А. П. Исаев1,2,a
Ключевые слова:
Некоммутативная дифференциальная геометрия, квантовые группы, уравнения Янга–Бакстера, диаграммы Фейнмана, спиновые цепочки, группы кос, теория узлов
Аннотация
Данный обзор является введением в новейший раздел теории симметрий — теории квантовых групп.
Основы теории квантовых групп рассматриваются с точки зрения возможности их использования для деформаций симметрий в физических моделях. Подробно обсуждается R матричный подход к теории квантовых групп, который положен в основу квантования классических групп Ли, а также некоторых супергрупп Ли. Мы начинаем с изложения основ некоммутативных и некокоммутативных алгебр Хопфа. Большое внимание уделено R-матрицам Гекке и Бирман-Мураками-Венцля (BMW) и связанным с ними квантовым матричным алгебрам. Обсуждается некоммутативная дифференциальная геометрия на квантовых группах специальных типов. Представлены тригонометрические решения уравнений Янга-Бакстера, связанных с квантовыми группами GL_q(N), SO_q(N), Sp_q(2n) и супергруппами GL_q(N|M), Osp_q(N|2m), а также рациональные (янгианские) пределы этих решений. Также рассматриваются рациональные R-матрицы для исключительных алгебр Ли и эллиптические решения уравнения Янга Бакстера. Изложены основные понятия групповой алгебры группы кос и ее конечномерных факторов (таких как алгебры Гекке и BMW). Дан набросок теорий представлений алгебр Гекке и BMW, включая методы нахождения идемпотентов (квантовых проекторов Юнга) и их квантовых размерностей. Кратко обсуждаются приложения теории квантовых групп и уравнений Янга-Бакстера в различных областях теоретической физики.
Это модифицированная версия обзорной статьи, опубликованной в 2004 году в виде
препринта Института математики Макса Планка в Бонне.
Поддерживающие организации
We would like to thank G. Arutyunov, L. Castellani, S. Derkachov, A. N. Kirillov, P. P. Ku-lish, A. I. Molev, O. V. Ogievetsky, Z. Popowicz, P. N. Pyatov, and V. O. Tarasov for valuablediscussions and comments. We are especially grateful to O. V. Ogievetsky for discussing thecontent of Section 4.
Библиографические ссылки
[1] R. J. Baxter, Exactly Solved Models in Statistical Mechanics, Academic Press, New York, 1982.
[2] C. N. Yang, Some exact results for the many-body problem in one dimension with repulsivedelta-function interaction, Phys. Rev. Lett. 19 (1967) 1312–1315.
[3] R. Baxter, Partition function of the eight-vertex lattice model, Ann. Phys. (N. Y.) 70 (1972)193–228 [reprinted in Ann. Phys. 281 (2000) 187].
[4] A. B. Zamolodchikov and Al. B. Zamolodchikov, Relativistic factorizedSmatrix in two-dimensions havingO(N)isotopic symmetry, Nucl. Phys. B 133 (1978) 525.
[5] A. B. Zamolodchikov and Al. B. Zamolodchikov, FactorizedS-matrices in two dimensions as theexact solutions of certain relativistic quantum field models, Ann. Phys. (N. Y.) 120 (1979) 253.
[6] E. K. Sklyanin and L. D. Faddeev, Quantum mechanical approach to completely integrable fieldtheory models, Sov. Phys. Dokl. 23 (1978) 902–904 [reprinted in Fifty Years of MathematicalPhysics: Selected Works of Ludwig Faddeev, 2016, pp. 290–292].
[7] E. K. Sklyanin, L. A. Takhtajan, and L. D. Faddeev, The quantum inverse problem method. 1,Theor. Math. Phys. 40 (1980) 688 [Teor. Mat. Fiz. 40 (1979) 194].
[8] L. A. Takhtajan and L. D. Faddeev, The quantum method of the inverse problem and theHeisenbergXY Zmodel, Usp. Mat. Nauk 34 (5) (1979) 11.
[9] L. D. Faddeev, Quantum completely integrable models in field theory, Math. Phys. Rev. Vol. 1,Soviet Sci. Rev. Sect. C: Math. Phys. Rev., Vol. 1, Harwood Academic, Chur, 1980, pp. 107–155 [L. D. Faddeev, 40 Years in Mathematical Physics, World Scientific Series in 20th CenturyMathematics, Vol. 2, 1995, pp. 187–235].
[10] V. G. Drinfeld, Quantum Groups, in: Proceedings of the International Congress of Mathematics,Vol. 1, Berkeley, 1986, p. 798.
[11] L. A. Takhtajan, A. Yu. Alekseev, I. Ya. Aref’eva, M. A. Semenov-Tian-Shansky, E. K. Sklyanin,F. A. Smirnov, S. L. Shatashvili, Scientific heritage of L. D. Faddeev. Survey of papers, Russ.Math. Surv., 72:6 (2017) 977–1081 [Usp. Mat. Nauk, 72:6 (438) (2017) 3–112].
[12] V. Pasquier and H. Saleur, Common structures between finite systems and conformal field the-ories through quantum groups, Nucl. Phys. B 330 (1990) 523.
[13] M. Karowski and A. Zapletal, Quantum group invariant integrableNstate vertex models withperiodic boundary conditions, Nucl. Phys. B 419 (1994) 567.
[14] P. P. Kulish, Quantum groups and quantum algebras as symmetries of dynamical systems, in:Quantum Symmetries, Proceedings of the International Workshop on Mathematical Physics,ASI, Clausthal, Germany, 15–20 July 1991, Eds: H.-D. Doebner and V. K. Dobrev, 1993.
[15] A. Connes, Geometrie Non Commutative, Intereditions, Paris, 1990.
[16] A. Connes and J. Lott, Particle models and noncommutative geometry (expanded version), Nucl.Phys. B Proc. Suppl. 18 (1991) 29.
[17] X. Calmet, B. Jurco, P. Schupp, J. Wess, and M. Wohlgenannt, The standard model on non-commutative space–time, Eur. Phys. J. C 23 (2002) 363.hep-ph/0111115.
[18] M. Kontsevich, Deformation quantization of Poisson manifolds, I, Lett. Math. Phys. 66 (2003)157.q-alg/9709040.
[19] A. Connes, M. R. Douglas, and A. Schwarz, Noncommutative geometry and matrix theory, J.High Energy Phys. 9802 (1998) 003.hep-th/9711162
[20] N. Seiberg and E. Witten, String theory and noncommutative geometry, J. High Energy Phys.9909 (1999) 032.hep-th/9908142.
[21] A. P. Isaev and Z. Popowicz, q-Trace for the quantum groups and q-deformed Yang–Mills theory,Phys. Lett. B 281 (1992) 271.
[22] A. M. Gavrilik, Quantum algebras in phenomenological description of particle properties, Nucl.Phys. Proc. Suppl. 102 (2001) 298.hep-ph/0103325; Quantum groups and Cabibbo mixing, in:Proceedings of 5th International Conference on Symmetry in Nonlinear Mathematical Physics(SYMMETRY 03), Kiev, Ukraine, 23–29 June 2003, pp. 759–766.hep-ph/0401086.
[23] O. Ogievetsky, W. B. Schmidke, J. Wess, and B. Zumino, q-Deformed Poincare algebra, Comm.Math. Phys. 150 (1992) 495.
[24] P. P. Kulish, Representations of Q-Minkowski space algebra, Algebra Anal. 6 (2) (1994) 195[English translation: St. Petersburg Math. J., 6 (2) (1995) 365].hep-th/9312139.
[25] J. A. De Azcarraga, P. P. Kulish, and F. Rodenas, Reflection equations and q Minkowski spacealgebras, Lett. Math. Phys. 32 (1994) 173.hep-th/9309036; J. A. De Azcarraga, P. P. Kulish,and F. Rodenas, On the physical contents of q-deformed Minkowski spaces, Phys. Lett. B 351(1–3) (1995) 123.arXiv:hep-th/9411121.
[26] W. Schmidke, J. Wess, and B. Zumino, A q-deformed Lorentz algebra, Z. Phys. C 52 (1991) 471.
[27] P. Podles and S. L. Woronowicz, Quantum deformation of Lorentz group, Comm. Math. Phys.130 (1990) 381.
[28] U. Carow-Watamura, M. Schlieker, M. Scholl, and S. Watamura, Tensor representation of thequantum groupSLq(2,C)and quantum Minkowski space, Z. Phys. C 48 (1990) 159; U. Carow-Watamura, et al., Quantum Lorentz group, Int. J. Mod. Phys. A 6 (1991) 3081.
[29] Sh. Majid, Braided momentum in the q-Poincar ́e group, J. Math. Phys. 34 (1993) 2045.
[30] J. Lukierski, A. Nowicki, H. Ruegg, and V. N. Tolstoy, Q deformation of Poincare algebra, Phys.Lett. B 264 (1991) 331; J. Lukierski, A. Nowicki, and H. Ruegg, New quantum Poincare algebraandκdeformed field theory, Phys. Lett. B 293 (1992) 344.
[31] B. L. Cerchiai and J. Wess, q-deformed Minkowski space based on a q-Lorentz algebra, Eur.Phys. J. C 5 (1998) 553.math.qa/9801104.
[32] L. Dolan, C. R. Nappi, and E. Witten, Yangian symmetry inD= 4superconformal Yang–Millstheory.hep-th/0401243.
[33] J. Drummond, J. Henn, J. Plefka, Yangian symmetry of scattering amplitudes inN= 4superYang–Mills theory, J. High Energy Phys. 05 (2009) 046.
[34] N. Beisert, On Yangian Symmetry in PlanarN= 4SYM.arxiv:1004.5423.
[35] T. Bargheer, N. Beisert, and F. Loebbert, Exact superconformal and Yangian symmetry ofscattering amplitudes, J. Phys. A 44 (2011) 454012.arxiv:1104.0700.
[36] D. Chicherin and R. Kirschner, Yangian symmetric correlators, Nucl. Phys. B 877 (2013) 484.arXiv:1306.0711 [math-ph].
[37] D. Chicherin, S. Derkachov, and R. Kirschner, Yang–Baxter operators and scattering ampli-tudes inN= 4super-Yang–Mills theory, Nucl. Phys. B 881 (2014) 467–501.arXiv:1309.5748[hep-th].
[38] A. Brandhuber, P. Heslop, G. Travaglini, and D. Young, Yangian Symmetry of Scattering Am-plitudes and the dilatation operator inN= 4supersymmetric Yang–Mills theory, Phys. Rev.Lett. 115 (2015) 141602.arxiv:1507.01504.
[39] D. Chicherin, V. Kazakov, F. Loebbert, D. M ̈uller, D. Zhong, Yangian symmetry for bi-scalarloop amplitudes, J. High Energy Phys. 05 (2018) 003.arXiv:1704.01967.
[40] V. Kazakov, F. Levkovich-Maslyuk, and V. Mishnyakov, Integrable Feynman Graphs and Yan-gian Symmetry on the Loom.arXiv:2304.04654v1 [hep-th].
[41] F. Loebbert, Lectures on Yangian symmetry, J. Phys. A 49 (2016) 323002.arxiv:1606.02947.
[42] L. D. Faddeev, N. Yu. Reshetikhin, and L. A. Takhtajan, Quantization of Lie groups and Liealgebras, Lengingrad Math. J. 1 (1990) 193 [Algebra Anal. 1 (1989) 178].
[43] E. K. Sklyanin, Some algebraic structures connected with the Yang–Baxter equation, Funct.Anal. Appl. 16 (1982) 263 [Funkt. Anal. Pril. 16 (4) (1982) 27].
[44] E. K. Sklyanin, Some algebraic structures connected with the Yang–Baxter equation. Represen-tations of quantum algebras, Funct. Anal. Appl. 17 (1983) 273 [Funkt. Anal. Pril. 17 (4) (1983)34].
[45] A. V. Odesskii, Elliptic algebras, Russ. Math. Surv. 75 (2002) 1127.math.QA/0303021.
[46] A. P. Isaev, Quantum groups and Yang–Baxter equations, Sov. J. Part. Nucl. 26 (1995) 501 [Fiz.Elem. Chastits i At. Yadra 26 (1995) 1204].
[47] A. P. Isaev, Quantum groups and Yang–Baxter equations, Preprint MPIM (Bonn), MPI 2004-132.http://webdoc.sub.gwdg.de/ebook/serien/e/mpi_mathematik/2004/132.pdf.
[48] A. P. Isaev, Lectures in Quantum Groups and Yang–Baxter Equations.arXiv:2206.08902[math.QA].
[49] E. Sweedler, Hopf Algebras, Benjamin, 1969.
[50] E. Abe, Hopf Algebras, Cambridge University Press, 1977.
[51] V. G. Drinfeld, Almost cocommutative Hopf algebras, Algebra Anal. 1 (2) (1989) 30.
[52] Sh. Majid, Quasitriangular Hopf algebras and Yang–Baxter equations, Int. J. Mod. Phys. A 5(1990) 1.
[53] A. A. Vladimirov, On the Hopf algebras generated by the Yang–BaxterRmatrices, Z. Phys. C58 (1993) 659.
[54] J.-P. Serre, Linear Representations of Finite Groups, Graduate Texts in Mathematics 42,Springer-Verlag, 1977.
[55] R. M. Kashaev, Heisenberg double and pentagon relation, Algebra Anal. 8 (4) (1996) 63.q-alg/9503005.
[56] S. Baaj, G. Skandalis, Unitaires multiplicatifs et dualit ́e pour les produits crois ́es deC∗-alg ́ebres,Annales scientifiques de l’Ecole normale sup ́erieure 26 (4) (1993) 425.
[57] N. Reshetikhin, Multiparameter quantum groups and twisted quasitriangular Hopf algebras,Lett. Math. Phys. 20 (1990) 331.
[58] N. Yu. Reshetikhin, V. G. Turaev, Ribbon graphs and their invariants derived from quantumgroups, Comm. Math. Phys. 127 (1990) 1–26.
[59] C. Kassel, M. Rosso, and V. Turaev, Quantum Groups and Knot Invariants, Panoramas etSynth`eses 5, Soci ́et ́e Math ́ematique de France, 1997.
[60] V. G. Drinfeld, Quasi-Hopf algebras, Algebra Anal. 1 (6) (1989) 114.
[61] V. G. Drinfeld, On quasitriangular quasi-Hopf algebras and a group related to Gal(Q/Q), AlgebraAnal. 2 (4) (1990) 149.
[62] M. Enock and J.-M. Schwartz, Kac Algebras and Duality of Locally Compact Groups, Springer-Verlag, 1992.
[63] L. Pontryagin, Topological Groups, Princeton University Press, 1946.
[64] P. Etingof, T. Schedler, and O. Schiffmann, Explicit quantization for dynamicalr-matrices forfinite dimensional semisimple Lie algebras, J. Amer. Math. Soc. 13 (2000) 595.math.QA/9912009.
[65] A. P. Isaev and O. V. Ogievetsky, On quantization ofr-matrices for Belavin–Drinfeld triples,Phys. At. Nucl. 64 (12) (2001) 2126.math.QA/0010190.
[66] P. P. Kulish, V. D. Lyakhovsky, and A. I. Mudrov, Extended Jordanian twists for Lie algebras,J. Math. Phys. 40 (1999) 4569.math.QA/9806014.
[67] N. Yu. Reshetikhin, Quasitriangular Hopf algebras and link invariants, Algebra Anal. 1 (2) (1989)169.
[68] O. Ogievetsky, Uses of Quantum Spaces, Lectures presented at the School “Quantum Symmetriesin Theoretical Physics and Mathematics”, Bariloche, 2000, Contemporary Mathematics, 294(2002) 161–231.
[69] A. A. Vladimirov, A Method for obtaining quantum doubles from the Yang–BaxterRmatrices,Mod. Phys. Lett. A 8 (1993) 1315.hep-th/9302042.
[70] S. L. Woronowicz, Differential calculus on compact matrix pseudogroups (quantum groups)Comm. Math. Phys. 122 (1989) 125.
[71] P. Aschieri and L. Castellani, An introduction to noncommutative differential geometry on quan-tum groups, Int. J. Mod. Phys. A 8 (1993) 1667.hep-th/9207084.
[72] A. Sudbery, Canonical differential calculus on quantum general linear groups and supergroups,Phys. Lett. B 284 (1–2) (1992) 61.
[73] Yu. I. Manin, Quantum groups and noncommutative geometry, Preprint CRM-1561, MontrealUniversity, 1989.
[74] Yu. I. Manin, Multiparametric quantum deformation of the general linear supergroup, Comm.Math. Phys. 123 (1989) 163.
[75] Yu. I. Manin, Notes on quantum groups and quantum de Rham complexes, Theor. Math. Phys.92 (3) (1992) 997 [Teor. Mat. Fiz. 92 (3) (1992) 425].
[76] G. Maltsiniotis, The language of spaces and of quantum groups, Comm. Math. Phys. 151 (1993)275; Groupes quantiques et structures diff ́erentielles, C. R. Acad. Sci. 311 S ́er. I (1990) 831.
[77] B. Tsygan, Notes on differential forms on quantum groups, Sel. Math. 12 (1993) 75.
[78] J. Wess and B. Zumino, Covariant differential calculus on the quantum hyperplane, Nucl. Phys.B Proc. Suppl. 18 (1990) 302.
[79] P. Schupp, P. Watts, B. Zumino, Differential geometry on linear quantum groups, Lett. Math.Phys. 25 (2) (1992) 139.arXiv:hep-th/9206029.
[80] B. Zumino, Differential calculus on quantum spaces and quantum groups, in: Proceedings ofXIX International Conference on Group Theoretical Methods in Physics, Salamanca, Spain,1992.arXiv:hep-th/9212093.
[81] T. Brzezi ́nski, Remarks on bicovariant differential calculi and exterior Hopf algebras, Lett. Math.Phys. 27 (1993) 287.
[82] A. P. Isaev and Z. Popowicz, Quantum group gauge theories and covariant quantum algebras,Phys. Lett. B 307 (1993) 353.arXiv:hep-th/9302090v2.
[83] A. P. Isaev, Quantum group covariant noncommutative geometry, J. Math. Phys. 35 (1994) 6784.arXiv:hep-th/9402060v2.
[84] A. P. Isaev,R-matrix approach to differential calculus on quantum groups, Phys. Part. Nucl. 28(3) (1997) 267 [Fiz. Elem. Chastits i At. Yadra 28 (3) (1997) 685].
[85] A. P. Isaev and P. N. Pyatov,Glq(N)-covariant quantum algebras and covariant differentialcalculus, Phys. Lett. A 179 (1993) 81.arXiv:hep-th/9211093.
[86] A. P. Isaev and P. N. Pyatov, Covariant differential complexes on quantum linear groups, J.Phys. A 28 (1995) 2227.arXiv:hep-th/9311112.
[87] L. D. Faddeev and P. N. Pyatov, The differential calculus on quantum linear groups, Am. Math.Soc. Transl. 175 (1996) 35.arXiv:hep-th/9402070.
[88] A. P. Isaev, Interrelations between quantum groups and reflection equation (braided) algebras,Lett. Math. Phys. 34 (1995) 333.arXiv:hep-th/9403154.
[89] A. P. Isaev and O. V. Ogievetsky, BRST operator for quantum Lie algebras and differentialcalculus on quantum groups, Theor. Math. Phys. 129 (2) (2001) 298.math.QA/0106206.
[90] A. P. Isaev and O. V. Ogievetsky, BRST and anti-BRST operators for quantum linear algebraUq(gl(N)), Nucl. Phys. Proc. Suppl. 102 (2001) 306.
[91] P. Schupp, P. Watts, B. Zumino, Bicovariant quantum algebras and quantum Lie algebras,Comm. Math. Phys. 157 (1993) 305–329.hep-th/9210150.
[92] B. Zumino, Wu Yong-Shi, and A. Zee, Chiral anomalies, higher dimensions, and differentialgeometry, Nucl. Phys. B 239 (1984) 477.
[93] D. Gurevich, P. Pyatov, and P. Saponov, Braided differential operators on quantum algebras, J.Geom. Phys. 61 (2011) 1485.arXiv:1004.4721 [math.QA].
[94] J. M. Maillet, Lax equations and quantum groups, Phys. Lett. B 245 (1990) 480.[95] N. Yu. Reshetikhin, M. A. Semenov-Tian-Shansky, QuantumR-matrices and factorization prob-lems, Geom. Phys. 5 (1988) 533–550.
[96] A. Isaev, O. Ogievetsky, P. Pyatov, and P. Saponov, Characteristic polynomials for quantummatrices, Lect. Notes Phys. Vol. 524, Springer, 1999, p. 322.
[97] A. Isaev, O. Ogievetsky, and P. Pyatov, On quantum matrix algebras satisfying the Cayley–Hamilton–Newton identities, J. Phys. A: Math. Gen. 32 (1999) L115–L121.math.QA/9809170.
[98] A. Yu. Alekseev and L. D. Faddeev,(T∗G)(T): A toy model for conformal field theory, Comm.Math. Phys. 141 (1991) 413.
[99] A. P. Isaev, O. V. Ogievetsky, and P. N. Pyatov, Cayley–Hamilton–Newton identities and qu-asitriangular Hopf algebras, in: Proceedings of the Internatinal Seminar “Supersymmetries andQuantum Symmetries” (SQS’99), Dubna, 2000.arXiv: math.QA/9912197.
[100] L. Freidel and J. Maillet, Quadratic algebras and integrable systems, Phys. Lett. B 262 (1991)278; L. Hlavaty and A. Kundu, Int. J. Mod. Phys. A 11 (1996) 2143.hep-th/9406215.
[101] A. I. Molev, E. Ragoucy, and P. Sorba, Coideal subalgebras in quantum affine algebras, Rev.Math. Phys. 15 (8) (2003) 789.math.QA/0208140.
[102] A. P. Isaev, A. I. Molev, and O. V. Ogievetsky, A new fusion procedure for the Brauer algebraand evaluation homomorphisms, Int. Math. Res. Not. 11 (2012) 2571–2606.arXiv:1101.1336[math.RT].
[103] N. Yu. Reshetikhin, Integrable models of quantum one-dimensional magnets withO(N)andSp(2k)symmetry, Theor. Math. Phys. 63 (1985) 555 [Teor. Mat. Fiz. 63 (1985) 347].
[104] A. P. Isaev, D. Karakhanyan, and R. Kirschner, Orthogonal and symplectic Yangians and Yang–BaxterR-operators, Nucl. Phys. B 904 (2016) 124–147.arXiv:1511.06152 [math-ph].
[105] A. M. Gavrilik and A. U. Klimyk, q-Deformed orthogonal and pseudo-orthogonal algebras andtheir representations, Lett. Math. Phys. 21 (1991) 215.e-Print:math.qa/0203201.
[106] M. Noumi, Macdonald’s symmetric polynomials as zonal spherical functions on quantum homo-geneous spaces, Adv. Math. 123 (1996) 16.math.QA/9503224.
[107] A. Isaev, O. Ogievetsky, Half-quantum linear algebra, in: Proceedings of the XXIX InternationalColloquium on Group-Theoretical Methods in Physics, Tianjin, China, 20–26 Aug. 2012, WorldScientific, 2013, p. 479.arXiv:1303.3991 [math.QA].
[108] A. Chervov, G. Falqui, and V. Rubtsov, Algebraic properties of Manin matrices 1, Adv. Appl.Math. 43 (3) (2009) 239.
[109] A. P. Isaev, Twisted Yang–Baxter equations for linear quantum (super)groups, J. Phys. A 29(1996) 6903.q-alg/9511006.
[110] P. Etingof and A. Varchenko, Solutions of the quantum dynamical Yang–Baxter equation anddynamical quantum groups, Comm. Math. Phys. 196 (1998) 591.q-alg/9708015.
[111] D. I. Gurevich, Algebraic aspects of the quantum Yang–Baxter equation, Algebra Anal. 2 (1990)119–148 [English translation: Leningrad Math. J. 2 (1991) 801–828].
[112] J. Ding and I. Frenkel, Isomorphism of two realizations of quantum affine algebraUq(ˆgl(n)),Comm. Math. Phys. 156 (1993) 277.
[113] M. Jimbo, A q-analogue ofU(gl(N+ 1)), Hecke algebra and the Yang–Baxter equation, Lett.Math. Phys. 11 (1986) 247.
[114] M. Jimbo, A Q-difference analog ofU(G)and the Yang–Baxter equation, Lett. Math. Phys. 10(1985) 63.
[115] M. Rosso, Finite dimensional representations of the quantum analog of the enveloping algebraof a complex simple Lie algebra, Comm. Math. Phys. 117 (1988) 581.
[116] Ya. Soibelman, The algebra of functions on compact quantum group and its representations,Algebra Anal. 2 (1) (1990) 190 [English translation: Leningrad Math. J. 2 (1) (1991)].
[117] J. Donin, P. P. Kulish, and A. I. Mudrov, On universal solution to reflection equation, Lett.Math. Phys. 63 (2003) 179.math.QA/0210242.
[118] P. A. Saponov, Weyl approach to representation theory of reflection equation algebra, J. Phys.A 37 (2004) 5021.math.QA/0307024.
[119] A. I. Mudrov, Characters ofUq(gl(n))-reflection equation algebra, Lett. Math. Phys. 60 (2002)283.math.QA/0204296.
[120] D. Algethami, A. Mudrov, and V. Stukopin, Solutions to graded reflection equation ofGL-type,Lett. Math. Phys. 114 (1) (2024) 22.arXiv:2310.15667 [math.QA].
[121] M. A. Semenov-Tian-Shansky, Poisson Lie groups, quantum duality principle, and the quantumdouble, Contemp. Math. 175 (1994) 219; Theor. Math. Phys. 93 (1992) 1292–1307.
[122] M. Rosso, An analogue of PBW theorem and the universalRmatrix forUhsl(N+ 1), Comm.Math. Phys. 124 (1989) 307.
[123] A. N. Kirillov and N. Reshetikhin, q-Weyl group and a multiplicative formula for universalR-matrices, Comm. Math. Phys. 134 (1990) 421.
[124] A. Yu. Alekseev and L. D. Faddeev, An involution and dynamics for the q-deformed quantumtop, Zap. Nauchn. Semin. LOMI 200 (1992) 3.hep-th/9406196.
[125] A. P. Isaev, O. V. Ogievetsky, and P. N. Pyatov, Generalized Cayley–Hamilton–Newton identi-ties, Czech. J. Phys. 48 (1998) 1369–1374.math.QA/9809047.
[126] M. Nazarov and V. Tarasov, Yangians and Gelfand–Zetlin basis, Publ. RIMS 30 (1994) 459.hep-th/9302102.
[127] P. N. Pyatov and P. A. Saponov, Characteristic relations for quantum matrices, J. Phys. A:Math. Gen. 28 (1995) 4415.q-alg/9502012.
[128] D. I. Gurevich, P. N. Pyatov, and P. A. Saponov, Hecke symmetries and characteristic relationson reflection equation algebras, Lett. Math. Phys. 41 (1997) 255.q-alg/9605048.
[129] A. P. Isaev and P. N. Pyatov, Spectral extension of the quantum group cotangent bundle, Comm.Math. Phys. 288 (2009) 1137.arXiv:0812.2225[math.QA].
[130] D. I. Gurevich, P. N. Pyatov, and P. A. Saponov, Cayley–Hamilton theorem for quantum matrixalgebras ofGL(m|n)type, Algebra Anal. 17 (1) (2005) 160 [St. Petersburg Math. J. 17 (1) (2006)119].math.QA/0412192.
[131] O. Ogievetsky and P. Pyatov, Orthogonal and symplectic quantum matrix algebras and Cayley–Hamilton theorem for them, Preprint MPIM2005-53, 2005.math.QA/0511618.[132] D. D. Demidov, Yu. I. Manin, E. E. Mukhin, and D. V. Zhdanovich, Nonstandard quantumdeformations ofGl(N)and constant solutions of the Yang–Baxter equation, Prog. Theor. Phys.Suppl. 102 (1990) 203.
[133] A. Sudbery, Consistent multiparameter quantization ofGl(N), J. Phys. A 23 (1990) L697.
[134] D. B. Fairlie and C. K. Zachos, Multiparameter associative generalizations of canonical commu-tation relations and quantized planes, Phys. Lett. B 256 (1991) 43.
[135] A. Schirrmacher, The multiparametric deformation ofGl(N)and the covariant differential cal-culus on the quantum vector space, Z. Phys. C 50 (1991) 321.
[136] A. P. Isaev, O. V. Ogievetsky, and P. N. Pyatov, OnR-matrix representations of Birman–Murakami–Wenzl algebras, Trudy MIAN 246 (2004) 147–153 (Memorial volume A. N. Tyurin).
[137] A. Schirrmacher, MultiparameterRmatrices and their quantum groups, J. Phys. A 24 (1991)L1249.
[138] H. Weyl, The Theory of Groups and Quantum Mechanics, Dover Publications, 1950.
[139] A. P. Isaev, V. A. Rubakov, Theory of Groups and Symmetries II. Representations of Groupsand Lie Algebras, Applications, World Scientific, 2021, 600 p.
[140] N. Yu. Reshetikhin and M. A. Semenov–Tian–Shansky, Central extensions of quantum currentgroups, Lett. Math. Phys. 19 (1990) 133.
[141] A. P. Isaev and R. P. Malik, Deformed traces and covariant quantum algebras for quantumgroupsGLqp(2)andGLqp(1|1), Phys. Lett. B 280 (1992) 219.hep-th/0309186.
[142] V. K. Dobrev, Duality for the matrix quantum groupGLp,q(2,C), J. Math. Phys. 33 (1992)3419.
[143] V. G. Drinfeld, Hopf algebras and the quantum Yang–Baxter equation, Sov. Math. Dokl. 32(1985) 254.
[144] A. I. Molev, Yangians and their applications, in: “Handbook of Algebra”, Vol. 3, M. Hazewinkel,ed., Elsevier, 2003.math.QA/0211288.
[145] A. Molev, M. Nazarov, and G. Olshanski, Yangians and classical Lie algebras, Russ. Math. Surv.51 (2) (1996) 205.
[146] P. P. Kulish and E. K. Sklyanin, Quantum spectral transform method. Recent developments, in:“Integrable Quantum Field Theories”, Lect. Notes Phys. 151 (1982) 61.
[147] R. M. Kashaev and N. Yu. Reshetikhin, Affine Toda field theory as a three-dimensional integrablesystem, Comm. Math. Phys. 188 (1997) 251.arXiv:hep-th/9507065.
[148] A. P. Isaev and S. M. Sergeev, Quantum Lax operators and discrete2+1-dimensional integrablemodels, Lett. Math. Phys. 64 (2003) 57.
[149] R. B. Zhang, Thegℓ(M|N)super Yangian and its finite dimensional representations, Lett. Math.Phys. 37 (1996) 419.q-alg/9507029.
[150] M. Chaichian and P. P. Kulish, Quantum Lie superalgebras and Q oscillators, Phys. Lett. B 234(1990) 72.
[151] E. V. Damaskinsky, P. P. Kulish, and M. A. Sokolov, Gauss decomposition for quantum groupsand supergroups, Zap. Nauchn. Semin. POMI 211 (1995) 11–45.q-alg/9505001.[152] Y. Leblanc and J. C. Wallet,Rmatrix andqcovariant oscillators forUq(sl(n|m)), Phys. Lett.B 304 (1993) 89.
[153] M. Chaichian, P. Kulish, and J. Lukierski, Supercovariant systems ofqoscillators andqsuper-symmetric Hamiltonians, Phys. Lett. B 262 (1991) 43.
[154] P. P. Kulish and E. K. Sklyanin, On the solution of the Yang–Baxter equation, J. Sov. Math. 19(1982) 1596 [Zap. Nauchn. Semin. LOMI 95 (1980) 129].
[155] M. L. Nazarov, Quantum Berezinian and the classical Capelli identity, Lett. Math. Phys. 21(1991) 123.
[156] V. Lyubashenko and A. Sudbery, Quantum supergroups ofGL(n|m)type: Differential forms,Koszul complexes and Berezinians, Duke Math. J. 90 (1) (1997) 1.hep-th/9311095.
[157] J. Schwenk, W. B. Schmidke, and S. P. Vokos, Properties of2×2quantum matrices inZ(2)graded spaces, Z. Phys. C 46 (1990) 643.
[158] W. B. Schmidke, S. P. Vokos, and B. Zumino, Differential geometry of the quantum supergroupGlq(1|1), Z. Phys. C 48 (1990) 249.
[159] P. P. Kulish, Two-parameters quantum group and gauge transformation, Zap. Nauchn. Semin.LOMI 180 (1990) 89.
[160] L. Dabrowski and L. Y. Wang, Two parameter quantum deformation ofGL(1|1), Phys. Lett. B266 (1991) 51.
[161] V. Bazhanov and A. Shadrikov, Quantum triangle equations and Lie superalgebras, Teor. Mat.Fiz. 73 (1987) 402.
[162] P. P. Kulish, Integrable graded magnets, J. Sov. Math. 35 (1986) 2648 [Zap. Nauchn. Semin.LOMI 145 (1985) 140].
[163] S. M. Khoroshkin and V. N. Tolstoy, UniversalR-matrix for quantized (super)algebras, Comm.Math. Phys. 141 (1991) 599.
[164] P. P. Kulish and N. Yu. Reshetikhin, UniversalRmatrix of the quantum superalgebraOsp(2|1),Lett. Math. Phys. 18 (1989) 143.
[165] A. Klimyk and K. Schm ̈udgen, Quantum Groups and Their Representations, Springer, 1997.
[166] B. Berg, M. Karowski, V. Kurak, and P. Weisz, FactorizedU(N)symmetricSmatrices intwo-dimensions, Nucl. Phys. B 134 (1978) 125.
[167] D. Arnaudon, J. Avan, N. Crampe, L. Frappat, E. Ragoucy,R-matrix presentation for (super)-YangiansY(g), J. Math. Phys. 44 (2003) 302.math.QA/0111325.
[168] D. Arnaudon, et al., Bethe Ansatz equations and exactSmatrices for theosp(M|2n)open superspin chain, Nucl. Phys. B 687 (3) (2004) 257.math-ph/0310042.
[169] V. V. Bazhanov, Integrable quantum systems and classical Lie algebras, Comm. Math. Phys.113 (1987) 471.
[170] V. V. Bazhanov, Trigonometric solution of triangle equations and classical Lie algebras, Phys.Lett. B 159 (1985) 321.
[171] M. Jimbo, QuantumRmatrix for the generalized Toda system, Comm. Math. Phys. 102 (1986)537.
[172] N. J. MacKay, RationalR-matrices in irreducible representations. J. Phys. A: Math. Gen. 24(17) (1991) 4017.
[173] N. J. MacKay, Introduction to Yangian symmetry in integrable field theory, Int. J. Mod. Phys.A 20 (30) (2005) 7189.
[174] A. Pressley and V. Chari, Fundamental representations of Yangians and singularities ofR-matrices, J. Reine Angew. Math. 417 (1991) 87–128.
[175] E. Ogievetsky and P. Wiegmann, FactorizedS-matrix and the Bethe ansatz for simple Lie groups,Phys. Lett. B 168 (4) (1986) 360.
[176] A. P. Isaev and S. O. Krivonos, Split Casimir operator for simple Lie algebras, solutions ofYang–Baxter equations and Vogel parameters, J. Math. Phys. 62 (8) (2021) 083503.e-Print:2102.08258 [math-ph].
[177] A. P. Isaev and S. O. Krivonos, Split Casimir operator and universal formulation of the simpleLie algebras, Symmetry 13 (6) (2021) 1046.e-Print: 2106.04470 [math-ph].
[178] N. Bourbaki, Lie Groups and Lie Algebras: Chapters 4–6, Vol. 3, Springer, 2008.
[179] N. Yamatsu, Finite-Dimensional Lie Algebras and Their Representations for Unified Model Build-ing.arXiv:1511.08771 [hep-ph].
[180] P. Cvitanovi ́c, Birdtracks, Lie’s, and Exceptional Groups, Princeton; Oxford: Princeton Univer-sity Press, 2008.http://cns.physics.gatech.edu/grouptheory/chapters/draft.pdf.
[181] M. Jimbo and T. Miwa, q-KZ equation with|q|= 1and correlation functions of theXXZmodelin the gapless regime, J. Phys. A 29 (1996) 2923.hep-th/9601135.
[182] H. E. Boos, V. E. Korepin, F. A. Smirnov, Connecting lattice and relativistic models via confor-mal field theory, Progress in Mathematics, Vol. 237, “Infinite Dimensional Algebras and QuantumIntegrable Systems”, 2005, p. 157.math-ph/0311020.
[183] H. E. Boos, V. E. Korepin, F. A. Smirnov, New formulae for solutions of quantum Knizhnik–Zamolodchikov equations on level-4, J. Phys. A 37 (2004) 323.hep-th/0304077.
[184] I. B. Frenkel and N. Yu. Reshetikhin, Quantum affine algebras and holonomic difference equa-tions, Comm. Math. Phys. 146 (1992) 1.
[185] V. Tarasov and A. Varchenko, Geometry ofq-hypergeometric functions as a bridge betweenYangians and quantum affine algebras, Invent. Math. 128 (1997) 501.q-alg/9604011.
[186] F. A. Smirnov, Dynamical symmetries of massive integrable models, 1. Form-factor bootstrapequations as a special case of deformed Knizhnik–Zamolodchikov equations, Int. J. Mod. Phys.A 7 (1992) S813.
[187] A. P. Isaev, A. N. Kirillov, and V. O. Tarasov, Bethe subalgebras in affine Birman–Murakami–Wenzl algebras and flat connections for q-KZ equations, J. Phys. A: Math. Theor. 49 (2016)204002.arXiv:1510.05374.
[188] I. Cherednik, Difference elliptic operators and root systems, Int. Math. Res. Not. (1995) 44–59.arXiv:hep-th/9410188.
[189] A. N. Kirillov, On Some Quadratic Algebras: Jucys–Murphy and Dunkl Elements, in: “Calogero–Moser–Sutherland Models”, Springer, New York, 2000, pp. 231–248.arXiv:q-alg/9705003.
[190] A. A. Belavin, Dynamical symmetry of integrable quantum systems, Nucl. Phys. B 180[FS2](1981) 189.
[191] R. J. Baxter, One-dimensional anisotropic Heisenberg chain, Ann. Phys. 70 (1972) 323.
[192] I. V. Cherednik, On the properties of factorizedSmatrices in elliptic functions, Sov. J. Nucl.Phys. 36 (1982) 320 [Yad. Fiz. 36 (1982) 549].
[193] C. A. Tracy, Embedded elliptic curves and the Yang–Baxter equations, Physica (Utrecht) D 16(1985) 203.
[194] A. Bovier, FactorizedSmatrices and generalized Baxter models, J. Math. Phys. 24 (1983) 631.
[195] G. Felder and V. Pasquier, A simple construction of ellipticRmatrices, Lett. Math. Phys. 32(1994) 167.
[196] L. A. Takhtajan, Solutions of the triangle equations withZn×Zn-symmetry and matrix analoguesof the Weierstrass zeta and sigma functions, Zap. Nauchn. Semin. LOMI 133 (1984) 258–276.
[197] V. V. Bazhanov, QuantumRmatrices and matrix generalizations of trigonometric functions,Theor. Math. Phys. 73 (1) (1987) 1035–1039 [Teor. Mat. Fiz. 73 (1) (1987) 26–32].
[198] M. Rosso, Quantum groups and quantum shuffles, Invent. Math. 133 (1998) 133.
[199] A. P. Isaev and O. V. Ogievetsky, Braids, shuffles and symmetrizers, J. Phys. A: Math. Theor.42 (2009) 304017.arXiv:0812.3974 [math.QA].
[200] V. F. R. Jones, Baxterization, Int. J. Mod. Phys. B 4 (5) (1990) 701.
[201] V. F. R. Jones, Hecke algebra representations of braid groups and link polynomials, Ann. Math.126 (1987) 335.
[202] G. E. Murphy, On the representation theory of the symmetric groups and associated Heckealgebras, J. Algebra 152 (1992) 287.
[203] H. Wenzl, Hecke algebras of typeAnand subfactors, Invent. Math. 92 (1988) 349.
[204] I. V. Cherednik, A new interpretation of Gelfand–Tzetlin bases, Duke Math. J. 54 (2) (1987)563.
[205] O. V. Ogievetsky and P. N. Pyatov, Lecture on Hecke Algebras, Preprint MPIM (Bonn), MPI2001-40.http://www.mpim-bonn.mpg.de/html/preprints/preprints.html.
[206] P. Martin, Potts Models and Related Problems in Statistical Mechanics, World Scientific, Sin-gapore, 1991.
[207] V. Chari and A. Pressley, A Guide to Quantum Groups, Cambrige University Press, 1994.
[208] A. P. Isaev and O. V. Ogievetsky, On representations of Hecke algebras, Czech. J. Phys. 55 (11)(2005) 1433.
[209] A. P. Isaev and O. V. Ogievetsky, Representations ofA-type Hecke algebras, in: Proceedingsof International Workshop “Supersymmetries and Quantum Symmetries”, Dubna, 2006.arXiv:0912.3701[math.QA].
[210] D. Levy, Algebraic structure of translation-invariant spin-1/2XXZand q-Potts quantum chains,Phys. Rev. Lett. 67 (15) (1991) 1971–1974.
[211] A. P. Isaev and O. V. Ogievetsky, On Baxterized solutions of reflection equation and integrablechain models, Nucl. Phys. B 760 (2007) 167.math-ph/0510078.
[212] A. P. Isaev, Functional equations for transfer-matrix operators in open Hecke chain models,Theor. Math. Phys. 150 (2) (2007) 187.arXive:1003.3385 [math-ph].
[213] A. P. Isaev, O. V. Ogievetsky, and A. F. Os’kin, Chain models on Hecke algebra for corner typerepresentations, Rep. Math. Phys. 61 (2) (2008) 309.arXiv: 0710.0261[math.QA].
[214] C. Burdik, J. Fuksa, A. P. Isaev, S. O. Krivonos, and O. Navratil, Remarks towards the spectrumof the Heisenberg spin chain type models, Phys. Part. Nucl. 46 (3) (2015) 277.arXiv: 1412.3999[math-ph].
[215] Y. K. Zhou, Row transfer matrix functional relations for Baxter’s eight-vertex and six-vertexmodels with open boundaries via more general reflection matrices, Nucl. Phys. B 458 (1996) 504.hep-th/9510095(1995).
[216] W. L. Yang, R. I. Nepomechie, and Y. Z. Zhang, Q-operator and T–Q relation from the fusionhierarchy, Phys. Lett. B 633 (2006) 664.hep-th/0511134(2005).
[217] D. Levy and P. Martin, Hecke algebra solutions to the reflection equation, J. Phys. A 27 (1994)L521.
[218] P. Martin, D. Woodcock, and D. Levy, A diagrammatic approach to Hecke algebras of thereflection equation, J. Phys. A 33 (2000) 1265.
[219] A. Doikou and P. Martin, Hecke algebraic approach to the reflection equation for spin chains, J.Phys. A 36 (2003) 2203.
[220] P. P. Kulish and A. I. Mudrov, Baxterization of solutions to reflection equation with HeckeR-matrix, Lett. Math. Phys. 75 (2) (2006) 151–170.arXive:math/0508289 [math.QA].
[221] M. Rosso and V. F. R. Jones, On the invariants of torus knots derived from quantum groups, J.Knot Theory Ramifications 2 (1993) 97.
[222] A. Mironov, R. Mkrtchyan, and A. Morozov, On universal knot polynomials, J. High EnergyPhys. 2016 (2) (2016) 1.arXiv:1510.05884.
[223] A. Morozov, Differential expansion and rectangular HOMFLY for the figure eight knot, Nucl.Phys. B 911 (2016) 582.arXiv:1605.09728 [hep-th].
[224] Ya. Kononov and A. Morozov, On rectangular HOMFLY for twist knots, Mod. Phys. Lett. A 31(38) (2016) 1650223.arXiv:1610.04778 [hep-th].
[225] H. Itoyama, A. Mironov, A. Morozov, and An. Morozov, Character expansion for HOMFLYpolynomials. III. All 3-strand braids in the first symmetric representation, Int. J. Mod. Phys. A27 (2012) 1250099.arXiv:1204.4785 [hep-th].
[226] P. Dunin-Barkowski, A. Mironov, A. Morozov, A. Sleptsov, and A. Smirnov, Superpolynomialsfor toric knots from evolution induced by cut-and-join operators, J. High Energy Phys. 03 (2013)021.arXiv:1106.4305.
[227] A. Anokhina and A. Morozov, TowardsR-matrix construction of Khovanov–Rozansky poly-nomials. I. Primary T-deformation of HOMFLY, J. High Energy Phys. 07 (2014) 063.arXiv:1403.8087 [hep-th].
[228] J. S. Birman and H. Wenzl, Braids, link polynomials and new algebra, Trans. Amer. Math. Soc.313 (1) (1989) 249.
[229] H. Wenzl, Quantum Groups and Subfactors of Type B, C and D, Comm. Math. Phys. 133 (1990)383.
[230] A. P. Isaev, A. I. Molev, and A. F. Os’kin, On the idempotents of Hecke algebras, Lett. Math.Phys. 85 (2008) 79.arXiv:0804.4214 [math.QA].
[231] A. Okounkov and A. Vershik, A new approach to representation theory of symmetric groups,Selecta Math. New Ser. 2 (4) (1996) 581–605.
[232] L. K. Hadjiivanov, A. P. Isaev, O. V. Ogievetsky, P. N. Pyatov, and I. T. Todorov, Hecke algebraicproperties of dynamicalR-matrices: Application to related quantum matrix algebras, J. Math.Phys. 40 (1999) 427.arXiv:q-alg/9712026.
[233] J.-L. Gervais and A. Neveu, Novel triangle relation and absence of tachyons in Liouville stringfield theory, Nucl. Phys. B 238 (1984) 125.
[234] G. Felder, Elliptic quantum groups, in: Proceedings of the International Congress of Mathemat-ical Physics, 1994.arXiv:hep-th/9412207.
[235] A. G. Bytsko and L. D. Faddeev, (TB)q, q-analog of model space and the Clebsch–Gordancoefficients generating matrices, J. Math. Phys. 37 (1996) 6324.arXiv:q-alg/9508022.
[236] P. Etingof and A. Varchenko, Solutions of the quantum dynamical Yang–Baxter equation anddynamical quantum groups, Comm. Math. Phys. 196 (3) (1998) 591.arXiv:q-alg/9708015.
[237] Y. Cheng, M. L. Ge, and K. Xue, Yang–Baxterization of braid group representations, Comm.Math. Phys. 136 (1) (1991) 195–208.
[238] V. F. R. Jones, On a certain value of the Kauffman polynomial, Comm. Math. Phys. 125 (1989)459.
[239] J. Murakami, Solvable lattice models and algebras of face operators, Adv. Stud. Pure Math. 19(1989) 399.
[240] I. Heckenberger and A. Sch ̈uler, Symmetrizer and antisymmetrizer of the Birman–Wenzl–Murakami algebras, Lett. Mat. Phys. 50 (1999) 45.math.QA/0002170.
[241] I. Tuba and H. Wenzl, On braided tensor categories of type BCD, Preprint, 2003.math.QA/0301142.
[242] A. P. Isaev and O. V. Ogievetsky, Jucys–Murphy elements for Birman–Murakami–Wenzl alge-bras, Phys. Part. Nucl. Lett. 8 (3) (2011) 234.arXiv:0912.4010 [math.QA].
[243] A. P. Isaev, A. I. Molev, and O. V. Ogievetsky, Idempotents for Birman–Murakami–Wenzl al-gebras and reflection equation, Adv. Theor. Math. Phys. 18 (1) (2014) 1–25.arXiv:1111.2502[math.RT].
[244] J. Murakami, The representations of the q-analogue of Brauer’s centralizer algebras and theKauffman polynomial of links, Publ. RIMS (Kyoto University) 26 (1990) 935.
[245] M. Nazarov, Young’s orthogonal form for Brauer’s centralizer algebra, J. Algebra 182 (1996) 664.
[246] A. Beliakova and Ch. Blanchet, Skein construction of idempotents in Birman–Murakami–Wenzlalgebras, Math. Ann. 321 (2) (2001) 347.arXiv: math.QA/0006143.
[247] A. Ram and H. Wenzl, Matrix units for centralizer algebras, J. Algebra 145 (1992) 378.
[248] R. Leduc and A. Ram, A ribbon Hopf algebra approach to the irreducible representations ofcentralizer algebras: The Brauer, Birman–Wenzl, and typeAIwahori–Hecke algebras, Adv.Math. 125 (AI971602) (1997) 1–94.
[249] V. A. Rubakov, Adler–Bell–Jackiw anomaly and fermion number breaking in the presence of amagnetic monopole, Nucl. Phys. B 203 (1982) 311.
[250] J. A. Minahan and K. Zarembo, The Bethe-ansatz forN= 4super Yang–Mills, J. High EnergyPhys. 0303 (2003) 013.hep-th/0212208.
[251] N. Beisert and M. Staudacher, TheN= 4SYM integrable super spin chain, Nucl. Phys. B 670(2003) 439.hep-th/0307042.
[252] L. N. Lipatov, High-energy asymptotics of multicolor QCD and exactly solvable lattice models,JETP Lett. 59 (1994) 596 [Pisma Zh. Eksp. Teor. Fiz. 59 (1994) 571].hep-th/9311037.
[253] L. N. Lipatov, High-energy asymptotics of multicolor QCD and two-dimensional conformal fieldtheories, Phys. Lett. B 309 (1993) 394.
[254] L. D. Faddeev and G. P. Korchemsky, High-energy QCD as a completely integrable model, Phys.Lett. B 342 (1995) 311.hep-th/9404173.
[255] A. V. Belitsky, A. S. Gorsky, and G. P. Korchemsky, Gauge/string duality for QCD conformaloperators, Nucl. Phys. B 667 (2003) 3.hep-th/0304028.
[256] L. D. Faddeev, How Algebraic Bethe Ansatz Works for Integrable Model, Les-Houches Lectures.hep-th/9605187.
[257] A. A. Vladimirov, Introduction to Quantum Integrable Systems. TheR-Matrix Method [in Rus-sian] (Lectures for Young Scientists, Vol. 32), R17-85-742, JINR, Dubna, 1985.
[258] N. Slavnov, Algebraic Bethe Ansatz and Correlation Functions. An Advanced Course, WorldScientific, 2022.
[259] P. P. Kulish and N. Yu. Reshetikhin, Diagonalization ofGl(N)invariant transfer matrices andquantumNwave system (Lee model), J. Phys. A 16 (1983) L591.
[260] M. L ̈usher, Dynamical charges in the quantized renormalized massive Thirring model, Nucl.Phys. B 117 (1976) 475.
[261] I. V. Cherednik, Factorizing particles on a half line and root systems, Theor. Math. Phys. 61(1984) 977 [Teor. Mat. Fiz. 61 (1984) 35].
[262] S. Ghoshal and A. B. Zamolodchikov, BoundarySmatrix and boundary state in two-dimensionalintegrable quantum field theory, Int. J. Mod. Phys. A 9 (1994) 3841.hep-th/9306002.
[263] E. K. Sklyanin, Boundary conditions for integrable quantum systems, J. Phys. A 21 (1988) 2375.
[264] P. P. Kulish and E. K. Sklyanin, Algebraic structures related to the reflection equations, J. Phys.A 25 (1992) 5963.
[265] P. P. Kulish and R. Sasaki, Covariance properties of reflection equation algebras, Prog. Theor.Phys. 89 (1993) 741.
[266] P. P. Kulish, Yang–Baxter equation and reflection equations in integrable models, in: ProceedingsSchladming School, Lect. Notes Phys. 469 (1996) 125–144.hep-th/9507070.
[267] N. J. MacKay and B. J. Short, Boundary scattering, symmetric spaces and the principal chiralmodel on the half-line, Comm. Math. Phys. 233 (2003) 313.hep-th/0104212.
[268] A. Liguori, M. Mintchev, and L. Zhao, Boundary exchange algebras and scattering on the halfline, Comm. Math. Phys. 194 (1998) 569.hep-th/9607085.
[269] L. Mezincescu, R. I. Nepomechie, and V. Rittenberg, Bethe ansatz solution of the Fateev–Zamolodchikov quantum spin chain with boundary terms, Phys. Lett. A 147 (1990) 70.
[270] L. Mezincescu, R. I. Nepomechie, Integrable open spin chains with nonsymmetricRmatrices, J.Phys. A 24 (1991) L17.
[271] L. Mezincescu, R. I. Nepomechie, Integrability of open spin chains with quantum algebra sym-metry, Int. J. Mod. Phys. A 6 (1991) 5231 [Addendum: ibid. A 7 (1992) 5657].hep-th/9206047.
[272] A. B. Zamolodchikov, Tetrahedra equations and integrable systems in three-dimensional space,Sov. Phys. JETP 52 (1980) 325 [Zh. Eksp. Teor. Fiz. 79 (1980) 641].
[273] A. B. Zamolodchikov, Tetrahedron equations and the relativisticSmatrix of straight strings in(2+1)-dimensions, Comm. Math. Phys. 79 (1981) 489.
[274] V. V. Bazhanov and R. J. Baxter, New solvable lattice models in three-dimensions, J. Stat. Phys.69 (1992) 453.
[275] V. V. Bazhanov and R. J. Baxter, Star triangle relation for a three-dimensional model, J. Stat.Phys. 71 (1993) 839.hep-th/9212050.
[276] S. M. Sergeev, Complex of three dimensional integrable models, J. Phys. A: Math. Gen. 34 (2001)10493.
[277] V. V. Bazhanov and S. M. Sergeev, Zamolodchikov’s tetrahedron equation and hidden structureof quantum groups, J. Phys. A 39 (2006) 3295.hep-th/0509181.
[278] V. V. Bazhanov, V. V. Mangazeev, and S. M. Sergeev, Quantum geometry of 3-dimensionallattices, J. Stat. Mech. 0807 (2008) P07004.arXiv:0801.0129 [hep-th].
[279] A. P. Isaev and P. P. Kulish, Tetrahedron reflection equation, Mod. Phys. Lett. A 12 (1997) 427.hep-th/9702013.
[280] A. Kuniba and M. Okado, Tetrahedron and 3D reflection equations from quantized algebra offunctions, J. Phys. A 45 (2012) 465206.arXiv: 1208.1586 [math-ph].
[281] A. Kuniba and V. Pasquier, Matrix product solutions to the reflection equation from three dimen-sional integrability, J. Phys. A: Math. Theor. 51 (2018) 255204.arXiv:1802.09164 [math-ph].
[282] A. Yoneyama, Tetrahedron and 3D reflection equation from PBW bases of the nilpotent subal-gebra of quantum superalgebras, Comm. Math. Phys. 387 (1) (2021) 481.
[283] Yu. I. Manin and V. V. Schechtman, Arrangements of hyperplanes, higher braid groups andhigher Bruhat orders, Adv. Stud. Pure Math. 17 (1990) 289.
[284] M. Kapranov and V. Voevodsky, Braided monoidal 2-categories and Manin–Schechtman higherbraid groups, J. Pure Appl. Algebra 92 (1994) 241.
[285] R. J. Lawrence, Algebras and triangle relations, J. Pure Appl. Algebra, 100 (1–3) (1995) 43.
[286] M. Kapranov and V. Voevodsky, Freen-categories generated by a cube, oriented matroids, andhigher Bruhat orders, Funct. Anal. Appl. 25 (1990) 50.
[287] J. C. Baez, An Introduction ton-Categories, in: International Conference on Category Theoryand Computer Science, Springer, Berlin, Heidelberg, 1997, pp. 1–33.q-alg/9705009.
[288] J. C. Baez and M. Shulman, Lectures onn-Categories and Cohomology. Towards Higher Cate-gories. Springer, New York, 2010, p. 1–68.arXiv: math/0608420 [math.CT].
[289] V. V. Bazhanov and Yu. G. Stroganov, On commutativity conditions for transfer matrices onmultidimensional lattice, Teor. Mat. Fiz. 52 (1982) 105.
[290] A. B. Zamolodchikov, Fishnet diagrams as a completely integrable system, Phys. Lett. B 97(1980) 63.
[291] D. I. Kazakov, The method of uniqueness, a new powerful technique for multiloop calculations,Phys. Lett. B 133 (1983) 406.
[292] V. V. Belokurov and N. I. Ussyukina, Calculation of ladder diagrams in arbitrary order, J. Phys.A 16 (1983) 2811.
[293] D. J. Broadhurst, Summation of an infinite series of ladder diagrams, Phys. Lett. B 307 (1993)132.
[294] N. I. Ussyukina and A. I. Davydychev, Exact results for three- and four-point ladder diagramswith an arbitrary number of rungs, Phys. Lett. B 305 (1993) 136.
[295] S. G. Gorishnii and A. P. Isaev, On an approach to the calculation of multiloop massless Feynmanintegrals, Theor. Math. Phys. 62 (1985) 232 [Teor. Mat. Fiz. 62 (1985) 345].
[296] P. A. Baikov and K. G. Chetyrkin, Four loop massless propagators: An algebraic evaluation ofall master integrals, Nucl. Phys. B 837 (2010) 186.arXiv:1004.1153.
[297] A. P. Isaev, Multi-loop Feynman integrals and conformal quantum mechanics, Nucl. Phys. B 662(2003) 461.hep-th/0303056.
[298] A. P. Isaev, Operator approach to analytical evaluation of Feynman diagrams, Phys. At. Nucl.71 (2008) 914.arXiv:0709.0419[hep-th].
[299] D. Chicherin, S. Derkachov, and A. P. Isaev, Conformal group:R-matrix and star-trianglerelation, J. High Energy Phys. 04 (2013) 020.arXiv:1206.4150 [math-ph].
[300] S. E. Derkachov, G. P. Korchemsky, and A. N. Manashov, Noncompact Heisenberg spin magnetsfrom high-energy QCD: I. Baxter Q-operator and separation of variables, Nucl. Phys. B 617(2001) 375-440.arXiv:hep-th/0107193.
[301] S. E. Derkachov and A. N. Manashov, Factorization ofR-matrix and Baxter Q-operators forgenericsl(N)spin chains, J. Phys. A 42 (2009) 075204.arXiv:0809.2050 [nlin.SI].
[302] D. T. Barfoot and D. J. Broadhurst,Z(2)×S(6)symmetry of the two loop diagram, Z. Phys. C41 (1988) 81.
[303] G. t’Hooft, Dimensional regularization and the renormalization group, Nucl. Phys. B 61 (1973)455.
[304] R. M. Kashaev, On discrete three-dimensional equations associated with the local Yang–Baxterrelation, Lett. Math. Phys. 38 (1996) 389.
[305] J. M. Maillet and F. Nijhoff, Integrability for multidimensional lattice models, Phys. Lett. B 224(1989) 389.
[306] E. S. Fradkin and M. Ya. Palchik, Recent developments in conformal invariant quantum fieldtheory, Phys. Rep. 44 (5) (1978) 249–349.
[307] D. Chicherin, S. Derkachov, and A. P. Isaev, The spinorialR-matrix, J. Phys. A 46 (2013)485201.arXiv:1303.4929 [math-ph].
[308] I. Mitra, On conformal invariant integrals involving spin one-half and spin-one particles, J. Phys.A 41 (2008) 315401.arXiv:0803.2630 [hep-th].
[309] V. Kazakov and E. Olivucci, Biscalar integrable conformal field theories in any dimension, Phys.Rev. Lett. 121 (2018) 131601.arXiv:1801.09844.
[310] B. Basso and L. J. Dixon, Gluing ladder Feynman diagrams into fishnets, Phys. Rev. Lett. 119(2017) 071601.arXiv:1705.03545.
[311] N. Gromov, V. Kazakov, and G. Korchemsky, Exact correlation functions in conformal fishnettheory, J. High Energy Phys. 08 (2019) 123.arXiv:1808.02688.
[312] S. Derkachov, E. Olivucci, Conformal quantum mechanics & the integrable spinning fishnet, J.High Energy Phys. 11 (2021) 060.e-Print: 2103.01940 [hep-th].
[313] S. Derkachov, G. Ferrando, and E. Olivucci, Mirror channel eigenvectors of thed-dimensionalfishnets, J. High Energy Phys. 12 (2021) 174.arXiv:2108.12620 [hep-th].
[314] S. E. Derkachov, A. P. Isaev, and L. A. Shumilov, Ladder and zig-zag Feynman diagrams, oper-ator formalism and conformal triangles, J. High Energy Phys. 06 (2023) 059.arXiv:2302.11238[hep-th].
[315] R. M. Iakhibbaev, Regge limit of correlation function in 6d biscalar fishnet models, Phys. Part.Nucl. Lett. 21 (4) (2024) 587–589.
[316] M. Alfimov, G. Ferrando, V. Kazakov, and E. Olivucci, Checkerboard CFT, J. High EnergyPhys. 01 (2025) 015.arXiv:2311.01437 [hep-th].
[317] E. Witten, Gauge theories, vertex models and quantum groups, Nucl. Phys. B 330 (2–3) (1990)285.
[318] L. Alvarez-Gaume, C. Gomez, and G. Sierra, Duality and quantum groups, Nucl. Phys. B 330(2–3) (1990) 347.
[319] V. V. Bazhanov, R. M. Kashaev, V. V. Mangazeev, and Yu. G. Stroganov,(ZN×)n−1general-ization of the chiral Potts model, Comm. Math. Phys. 138 (1991) 393.
[320] E. Date, M. Jimbo, K. Miki, and T. Miwa, Generalized chiral Potts models and minimal cyclicrepresentations ofUq(ˆgl(n,C)), Comm. Math. Phys. 137 (1991) 133.
[321] P. Furlan, A. Ganchev, and V. P. Petkova, Quantum groups and fusion rules multiplicities, Nucl.Phys. B 343 (1990) 205.
[322] A. T. Filippov, A. P. Isaev, and A. B. Kurdikov, Paragrassmann analysis and quantum groups,Mod. Phys. Lett. A 7 (1992) 2129.hep-th/9204089.
[323] P. Furlan, L. K. Hadjiivanov, A. P. Isaev, O. V. Ogievetsky, P. N. Pyatov, I. T. Todorov, Quantummatrix algebra for theSU(n)WZNW model, J. Phys. A 36 (2003) 5497.hep-th/0003210.
[324] V. Chari, A. Pressley, Quantum affine algebras at roots of unity, Represent. Theor. Amer. Math.Soc. 1 (12) (1997) 280.
[325] B. L. Feigin, A. M. Gainutdinov, A. M. Semikhatov, I. Yu. Tipunin, Modular group representa-tions and fusion in logarithmic conformal field theories and in the quantum group center, Comm.Math. Phys. 265 (2006) 47.hep-th/0504093.
[326] B. L. Feigin, A. M. Gainutdinov, A. M. Semikhatov, I. Y. Tipunin, Kazhdan–Lusztig-dual quan-tum group for logarithimic extensions of Virasoro minimal models, J. Math. Phys. 48 (3) (2007).
[2] C. N. Yang, Some exact results for the many-body problem in one dimension with repulsivedelta-function interaction, Phys. Rev. Lett. 19 (1967) 1312–1315.
[3] R. Baxter, Partition function of the eight-vertex lattice model, Ann. Phys. (N. Y.) 70 (1972)193–228 [reprinted in Ann. Phys. 281 (2000) 187].
[4] A. B. Zamolodchikov and Al. B. Zamolodchikov, Relativistic factorizedSmatrix in two-dimensions havingO(N)isotopic symmetry, Nucl. Phys. B 133 (1978) 525.
[5] A. B. Zamolodchikov and Al. B. Zamolodchikov, FactorizedS-matrices in two dimensions as theexact solutions of certain relativistic quantum field models, Ann. Phys. (N. Y.) 120 (1979) 253.
[6] E. K. Sklyanin and L. D. Faddeev, Quantum mechanical approach to completely integrable fieldtheory models, Sov. Phys. Dokl. 23 (1978) 902–904 [reprinted in Fifty Years of MathematicalPhysics: Selected Works of Ludwig Faddeev, 2016, pp. 290–292].
[7] E. K. Sklyanin, L. A. Takhtajan, and L. D. Faddeev, The quantum inverse problem method. 1,Theor. Math. Phys. 40 (1980) 688 [Teor. Mat. Fiz. 40 (1979) 194].
[8] L. A. Takhtajan and L. D. Faddeev, The quantum method of the inverse problem and theHeisenbergXY Zmodel, Usp. Mat. Nauk 34 (5) (1979) 11.
[9] L. D. Faddeev, Quantum completely integrable models in field theory, Math. Phys. Rev. Vol. 1,Soviet Sci. Rev. Sect. C: Math. Phys. Rev., Vol. 1, Harwood Academic, Chur, 1980, pp. 107–155 [L. D. Faddeev, 40 Years in Mathematical Physics, World Scientific Series in 20th CenturyMathematics, Vol. 2, 1995, pp. 187–235].
[10] V. G. Drinfeld, Quantum Groups, in: Proceedings of the International Congress of Mathematics,Vol. 1, Berkeley, 1986, p. 798.
[11] L. A. Takhtajan, A. Yu. Alekseev, I. Ya. Aref’eva, M. A. Semenov-Tian-Shansky, E. K. Sklyanin,F. A. Smirnov, S. L. Shatashvili, Scientific heritage of L. D. Faddeev. Survey of papers, Russ.Math. Surv., 72:6 (2017) 977–1081 [Usp. Mat. Nauk, 72:6 (438) (2017) 3–112].
[12] V. Pasquier and H. Saleur, Common structures between finite systems and conformal field the-ories through quantum groups, Nucl. Phys. B 330 (1990) 523.
[13] M. Karowski and A. Zapletal, Quantum group invariant integrableNstate vertex models withperiodic boundary conditions, Nucl. Phys. B 419 (1994) 567.
[14] P. P. Kulish, Quantum groups and quantum algebras as symmetries of dynamical systems, in:Quantum Symmetries, Proceedings of the International Workshop on Mathematical Physics,ASI, Clausthal, Germany, 15–20 July 1991, Eds: H.-D. Doebner and V. K. Dobrev, 1993.
[15] A. Connes, Geometrie Non Commutative, Intereditions, Paris, 1990.
[16] A. Connes and J. Lott, Particle models and noncommutative geometry (expanded version), Nucl.Phys. B Proc. Suppl. 18 (1991) 29.
[17] X. Calmet, B. Jurco, P. Schupp, J. Wess, and M. Wohlgenannt, The standard model on non-commutative space–time, Eur. Phys. J. C 23 (2002) 363.hep-ph/0111115.
[18] M. Kontsevich, Deformation quantization of Poisson manifolds, I, Lett. Math. Phys. 66 (2003)157.q-alg/9709040.
[19] A. Connes, M. R. Douglas, and A. Schwarz, Noncommutative geometry and matrix theory, J.High Energy Phys. 9802 (1998) 003.hep-th/9711162
[20] N. Seiberg and E. Witten, String theory and noncommutative geometry, J. High Energy Phys.9909 (1999) 032.hep-th/9908142.
[21] A. P. Isaev and Z. Popowicz, q-Trace for the quantum groups and q-deformed Yang–Mills theory,Phys. Lett. B 281 (1992) 271.
[22] A. M. Gavrilik, Quantum algebras in phenomenological description of particle properties, Nucl.Phys. Proc. Suppl. 102 (2001) 298.hep-ph/0103325; Quantum groups and Cabibbo mixing, in:Proceedings of 5th International Conference on Symmetry in Nonlinear Mathematical Physics(SYMMETRY 03), Kiev, Ukraine, 23–29 June 2003, pp. 759–766.hep-ph/0401086.
[23] O. Ogievetsky, W. B. Schmidke, J. Wess, and B. Zumino, q-Deformed Poincare algebra, Comm.Math. Phys. 150 (1992) 495.
[24] P. P. Kulish, Representations of Q-Minkowski space algebra, Algebra Anal. 6 (2) (1994) 195[English translation: St. Petersburg Math. J., 6 (2) (1995) 365].hep-th/9312139.
[25] J. A. De Azcarraga, P. P. Kulish, and F. Rodenas, Reflection equations and q Minkowski spacealgebras, Lett. Math. Phys. 32 (1994) 173.hep-th/9309036; J. A. De Azcarraga, P. P. Kulish,and F. Rodenas, On the physical contents of q-deformed Minkowski spaces, Phys. Lett. B 351(1–3) (1995) 123.arXiv:hep-th/9411121.
[26] W. Schmidke, J. Wess, and B. Zumino, A q-deformed Lorentz algebra, Z. Phys. C 52 (1991) 471.
[27] P. Podles and S. L. Woronowicz, Quantum deformation of Lorentz group, Comm. Math. Phys.130 (1990) 381.
[28] U. Carow-Watamura, M. Schlieker, M. Scholl, and S. Watamura, Tensor representation of thequantum groupSLq(2,C)and quantum Minkowski space, Z. Phys. C 48 (1990) 159; U. Carow-Watamura, et al., Quantum Lorentz group, Int. J. Mod. Phys. A 6 (1991) 3081.
[29] Sh. Majid, Braided momentum in the q-Poincar ́e group, J. Math. Phys. 34 (1993) 2045.
[30] J. Lukierski, A. Nowicki, H. Ruegg, and V. N. Tolstoy, Q deformation of Poincare algebra, Phys.Lett. B 264 (1991) 331; J. Lukierski, A. Nowicki, and H. Ruegg, New quantum Poincare algebraandκdeformed field theory, Phys. Lett. B 293 (1992) 344.
[31] B. L. Cerchiai and J. Wess, q-deformed Minkowski space based on a q-Lorentz algebra, Eur.Phys. J. C 5 (1998) 553.math.qa/9801104.
[32] L. Dolan, C. R. Nappi, and E. Witten, Yangian symmetry inD= 4superconformal Yang–Millstheory.hep-th/0401243.
[33] J. Drummond, J. Henn, J. Plefka, Yangian symmetry of scattering amplitudes inN= 4superYang–Mills theory, J. High Energy Phys. 05 (2009) 046.
[34] N. Beisert, On Yangian Symmetry in PlanarN= 4SYM.arxiv:1004.5423.
[35] T. Bargheer, N. Beisert, and F. Loebbert, Exact superconformal and Yangian symmetry ofscattering amplitudes, J. Phys. A 44 (2011) 454012.arxiv:1104.0700.
[36] D. Chicherin and R. Kirschner, Yangian symmetric correlators, Nucl. Phys. B 877 (2013) 484.arXiv:1306.0711 [math-ph].
[37] D. Chicherin, S. Derkachov, and R. Kirschner, Yang–Baxter operators and scattering ampli-tudes inN= 4super-Yang–Mills theory, Nucl. Phys. B 881 (2014) 467–501.arXiv:1309.5748[hep-th].
[38] A. Brandhuber, P. Heslop, G. Travaglini, and D. Young, Yangian Symmetry of Scattering Am-plitudes and the dilatation operator inN= 4supersymmetric Yang–Mills theory, Phys. Rev.Lett. 115 (2015) 141602.arxiv:1507.01504.
[39] D. Chicherin, V. Kazakov, F. Loebbert, D. M ̈uller, D. Zhong, Yangian symmetry for bi-scalarloop amplitudes, J. High Energy Phys. 05 (2018) 003.arXiv:1704.01967.
[40] V. Kazakov, F. Levkovich-Maslyuk, and V. Mishnyakov, Integrable Feynman Graphs and Yan-gian Symmetry on the Loom.arXiv:2304.04654v1 [hep-th].
[41] F. Loebbert, Lectures on Yangian symmetry, J. Phys. A 49 (2016) 323002.arxiv:1606.02947.
[42] L. D. Faddeev, N. Yu. Reshetikhin, and L. A. Takhtajan, Quantization of Lie groups and Liealgebras, Lengingrad Math. J. 1 (1990) 193 [Algebra Anal. 1 (1989) 178].
[43] E. K. Sklyanin, Some algebraic structures connected with the Yang–Baxter equation, Funct.Anal. Appl. 16 (1982) 263 [Funkt. Anal. Pril. 16 (4) (1982) 27].
[44] E. K. Sklyanin, Some algebraic structures connected with the Yang–Baxter equation. Represen-tations of quantum algebras, Funct. Anal. Appl. 17 (1983) 273 [Funkt. Anal. Pril. 17 (4) (1983)34].
[45] A. V. Odesskii, Elliptic algebras, Russ. Math. Surv. 75 (2002) 1127.math.QA/0303021.
[46] A. P. Isaev, Quantum groups and Yang–Baxter equations, Sov. J. Part. Nucl. 26 (1995) 501 [Fiz.Elem. Chastits i At. Yadra 26 (1995) 1204].
[47] A. P. Isaev, Quantum groups and Yang–Baxter equations, Preprint MPIM (Bonn), MPI 2004-132.http://webdoc.sub.gwdg.de/ebook/serien/e/mpi_mathematik/2004/132.pdf.
[48] A. P. Isaev, Lectures in Quantum Groups and Yang–Baxter Equations.arXiv:2206.08902[math.QA].
[49] E. Sweedler, Hopf Algebras, Benjamin, 1969.
[50] E. Abe, Hopf Algebras, Cambridge University Press, 1977.
[51] V. G. Drinfeld, Almost cocommutative Hopf algebras, Algebra Anal. 1 (2) (1989) 30.
[52] Sh. Majid, Quasitriangular Hopf algebras and Yang–Baxter equations, Int. J. Mod. Phys. A 5(1990) 1.
[53] A. A. Vladimirov, On the Hopf algebras generated by the Yang–BaxterRmatrices, Z. Phys. C58 (1993) 659.
[54] J.-P. Serre, Linear Representations of Finite Groups, Graduate Texts in Mathematics 42,Springer-Verlag, 1977.
[55] R. M. Kashaev, Heisenberg double and pentagon relation, Algebra Anal. 8 (4) (1996) 63.q-alg/9503005.
[56] S. Baaj, G. Skandalis, Unitaires multiplicatifs et dualit ́e pour les produits crois ́es deC∗-alg ́ebres,Annales scientifiques de l’Ecole normale sup ́erieure 26 (4) (1993) 425.
[57] N. Reshetikhin, Multiparameter quantum groups and twisted quasitriangular Hopf algebras,Lett. Math. Phys. 20 (1990) 331.
[58] N. Yu. Reshetikhin, V. G. Turaev, Ribbon graphs and their invariants derived from quantumgroups, Comm. Math. Phys. 127 (1990) 1–26.
[59] C. Kassel, M. Rosso, and V. Turaev, Quantum Groups and Knot Invariants, Panoramas etSynth`eses 5, Soci ́et ́e Math ́ematique de France, 1997.
[60] V. G. Drinfeld, Quasi-Hopf algebras, Algebra Anal. 1 (6) (1989) 114.
[61] V. G. Drinfeld, On quasitriangular quasi-Hopf algebras and a group related to Gal(Q/Q), AlgebraAnal. 2 (4) (1990) 149.
[62] M. Enock and J.-M. Schwartz, Kac Algebras and Duality of Locally Compact Groups, Springer-Verlag, 1992.
[63] L. Pontryagin, Topological Groups, Princeton University Press, 1946.
[64] P. Etingof, T. Schedler, and O. Schiffmann, Explicit quantization for dynamicalr-matrices forfinite dimensional semisimple Lie algebras, J. Amer. Math. Soc. 13 (2000) 595.math.QA/9912009.
[65] A. P. Isaev and O. V. Ogievetsky, On quantization ofr-matrices for Belavin–Drinfeld triples,Phys. At. Nucl. 64 (12) (2001) 2126.math.QA/0010190.
[66] P. P. Kulish, V. D. Lyakhovsky, and A. I. Mudrov, Extended Jordanian twists for Lie algebras,J. Math. Phys. 40 (1999) 4569.math.QA/9806014.
[67] N. Yu. Reshetikhin, Quasitriangular Hopf algebras and link invariants, Algebra Anal. 1 (2) (1989)169.
[68] O. Ogievetsky, Uses of Quantum Spaces, Lectures presented at the School “Quantum Symmetriesin Theoretical Physics and Mathematics”, Bariloche, 2000, Contemporary Mathematics, 294(2002) 161–231.
[69] A. A. Vladimirov, A Method for obtaining quantum doubles from the Yang–BaxterRmatrices,Mod. Phys. Lett. A 8 (1993) 1315.hep-th/9302042.
[70] S. L. Woronowicz, Differential calculus on compact matrix pseudogroups (quantum groups)Comm. Math. Phys. 122 (1989) 125.
[71] P. Aschieri and L. Castellani, An introduction to noncommutative differential geometry on quan-tum groups, Int. J. Mod. Phys. A 8 (1993) 1667.hep-th/9207084.
[72] A. Sudbery, Canonical differential calculus on quantum general linear groups and supergroups,Phys. Lett. B 284 (1–2) (1992) 61.
[73] Yu. I. Manin, Quantum groups and noncommutative geometry, Preprint CRM-1561, MontrealUniversity, 1989.
[74] Yu. I. Manin, Multiparametric quantum deformation of the general linear supergroup, Comm.Math. Phys. 123 (1989) 163.
[75] Yu. I. Manin, Notes on quantum groups and quantum de Rham complexes, Theor. Math. Phys.92 (3) (1992) 997 [Teor. Mat. Fiz. 92 (3) (1992) 425].
[76] G. Maltsiniotis, The language of spaces and of quantum groups, Comm. Math. Phys. 151 (1993)275; Groupes quantiques et structures diff ́erentielles, C. R. Acad. Sci. 311 S ́er. I (1990) 831.
[77] B. Tsygan, Notes on differential forms on quantum groups, Sel. Math. 12 (1993) 75.
[78] J. Wess and B. Zumino, Covariant differential calculus on the quantum hyperplane, Nucl. Phys.B Proc. Suppl. 18 (1990) 302.
[79] P. Schupp, P. Watts, B. Zumino, Differential geometry on linear quantum groups, Lett. Math.Phys. 25 (2) (1992) 139.arXiv:hep-th/9206029.
[80] B. Zumino, Differential calculus on quantum spaces and quantum groups, in: Proceedings ofXIX International Conference on Group Theoretical Methods in Physics, Salamanca, Spain,1992.arXiv:hep-th/9212093.
[81] T. Brzezi ́nski, Remarks on bicovariant differential calculi and exterior Hopf algebras, Lett. Math.Phys. 27 (1993) 287.
[82] A. P. Isaev and Z. Popowicz, Quantum group gauge theories and covariant quantum algebras,Phys. Lett. B 307 (1993) 353.arXiv:hep-th/9302090v2.
[83] A. P. Isaev, Quantum group covariant noncommutative geometry, J. Math. Phys. 35 (1994) 6784.arXiv:hep-th/9402060v2.
[84] A. P. Isaev,R-matrix approach to differential calculus on quantum groups, Phys. Part. Nucl. 28(3) (1997) 267 [Fiz. Elem. Chastits i At. Yadra 28 (3) (1997) 685].
[85] A. P. Isaev and P. N. Pyatov,Glq(N)-covariant quantum algebras and covariant differentialcalculus, Phys. Lett. A 179 (1993) 81.arXiv:hep-th/9211093.
[86] A. P. Isaev and P. N. Pyatov, Covariant differential complexes on quantum linear groups, J.Phys. A 28 (1995) 2227.arXiv:hep-th/9311112.
[87] L. D. Faddeev and P. N. Pyatov, The differential calculus on quantum linear groups, Am. Math.Soc. Transl. 175 (1996) 35.arXiv:hep-th/9402070.
[88] A. P. Isaev, Interrelations between quantum groups and reflection equation (braided) algebras,Lett. Math. Phys. 34 (1995) 333.arXiv:hep-th/9403154.
[89] A. P. Isaev and O. V. Ogievetsky, BRST operator for quantum Lie algebras and differentialcalculus on quantum groups, Theor. Math. Phys. 129 (2) (2001) 298.math.QA/0106206.
[90] A. P. Isaev and O. V. Ogievetsky, BRST and anti-BRST operators for quantum linear algebraUq(gl(N)), Nucl. Phys. Proc. Suppl. 102 (2001) 306.
[91] P. Schupp, P. Watts, B. Zumino, Bicovariant quantum algebras and quantum Lie algebras,Comm. Math. Phys. 157 (1993) 305–329.hep-th/9210150.
[92] B. Zumino, Wu Yong-Shi, and A. Zee, Chiral anomalies, higher dimensions, and differentialgeometry, Nucl. Phys. B 239 (1984) 477.
[93] D. Gurevich, P. Pyatov, and P. Saponov, Braided differential operators on quantum algebras, J.Geom. Phys. 61 (2011) 1485.arXiv:1004.4721 [math.QA].
[94] J. M. Maillet, Lax equations and quantum groups, Phys. Lett. B 245 (1990) 480.[95] N. Yu. Reshetikhin, M. A. Semenov-Tian-Shansky, QuantumR-matrices and factorization prob-lems, Geom. Phys. 5 (1988) 533–550.
[96] A. Isaev, O. Ogievetsky, P. Pyatov, and P. Saponov, Characteristic polynomials for quantummatrices, Lect. Notes Phys. Vol. 524, Springer, 1999, p. 322.
[97] A. Isaev, O. Ogievetsky, and P. Pyatov, On quantum matrix algebras satisfying the Cayley–Hamilton–Newton identities, J. Phys. A: Math. Gen. 32 (1999) L115–L121.math.QA/9809170.
[98] A. Yu. Alekseev and L. D. Faddeev,(T∗G)(T): A toy model for conformal field theory, Comm.Math. Phys. 141 (1991) 413.
[99] A. P. Isaev, O. V. Ogievetsky, and P. N. Pyatov, Cayley–Hamilton–Newton identities and qu-asitriangular Hopf algebras, in: Proceedings of the Internatinal Seminar “Supersymmetries andQuantum Symmetries” (SQS’99), Dubna, 2000.arXiv: math.QA/9912197.
[100] L. Freidel and J. Maillet, Quadratic algebras and integrable systems, Phys. Lett. B 262 (1991)278; L. Hlavaty and A. Kundu, Int. J. Mod. Phys. A 11 (1996) 2143.hep-th/9406215.
[101] A. I. Molev, E. Ragoucy, and P. Sorba, Coideal subalgebras in quantum affine algebras, Rev.Math. Phys. 15 (8) (2003) 789.math.QA/0208140.
[102] A. P. Isaev, A. I. Molev, and O. V. Ogievetsky, A new fusion procedure for the Brauer algebraand evaluation homomorphisms, Int. Math. Res. Not. 11 (2012) 2571–2606.arXiv:1101.1336[math.RT].
[103] N. Yu. Reshetikhin, Integrable models of quantum one-dimensional magnets withO(N)andSp(2k)symmetry, Theor. Math. Phys. 63 (1985) 555 [Teor. Mat. Fiz. 63 (1985) 347].
[104] A. P. Isaev, D. Karakhanyan, and R. Kirschner, Orthogonal and symplectic Yangians and Yang–BaxterR-operators, Nucl. Phys. B 904 (2016) 124–147.arXiv:1511.06152 [math-ph].
[105] A. M. Gavrilik and A. U. Klimyk, q-Deformed orthogonal and pseudo-orthogonal algebras andtheir representations, Lett. Math. Phys. 21 (1991) 215.e-Print:math.qa/0203201.
[106] M. Noumi, Macdonald’s symmetric polynomials as zonal spherical functions on quantum homo-geneous spaces, Adv. Math. 123 (1996) 16.math.QA/9503224.
[107] A. Isaev, O. Ogievetsky, Half-quantum linear algebra, in: Proceedings of the XXIX InternationalColloquium on Group-Theoretical Methods in Physics, Tianjin, China, 20–26 Aug. 2012, WorldScientific, 2013, p. 479.arXiv:1303.3991 [math.QA].
[108] A. Chervov, G. Falqui, and V. Rubtsov, Algebraic properties of Manin matrices 1, Adv. Appl.Math. 43 (3) (2009) 239.
[109] A. P. Isaev, Twisted Yang–Baxter equations for linear quantum (super)groups, J. Phys. A 29(1996) 6903.q-alg/9511006.
[110] P. Etingof and A. Varchenko, Solutions of the quantum dynamical Yang–Baxter equation anddynamical quantum groups, Comm. Math. Phys. 196 (1998) 591.q-alg/9708015.
[111] D. I. Gurevich, Algebraic aspects of the quantum Yang–Baxter equation, Algebra Anal. 2 (1990)119–148 [English translation: Leningrad Math. J. 2 (1991) 801–828].
[112] J. Ding and I. Frenkel, Isomorphism of two realizations of quantum affine algebraUq(ˆgl(n)),Comm. Math. Phys. 156 (1993) 277.
[113] M. Jimbo, A q-analogue ofU(gl(N+ 1)), Hecke algebra and the Yang–Baxter equation, Lett.Math. Phys. 11 (1986) 247.
[114] M. Jimbo, A Q-difference analog ofU(G)and the Yang–Baxter equation, Lett. Math. Phys. 10(1985) 63.
[115] M. Rosso, Finite dimensional representations of the quantum analog of the enveloping algebraof a complex simple Lie algebra, Comm. Math. Phys. 117 (1988) 581.
[116] Ya. Soibelman, The algebra of functions on compact quantum group and its representations,Algebra Anal. 2 (1) (1990) 190 [English translation: Leningrad Math. J. 2 (1) (1991)].
[117] J. Donin, P. P. Kulish, and A. I. Mudrov, On universal solution to reflection equation, Lett.Math. Phys. 63 (2003) 179.math.QA/0210242.
[118] P. A. Saponov, Weyl approach to representation theory of reflection equation algebra, J. Phys.A 37 (2004) 5021.math.QA/0307024.
[119] A. I. Mudrov, Characters ofUq(gl(n))-reflection equation algebra, Lett. Math. Phys. 60 (2002)283.math.QA/0204296.
[120] D. Algethami, A. Mudrov, and V. Stukopin, Solutions to graded reflection equation ofGL-type,Lett. Math. Phys. 114 (1) (2024) 22.arXiv:2310.15667 [math.QA].
[121] M. A. Semenov-Tian-Shansky, Poisson Lie groups, quantum duality principle, and the quantumdouble, Contemp. Math. 175 (1994) 219; Theor. Math. Phys. 93 (1992) 1292–1307.
[122] M. Rosso, An analogue of PBW theorem and the universalRmatrix forUhsl(N+ 1), Comm.Math. Phys. 124 (1989) 307.
[123] A. N. Kirillov and N. Reshetikhin, q-Weyl group and a multiplicative formula for universalR-matrices, Comm. Math. Phys. 134 (1990) 421.
[124] A. Yu. Alekseev and L. D. Faddeev, An involution and dynamics for the q-deformed quantumtop, Zap. Nauchn. Semin. LOMI 200 (1992) 3.hep-th/9406196.
[125] A. P. Isaev, O. V. Ogievetsky, and P. N. Pyatov, Generalized Cayley–Hamilton–Newton identi-ties, Czech. J. Phys. 48 (1998) 1369–1374.math.QA/9809047.
[126] M. Nazarov and V. Tarasov, Yangians and Gelfand–Zetlin basis, Publ. RIMS 30 (1994) 459.hep-th/9302102.
[127] P. N. Pyatov and P. A. Saponov, Characteristic relations for quantum matrices, J. Phys. A:Math. Gen. 28 (1995) 4415.q-alg/9502012.
[128] D. I. Gurevich, P. N. Pyatov, and P. A. Saponov, Hecke symmetries and characteristic relationson reflection equation algebras, Lett. Math. Phys. 41 (1997) 255.q-alg/9605048.
[129] A. P. Isaev and P. N. Pyatov, Spectral extension of the quantum group cotangent bundle, Comm.Math. Phys. 288 (2009) 1137.arXiv:0812.2225[math.QA].
[130] D. I. Gurevich, P. N. Pyatov, and P. A. Saponov, Cayley–Hamilton theorem for quantum matrixalgebras ofGL(m|n)type, Algebra Anal. 17 (1) (2005) 160 [St. Petersburg Math. J. 17 (1) (2006)119].math.QA/0412192.
[131] O. Ogievetsky and P. Pyatov, Orthogonal and symplectic quantum matrix algebras and Cayley–Hamilton theorem for them, Preprint MPIM2005-53, 2005.math.QA/0511618.[132] D. D. Demidov, Yu. I. Manin, E. E. Mukhin, and D. V. Zhdanovich, Nonstandard quantumdeformations ofGl(N)and constant solutions of the Yang–Baxter equation, Prog. Theor. Phys.Suppl. 102 (1990) 203.
[133] A. Sudbery, Consistent multiparameter quantization ofGl(N), J. Phys. A 23 (1990) L697.
[134] D. B. Fairlie and C. K. Zachos, Multiparameter associative generalizations of canonical commu-tation relations and quantized planes, Phys. Lett. B 256 (1991) 43.
[135] A. Schirrmacher, The multiparametric deformation ofGl(N)and the covariant differential cal-culus on the quantum vector space, Z. Phys. C 50 (1991) 321.
[136] A. P. Isaev, O. V. Ogievetsky, and P. N. Pyatov, OnR-matrix representations of Birman–Murakami–Wenzl algebras, Trudy MIAN 246 (2004) 147–153 (Memorial volume A. N. Tyurin).
[137] A. Schirrmacher, MultiparameterRmatrices and their quantum groups, J. Phys. A 24 (1991)L1249.
[138] H. Weyl, The Theory of Groups and Quantum Mechanics, Dover Publications, 1950.
[139] A. P. Isaev, V. A. Rubakov, Theory of Groups and Symmetries II. Representations of Groupsand Lie Algebras, Applications, World Scientific, 2021, 600 p.
[140] N. Yu. Reshetikhin and M. A. Semenov–Tian–Shansky, Central extensions of quantum currentgroups, Lett. Math. Phys. 19 (1990) 133.
[141] A. P. Isaev and R. P. Malik, Deformed traces and covariant quantum algebras for quantumgroupsGLqp(2)andGLqp(1|1), Phys. Lett. B 280 (1992) 219.hep-th/0309186.
[142] V. K. Dobrev, Duality for the matrix quantum groupGLp,q(2,C), J. Math. Phys. 33 (1992)3419.
[143] V. G. Drinfeld, Hopf algebras and the quantum Yang–Baxter equation, Sov. Math. Dokl. 32(1985) 254.
[144] A. I. Molev, Yangians and their applications, in: “Handbook of Algebra”, Vol. 3, M. Hazewinkel,ed., Elsevier, 2003.math.QA/0211288.
[145] A. Molev, M. Nazarov, and G. Olshanski, Yangians and classical Lie algebras, Russ. Math. Surv.51 (2) (1996) 205.
[146] P. P. Kulish and E. K. Sklyanin, Quantum spectral transform method. Recent developments, in:“Integrable Quantum Field Theories”, Lect. Notes Phys. 151 (1982) 61.
[147] R. M. Kashaev and N. Yu. Reshetikhin, Affine Toda field theory as a three-dimensional integrablesystem, Comm. Math. Phys. 188 (1997) 251.arXiv:hep-th/9507065.
[148] A. P. Isaev and S. M. Sergeev, Quantum Lax operators and discrete2+1-dimensional integrablemodels, Lett. Math. Phys. 64 (2003) 57.
[149] R. B. Zhang, Thegℓ(M|N)super Yangian and its finite dimensional representations, Lett. Math.Phys. 37 (1996) 419.q-alg/9507029.
[150] M. Chaichian and P. P. Kulish, Quantum Lie superalgebras and Q oscillators, Phys. Lett. B 234(1990) 72.
[151] E. V. Damaskinsky, P. P. Kulish, and M. A. Sokolov, Gauss decomposition for quantum groupsand supergroups, Zap. Nauchn. Semin. POMI 211 (1995) 11–45.q-alg/9505001.[152] Y. Leblanc and J. C. Wallet,Rmatrix andqcovariant oscillators forUq(sl(n|m)), Phys. Lett.B 304 (1993) 89.
[153] M. Chaichian, P. Kulish, and J. Lukierski, Supercovariant systems ofqoscillators andqsuper-symmetric Hamiltonians, Phys. Lett. B 262 (1991) 43.
[154] P. P. Kulish and E. K. Sklyanin, On the solution of the Yang–Baxter equation, J. Sov. Math. 19(1982) 1596 [Zap. Nauchn. Semin. LOMI 95 (1980) 129].
[155] M. L. Nazarov, Quantum Berezinian and the classical Capelli identity, Lett. Math. Phys. 21(1991) 123.
[156] V. Lyubashenko and A. Sudbery, Quantum supergroups ofGL(n|m)type: Differential forms,Koszul complexes and Berezinians, Duke Math. J. 90 (1) (1997) 1.hep-th/9311095.
[157] J. Schwenk, W. B. Schmidke, and S. P. Vokos, Properties of2×2quantum matrices inZ(2)graded spaces, Z. Phys. C 46 (1990) 643.
[158] W. B. Schmidke, S. P. Vokos, and B. Zumino, Differential geometry of the quantum supergroupGlq(1|1), Z. Phys. C 48 (1990) 249.
[159] P. P. Kulish, Two-parameters quantum group and gauge transformation, Zap. Nauchn. Semin.LOMI 180 (1990) 89.
[160] L. Dabrowski and L. Y. Wang, Two parameter quantum deformation ofGL(1|1), Phys. Lett. B266 (1991) 51.
[161] V. Bazhanov and A. Shadrikov, Quantum triangle equations and Lie superalgebras, Teor. Mat.Fiz. 73 (1987) 402.
[162] P. P. Kulish, Integrable graded magnets, J. Sov. Math. 35 (1986) 2648 [Zap. Nauchn. Semin.LOMI 145 (1985) 140].
[163] S. M. Khoroshkin and V. N. Tolstoy, UniversalR-matrix for quantized (super)algebras, Comm.Math. Phys. 141 (1991) 599.
[164] P. P. Kulish and N. Yu. Reshetikhin, UniversalRmatrix of the quantum superalgebraOsp(2|1),Lett. Math. Phys. 18 (1989) 143.
[165] A. Klimyk and K. Schm ̈udgen, Quantum Groups and Their Representations, Springer, 1997.
[166] B. Berg, M. Karowski, V. Kurak, and P. Weisz, FactorizedU(N)symmetricSmatrices intwo-dimensions, Nucl. Phys. B 134 (1978) 125.
[167] D. Arnaudon, J. Avan, N. Crampe, L. Frappat, E. Ragoucy,R-matrix presentation for (super)-YangiansY(g), J. Math. Phys. 44 (2003) 302.math.QA/0111325.
[168] D. Arnaudon, et al., Bethe Ansatz equations and exactSmatrices for theosp(M|2n)open superspin chain, Nucl. Phys. B 687 (3) (2004) 257.math-ph/0310042.
[169] V. V. Bazhanov, Integrable quantum systems and classical Lie algebras, Comm. Math. Phys.113 (1987) 471.
[170] V. V. Bazhanov, Trigonometric solution of triangle equations and classical Lie algebras, Phys.Lett. B 159 (1985) 321.
[171] M. Jimbo, QuantumRmatrix for the generalized Toda system, Comm. Math. Phys. 102 (1986)537.
[172] N. J. MacKay, RationalR-matrices in irreducible representations. J. Phys. A: Math. Gen. 24(17) (1991) 4017.
[173] N. J. MacKay, Introduction to Yangian symmetry in integrable field theory, Int. J. Mod. Phys.A 20 (30) (2005) 7189.
[174] A. Pressley and V. Chari, Fundamental representations of Yangians and singularities ofR-matrices, J. Reine Angew. Math. 417 (1991) 87–128.
[175] E. Ogievetsky and P. Wiegmann, FactorizedS-matrix and the Bethe ansatz for simple Lie groups,Phys. Lett. B 168 (4) (1986) 360.
[176] A. P. Isaev and S. O. Krivonos, Split Casimir operator for simple Lie algebras, solutions ofYang–Baxter equations and Vogel parameters, J. Math. Phys. 62 (8) (2021) 083503.e-Print:2102.08258 [math-ph].
[177] A. P. Isaev and S. O. Krivonos, Split Casimir operator and universal formulation of the simpleLie algebras, Symmetry 13 (6) (2021) 1046.e-Print: 2106.04470 [math-ph].
[178] N. Bourbaki, Lie Groups and Lie Algebras: Chapters 4–6, Vol. 3, Springer, 2008.
[179] N. Yamatsu, Finite-Dimensional Lie Algebras and Their Representations for Unified Model Build-ing.arXiv:1511.08771 [hep-ph].
[180] P. Cvitanovi ́c, Birdtracks, Lie’s, and Exceptional Groups, Princeton; Oxford: Princeton Univer-sity Press, 2008.http://cns.physics.gatech.edu/grouptheory/chapters/draft.pdf.
[181] M. Jimbo and T. Miwa, q-KZ equation with|q|= 1and correlation functions of theXXZmodelin the gapless regime, J. Phys. A 29 (1996) 2923.hep-th/9601135.
[182] H. E. Boos, V. E. Korepin, F. A. Smirnov, Connecting lattice and relativistic models via confor-mal field theory, Progress in Mathematics, Vol. 237, “Infinite Dimensional Algebras and QuantumIntegrable Systems”, 2005, p. 157.math-ph/0311020.
[183] H. E. Boos, V. E. Korepin, F. A. Smirnov, New formulae for solutions of quantum Knizhnik–Zamolodchikov equations on level-4, J. Phys. A 37 (2004) 323.hep-th/0304077.
[184] I. B. Frenkel and N. Yu. Reshetikhin, Quantum affine algebras and holonomic difference equa-tions, Comm. Math. Phys. 146 (1992) 1.
[185] V. Tarasov and A. Varchenko, Geometry ofq-hypergeometric functions as a bridge betweenYangians and quantum affine algebras, Invent. Math. 128 (1997) 501.q-alg/9604011.
[186] F. A. Smirnov, Dynamical symmetries of massive integrable models, 1. Form-factor bootstrapequations as a special case of deformed Knizhnik–Zamolodchikov equations, Int. J. Mod. Phys.A 7 (1992) S813.
[187] A. P. Isaev, A. N. Kirillov, and V. O. Tarasov, Bethe subalgebras in affine Birman–Murakami–Wenzl algebras and flat connections for q-KZ equations, J. Phys. A: Math. Theor. 49 (2016)204002.arXiv:1510.05374.
[188] I. Cherednik, Difference elliptic operators and root systems, Int. Math. Res. Not. (1995) 44–59.arXiv:hep-th/9410188.
[189] A. N. Kirillov, On Some Quadratic Algebras: Jucys–Murphy and Dunkl Elements, in: “Calogero–Moser–Sutherland Models”, Springer, New York, 2000, pp. 231–248.arXiv:q-alg/9705003.
[190] A. A. Belavin, Dynamical symmetry of integrable quantum systems, Nucl. Phys. B 180[FS2](1981) 189.
[191] R. J. Baxter, One-dimensional anisotropic Heisenberg chain, Ann. Phys. 70 (1972) 323.
[192] I. V. Cherednik, On the properties of factorizedSmatrices in elliptic functions, Sov. J. Nucl.Phys. 36 (1982) 320 [Yad. Fiz. 36 (1982) 549].
[193] C. A. Tracy, Embedded elliptic curves and the Yang–Baxter equations, Physica (Utrecht) D 16(1985) 203.
[194] A. Bovier, FactorizedSmatrices and generalized Baxter models, J. Math. Phys. 24 (1983) 631.
[195] G. Felder and V. Pasquier, A simple construction of ellipticRmatrices, Lett. Math. Phys. 32(1994) 167.
[196] L. A. Takhtajan, Solutions of the triangle equations withZn×Zn-symmetry and matrix analoguesof the Weierstrass zeta and sigma functions, Zap. Nauchn. Semin. LOMI 133 (1984) 258–276.
[197] V. V. Bazhanov, QuantumRmatrices and matrix generalizations of trigonometric functions,Theor. Math. Phys. 73 (1) (1987) 1035–1039 [Teor. Mat. Fiz. 73 (1) (1987) 26–32].
[198] M. Rosso, Quantum groups and quantum shuffles, Invent. Math. 133 (1998) 133.
[199] A. P. Isaev and O. V. Ogievetsky, Braids, shuffles and symmetrizers, J. Phys. A: Math. Theor.42 (2009) 304017.arXiv:0812.3974 [math.QA].
[200] V. F. R. Jones, Baxterization, Int. J. Mod. Phys. B 4 (5) (1990) 701.
[201] V. F. R. Jones, Hecke algebra representations of braid groups and link polynomials, Ann. Math.126 (1987) 335.
[202] G. E. Murphy, On the representation theory of the symmetric groups and associated Heckealgebras, J. Algebra 152 (1992) 287.
[203] H. Wenzl, Hecke algebras of typeAnand subfactors, Invent. Math. 92 (1988) 349.
[204] I. V. Cherednik, A new interpretation of Gelfand–Tzetlin bases, Duke Math. J. 54 (2) (1987)563.
[205] O. V. Ogievetsky and P. N. Pyatov, Lecture on Hecke Algebras, Preprint MPIM (Bonn), MPI2001-40.http://www.mpim-bonn.mpg.de/html/preprints/preprints.html.
[206] P. Martin, Potts Models and Related Problems in Statistical Mechanics, World Scientific, Sin-gapore, 1991.
[207] V. Chari and A. Pressley, A Guide to Quantum Groups, Cambrige University Press, 1994.
[208] A. P. Isaev and O. V. Ogievetsky, On representations of Hecke algebras, Czech. J. Phys. 55 (11)(2005) 1433.
[209] A. P. Isaev and O. V. Ogievetsky, Representations ofA-type Hecke algebras, in: Proceedingsof International Workshop “Supersymmetries and Quantum Symmetries”, Dubna, 2006.arXiv:0912.3701[math.QA].
[210] D. Levy, Algebraic structure of translation-invariant spin-1/2XXZand q-Potts quantum chains,Phys. Rev. Lett. 67 (15) (1991) 1971–1974.
[211] A. P. Isaev and O. V. Ogievetsky, On Baxterized solutions of reflection equation and integrablechain models, Nucl. Phys. B 760 (2007) 167.math-ph/0510078.
[212] A. P. Isaev, Functional equations for transfer-matrix operators in open Hecke chain models,Theor. Math. Phys. 150 (2) (2007) 187.arXive:1003.3385 [math-ph].
[213] A. P. Isaev, O. V. Ogievetsky, and A. F. Os’kin, Chain models on Hecke algebra for corner typerepresentations, Rep. Math. Phys. 61 (2) (2008) 309.arXiv: 0710.0261[math.QA].
[214] C. Burdik, J. Fuksa, A. P. Isaev, S. O. Krivonos, and O. Navratil, Remarks towards the spectrumof the Heisenberg spin chain type models, Phys. Part. Nucl. 46 (3) (2015) 277.arXiv: 1412.3999[math-ph].
[215] Y. K. Zhou, Row transfer matrix functional relations for Baxter’s eight-vertex and six-vertexmodels with open boundaries via more general reflection matrices, Nucl. Phys. B 458 (1996) 504.hep-th/9510095(1995).
[216] W. L. Yang, R. I. Nepomechie, and Y. Z. Zhang, Q-operator and T–Q relation from the fusionhierarchy, Phys. Lett. B 633 (2006) 664.hep-th/0511134(2005).
[217] D. Levy and P. Martin, Hecke algebra solutions to the reflection equation, J. Phys. A 27 (1994)L521.
[218] P. Martin, D. Woodcock, and D. Levy, A diagrammatic approach to Hecke algebras of thereflection equation, J. Phys. A 33 (2000) 1265.
[219] A. Doikou and P. Martin, Hecke algebraic approach to the reflection equation for spin chains, J.Phys. A 36 (2003) 2203.
[220] P. P. Kulish and A. I. Mudrov, Baxterization of solutions to reflection equation with HeckeR-matrix, Lett. Math. Phys. 75 (2) (2006) 151–170.arXive:math/0508289 [math.QA].
[221] M. Rosso and V. F. R. Jones, On the invariants of torus knots derived from quantum groups, J.Knot Theory Ramifications 2 (1993) 97.
[222] A. Mironov, R. Mkrtchyan, and A. Morozov, On universal knot polynomials, J. High EnergyPhys. 2016 (2) (2016) 1.arXiv:1510.05884.
[223] A. Morozov, Differential expansion and rectangular HOMFLY for the figure eight knot, Nucl.Phys. B 911 (2016) 582.arXiv:1605.09728 [hep-th].
[224] Ya. Kononov and A. Morozov, On rectangular HOMFLY for twist knots, Mod. Phys. Lett. A 31(38) (2016) 1650223.arXiv:1610.04778 [hep-th].
[225] H. Itoyama, A. Mironov, A. Morozov, and An. Morozov, Character expansion for HOMFLYpolynomials. III. All 3-strand braids in the first symmetric representation, Int. J. Mod. Phys. A27 (2012) 1250099.arXiv:1204.4785 [hep-th].
[226] P. Dunin-Barkowski, A. Mironov, A. Morozov, A. Sleptsov, and A. Smirnov, Superpolynomialsfor toric knots from evolution induced by cut-and-join operators, J. High Energy Phys. 03 (2013)021.arXiv:1106.4305.
[227] A. Anokhina and A. Morozov, TowardsR-matrix construction of Khovanov–Rozansky poly-nomials. I. Primary T-deformation of HOMFLY, J. High Energy Phys. 07 (2014) 063.arXiv:1403.8087 [hep-th].
[228] J. S. Birman and H. Wenzl, Braids, link polynomials and new algebra, Trans. Amer. Math. Soc.313 (1) (1989) 249.
[229] H. Wenzl, Quantum Groups and Subfactors of Type B, C and D, Comm. Math. Phys. 133 (1990)383.
[230] A. P. Isaev, A. I. Molev, and A. F. Os’kin, On the idempotents of Hecke algebras, Lett. Math.Phys. 85 (2008) 79.arXiv:0804.4214 [math.QA].
[231] A. Okounkov and A. Vershik, A new approach to representation theory of symmetric groups,Selecta Math. New Ser. 2 (4) (1996) 581–605.
[232] L. K. Hadjiivanov, A. P. Isaev, O. V. Ogievetsky, P. N. Pyatov, and I. T. Todorov, Hecke algebraicproperties of dynamicalR-matrices: Application to related quantum matrix algebras, J. Math.Phys. 40 (1999) 427.arXiv:q-alg/9712026.
[233] J.-L. Gervais and A. Neveu, Novel triangle relation and absence of tachyons in Liouville stringfield theory, Nucl. Phys. B 238 (1984) 125.
[234] G. Felder, Elliptic quantum groups, in: Proceedings of the International Congress of Mathemat-ical Physics, 1994.arXiv:hep-th/9412207.
[235] A. G. Bytsko and L. D. Faddeev, (TB)q, q-analog of model space and the Clebsch–Gordancoefficients generating matrices, J. Math. Phys. 37 (1996) 6324.arXiv:q-alg/9508022.
[236] P. Etingof and A. Varchenko, Solutions of the quantum dynamical Yang–Baxter equation anddynamical quantum groups, Comm. Math. Phys. 196 (3) (1998) 591.arXiv:q-alg/9708015.
[237] Y. Cheng, M. L. Ge, and K. Xue, Yang–Baxterization of braid group representations, Comm.Math. Phys. 136 (1) (1991) 195–208.
[238] V. F. R. Jones, On a certain value of the Kauffman polynomial, Comm. Math. Phys. 125 (1989)459.
[239] J. Murakami, Solvable lattice models and algebras of face operators, Adv. Stud. Pure Math. 19(1989) 399.
[240] I. Heckenberger and A. Sch ̈uler, Symmetrizer and antisymmetrizer of the Birman–Wenzl–Murakami algebras, Lett. Mat. Phys. 50 (1999) 45.math.QA/0002170.
[241] I. Tuba and H. Wenzl, On braided tensor categories of type BCD, Preprint, 2003.math.QA/0301142.
[242] A. P. Isaev and O. V. Ogievetsky, Jucys–Murphy elements for Birman–Murakami–Wenzl alge-bras, Phys. Part. Nucl. Lett. 8 (3) (2011) 234.arXiv:0912.4010 [math.QA].
[243] A. P. Isaev, A. I. Molev, and O. V. Ogievetsky, Idempotents for Birman–Murakami–Wenzl al-gebras and reflection equation, Adv. Theor. Math. Phys. 18 (1) (2014) 1–25.arXiv:1111.2502[math.RT].
[244] J. Murakami, The representations of the q-analogue of Brauer’s centralizer algebras and theKauffman polynomial of links, Publ. RIMS (Kyoto University) 26 (1990) 935.
[245] M. Nazarov, Young’s orthogonal form for Brauer’s centralizer algebra, J. Algebra 182 (1996) 664.
[246] A. Beliakova and Ch. Blanchet, Skein construction of idempotents in Birman–Murakami–Wenzlalgebras, Math. Ann. 321 (2) (2001) 347.arXiv: math.QA/0006143.
[247] A. Ram and H. Wenzl, Matrix units for centralizer algebras, J. Algebra 145 (1992) 378.
[248] R. Leduc and A. Ram, A ribbon Hopf algebra approach to the irreducible representations ofcentralizer algebras: The Brauer, Birman–Wenzl, and typeAIwahori–Hecke algebras, Adv.Math. 125 (AI971602) (1997) 1–94.
[249] V. A. Rubakov, Adler–Bell–Jackiw anomaly and fermion number breaking in the presence of amagnetic monopole, Nucl. Phys. B 203 (1982) 311.
[250] J. A. Minahan and K. Zarembo, The Bethe-ansatz forN= 4super Yang–Mills, J. High EnergyPhys. 0303 (2003) 013.hep-th/0212208.
[251] N. Beisert and M. Staudacher, TheN= 4SYM integrable super spin chain, Nucl. Phys. B 670(2003) 439.hep-th/0307042.
[252] L. N. Lipatov, High-energy asymptotics of multicolor QCD and exactly solvable lattice models,JETP Lett. 59 (1994) 596 [Pisma Zh. Eksp. Teor. Fiz. 59 (1994) 571].hep-th/9311037.
[253] L. N. Lipatov, High-energy asymptotics of multicolor QCD and two-dimensional conformal fieldtheories, Phys. Lett. B 309 (1993) 394.
[254] L. D. Faddeev and G. P. Korchemsky, High-energy QCD as a completely integrable model, Phys.Lett. B 342 (1995) 311.hep-th/9404173.
[255] A. V. Belitsky, A. S. Gorsky, and G. P. Korchemsky, Gauge/string duality for QCD conformaloperators, Nucl. Phys. B 667 (2003) 3.hep-th/0304028.
[256] L. D. Faddeev, How Algebraic Bethe Ansatz Works for Integrable Model, Les-Houches Lectures.hep-th/9605187.
[257] A. A. Vladimirov, Introduction to Quantum Integrable Systems. TheR-Matrix Method [in Rus-sian] (Lectures for Young Scientists, Vol. 32), R17-85-742, JINR, Dubna, 1985.
[258] N. Slavnov, Algebraic Bethe Ansatz and Correlation Functions. An Advanced Course, WorldScientific, 2022.
[259] P. P. Kulish and N. Yu. Reshetikhin, Diagonalization ofGl(N)invariant transfer matrices andquantumNwave system (Lee model), J. Phys. A 16 (1983) L591.
[260] M. L ̈usher, Dynamical charges in the quantized renormalized massive Thirring model, Nucl.Phys. B 117 (1976) 475.
[261] I. V. Cherednik, Factorizing particles on a half line and root systems, Theor. Math. Phys. 61(1984) 977 [Teor. Mat. Fiz. 61 (1984) 35].
[262] S. Ghoshal and A. B. Zamolodchikov, BoundarySmatrix and boundary state in two-dimensionalintegrable quantum field theory, Int. J. Mod. Phys. A 9 (1994) 3841.hep-th/9306002.
[263] E. K. Sklyanin, Boundary conditions for integrable quantum systems, J. Phys. A 21 (1988) 2375.
[264] P. P. Kulish and E. K. Sklyanin, Algebraic structures related to the reflection equations, J. Phys.A 25 (1992) 5963.
[265] P. P. Kulish and R. Sasaki, Covariance properties of reflection equation algebras, Prog. Theor.Phys. 89 (1993) 741.
[266] P. P. Kulish, Yang–Baxter equation and reflection equations in integrable models, in: ProceedingsSchladming School, Lect. Notes Phys. 469 (1996) 125–144.hep-th/9507070.
[267] N. J. MacKay and B. J. Short, Boundary scattering, symmetric spaces and the principal chiralmodel on the half-line, Comm. Math. Phys. 233 (2003) 313.hep-th/0104212.
[268] A. Liguori, M. Mintchev, and L. Zhao, Boundary exchange algebras and scattering on the halfline, Comm. Math. Phys. 194 (1998) 569.hep-th/9607085.
[269] L. Mezincescu, R. I. Nepomechie, and V. Rittenberg, Bethe ansatz solution of the Fateev–Zamolodchikov quantum spin chain with boundary terms, Phys. Lett. A 147 (1990) 70.
[270] L. Mezincescu, R. I. Nepomechie, Integrable open spin chains with nonsymmetricRmatrices, J.Phys. A 24 (1991) L17.
[271] L. Mezincescu, R. I. Nepomechie, Integrability of open spin chains with quantum algebra sym-metry, Int. J. Mod. Phys. A 6 (1991) 5231 [Addendum: ibid. A 7 (1992) 5657].hep-th/9206047.
[272] A. B. Zamolodchikov, Tetrahedra equations and integrable systems in three-dimensional space,Sov. Phys. JETP 52 (1980) 325 [Zh. Eksp. Teor. Fiz. 79 (1980) 641].
[273] A. B. Zamolodchikov, Tetrahedron equations and the relativisticSmatrix of straight strings in(2+1)-dimensions, Comm. Math. Phys. 79 (1981) 489.
[274] V. V. Bazhanov and R. J. Baxter, New solvable lattice models in three-dimensions, J. Stat. Phys.69 (1992) 453.
[275] V. V. Bazhanov and R. J. Baxter, Star triangle relation for a three-dimensional model, J. Stat.Phys. 71 (1993) 839.hep-th/9212050.
[276] S. M. Sergeev, Complex of three dimensional integrable models, J. Phys. A: Math. Gen. 34 (2001)10493.
[277] V. V. Bazhanov and S. M. Sergeev, Zamolodchikov’s tetrahedron equation and hidden structureof quantum groups, J. Phys. A 39 (2006) 3295.hep-th/0509181.
[278] V. V. Bazhanov, V. V. Mangazeev, and S. M. Sergeev, Quantum geometry of 3-dimensionallattices, J. Stat. Mech. 0807 (2008) P07004.arXiv:0801.0129 [hep-th].
[279] A. P. Isaev and P. P. Kulish, Tetrahedron reflection equation, Mod. Phys. Lett. A 12 (1997) 427.hep-th/9702013.
[280] A. Kuniba and M. Okado, Tetrahedron and 3D reflection equations from quantized algebra offunctions, J. Phys. A 45 (2012) 465206.arXiv: 1208.1586 [math-ph].
[281] A. Kuniba and V. Pasquier, Matrix product solutions to the reflection equation from three dimen-sional integrability, J. Phys. A: Math. Theor. 51 (2018) 255204.arXiv:1802.09164 [math-ph].
[282] A. Yoneyama, Tetrahedron and 3D reflection equation from PBW bases of the nilpotent subal-gebra of quantum superalgebras, Comm. Math. Phys. 387 (1) (2021) 481.
[283] Yu. I. Manin and V. V. Schechtman, Arrangements of hyperplanes, higher braid groups andhigher Bruhat orders, Adv. Stud. Pure Math. 17 (1990) 289.
[284] M. Kapranov and V. Voevodsky, Braided monoidal 2-categories and Manin–Schechtman higherbraid groups, J. Pure Appl. Algebra 92 (1994) 241.
[285] R. J. Lawrence, Algebras and triangle relations, J. Pure Appl. Algebra, 100 (1–3) (1995) 43.
[286] M. Kapranov and V. Voevodsky, Freen-categories generated by a cube, oriented matroids, andhigher Bruhat orders, Funct. Anal. Appl. 25 (1990) 50.
[287] J. C. Baez, An Introduction ton-Categories, in: International Conference on Category Theoryand Computer Science, Springer, Berlin, Heidelberg, 1997, pp. 1–33.q-alg/9705009.
[288] J. C. Baez and M. Shulman, Lectures onn-Categories and Cohomology. Towards Higher Cate-gories. Springer, New York, 2010, p. 1–68.arXiv: math/0608420 [math.CT].
[289] V. V. Bazhanov and Yu. G. Stroganov, On commutativity conditions for transfer matrices onmultidimensional lattice, Teor. Mat. Fiz. 52 (1982) 105.
[290] A. B. Zamolodchikov, Fishnet diagrams as a completely integrable system, Phys. Lett. B 97(1980) 63.
[291] D. I. Kazakov, The method of uniqueness, a new powerful technique for multiloop calculations,Phys. Lett. B 133 (1983) 406.
[292] V. V. Belokurov and N. I. Ussyukina, Calculation of ladder diagrams in arbitrary order, J. Phys.A 16 (1983) 2811.
[293] D. J. Broadhurst, Summation of an infinite series of ladder diagrams, Phys. Lett. B 307 (1993)132.
[294] N. I. Ussyukina and A. I. Davydychev, Exact results for three- and four-point ladder diagramswith an arbitrary number of rungs, Phys. Lett. B 305 (1993) 136.
[295] S. G. Gorishnii and A. P. Isaev, On an approach to the calculation of multiloop massless Feynmanintegrals, Theor. Math. Phys. 62 (1985) 232 [Teor. Mat. Fiz. 62 (1985) 345].
[296] P. A. Baikov and K. G. Chetyrkin, Four loop massless propagators: An algebraic evaluation ofall master integrals, Nucl. Phys. B 837 (2010) 186.arXiv:1004.1153.
[297] A. P. Isaev, Multi-loop Feynman integrals and conformal quantum mechanics, Nucl. Phys. B 662(2003) 461.hep-th/0303056.
[298] A. P. Isaev, Operator approach to analytical evaluation of Feynman diagrams, Phys. At. Nucl.71 (2008) 914.arXiv:0709.0419[hep-th].
[299] D. Chicherin, S. Derkachov, and A. P. Isaev, Conformal group:R-matrix and star-trianglerelation, J. High Energy Phys. 04 (2013) 020.arXiv:1206.4150 [math-ph].
[300] S. E. Derkachov, G. P. Korchemsky, and A. N. Manashov, Noncompact Heisenberg spin magnetsfrom high-energy QCD: I. Baxter Q-operator and separation of variables, Nucl. Phys. B 617(2001) 375-440.arXiv:hep-th/0107193.
[301] S. E. Derkachov and A. N. Manashov, Factorization ofR-matrix and Baxter Q-operators forgenericsl(N)spin chains, J. Phys. A 42 (2009) 075204.arXiv:0809.2050 [nlin.SI].
[302] D. T. Barfoot and D. J. Broadhurst,Z(2)×S(6)symmetry of the two loop diagram, Z. Phys. C41 (1988) 81.
[303] G. t’Hooft, Dimensional regularization and the renormalization group, Nucl. Phys. B 61 (1973)455.
[304] R. M. Kashaev, On discrete three-dimensional equations associated with the local Yang–Baxterrelation, Lett. Math. Phys. 38 (1996) 389.
[305] J. M. Maillet and F. Nijhoff, Integrability for multidimensional lattice models, Phys. Lett. B 224(1989) 389.
[306] E. S. Fradkin and M. Ya. Palchik, Recent developments in conformal invariant quantum fieldtheory, Phys. Rep. 44 (5) (1978) 249–349.
[307] D. Chicherin, S. Derkachov, and A. P. Isaev, The spinorialR-matrix, J. Phys. A 46 (2013)485201.arXiv:1303.4929 [math-ph].
[308] I. Mitra, On conformal invariant integrals involving spin one-half and spin-one particles, J. Phys.A 41 (2008) 315401.arXiv:0803.2630 [hep-th].
[309] V. Kazakov and E. Olivucci, Biscalar integrable conformal field theories in any dimension, Phys.Rev. Lett. 121 (2018) 131601.arXiv:1801.09844.
[310] B. Basso and L. J. Dixon, Gluing ladder Feynman diagrams into fishnets, Phys. Rev. Lett. 119(2017) 071601.arXiv:1705.03545.
[311] N. Gromov, V. Kazakov, and G. Korchemsky, Exact correlation functions in conformal fishnettheory, J. High Energy Phys. 08 (2019) 123.arXiv:1808.02688.
[312] S. Derkachov, E. Olivucci, Conformal quantum mechanics & the integrable spinning fishnet, J.High Energy Phys. 11 (2021) 060.e-Print: 2103.01940 [hep-th].
[313] S. Derkachov, G. Ferrando, and E. Olivucci, Mirror channel eigenvectors of thed-dimensionalfishnets, J. High Energy Phys. 12 (2021) 174.arXiv:2108.12620 [hep-th].
[314] S. E. Derkachov, A. P. Isaev, and L. A. Shumilov, Ladder and zig-zag Feynman diagrams, oper-ator formalism and conformal triangles, J. High Energy Phys. 06 (2023) 059.arXiv:2302.11238[hep-th].
[315] R. M. Iakhibbaev, Regge limit of correlation function in 6d biscalar fishnet models, Phys. Part.Nucl. Lett. 21 (4) (2024) 587–589.
[316] M. Alfimov, G. Ferrando, V. Kazakov, and E. Olivucci, Checkerboard CFT, J. High EnergyPhys. 01 (2025) 015.arXiv:2311.01437 [hep-th].
[317] E. Witten, Gauge theories, vertex models and quantum groups, Nucl. Phys. B 330 (2–3) (1990)285.
[318] L. Alvarez-Gaume, C. Gomez, and G. Sierra, Duality and quantum groups, Nucl. Phys. B 330(2–3) (1990) 347.
[319] V. V. Bazhanov, R. M. Kashaev, V. V. Mangazeev, and Yu. G. Stroganov,(ZN×)n−1general-ization of the chiral Potts model, Comm. Math. Phys. 138 (1991) 393.
[320] E. Date, M. Jimbo, K. Miki, and T. Miwa, Generalized chiral Potts models and minimal cyclicrepresentations ofUq(ˆgl(n,C)), Comm. Math. Phys. 137 (1991) 133.
[321] P. Furlan, A. Ganchev, and V. P. Petkova, Quantum groups and fusion rules multiplicities, Nucl.Phys. B 343 (1990) 205.
[322] A. T. Filippov, A. P. Isaev, and A. B. Kurdikov, Paragrassmann analysis and quantum groups,Mod. Phys. Lett. A 7 (1992) 2129.hep-th/9204089.
[323] P. Furlan, L. K. Hadjiivanov, A. P. Isaev, O. V. Ogievetsky, P. N. Pyatov, I. T. Todorov, Quantummatrix algebra for theSU(n)WZNW model, J. Phys. A 36 (2003) 5497.hep-th/0003210.
[324] V. Chari, A. Pressley, Quantum affine algebras at roots of unity, Represent. Theor. Amer. Math.Soc. 1 (12) (1997) 280.
[325] B. L. Feigin, A. M. Gainutdinov, A. M. Semikhatov, I. Yu. Tipunin, Modular group representa-tions and fusion in logarithmic conformal field theories and in the quantum group center, Comm.Math. Phys. 265 (2006) 47.hep-th/0504093.
[326] B. L. Feigin, A. M. Gainutdinov, A. M. Semikhatov, I. Y. Tipunin, Kazhdan–Lusztig-dual quan-tum group for logarithimic extensions of Virasoro minimal models, J. Math. Phys. 48 (3) (2007).