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JOURNALS // Uspekhi Matematicheskikh Nauk

Uspekhi Mat. Nauk, 1995, Volume 50, Issue 6(306), Pages 3–56 (Mi rm1121)

Spin generalization of the Ruijsenaars–Schneider model, the non-Abelian Toda chain, and representations of the Sklyanin algebra
I. M. Krichever, A. V. Zabrodin

This publication is cited in the following articles:
  1. Martin Hallnäs, Edwin Langmann, Encyclopedia of Mathematical Physics, 2025, 83  crossref
  2. Maxime Fairon, “Integrable systems on multiplicative quiver varieties from cyclic quivers”, J. Phys. A: Math. Theor., 58:4 (2025), 045202  crossref
  3. M. Matushko, A. Zotov, “Supersymmetric generalization of $q$-deformed long-range spin chains of Haldane–Shastry type and trigonometric $\mathrm{GL}(N|M)$ solution of associative Yang–Baxter equation”, Nuclear Phys. B, 1001 (2024), 116499–14  mathnet  crossref
  4. Jan Felipe van Diejen, “Inhomogeneous isotropic quantum spin chain associated with the difference Lamé equation”, Forum of Mathematics, Sigma, 12 (2024)  crossref
  5. A. V. Zabrodin, “On integrability of the deformed Ruijsenaars–Schneider system”, Russian Math. Surveys, 78:2 (2023), 349–386  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
  6. L. Fehér, “Poisson Reductions of Master Integrable Systems on Doubles of Compact Lie Groups”, Ann. Henri Poincaré, 24:6 (2023), 1823  crossref
  7. M. Matushko, Andrei Zotov, “Anisotropic spin generalization of elliptic Macdonald–Ruijsenaars operators and $R$-matrix identities”, Ann. Henri Poincaré, 24 (2023), 3373–3419  mathnet  crossref
  8. M. Fairon, L. Fehér, “Integrable Multi-Hamiltonian Systems from Reduction of an Extended Quasi-Poisson Double of ${\text {U}}(n)$”, Ann. Henri Poincaré, 24:10 (2023), 3461  crossref
  9. A. V. Zabrodin, “Elliptic families of solutions of the constrained Toda hierarchy”, Theoret. and Math. Phys., 213:1 (2022), 1362–1368  mathnet  crossref  crossref  mathscinet  adsnasa
  10. L Fehér, “Bi-Hamiltonian structure of Sutherland models coupled to two
    u(n)*
    -valued spins from Poisson reduction”, Nonlinearity, 35:6 (2022), 2971  crossref
  11. E. Trunina, A. Zotov, “Lax equations for relativistic $\mathrm{G}\mathrm{L}(NM,\mathbb{C})$ Gaudin models on elliptic curve”, J. Phys. A, 55:39 (2022), 395202–31  mathnet  crossref
  12. Bjorn K. Berntson, Rob Klabbers, Edwin Langmann, “The non-chiral intermediate Heisenberg ferromagnet equation”, J. High Energ. Phys., 2022:3 (2022)  crossref
  13. Jan Felipe van Diejen, Tamás Görbe, “Elliptic Kac–Sylvester Matrix from Difference Lamé Equation”, Ann. Henri Poincaré, 23:1 (2022), 49  crossref
  14. Fairon M., Feher L., Marshall I., “Trigonometric Real Form of the Spin Rs Model of Krichever and Zabrodin”, Ann. Henri Poincare, 22:2 (2021), 615–675  crossref  isi
  15. V. V. Prokofev, A. V. Zabrodin, “Elliptic solutions of the Toda lattice hierarchy and the elliptic Ruijsenaars–Schneider model”, Theoret. and Math. Phys., 208:2 (2021), 1093–1115  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
  16. M. Fairon, L. Fehér, “A decoupling property of some Poisson structures on Matn×d(C)×Matd×n(C) supporting GL(n,C)×GL(d,C) Poisson–Lie symmetry”, Journal of Mathematical Physics, 62:3 (2021)  crossref
  17. Gamal Mograby, Maxim Derevyagin, Gerald V. Dunne, Alexander Teplyaev, “Hamiltonian systems, Toda lattices, solitons, Lax pairs on weighted Z-graded graphs”, Journal of Mathematical Physics, 62:4 (2021)  crossref
  18. Igor Krichever, Alexander Varchenko, Proceedings of Symposia in Pure Mathematics, 103.1, Integrability, Quantization, and Geometry, 2021, 239  crossref
  19. Gleb E. Arutyunov, Enrico Olivucci, “Hyperbolic Spin Ruijsenaars–Schneider Model from Poisson Reduction”, Proc. Steklov Inst. Math., 309 (2020), 31–45  mathnet  crossref  crossref  mathscinet  isi  elib
  20. D. S. Rudneva, A. V. Zabrodin, “Elliptic solutions of the semidiscrete B-version of the Kadomtsev–Petviashvili equation”, Theoret. and Math. Phys., 204:3 (2020), 1209–1215  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
  21. I. A. Sechin, A. V. Zotov, “Integrable system of generalized relativistic interacting tops”, Theoret. and Math. Phys., 205:1 (2020), 1291–1302  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
  22. Chalykh O., Fairon M., “On the Hamiltonian Formulation of the Trigonometric Spin Ruijsenaars-Schneider System”, Lett. Math. Phys., 110:11 (2020), 2893–2940  crossref  isi
  23. D Rudneva, A Zabrodin, “Dynamics of poles of elliptic solutions to the BKP equation”, J. Phys. A: Math. Theor., 53:7 (2020), 075202  crossref
  24. Hidetoshi Awata, Hiroaki Kanno, Andrei Mironov, Alexei Morozov, “On a complete solution of the quantum Dell system”, J. High Energ. Phys., 2020:4 (2020)  crossref
  25. L. Fehér, “Reduction of a bi-Hamiltonian hierarchy on $T^*\mathrm{U}(n)$ to spin Ruijsenaars–Sutherland models”, Lett Math Phys, 110:5 (2020), 1057  crossref
  26. G. Arutyunov, “Spin Ruijsenaars–Schneider Models from Reduction”, Phys. Part. Nuclei Lett., 17:5 (2020), 730  crossref
  27. A. V. Zabrodin, “Matrix modified Kadomtsev–Petviashvili hierarchy”, Theoret. and Math. Phys., 199:3 (2019), 771–783  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
  28. A. V. Zotov, “Relativistic interacting integrable elliptic tops”, Theoret. and Math. Phys., 201:2 (2019), 1565–1580  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
  29. L. Fehér, “Poisson-Lie analogues of spin Sutherland models”, Nuclear Physics B, 949 (2019), 114807  crossref
  30. Oleg Chalykh, “Quantum Lax Pairs via Dunkl and Cherednik Operators”, Commun. Math. Phys., 369:1 (2019), 261  crossref
  31. Gleb Arutyunov, UNITEXT for Physics, Elements of Classical and Quantum Integrable Systems, 2019, 69  crossref
  32. L Fehér, “Bi-Hamiltonian structure of a dynamical system introduced by Braden and Hone”, Nonlinearity, 32:11 (2019), 4377  crossref
  33. Mauleshova G.S. Mironov A.E., “One-Point Commuting Difference Operators of Rank One and Their Relation With Finite-Gap Schrodinger Operators”, Dokl. Math., 97:1 (2018), 62–64  crossref  isi
  34. A. K. Pogrebkov, “Commutator identities on associative algebras, the non-Abelian Hirota difference equation and its reductions”, Theoret. and Math. Phys., 187:3 (2016), 823–834  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib  elib
  35. A. Mironov, A. Morozov, Y. Zenkevich, “Spectral duality in elliptic systems, six-dimensional gauge theories and topological strings”, J. High Energ. Phys., 2016:5 (2016)  crossref
  36. A. Zabrodin, A. Zotov, “Classical-Quantum Correspondence and Functional Relations for Painlevé Equations”, Constr Approx, 2015  crossref
  37. Tsuboi Z. Zabrodin A. Zotov A., “Supersymmetric Quantum Spin Chains and Classical Integrable Systems”, no. 5, 2015, 086  crossref  isi
  38. Anton Zabrodin, “The Master $T$-Operator for Inhomogeneous $XXX$ Spin Chain and mKP Hierarchy”, SIGMA, 10 (2014), 006, 18 pp.  mathnet  crossref  mathscinet
  39. A. Levin, M. Olshanetsky, A. Zotov, “Relativistic classical integrable tops and quantum R-matrices”, J. High Energ. Phys, 2014:7 (2014)  crossref
  40. G Aminov, S Arthamonov, A Smirnov, A Zotov, “Rational top and its classicalr-matrix”, J. Phys. A: Math. Theor, 47:30 (2014), 305207  crossref
  41. A. Levin, M. Olshanetsky, A. Zotov, “Classical integrable systems and soliton equations related to eleven-vertex R-matrix”, Nuclear Physics B, 2014  crossref
  42. A Levin, M Olshanetsky, A Smirnov, A Zotov, “Characteristic classes of ${\rm SL}(N, {\mathbb {C}})$-bundles and quantum dynamical elliptic R-matrices”, J. Phys. A: Math. Theor, 46:3 (2013), 035201  crossref
  43. A. V. Zabrodin, “The master $T$-operator for vertex models with trigonometric $R$-matrices as a classical $\tau$-function”, Theoret. and Math. Phys., 174:1 (2013), 52–67  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
  44. Eric Rains, Simon Ruijsenaars, “Difference Operators of Sklyanin and van Diejen Type”, Commun. Math. Phys, 2013  crossref
  45. Alexandrov A., Kazakov V., Leurent S., Tsuboi Z., Zabrodin A., “Classical Tau-Function for Quantum Spin Chains”, J. High Energy Phys., 2013, no. 9, 064  crossref  isi
  46. Zabrodin A., “Intertwining operators for Sklyanin algebra and elliptic hypergeometric series”, J Geom Phys, 61:9 (2011), 1733–1754  crossref  isi
  47. Igor Krichever, “Characterizing Jacobians via trisecants of the Kummer variety”, Ann of Math, 172:1 (2010), 485  crossref
  48. Krichever I., “Characterizing Jacobians via Trisecants of the Kummer Variety”, Ann. Math., 172:1 (2010), 485–516  isi
  49. F. L. Soloviev, “On the Hamiltonian form of equations of the elliptic spin Ruijsenaars–Schneider model”, Russian Math. Surveys, 64:6 (2009), 1142–1144  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
  50. I. M. Krichever, “Abelian solutions of the soliton equations and Riemann–Schottky problems”, Russian Math. Surveys, 63:6 (2008), 1011–1022  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
  51. Chalykh, O, “Algebro-geometric Schrodinger operators in many dimensions”, Philosophical Transactions of the Royal Society A-Mathematical Physical and Engineering Sciences, 366:1867 (2008), 947  crossref  mathscinet  zmath  adsnasa  isi
  52. Oleg Chalykh, “Bethe Ansatz for the Ruijsenaars Model of $BC_1$-Type”, SIGMA, 3 (2007), 028, 9 pp.  mathnet  crossref  mathscinet  zmath
  53. Plamen Iliev, “Rational Ruijsenaars–Schneider hierarchy and bispectral difference operators”, Physica D: Nonlinear Phenomena, 229:2 (2007), 184  crossref
  54. Luen-Chau Li, “Poisson Involutions, Spin Calogero–Moser Systems Associated with Symmetric Lie Subalgebras and the Symmetric Space Spin Ruijsenaars-Schneider Models”, Comm Math Phys, 265:2 (2006), 333  crossref  mathscinet  zmath  adsnasa  isi
  55. V. M. Buchstaber, I. M. Krichever, “Integrable equations, addition theorems, and the Riemann–Schottky problem”, Russian Math. Surveys, 61:1 (2006), 19–78  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
  56. A. V. Marshakov, “Semiclassical geometry and integrability of the AdS/CFT correspondence”, Theor Math Phys, 142:2 (2005), 222  crossref
  57. Chalykh, O, “Generalized Lame operators”, Communications in Mathematical Physics, 239:1–2 (2003), 115  crossref  mathscinet  zmath  adsnasa  isi
  58. Treibich, A, “Difference analogs of elliptic KdV solitons and Schrodinger operators”, International Mathematics Research Notices, 2003, no. 6, 313  crossref  mathscinet  zmath  isi
  59. D. V. Talalaev, “The Elliptic Gaudin System with Spin”, Theoret. and Math. Phys., 130:3 (2002), 361–374  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
  60. A. V. Odesskii, V. N. Rubtsov, “Polynomial Poisson Algebras with Regular Structure of Symplectic Leaves”, Theoret. and Math. Phys., 133:1 (2002), 1321–1337  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
  61. A. A. Akhmetshin, Yu. S. Vol'vovskii, I. M. Krichever, “Elliptic Families of Solutions of the Kadomtsev–Petviashvili Equation and the Field Elliptic Calogero–Moser System”, Funct. Anal. Appl., 36:4 (2002), 253–266  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
  62. A. V. Odesskii, “Elliptic algebras”, Russian Math. Surveys, 57:6 (2002), 1127–1162  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
  63. H.W. Braden, A. Gorsky, A. Odesskii, V. Rubtsov, “Double-elliptic dynamical systems from generalized Mukai–Sklyanin algebras”, Nuclear Physics B, 633:3 (2002), 414  crossref
  64. S N M Ruijsenaars, J Phys A Math Gen, 34:48 (2001), 10595  crossref  mathscinet  zmath  adsnasa  isi
  65. S N M Ruijsenaars, “Reflectionless Analytic Difference Operators II. Relations to Soliton Systems”, Journal of Nonlinear Mathematical Physics, 8:2 (2001), 256  crossref
  66. A. Gorsky, V. Rubtsov, Integrable Structures of Exactly Solvable Two-Dimensional Models of Quantum Field Theory, 2001, 173  crossref
  67. A. Gorsky, A. Mironov, Integrable Hierarchies and Modern Physical Theories, 2001, 33  crossref
  68. Ruijsenaars, SNM, “Hilbert space theory for reflectionless relativistic potentials”, Publications of the Research Institute For Mathematical Sciences, 36:6 (2000), 707  crossref  mathscinet  zmath  isi
  69. A. Marshakov, “Duality in integrable systems and generating functions for new Hamiltonians”, Physics Letters B, 476:3-4 (2000), 420  crossref
  70. J. Avan, Calogero—Moser— Sutherland Models, 2000, 1  crossref
  71. H. W. Braden, Calogero—Moser— Sutherland Models, 2000, 77  crossref
  72. S. N. M. Ruijsenaars, Calogero—Moser— Sutherland Models, 2000, 421  crossref
  73. I. Krichever, Calogero—Moser— Sutherland Models, 2000, 249  crossref
  74. S N M Ruijsenaars, J Phys A Math Gen, 32:9 (1999), 1737  crossref  mathscinet  zmath  adsnasa  isi
  75. S. N. M. Ruijsenaars, “Generalized Lamé functions. I. The elliptic case”, J Math Phys (N Y ), 40:3 (1999), 1595  crossref  mathscinet  zmath  adsnasa  isi
  76. H.W. Braden, A. Marshakov, A. Mironov, A. Morozov, “The Ruijsenaars-Schneider model in the context of Seiberg-Witten theory”, Nuclear Physics B, 558:1-2 (1999), 371  crossref
  77. A. V. Zabrodin, “Hirota equation and Bethe ansatz”, Theor Math Phys, 116:1 (1998), 782  mathnet  crossref  mathscinet  zmath  isi  elib
  78. Fritz Gesztesy, Rudi Weikard, “Elliptic algebro-geometric solutions of the KdV and AKNS hierarchies - an analytic approach”, Bull. Amer. Math. Soc., 35:4 (1998), 271  crossref
  79. G. E. Arutyunov, L. O. Chekhov, S. A. Frolov, “ R-Matrix quantization of the elliptic Ruijsenaars-Schneider model”, Theor Math Phys, 111:2 (1997), 536  mathnet  crossref  mathscinet  zmath  isi  elib
  80. S.N.M. Ruijsenaars, “Integrable Particle Systems vs Solutions to the KP and 2D Toda Equations”, Annals of Physics, 256:2 (1997), 226  crossref


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