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TMF, 2015, Volume 184, Number 1, Pages 41–56 (Mi tmf8877)  

This article is cited in 15 scientific papers (total in 15 papers)

Quantum Baxter–Belavin $R$-matrices and multidimensional Lax pairs for Painlevé VI

A. M. Levinab, M. A. Olshanetskyca, A. V. Zotovd

a Institute for Theoretical and Experimental Physics, Moscow, Russia
b Department of Mathematics, National Research University Higher School of Economics, Moscow, Russia
c Moscow Institute of Physics and Technology, Dolgoprudny, Moscow Oblast, Russia
d Steklov Mathematical Institute of RAS, Moscow, Russia

Abstract: Quantum elliptic $R$-matrices satisfy the associative Yang–Baxter equation in $\mathrm{Mat}(N)^{\otimes 2}$, which can be regarded as a noncommutative analogue of the Fay identity for the scalar Kronecker function. We present a broader list of $R$-matrix-valued identities for elliptic functions. In particular, we propose an analogue of the Fay identities in $\mathrm{Mat}(N)^{\otimes 2}$. As an application, we use the $\mathbb{Z}_N\times\mathbb{Z}_N$ elliptic $R$-matrix to construct $R$-matrix-valued $2N^2\times 2N^2$ Lax pairs for the Painlevé VI equation {(}in the elliptic form{\rm)} with four free constants. More precisely, the case with four free constants corresponds to odd $N$, and even $N$ corresponds to the case with a single constant in the equation.

Keywords: quantum $R$-matrix, multidimensional Lax pair, Painlevé equation

Funding Agency Grant Number
Russian Science Foundation 14-50-00005


DOI: https://doi.org/10.4213/tmf8877

Full text: PDF file (607 kB)
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English version:
Theoretical and Mathematical Physics, 2015, 184:1, 924–939

Bibliographic databases:

ArXiv: 1501.07351
Received: 26.02.2015

Citation: A. M. Levin, M. A. Olshanetsky, A. V. Zotov, “Quantum Baxter–Belavin $R$-matrices and multidimensional Lax pairs for Painlevé VI”, TMF, 184:1 (2015), 41–56; Theoret. and Math. Phys., 184:1 (2015), 924–939

Citation in format AMSBIB
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    Erratum

    This publication is cited in the following articles:
    1. M. Beketov, A. Liashyk, A. Zabrodin, A. Zotov, “Trigonometric version of quantumclassical duality in integrable systems”, Nuclear Phys. B, 903 (2016), 150–163 , arXiv: 1510.07509  mathnet  crossref  mathscinet  zmath  adsnasa  isi  scopus
    2. A. Levin, M. Olshanetsky, A. Zotov, “Yang–Baxter equations with two Planck constants”, J. Phys. A, 49:1 (2016), 014003 , 19 pp., Exactly Solved Models and Beyond: a special issue in honour of R. J. Baxter's 75th birthday, arXiv: 1507.02617  mathnet  crossref  mathscinet  zmath  isi  scopus
    3. A. V. Zotov, “Higher-order analogues of the unitarity condition for quantum $R$-matrices”, Theoret. and Math. Phys., 189:2 (2016), 1554–1562  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    4. 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  crossref  mathscinet  zmath  isi  elib  scopus
    5. I. Sechin, A. Zotov, “Associative Yang–Baxter equation for quantum (semi-)dynamical $R$-matrices”, J. Math. Phys., 57:5 (2016), 053505  crossref  mathscinet  zmath  isi  elib  scopus
    6. A. Zotov, “Relativistic elliptic matrix tops and finite Fourier transformations”, Mod. Phys. Lett. A, 32:32 (2017), 1750169  crossref  mathscinet  zmath  isi  scopus
    7. A. Grekov, A. Zotov, “On $R$-matrix valued Lax pairs for Calogero–Moser models”, J. Phys. A-Math. Theor., 51:31 (2018), 315202  crossref  isi  scopus
    8. I. Sechin, A. Zotov, “$R$ -matrix-valued Lax pairs and long-range spin chains”, Phys. Lett. B, 781 (2018), 1–7  crossref  mathscinet  isi  scopus
    9. A. V. Zotov, “Calogero–Moser model and $R$-matrix identities”, Theoret. and Math. Phys., 197:3 (2018), 1755–1770  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    10. T. Krasnov, A. Zotov, “Trigonometric integrable tops from solutions of associative Yang-Baxter equation”, Ann. Henri Poincare, 20:8 (2019), 2671–2697  crossref  isi
    11. A. Grekov, I. Sechin, A. Zotov, “Generalized model of interacting integrable tops”, J. High Energy Phys., 2019, no. 10, 081  crossref  mathscinet  isi
    12. 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  isi  elib
    13. A. Levin, M. Olshanetsky, A. Zotov, “Odd supersymmetric Kronecker elliptic function and Yang-Baxter equations”, J. Math. Phys., 61:10 (2020), 103504  crossref  mathscinet  isi
    14. A. Levin, M. Olshanetsky, A. Zotov, “Odd supersymmetrization of elliptic r-matrices”, J. Phys. A-Math. Theor., 53:18 (2020), 185202  crossref  mathscinet  isi
    15. E. S. Trunina, A. V. Zotov, “Multi-pole extension of the elliptic models of interacting integrable tops”, Theoret. and Math. Phys., 209:1 (2021), 1331–1356  mathnet  crossref  crossref  isi
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