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

This article is cited in 10 scientific papers (total in 10 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  adsnasa  isi  elib
    10. Krasnov T. Zotov A., “Trigonometric Integrable Tops From Solutions of Associative Yang-Baxter Equation”, Ann. Henri Poincare, 20:8 (2019), 2671–2697  crossref  isi
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