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Set-theoretical solutions of the Zamolodchikov tetrahedron equation on associative rings and Liouville integrability

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Abstract

This paper is devoted to tetrahedron maps, which are set-theoretical solutions of the Zamolodchikov tetrahedron equation. We construct a family of tetrahedron maps on associative rings. The obtained maps are new to our knowledge. We show that matrix tetrahedron maps derived previously are a particular case of our construction. This provides an algebraic explanation of the fact that the matrix maps satisfy the tetrahedron equation. Also, Liouville integrability is established for some of the constructed maps.

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References

  1. A. B. Zamolodchikov, “Tetrahedra equations and integrable systems in three-dimensional space,” Sov. Phys. JETP, 52, 325–336 (1980).

    ADS  Google Scholar 

  2. A. B. Zamolodchikov, “Tetrahedron equations and the relativistic \(S\)-matrix of straight- strings in \((2+1)\)-dimensions,” Commun. Math. Phys., 79, 489–505 (1981).

    Article  ADS  MathSciNet  Google Scholar 

  3. V. V. Bazhanov and S. M. Sergeev, “Zamolodchikov’s tetrahedron equation and hidden structure of quantum groups,” J. Phys. A: Math. Theor., 39, 3295–3310 (2006).

    Article  ADS  MathSciNet  Google Scholar 

  4. V. V. Bazhanov, V. V. Mangazeev, and S. M. Sergeev, “Quantum geometry of three-dimensional lattices,” J. Stat. Mech., 2008, P07004, 27 pp. (2008).

    MathSciNet  MATH  Google Scholar 

  5. R. M. Kashaev, I. G. Korepanov, and S. M. Sergeev, “Functional tetrahedron equation,” Theoret. and Math. Phys., 117, 1402–1413 (1998).

    Article  ADS  MathSciNet  Google Scholar 

  6. P. Kassotakis, M. Nieszporski, V. Papageorgiou, and A. Tongas, “Tetrahedron maps and symmetries of three dimensional integrable discrete equations,” J. Math. Phys., 60, 123503, 18 pp. (2019).

    Article  ADS  MathSciNet  Google Scholar 

  7. J. M. Maillet and F. Nijhoff, “The tetrahedron equation and the four-simplex equation,” Phys. Lett. A, 134, 221–228 (1989).

    Article  ADS  MathSciNet  Google Scholar 

  8. D. V. Talalaev, “Tetrahedron equation: algebra, topology, and integrability,” Russian Math. Surveys, 76, 685–721 (2021).

    Article  ADS  MathSciNet  Google Scholar 

  9. A. Doliwa and R. M. Kashaev, “Non-commutative birational maps satisfying Zamolodchikov equation, and Desargues lattices,” J. Math. Phys., 61, 092704, 23 pp. (2020).

    Article  ADS  MathSciNet  Google Scholar 

  10. S. Igonin, S. Konstantinou-Rizos, “Algebraic and differential-geometric constructions of set-theoretical solutions to the Zamolodchikov tetrahedron equation,” arXiv: 2110.05998.

  11. I. G. Korepanov, G. I. Sharygin, D. V. Talalaev, “Cohomologies of \(n\)-simplex relations,” Math. Proc. Cambridge Philos. Soc., 161, 203–222 (2016).

    Article  ADS  MathSciNet  Google Scholar 

  12. A. Yoneyama, “Boundary from bulk integrability in three dimensions: 3D reflection maps from tetrahedron maps,” Math. Phys. Anal. Geom., 24, 21, 19 pp. (2021).

    Article  MathSciNet  Google Scholar 

  13. A. P. Fordy, “Periodic cluster mutations and related integrable maps,” J. Phys. A: Math. Theor., 47, 474003, 44 pp. (2014).

    Article  MathSciNet  Google Scholar 

  14. S. Konstantinou-Rizos and A. V. Mikhailov, “Darboux transformations, finite reduction groups and related Yang–Baxter maps,” J. Phys. A: Math. Theor., 46, 425201, 16 pp. (2013).

    Article  MathSciNet  Google Scholar 

  15. A. P. Veselov, “Integrable maps,” Russian Math. Surveys, 46, 1–51 (1991).

    Article  ADS  MathSciNet  Google Scholar 

  16. S. M. Sergeev, “Solutions of the functional tetrahedron equation connected with the local Yang–Baxter equation for the ferro-electric condition,” Lett. Math. Phys., 45, 113–119 (1998).

    Article  MathSciNet  Google Scholar 

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Acknowledgments

The author thanks S. Konstantinou-Rizos, S. M. Sergeev, and D. V. Talalaev for the useful discussions.

Funding

The work on Sections 1 and 2 was supported by the Russian Science Foundation grant No. 21-71-30011. The work on Section 3 was carried out within the framework of a development program for the Regional Scientific and Educational Mathematical Center of the Demidov Yaroslavl State University with financial support from the Ministry of Science and Higher Education of the Russian Federation (Agreement on provision of subsidy from the federal budget No. 075-02-2022-886).

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Authors and Affiliations

  1. Center of Integrable Systems, Demidov Yaroslavl State University, Yaroslavl, Russia

    S. A. Igonin

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  1. S. A. Igonin
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Correspondence to S. A. Igonin.

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Translated from Teoreticheskaya i Matematicheskaya Fizika, 2022, Vol. 212, pp. 263–272 https://doi.org/10.4213/tmf10275.

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Igonin, S.A. Set-theoretical solutions of the Zamolodchikov tetrahedron equation on associative rings and Liouville integrability. Theor Math Phys 212, 1116–1124 (2022). https://doi.org/10.1134/S0040577922080074

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  • DOI: https://doi.org/10.1134/S0040577922080074

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