publ., Teaneck, NJ, 1990, 8–33 | MR | MR
[17] Gaudin M., Pasquier V., “The periodic Toda chain and a matrix generalization of the Bessel function recursion relations”, J. Phys. A, 25 (1992), 5243–5252 | DOI | MR | Zbl
[18] Lipatov L. N., “Asimptotika mnogotsvetnoi KKhD pri bolshikh energiyakh i tochno reshaemye spinoye modeli”, Pisma v ZhETF, 59:9–10 (1994), 571–574
[19] Lipatov L. N., “Duality symmetry of reggeon interactions in multicolor $QCD$”, Nuclear Phys. B, 548 (1999), 328 | DOI
[20] Faddeev L. D., Korchemsky G. P., “High-energy $QCD$ as a completely integrable model”, Phys. Lett. B, 342 (1995), 311 | DOI | MR
[21] Karakhanyan D., Kirschner R., Conserved currents of the three-reggeon interaction, ; “High-energy scattering in gauge theories and integrable spin chains”, Fortschr. Phys., 48 (2000), 139–142 ; arXiv: hep-th/9902147arXiv: hep-th/9902031 | 3.0.CO;2-S class='badge bg-secondary rounded-pill ref-badge extid-badge'>DOI | MR | Zbl
[22] Derkachov S. E., Korchemsky G. P., Manashov A. N., “Noncompact Heisenberg spin magnets from high-energy $QCD$. I. Baxter $Q$-operator and separation of variables”, Nuclear Phys. B, 617 (2001), 375–440 ; arXiv: hep-th/0107193 | DOI | MR | Zbl
[23] Kirch M., Manashov A. N., “Noncompact $SL(2,R)$ spin chain”, J. High Energy Phys., 2004:6, 035, 31 pp., (electronic) | DOI | MR
[24] Bytsko A. G., Teschner J., “Quantization of models with non-compact quantum group symmetry: modular XXZ magnet and lattice sinh-Gordon model”, J. Phys. A, 39:41 (2006), 12927–12981 ; arXiv: hep-th/0602093 | DOI | MR | Zbl
[25] Derkachov S. E., “Baxter's $Q$-operator for the homogeneous XXX spin chain”, J. Phys. A, 32 (1999), 5299–5316 ; arXiv: solv-int/9902015 | DOI | MR | Zbl
[26] Kuznetsov V. B., Salerno M., Sklyanin E. K., “Quantum Bäcklund transformation for the integrable DST model”, J. Phys. A, 33 (2000), 171–189 ; arXiv: solv-int/9908002 | DOI | MR | Zbl
[27] Pronko G. P., “On the Baxter's $Q$-operator for the XXX spin chain”, Comm. Math. Phys., 212 (2000), 687–701 ; arXiv: hep-th/9908179 | DOI | MR | Zbl
[28] Kovalsky A. E., Pronko G. P., Baxter $Q$-operators for integrable $DST$ chain, arXiv: nlin.SI/0203030
[29] Kovalsky A. E., Pronko G. P., Baxter's $Q$-operators for the simplest $q$-deformed model, arXiv: nlin.SI/0307040
[30] Volkov A. Y., “Quantum lattice $KdV$ equation”, Lett. Math. Phys., 39 (1997), 313–329 | DOI | MR | Zbl
[31] Antonov A., Feigin B., “Quantum group representations and Baxter equation”, Phys. Lett. B, 392 (1997), 115–122 ; arXiv: hep-th/9603105 | DOI | MR
[32] Belavin A. A., Odessky A. V., Usmanov R. A., New relations in the algebra of the Baxter $Q$-operators, ; Teor. mat. fiz., 130:3 (2002), 383–413 arXiv: hep-th/0110126 | MR | Zbl
[33] Rossi M., Weston R., “A generalized $Q$-operator for $U_q(\widehat{sl_2})$ vertex models”, J. Phys. A, 35 (2002), 10015–10032 ; arXiv: math-ph/0207004 | DOI | MR | Zbl
[34] Zabrodin A., “Commuting difference operators with elliptic coefficients from Baxter's vacuum vectors”, J. Phys. A, 33 (2000), 3825–3850 ; arXiv: math.QA/9912218 | DOI | MR | Zbl
[35] Korff C., Representation theory and Baxter's $TQ$ equation for the six-vertex model. A pedagogical overview, arXiv: cond-mat/0411758
[36] Bazhanov V. V., Lukyanov S. L., Zamolodchikov A. B., “Integrable structure of conformal field theory. I”, Comm. Math. Phys., 177 (1996), 381–398 ; ; “II”, Comm. Math. Phys., 190 (1997), 247–278 ; ; “III”, Comm. Math. Phys., 200 (1999), 297–324 ; arXiv: hep-th/9412229arXiv: hep-th/9604044arXiv: hep-th/9805008 | DOI | MR | Zbl | DOI | MR | Zbl | DOI | MR | Zbl
[37] Bazhanov V. V., Hibberd A. N., Khoroshkin S. M., “Integrable structure of $W(3)$ conformal field theory, quantum Boussinesq theory and boundary affine Toda theory”, Nuclear Phys. B, 622 (2002), 475–547 ; arXiv: hep-th/0105177 | DOI | MR | Zbl
[38] Bazhanov V. V., Tsuboi Z., “Baxter's $Q$-operators for supersymmetric spin chains”, Nuclear Phys. B, 805 (2008), 451–516 ; arXiv: 0805.4274[hep-th] | DOI | MR
[39] Khoroshkin S. M., Tolstoy V. N., “Universal $R$-matrix for quantized (super)algebras”, Comm. Math. Phys., 141 (1991), 599–617 | DOI | MR | Zbl
[40] Khoroshkin S. M., Tolstoy V. N., “The uniqueness theorem for the universal $R$-matrix”, Lett. Math. Phys., 24 (1992), 231–244 | DOI | MR | Zbl
[41] Khoroshkin S. M., Stolin A. A., Tolstoy V. N., “Generalized Gauss decomposition of trigonometric $R$-matrices”, Modern Phys. Lett. A, 10 (1995), 1375–1392 | DOI | MR | Zbl
[42] Kojima T., The Baxter's Q-operator for the $W$-algebra $W_N$, , 2008 arXiv: 0803.3505[nlin.SI] | MR
[43] Derkachov S. E., “Factorization of the $R$-matrix newline and Baxter's $Q$-operator”, Zap. nauch. semin. POMI, 347, 2007, 144–166 ; arXiv: math.QA/0507252 | MR
[44] Derkachov S. E., Manashov A. N., “Factorization of the transfer matrices for the quantum $s\ell(2)$ spin chains and Baxter equation”, J. Phys. A, 39 (2006), 4147–4159 ; arXiv: nlin/0512047[nlin.SI] | DOI | MR | Zbl
[45] Derkachov S. E., Manashov A. N., “Baxter operators for the noncompact quantum $s\ell(3)$ invariant spin chain”, J. Phys. A, 39 (2006), 13171–13190 ; arXiv: nlin/0604018[nlin.SI] | DOI | MR | Zbl
[46] Belitsky A. V., Derkachov S. E., Korchemsky G. P., Manashov A. N., “Baxter $Q$-operator for the graded $SL(2|1)$ spin chain”, J. Stat. Mech. Theory Exp., 2007, no. 1, PO1005, 63 pp., (electronic) ; arXiv: hep-th/0610332 | MR
[47] Derkachov S. E., Manashov A. N., “$R$-matrix and Baxter $Q$-operators for the noncompact $SL(N,\mathbb C)$ invariant spin chain”, SIGMA, 2 (2006), Paper 084 ; arXiv: nlin/0612003[nlin.SI] | MR | Zbl
[48] Derkachov S. É., “Factorization of the R-matrix. I”, Zap. nauch. semin. POMI, 335, 2006, 134–163 ; ; “II”, Зап. науч. семин. ПОМИ, 335, 2006, 164–187 ; arXiv: math.QA/0503396arXiv: math.QA/0503396 | MR | Zbl | MR | Zbl
[49] Gelfand I. M., Shilov G. E., Obobschënnye funktsii. I. Obobschënnye funktsii i deistviya nad nimi, Fizmatgiz, M., 1959
[50] Gelfand I. M., Naimark M. I., Unitarnye predstavleniya klassicheskikh grupp, Tr. Mat. in-ta AN SSSR, 36, 1950, 288 pp. | MR | Zbl
[51] Gelfand I. M., Naimark M. I., Vilenkin N. Ya., Obobschënnye funktsii. V. Integralnaya geometriya i svyazannye s nei voprosy teorii predstavlenii, Fizmatgiz, M., 1962
[52] Zhelobenko D. P., “Klassicheskie gruppy. Spektralnyi analiz konechnomernykh predstavlenii”, Uspekhi mat. nauk, 17:1 (1962), 27–120 | MR | Zbl
[53] Zhelobenko D. P., Kompaktnye gruppy Li i ikh predstavleniya, Nauka, M., 1970 | MR | Zbl
[54] Naimark M. A., Teoriya predstavlenii grupp, Nauka, M., 1976 | MR
[55] Kokster G. S. M., Mozer U. O. Dzh., Porozhdayuschie elementy i opredelyayuschie sootnosheniya diskretnykh grupp, Nauka, M., 1980 | MR
[56] Knapp A. W., Representation theory of semisimple groups. An overview based on examples, Princeton Math. Ser., 36, Princeton Univ. Press, Princeton, NJ, 1986 | MR | Zbl
[57] Kulish P. P., Reshetikhin N. Yu., Sklyanin E. K., “Yang–Baxter equations and representation theory. I”, Lett. Math. Phys., 5 (1981), 393–403 | DOI | MR | Zbl
[58] Vasilev A. N., Kvantovopolevaya renormgruppa v teorii kritichesskogo povedeniya i stokhasticheskoi dinamike, PIYaF, SPb., 1998
[59] Volkov A. Yu., “Noncommutative hypergeometry”, Comm. Math. Phys., 258 (2005), 257–273 ; arXiv: math/0312084 | DOI | MR | Zbl
[60] Isaev A. P., “Multi-loop Feynman integrals and conformal quantum mechanics”, Nuclear Phys. B, 662 (2003), 461–475 ; arXiv: hep-th/0303056 | DOI | MR | Zbl
[61] Sklyanin E. K., “O klassicheskikh predelakh $SU(2)$-invariantnykh reshenii uravneniya Yanga–Bakstera”, Zap. nauch. semin. LOMI, 146, 1985, 119–136 | MR | Zbl
[62] Gasper G., “Elementary derivations of summation and transformation formulas for $q$-series”, Spectral Functions, $q$-Series and Related Topics (Toronto, ON, 1995), Fields Inst. Commun., 14, Amer. Math. Soc., Providence, RI, 1997, 55–70 | MR | Zbl
[63] Gasper G., Rahman M., Basic hypergeometric series, Encyclopedia Math. Appl., 35, Cambridge Univ. Press, Cambridge, 1990 | MR | Zbl