Geodesic
Integrable system of generalized relativistic interacting tops
Teoretičeskaâ i matematičeskaâ fizika, Tome 205 (2020) no. 1, pp. 55-67 Cet article a éte moissonné depuis la source Math-Net.Ru

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We describe a family of integrable $GL(NM)$ models generalizing classical spin Ruijsenaars–Schneider systems (the case $N=1$) on one hand and relativistic integrable tops on the $GL(N)$ Lie group (the case $M=1$) on the other hand. We obtain the described models using the Lax pair with a spectral parameter and derive the equations of motion. To construct the Lax representation, we use the $GL(N)$ $R$-matrix in the fundamental representation of $GL(N)$.
Keywords: elliptic integrable system, spin Ruijsenaars–Schneider model, integrable interacting tops. 
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I. A. Sechin; A. V. Zotov. Integrable system of generalized relativistic interacting tops. Teoretičeskaâ i matematičeskaâ fizika, Tome 205 (2020) no. 1, pp. 55-67. http://geodesic.mathdoc.fr/item/TMF_2020_205_1_a3/

[1] A. Grekov, I. Sechin, A. Zotov, “Generalized model of interacting integrable tops”, JHEP, 10 (2019), 081, 33 pp., arXiv: 1905.07820 | DOI

[2] I. Sechin, A. Zotov, “$R$-matrix-valued Lax pairs and long-range spin chains”, Phys. Lett. B, 781 (2018), 1–7, arXiv: ; A. Grekov, A. Zotov, “On $R$-matrix valued Lax pairs for Calogero–Moser models”, J. Phys. A: Math. Theor., 51:31 (2018), 315202, 26 pp., arXiv: ; И. А. Сечин, А. В. Зотов, “${\rm GL}_{NM}$-значная квантовая динамическая $R$-матрица, построенная по решению ассоциативного уравнения Янга–Бакстера”, УМН, 74:4(448) (2019), 189–190, arXiv: 1801.089081801.002451905.08724 | DOI | MR | DOI | MR | DOI | DOI | MR

[3] A. V. Zotov, “Relyativistskie vzaimodeistvuyuschie integriruemye ellipticheskie volchki”, TMF, 201:2 (2019), 175–192, arXiv: 1910.08246 | DOI | DOI | MR

[4] A. Veil, Ellipticheskie funktsii po Eizenshteinu i Kronekeru, Mir, M., 1978 ; Д. Мамфорд, Лекции о тэта-функциях, Мир, М., 1998 | MR | MR | MR | Zbl

[5] S. Fomin, A. N. Kirillov, “Quadratic algebras, Dunkl elements, and Schubert calculus”, Advances in Geometry, Progress in Mathematics, 172, eds. J.-L. Brylinski, R. Brylinski, V. Nistor, B. Tsygan, P. Xu, Birkhäuser, Boston, MA, 1999, 147–182 ; A. Polishchuk, “Classical Yang–Baxter equation and the $A^\infty$-constraint”, Adv. Math., 168:1 (2002), 56–95 ; А. М. Левин, М. А. Ольшанецкий, А. В. Зотов, “Квантовые $R$-матрицы Бакстера–Белавина и многомерные пары Лакса для уравнения Пенлеве VI”, ТМФ, 184:1 (2015), 41–56, arXiv: 1501.07351 | DOI | MR | Zbl | DOI | MR | DOI | DOI | MR

[6] G. Aminov, S. Arthamonov, A. Smirnov, A. Zotov, “Rational top and its classical $R$-matrix”, J. Phys. A: Math. Theor., 47:30 (2014), 305207, 19 pp., arXiv: ; A. Levin, M. Olshanetsky, A. Zotov, “Noncommutative extensions of elliptic integrable Euler–Arnold tops and Painlevé VI equation”, J. Phys. A: Math. Theor., 49:39 (2016), 395202, 24 pp., arXiv: 1402.31891603.06101 | DOI | MR | DOI | MR

[7] A. Levin, M. Olshanetsky, A. Zotov, “Relativistic classical integrable tops and quantum $R$-matrices”, JHEP, 07 (2014), 012, 39 pp., arXiv: ; T. Krasnov, A. Zotov, “Trigonometric integrable tops from solutions of associative Yang-Baxter equation”, Ann. Henri Poincaré, 20:8 (2019), 2671–2697, arXiv: 1405.75231812.04209 | DOI | DOI | MR

[8] A. V. Zotov, “Model Kalodzhero–Mozera i $R$-matrichnye tozhdestva”, TMF, 197:3 (2018), 417–434 ; “Старшие аналоги условия унитарности для квантовых $R$-матриц”, ТМФ, 189:2 (2016), 176–185 ; А. М. Левин, М. А. Ольшанецкий, А. В. Зотов, “Квантовые $R$-матрицы Бакстера–Белавина и многомерные пары Лакса для уравнения Пенлеве VI”, ТМФ, 184:1 (2015), 41–56, arXiv: 1501.07351 | DOI | DOI | MR | DOI | DOI | MR | DOI | DOI | MR

[9] E. K. Sklyanin, “O nekotorykh algebraicheskikh strukturakh, svyazannykh s uravneniem Yanga–Bakstera”, Funkts. analiz i ego pril., 16:4 (1982), 27–34 | DOI | MR | Zbl

[10] S. N. M. Ruijsenaars, “Complete integrability of relativistic Calogero–Moser systems and elliptic function identities”, Commun. Math. Phys., 110:2 (1987), 191–213 | DOI | MR | Zbl

[11] I. M. Krichever, A. V. Zabrodin, “Spinovoe obobschenie modeli Reisenarsa–Shnaidera, neabeleva dvumerizovannaya tsepochka Toda i predstavleniya algebry Sklyanina”, UMN, 50:6(306) (1995), 3–56, arXiv: hep-th/9505039 | DOI | MR | Zbl

[12] G. E. Arutyunov, S. A. Frolov, “On Hamiltonian structure of the spin Ruijsenaars–Schneider model”, J. Phys. A: Math. Gen., 31:18 (1998), 4203–4216, arXiv: ; Г. Э. Арутюнов, Э. Оливуччи, “Гиперболическая спиновая модель Рёйсенарса–Шнайдера на основе пуассоновой редукции”, Тр. МИАН, 309 (2020), 38–53, arXiv: hep-th/97031191906.02619 | DOI | MR | DOI | DOI

[13] N. Reshetikhin, “Degenerately integrable systems”, Zap. nauchn. sem. POMI, 433 (2015), 224–245, arXiv: 1509.00730 | DOI | MR

[14] L. Fehér, “Poisson–Lie analogues of spin Sutherland models”, Nucl. Phys. B, 949 (2019), 114807, 26 pp., arXiv: 1809.01529 | DOI | MR

[15] O. Chalykh, M. Fairon, On the Hamiltonian formulation of the trigonometric spin Ruijsenaars–Schneider system, arXiv: ; M. Fairon, “Spin versions of the complex trigonometric Ruijsenaars–Schneider model from cyclic quivers”, J. Integrable Syst., 4:1 (2019), xyz008, 55 pp., arXiv: 1811.087271811.08717 | DOI | MR

[16] A. P. Polychronakos, “Calogero–Moser models with noncommutative spin interactions”, Phys. Rev. Lett., 89:12 (2002), 126403, 4 pp., arXiv: ; “The physics and mathematics of Calogero particles”, J. Phys. A: Math. Gen., 39:41 (2006), 12793–12945, arXiv: hep-th/0112141hep-th/0607033 | DOI | DOI | MR

[17] A. Levin, M. Olshanetsky, A. Smirnov, A. Zotov, “Characteristic classes of $SL(N)$-bundles and quantum dynamical elliptic $R$-matrices”, J. Phys. A: Math. Theor., 46:3 (2013), 035201, 25 pp., arXiv: ; А. В. Зотов, А. М. Левин, “Интегрируемая система взаимодействующих эллиптических волчков”, ТМФ, 146:1 (2006), 55–64 ; А. В. Зотов, А. В. Смирнов, “Модификации расслоений, эллиптические интегрируемые системы и связанные задачи”, ТМФ, 177:1 (2013), 3–67 ; A. Levin, M. Olshanetsky, A. Smirnov, A. Zotov, “Characteristic classes and Hitchin systems. General construction”, Commun. Math. Phys., 316:1 (2012), 1–44 ; “Calogero–Moser systems for simple Lie groups and characteristic classes of bundles”, J. Geom. Phys., 62:8 (2012), 1810–1850 1208.5750 | DOI | MR | DOI | DOI | MR | Zbl | DOI | DOI | MR | Zbl | DOI | MR | Zbl | DOI | MR