Skip to main content
SpringerLink
Log in

Integrable system of generalized relativistic interacting tops

  • Research Articles
  • Published:
Theoretical and Mathematical Physics Aims and scope Submit manuscript

Abstract

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 \(GL(N)\).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. In [3], the elliptic case was described in a slightly different normalization. It differs from the one used here by \(q_j\to q_j/N\), which leads to the additional factor \(1/N\) in the equations of motion in [3].

References

  1. A. Grekov, I. Sechin, and A. Zotov, “Generalized model of interacting integrable tops,” JHEP, 1910, 081 (2019); arXiv:1905.07820v2 [math-ph] (2019).

    Article  ADS  MathSciNet  Google Scholar 

  2. I. Sechin and A. Zotov, “\(R\)-matrix-valued Lax pairs and long-range spin chains,” Phys. Lett. B, 781, 1–7 (2018); arXiv:1801.08908v3 [math-ph] (2018); A. Grekov and A. Zotov, “On \(R\)-matrix valued Lax pairs for Calogero–Moser models,” J. Phys. A: Math. Theor., 51, 315202 (2018); arXiv:1801.00245v2 [math-ph] (2018); I. A. Sechin and A. V. Zotov, “\({\rm GL}_{NM}\) quantum dynamical \(R\)-matrix based on solution of the associative Yang–Baxter equation,” Russian Math. Surveys, 74, 767–769 (2019); arXiv:1905.08724v2 [math.QA] (2019).

    Article  ADS  MathSciNet  Google Scholar 

  3. A. V. Zotov, “Relativistic interacting integrable elliptic tops,” Theor. Math. Phys., 201, 1565–1580 (2019); arXiv:1910.08246v1 [math-ph] (2019).

    Article  MathSciNet  Google Scholar 

  4. A. Weil, Elliptic Functions According to Eisenstein and Kronecker, Springer, Berlin (1976); D. Mumford, Tata Lectures on Theta I, II (Progr. Math., Vol. 43), Birkhäuser, Boston (1984).

    Book  Google Scholar 

  5. S. Fomin and A. N. Kirillov, “Quadratic algebras, Dunkl elements, and Schubert calculus,” in: Advances in Geometry (Progr. Math., Vol. 172, J.-L. Brylinski, R. Brylinski, V. Nistor, B. Tsygan, and P. Xu, eds.), Birkhäuser, Boston, Mass. (1999), pp. 147–182; A. Polishchuk, “Classical Yang–Baxter equation and the \(A^\infty\)-constraint,” Adv. Math., 168, 56–95 (2002); A. M. Levin, M. A. Olshanetsky, and A. V. Zotov, “Quantum Baxter–Belavin \(R\)-matrices and multidimensional Lax pairs for Painlevé VI,” Theor. Math. Phys., 184, 924–939 (2015); arXiv:1501.07351v3 [math-ph] (2015).

    Article  MathSciNet  Google Scholar 

  6. G. Aminov, S. Arthamonov, A. Smirnov, and A. Zotov, “Rational top and its classical \(R\)-matrix,” J. Phys. A: Math. Theor., 47, 305207 (2014); arXiv:1402.3189v3 [hep-th] (2014); A. Levin, M. Olshanetsky, and A. Zotov, “Noncommutative extensions of elliptic integrable Euler–Arnold tops and Painlevé VI equation,” J. Phys. A: Math. Theor., 49, 395202 (2016); arXiv:1603.06101v2 [math-ph] (2016).

    Article  MathSciNet  Google Scholar 

  7. A. Levin, M. Olshanetsky, and A. Zotov, “Relativistic classical integrable tops and quantum \(R\)-matrices,” JHEP, 1407, 012 (2014); arXiv:1405.7523v3 [hep-th] (2014); T. Krasnov and A. Zotov, “Trigonometric integrable tops from solutions of associative Yang–Baxter equation,” Ann. Henri Poincaré, 20, 2671–2697 (2019); arXiv:1812.04209v3 [math-ph] (2018).

    Article  ADS  Google Scholar 

  8. A. V. Zotov, “Calogero–Moser model and \(R\)-matrix identities,” Theor. Math. Phys., 197, 1755–1770 (2018); “Higher-order analogues of the unitarity condition for quantum \(R\)-matrices,” Theor. Math. Phys., 189, 1554–1562 (2016); A. M. Levin, M. A. Olshanetsky, and A. V. Zotov, “Quantum Baxter–Belavin \(R\)-matrices and multidimensional Lax pairs for Painlevé VI,” Theor. Math. Phys., 184, 924–939 (2015); arXiv:1501.07351v3 [math-ph] (2015).

    Article  MathSciNet  Google Scholar 

  9. E. K. Sklyanin, “Some algebraic structures connected with the Yang–Baxter equation,” Funct. Anal. Appl., 16, 263–270 (1982).

    Article  MathSciNet  Google Scholar 

  10. S. N. M. Ruijsenaars, “Complete integrability of relativistic Calogero–Moser systems and elliptic function identities,” Commun. Math. Phys., 110, 191–213 (1987).

    Article  ADS  MathSciNet  Google Scholar 

  11. I. M. Krichever and A. V. Zabrodin, “Spin generalization of the Ruijsenaars–Schneider model, the non-Abelian Toda chain, and representations of the Sklyanin algebra,” Russian Math. Surveys, 50, 1101–1150 (1995); arXiv:hep-th/9505039v1 (1995).

    Article  ADS  MathSciNet  Google Scholar 

  12. G. E. Arutyunov and S. A. Frolov, “On Hamiltonian structure of the spin Ruijsenaars–Schneider model,” J. Phys. A: Math. Gen., 31, 4203–4216 (1998); arXiv:hep-th/9703119v2 (1997); G. E. Arutyunov and E. Olivucci, “Hyperbolic spin Ruijsenaars–Schneider model from Poisson reduction,” Proc. Steklov Inst. Math., 309, 31–45 (2020); arXiv:1906.02619v2 [hep-th] (2019).

    Article  ADS  MathSciNet  Google Scholar 

  13. N. Reshetikhin, “Degenerately integrable systems,” J. Math. Sci. (N. Y.), 213, 769–785 (2016); arXiv:1509.00730v1 [math-ph] (2015).

    Article  MathSciNet  Google Scholar 

  14. L. Fehér, “Poisson–Lie analogues of spin Sutherland models,” Nucl. Phys. B, 949, 114807 (2019); arXiv:1809.01529v3 [math-ph] (2018); “Bi-Hamiltonian structure of a dynamical system introduced by Braden and Hone,” Nonlinearity, 32, 4377–4394 (2019); arXiv:1901.03558v2 [math-ph] (2019).

    Article  MathSciNet  Google Scholar 

  15. O. Chalykh and M. Fairon, “On the Hamiltonian formulation of the trigonometric spin Ruijsenaars–Schneider system,” arXiv:1811.08727v3 [math-ph] (2018); M. Fairon, “Spin versions of the complex trigonometric Ruijsenaars–Schneider model from cyclic quivers,” J. Integrable Syst., 4, xyz008 (2019); arXiv:1811.08717v2 [math-ph] (2018).

  16. A. P. Polychronakos, “Calogero–Moser models with noncommutative spin interactions,” Phys. Rev. Lett., 89, 126403 (2002); arXiv:hep-th/0112141v3 (2001); “The physics and mathematics of Calogero particles,” J. Phys. A: Math. Gen., 39, 12793–12945 (2006); arXiv:hep-th/0607033v2 (2006).

    Article  ADS  Google Scholar 

  17. A. Levin, M. Olshanetsky, A. Smirnov, and A. Zotov, “Characteristic classes of \(SL(N)\)-bundles and quantum dynamical elliptic \(R\)-matrices,” J. Phys. A: Math. Theor., 46, 035201 (2013); arXiv:1208.5750v1 [math-ph] (2012); A. V. Zotov and A. M. Levin, “Integrable model of interacting elliptic tops,” Theor. Math. Phys., 146, 45–52 (2006); A. V. Zotov and A. V. Smirnov, “Modifications of bundles, elliptic integrable systems, and related problems,” Theor. Math. Phys., 177, 1281–1338 (2013); A. Levin, M. Olshanetsky, A. Smirnov, and A. Zotov, “Characteristic classes and Hitchin systems: General construction,” Commun. Math. Phys., 316, 1–44 (2012); “Calogero–Moser systems for simple Lie groups and characteristic classes of bundles,” J. Geom. Phys., 62, 1810–1850 (2012).

    Article  ADS  MathSciNet  Google Scholar 

Download references

Funding

This research (including the results in Sec. 2) was performed at the Steklov Mathematical Institute of Russian Academy of Sciences and is supported by a grant from the Russian Science Foundation (Project No. 19-11-00062).

Author information

Authors and Affiliations

  1. Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia

    I. A. Sechin & A. V. Zotov

  2. Center for Advanced Studies, Skolkovo Institute of Science and Technology, Moscow, Russia

    I. A. Sechin

Authors
  1. I. A. Sechin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. V. Zotov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. V. Zotov.

Ethics declarations

The authors declare no conflicts of interest.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sechin, I.A., Zotov, A.V. Integrable system of generalized relativistic interacting tops. Theor Math Phys 205, 1291–1302 (2020). https://doi.org/10.1134/S0040577920100049

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0040577920100049

Keywords

Search

Navigation