Classification of isomonodromy problems on elliptic curves

A. M. Levin   ; M. A. Olshanetsky   ; A. V. Zotov

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 69 (2014) no. 1, pp. 35-118

Résumé

This paper describes isomonodromy problems in terms of flat $G$-bundles over punctured elliptic curves $\Sigma_\tau$ and connections with regular singularities at marked points. The bundles are classified by their characteristic classes, which are elements of the second cohomology group $H^2(\Sigma_\tau,{\mathscr Z}(G))$, where ${\mathscr Z}(G)$ is the centre of $G$. For any complex simple Lie group $G$ and any characteristic class the moduli space of flat connections is defined, and for them the monodromy-preserving deformation equations are given in Hamiltonian form together with the corresponding Lax representation. In particular, they include the Painlevé VI equation, its multicomponent generalizations, and the elliptic Schlesinger equations. The general construction is described for punctured complex curves of arbitrary genus. The Drinfeld–Simpson (double coset) description of the moduli space of Higgs bundles is generalized to the case of the space of flat connections. This local description makes it possible to establish the Symplectic Hecke Correspondence for a wide class of monodromy-preserving problems classified by the characteristic classes of the underlying bundles. In particular, the Painlevé VI equation can be described in terms of $\operatorname{SL}(2,{\mathbb C})$-bundles. Since ${\mathscr Z}(\operatorname{SL}(2,{\mathbb C}))={\mathbb Z}_2$, the Painlevé VI equation has two representations related by the Hecke transformation: 1) as the well-known elliptic form of the Painlevé VI equation (for trivial bundles); 2) as the non-autonomous Zhukovsky–Volterra gyrostat (for non-trivial bundles). Bibliography: 123 titles.

Détail

Keywords: monodromy-preserving deformations, flat connections, Schlesinger systems, Higgs bundles.
Mots-clés : Painlevé equations

Bibliographie

[1] P. Painlevé, “Mémoire sur les équations différentielles dont l'intégrale générale est uniforme”, Bull. Soc. Math. France, 28 (1900), 201–261 | MR | Zbl

[2] P. Painlevé, “Sur les équations différentielles du second ordre et d'ordre supérieur dont l'intégrale générale est uniforme”, Acta Math., 25:1 (1902), 1–85 | DOI | MR | Zbl

[3] B. Gambier, “Sur les équations différentielles du second ordre et du premier degré dont l'intégrale générale est à points critique fixes”, Acta Math., 33:1 (1910), 1–55 | DOI | MR | Zbl

[4] R. Fuchs, “Über lineare homogene Differentialgleichungen zweiter Ordnung mit drei im Endlichen gelegenen wesentlich singulären Stellen”, Math. Ann., 63:3 (1907), 301–321 | DOI | MR | Zbl

[5] R. Garnier, “Sur des équations différentielles du troisième ordre dont l'intégrale générale est uniforme et sur une classe d'équations nouvelles d'ordre supérieur dont l'intégrale générale a ses points critique fixés”, Ann. Sci. École Norm. Sup. (3), 29 (1912), 1–126 | MR | Zbl

[6] L. Schlesinger, “Über eine Klasse von Differentialsystemen beliebiger Ordnung mit festen kritischen Punkten”, J. Reine Angew. Math., 141 (1912), 96–145 | DOI | Zbl

[7] H. Flaschka, A. C. Newell, “Monodromy- and spectrum-preserving deformations. I”, Comm. Math. Phys., 76:1 (1980), 65–116 | DOI | MR | Zbl

[8] H. Flaschka, A. C. Newell, “Multiphase similarity solutions of integrable evolution equations”, Phys. D, 3:1-2 (1981), 203–221 | DOI | Zbl

[9] M. Jimbo, T. Miwa, K. Ueno, “Monodromy preserving deformation of linear ordinary differential equations with rational coefficients: I. General theory and $\tau$-function”, Phys. D, 2:2 (1981), 306–352 | DOI | MR | Zbl

[10] M. Jimbo, T. Miwa, “Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. II”, Phys. D, 2:3 (1981), 407–448 | DOI | MR | Zbl

[11] M. Jimbo, T. Miwa, “Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. III”, Phys. D, 4:1 (1981), 26–46 | DOI | MR | Zbl

[12] E. L. Ince, Ordinary differential equations, Dover Publications, New York, 1956, viii+558 pp. | MR | Zbl

[13] K. Iwasaki, H. Kimura, S. Shimomura, M. Yoshida, From Gauss to Painlevé. A modern theory of special functions, Aspects Math., E16, Friedr. Vieweg Sohn, Braunschweig, 1991, xii+347 pp. | DOI | MR | Zbl

[14] The Painlevé property. One century later, CRM Ser. Math. Phys., ed. R. Conte, Springer-Verlag, New York, 1999, xxvi+810 pp. | DOI | MR | Zbl

[15] A. R. Its, V. Yu. Novokshenov, The isomonodromic deformation method in the theory of Painlevé equations, Lecture Notes in Math., 1191, Springer-Verlag, Berlin, 1986, iv+313 pp. | DOI | MR | Zbl

[16] A. S. Fokas, A. R. Its, A. A. Kapaev, V. Yu. Novokshenov, Painlevé transcendents. The Riemann–Hilbert approach, Math. Surveys Monogr., 128, Amer. Math. Soc., Providence, RI, 2006, xii+553 pp. | DOI | MR | Zbl

[17] Painlevé transcedents. Their asymptotics and physical applications, Proceedings of the NATO Advanced Research Workshop (Sainte-Adèle, Canada, 1990), NATO Adv. Sci. Inst. Ser. B: Phys., 278, eds. D. Levi, P. Winternitz, Springer-Verlag, New York, NY, 1992, xxvi+446 pp. | DOI | Zbl

[18] B. Dubrovin, “Geometry of 2D topological field theories”, Integrable systems and quantum groups (Montecatini Terme, 1993), Lecture Notes in Math., 1620, Springer, Berlin, 1996, 120–348 | DOI | MR | Zbl

[19] A. A. Bolibrukh, Fuksovy differentsialnye uravneniya i golomorfnye rassloeniya, MTsNMO, M., 2000, 127 pp.

[20] A. A. Bolibruch, “On isomonodromic deformations of Fuchsian systems”, J. Dynam. Control Syst., 3:4 (1997), 589–604 | DOI | MR | Zbl

[21] S. Yu. Slavyanov, W. Lay, Special functions. A unified theory based on singularities, with a foreword by A. Seeger, Oxford Math. Monogr., Oxford Univ. Press, Oxford, 2000, xvi+293 pp. | MR | Zbl

[22] I. Krichever, “Isomonodromy equations on algebraic curves, canonical transformations and Whitham equations”, Mosc. Math. J., 2:4 (2002), 717–752 | MR | Zbl

[23] P. Boalch, “From Klein to Painlevé via Fourier, Laplace and Jimbo”, Proc. London Math. Soc. (3), 90:1 (2005), 167–208 | DOI | MR | Zbl

[24] R. R. Gontsov, V. A. Poberezhnyi, G. F. Khel'minck, “On deformations of linear differential systems”, Russian Math. Surveys, 66:1 (2011), 63–105 | DOI | DOI | MR | Zbl

[25] K. Okamoto, “On Fuchs' problem on a torus”, Japan–United States seminar on ordinary differential and functional equations (Kyoto, 1971), Lecture Notes in Math., 243, Springer, Berlin, 1971, 277–280 | DOI | MR | Zbl

[26] K. Iwasaki, “Moduli and deformation for Fuchsian projective connections on a Riemann surface”, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 38:3 (1991), 431–531 | MR | Zbl

[27] S. Kawai, Deformation of complex structures on a torus and monodromy preserving deformation, preprint, University of Tokyo, 1995

[28] A. M. Levin, M. A. Olshanetsky, Classical limit of the Knizhnik–Zamolodchikov–Bernard equations as hierarchy of isomonodromic deformations. Free fields approach, 1997 (v3 – 1998), 40 pp., arXiv: hep-th/9709207

[29] M. A. Olshanetsky, “Generalized Hitchin systems and the Knizhnik–Zamolodchikov–Bernard equation on ellipic curves”, Lett. Math. Phys., 42:1 (1997), 59–71 ; (1995), 11 pp., arXiv: hep-th/9510143 | DOI | MR | Zbl

[30] A. Levin, M. Olshanetsky, “Hierarchies of isomonodromic deformations and Hitchin systems”, Moscow seminar in mathematical physics, Amer. Math. Soc. Transl. Ser. 2, 191, Amer. Math. Soc., Providence, RI, 1999, 223–262 | MR | Zbl

[31] N. Hitchin, “Stable bundles and integrable systems”, Duke Math. J., 54:1 (1987), 91–114 | DOI | MR | Zbl

[32] N. Nekrasov, “Holomorphic bundles and many-body systems”, Comm. Math. Phys., 180:3 (1996), 587–603 ; (1995), 21 pp., arXiv: hep-th/9503157 | DOI | MR | Zbl

[33] A. Gorsky, N. Nekrasov, Elliptic Calogero–Moser system from two dimensional current algebra, 1994, 8 pp., arXiv: hep-th/9401021

[34] B. Enriquez, V. Rubtsov, “Hitchin systems, higher Gaudin operators and $r$-matrices”, Math. Res. Lett., 3:3 (1996), 343–357 | DOI | MR | Zbl

[35] A. Levin, M. Olshanetsky, “Double coset construction of moduli space of holomorphic bundles and Hitchin systems”, Commun. Math. Phys., 188:2 (1997), 449–466 ; (1996), 19 pp., arXiv: alg-geom/9605005 | DOI | MR | Zbl

[36] D. Korotkin, H. Samtleben, “On the quantization of isomonodromic deformations on the torus”, Internat. J. Modern Phys. A, 12:11 (1997), 2013–2029 | DOI | MR | Zbl

[37] G. Kuroki, T. Takebe, “Twisted Wess–Zumino–Witten models on elliptic curves”, Comm. Math. Phys., 190:1 (1997), 1–56 ; (1996 (v4 – 1997)), 55 pp., arXiv: q-alg/9612033 | DOI | MR | Zbl

[38] K. Takasaki, “Gaudin model, KZ equation, and isomonodromic problem on torus”, Lett. Math. Phys., 44:2 (1998), 143–156 ; (1997), 15 pp., arXiv: hep-th/9711058 | DOI | MR | Zbl

[39] J. Harnad, M. A. Wisse, “Loop algebra moment maps and Hamiltonian models for the Painlevé transcendants”, Mechanics day (Waterloo, ON, 1992), Fields Inst. Commun., 7, Amer. Math. Soc., Providence, RI, 1996, 155–169 ; 1993, 14 pp., arXiv: hep-th/9305027 | MR | Zbl

[40] A. G. Reiman, M. A. Semenov-Tian-Schanskii, “Lie algebras and Lax equations with spectral parameter on elliptic curve”, J. Soviet Math., 46:1 (1989), 1631–1640 | DOI | MR | Zbl

[41] I. M. Krichever, “Elliptic solutions of the Kadomtsev–Petviashvili equation and integrable systems of particles”, Funct. Anal. Appl., 14:4 (1980), 282–290 | DOI | MR | Zbl

[42] Yu. Chernyakov, A. M. Levin, M. Olshanetsky, A. Zotov, “Elliptic Schlesinger system and Painlevé VI”, J. Phys. A, 39:39 (2006), 12083–12101 | DOI | MR | Zbl

[43] M. A. Olshanetsky, A. V. Zotov, “Isomonodromic problems on elliptic curve, rigid tops and reflection equations”, Elliptic integrable systems, Rokko Lectures in Math., 18, Kobe Univ., Kobe, Japan, 2005, 149–171

[44] A. M. Levin, M. A. Olshanetsky, “Painlevé–Calogero correspondence”, Calogero–Moser–Sutherland models (Montréal, QC, 1997), CRM Ser. Math. Phys., Springer, New York, 2000, 313–332 ; 1997, 17 pp., arXiv: alg-geom/9706010 | MR

[45] A. Levin, M. Olshanetsky, A. Smirnov, A. Zotov, “Characteristic classes and Hitchin systems. General construction”, Comm. Math. Phys., 316:1 (2012), 1–44 ; Characteristic classes and integrable systems. General construction, 2010, 52 pp., arXiv: 1006.0702 | DOI | MR | Zbl

[46] A. Levin, M. Olshanetsky, A. Smirnov, A. Zotov, “Calogero–Moser systems for simple Lie groups and characteristic classes of bundles”, J. Geom. Phys., 62:8 (2012), 1810–1850 ; Characteristic classes and integrable systems for simple Lie groups, 2010, 51 pp., arXiv: 1007.4127 | DOI | MR | Zbl

[47] A. V. Zotov, A. V. Smirnov, “Modifications of bundles, elliptic integrable systems, and related problems”, Theoret. and Math. Phys., 177:1 (2013), 1281–1338 | DOI | DOI

[48] A. M. Levin, M. A. Olshanetsky, A. V. Smirnov, A. V. Zotov, “Hecke transformations of conformal blocks in WZW theory. I. KZB equations for non-trivial bundles”, SIGMA, 8 (2012), 095, 37 pp. | DOI | MR | Zbl

[49] A. M. Levin, M. A. Olshanetsky, A. V. Zotov, “Monopoles and modifications of bundles over elliptic curves”, SIGMA, 5 (2009), 022, 12 pp. ; (2008 (v2 – 2009)), 22 pp., arXiv: 0811.3056 | DOI | MR | Zbl

[50] M. F. Atiyah, “Vector bundles over an elliptic curve”, Proc. London Math. Soc. (3), 7 (1957), 414–452 | DOI | MR | Zbl

[51] A. V. Levin, A. M. Zotov, “Integrable model of interacting elliptic tops”, Theoret. and Math. Phys., 146:1 (2006), 45–52 | DOI | DOI | MR | Zbl

[52] A. M. Levin, M. A. Olshanetsky, A. V. Smirnov, A. V. Zotov, “Characteristic classes of $\mathrm{SL}(N,\mathbb C)$-bundles and quantum dynamical elliptic $\mathrm R$-matrices”, J. Phys. A, 46:3 (2013), 035201, 25 pp. ; (2012), 27 pp., arXiv: 1208.5750 | DOI | MR | Zbl

[53] E. Franco, O. Garcia-Prada, P. E. Newstead, Higgs bundles over elliptic curves, 2013, 41 pp., arXiv: 1302.2881

[54] N. Hitchin, Higgs bundles and characteristic classes, 2013, 18 pp., arXiv: 1308.4603

[55] D. Baraglia, L. P. Schaposnik, Real structures on moduli spaces of Higgs bundles, 2013, 19 pp., arXiv: 1309.1195

[56] M. F. Atiyah, R. Bott, “The Yang–Mills equations over Riemann surfaces”, Philos. Trans. Roy. Soc. London Ser. A, 308:1505 (1983), 523–615 | DOI | MR | Zbl

[57] V. G. Drinfeld, C. Simpson, “$B$-structures on $G$-bundles and local triviality”, Math. Res. Lett., 2:6 (1995), 823–829 | DOI | MR | Zbl

[58] A. Beauville, Y. Laszlo, “Un lemme de descente”, C. R. Acad. Sci. Paris Sér. I Math., 320:3 (1995), 335–340 | MR | Zbl

[59] D. Arinkin, S. Lysenko, “Isomorphisms between moduli spaces of $SL(2)$-bundles with connections on ${\mathbb P}^1\setminus\{x_1,\dots,x_4\}$”, Math. Res. Lett., 4:2 (1997), 181–190 | DOI | MR | Zbl

[60] D. Arinkin, S. Lysenko, “On the moduli spaces of SL(2)-bundles with connections on ${\mathbf P}^1\setminus\{x_1,\dots,x_4\}$”, Int. Math. Res. Not., 1997:19 (1997), 983–999 | DOI | MR | Zbl

[61] A. M. Levin, M. A. Olshanetsky, A. Zotov, “Hitchin systems–symplectic Hecke correspondence and two-dimensional version”, Comm. Math. Phys., 236:1 (2003), 93–133 ; (2001 (v3 – 2002)), 39 pp., arXiv: nlin/0110045 | DOI | MR | Zbl

[62] P. Painlevé, “Sur les équations différentielles du second ordre à points critiques fixes”, C. R. Acad. Sci. (Paris), 143 (1906), 1111–1117 | Zbl

[63] Yu. I. Manin, “Sixth Painlevé equation, universal elliptic curve, and mirror of ${\mathbf{P}} ^2$”, Geometry of differential equations, Amer. Math. Soc. Transl. Ser. 2, 186, Amer. Math. Soc., Providence, RI, 1998, 131–151 | MR | Zbl

[64] A. M. Levin, M. A. Olshanetsky, A. V. Zotov, “Painlevé VI, rigid tops and reflection equation”, Comm. Math. Phys., 268:1 (2006), 67–103 ; (2005 (v2 – 2006)), 32 pp., arXiv: math/0508058 | DOI | MR | Zbl

[65] A. Zotov, “Elliptic linear problem for Calogero–Inozemtsev model and Painlevé VI equation”, Lett. Math. Phys., 67:2 (2004), 153–165 ; (2003), 13 pp., arXiv: hep-th/0310260 | DOI | MR | Zbl

[66] P. Boutroux, “Recherches sur les transcendantes de M. Painlevé et l'étude asymptotique des équations différentielles du second ordre”, Ann. Sci. École Norm. Super. (3), 30 (1913), 255–375 ; suite, 31 (1914), 99–159 | MR | Zbl | MR | Zbl

[67] R. Garnier, “Étude de l'intégrale générale de l'équation VI de M. Painlevé dans le voisinage de ses singularités transcendantes”, Ann. Sci. École Norm. Sup. (3), 34 (1917), 239–353 | MR | Zbl

[68] I. M. Krichever, “Isomonodromy equations on algebraic curves, canonical transformations and Whitham equations”, Mosc. Math. J., 2:4 (2002), 717–752 ; (2001), 38 pp., arXiv: hep-th/0112096 | MR | Zbl

[69] C. T. Simpson, “Harmonic bundles on noncompact curves”, J. Amer. Math. Soc., 3:3 (1990), 713–770 | DOI | MR | Zbl

[70] A. Pressley, G. Segal, Loop groups, Oxford Math. Monogr., Oxford Univ. Press, New York, 1986, viii+318 pp. | MR | MR | Zbl | Zbl

[71] V. G. Kac, Infinite-dimensional Lie algebras, 3rd ed., Cambridge Univ. Press, Cambridge, 1990, 400 pp. | DOI | MR | MR | Zbl | Zbl

[72] D. Arinkin, Moduli of connections with a small parameter on a curve, 2004, 19 pp., arXiv: math/0409373

[73] V. Arnold, Mathematical methods of classical mechanics, Grad. Texts in Math., 60, Springer-Verlag, New York–Heidelberg, 1978, x+462 pp. | DOI | MR | MR | Zbl | Zbl

[74] I. M. Krichever, “Method of averaging for two-dimensional “integrable” equations”, Funct. Anal. Appl., 22:3 (1988), 200–213 | DOI | MR | Zbl

[75] K. Okamoto, “On the $\tau$-function of the Painlevé equations”, Phys. D, 2:3 (1981), 525–535 | DOI | MR | Zbl

[76] K. Takasaki, “Spectral curves and Whitham equations in isomonodromic problems of Schlesinger type”, Asian J. Math., 2:4 (1998), 1049–1078 ; (1997), 41 pp., arXiv: solv-int/9704004 | DOI | MR | Zbl

[77] E. Looijenga, “Root systems and elliptic curves”, Invent. Math., 38:1 (1976/77), 17–32 | DOI | MR | Zbl

[78] I. N. Bernstein, O. V. Schwarzman, “Chevalley's theorem for complex crystallographic Coxeter groups”, Funct. Anal. Appl., 12:4 (1978), 308–310 | DOI | MR | Zbl

[79] J. Bernstein, O. Schwarzman, “Complex crystallographic Coxeter groups and affine root systems”, J. Nonlinear Math. Phys., 13:2 (2006), 163–182 | DOI | MR | Zbl

[80] C. Schweigert, “On moduli spaces of flat connections with non-simply connected structure group”, Nuclear Phys. B, 492:3 (1997), 743–755 | DOI | MR | Zbl

[81] R. Friedman, J. W. Morgan, Holomorphic principal bundles over elliptic curves, 1998, 68 pp., arXiv: math/9811130

[82] M. S. Narasimhan, C. S. Seshadri, “Stable and unitary vector bundles on a compact Riemann surface”, Ann. of Math. (2), 82:3 (1965), 540–567 | DOI | MR | Zbl

[83] N. Bourbaki, Lie groups and Lie algebras. Chapters 4–6, transl. from the 1968 French original by A. Pressley, Elem. Math. (Berlin), Springer-Verlag, Berlin, 2002, xii+300 pp. | MR | MR | Zbl | Zbl

[84] P. A. M. Dirac, Lectures on quantum mechanics, Belfer Grad. Sch. Sci. Monogr. Ser., 2, Academic Press, Inc., New York, 1967, v+87 pp. | MR | MR

[85] H. W. Braden, V. A. Dolgushev, M. A. Olshanetsky, A. V. Zotov, “Classical $r$-matrices and the Feigin–Odesskii algebra via Hamiltonian and Poisson reductions”, J. Phys. A, 36:5 (2003), 6979–7000 | DOI | MR | Zbl

[86] P. Etingof, A. Varchenko, “Geometry and classification of solutions of the classical dynamical Yang–Baxter equation”, Comm. Math. Phys., 192:1 (1998), 77–120 | DOI | MR | Zbl

[87] O. Schiffmann, “On classification of dynamical r-matrices”, Math. Res. Lett., 5:1-2 (1998), 13–30 | DOI | MR | Zbl

[88] P. Etingof, O. Schiffmann, “Twisted traces of intertwiners for Kac–Moody algebras and classical dynamical r-matrices corresponding to generalized Belavin–Drinfeld triples”, Math. Res. Lett., 6:5-6 (1999), 593–612 | DOI | MR | Zbl

[89] L. Fehér, B. G. Pusztai, “Generalizations of Felder's elliptic dynamical $r$-matrices associated with twisted loop algebras of self-dual Lie algebras”, Nuclear Phys. B, 621:3 (2002), 622–642 ; (2001), 22 pp., arXiv: math/0109132 | DOI | MR | Zbl

[90] O. Babelon, C.-M. Viallet, “Hamiltonian structures and Lax equations”, Phys. Lett. B, 237:3-4 (1990), 411–416 | DOI | MR

[91] E. Billey, J. Avan, O. Babelon, “The r-matrix structure of the Euler–Calogero–Moser model”, Phys. Lett. A, 186:1-2 (1994), 114–118 | DOI | MR | Zbl

[92] K. Takasaki, “Painlevé–Calogero correspondence revisited”, J. Math. Phys., 42:3 (2001), 1443–1473 | DOI | MR | Zbl

[93] B. I. Suleimanov, “The Hamilton property of Painlevé equations and the method of isomonodromic deformations”, Differential Equations, 30:5 (1994), 726–732 | MR | Zbl

[94] A. Zabrodin, A. Zotov, “Quantum Painlevé–Calogero correspondence”, J. Math. Phys., 53:7 (2012), 073507, 19 pp. ; (2011), 55 pp., arXiv: 1107.5672 | DOI | MR | Zbl

[95] A. Zabrodin, A. Zotov, “Quantum Painlevé–Calogero correspondence for Painlevé VI”, J. Math. Phys., 53:7 (2012), 073508, 19 pp. | DOI | MR | Zbl

[96] A. Zabrodin, A. Zotov, Classical-quantum correspondence and functional relations for Painlevé equations, 2012, 38 pp., arXiv: 1212.5813

[97] G. Aminov, S. Arthamonov, A. Smirnov, A. Zotov, Modifications of bundles as generating functions of Lax operators, Preprint ITEP-TH-27/13, 2013

[98] G. Felder, C. Wieczerkowski, “Conformal blocks on elliptic curves and the Knizhnik–Zamolodchikov–Bernard equations”, Comm. Math. Phys., 176:1 (1996), 133–161 ; (1994), 32 pp., arXiv: hep-th/9411004 | DOI | MR | Zbl

[99] N. J. Hitchin, “Flat connections and geometric quantization”, Comm. Math. Phys., 131:2 (1990), 347–380 | DOI | MR | Zbl

[100] G. Felder, “The KZB equations on Riemann surfaces”, Symétries quantiques (Les Houches, 1995), North-Holland, Amsterdam, 1998, 687–725 ; 1996, 29 pp., arXiv: hep-th/9609153 | MR | Zbl

[101] D. A. Ivanov, “Knizhnik–Zamolodchikov–Bernard equations on Riemann surfaces”, Internat. J. Modern Phys. A, 10:17 (1995), 2507–2536 ; (1994), 23 pp., arXiv: hep-th/9410091 | DOI | MR | Zbl

[102] N. Reshetikhin, “The Knizhnik–Zamolodchikov system as a deformation of the isomonodromy problem”, Lett. Math. Phys., 26:3 (1992), 167–177 | DOI | MR | Zbl

[103] J. Harnard, “Quantum isomonodromic deformations and the Knizhnik–Zamolodchikov equations”, Symmetries and integrability of difference equations (Estérel, PQ, 1994), CRM Proc. Lecture Notes, 9, Amer. Math. Soc., Providence, RI, 1996, 155–161 ; 1994, 9 pp., arXiv: hep-th/9406078 | MR | Zbl

[104] L. D. Landau, E. M. Lifshitz, Course of theoretical physics, v. 3, Addison-Wesley Series in Advanced Physics, Quantum mechanics: non-relativistic theory, Pergamon Press Ltd., London–Paris; Addison-Wesley Publishing Co., Inc., Reading, MA, 1958, xii+515 pp. | MR | MR | Zbl

[105] A. V. Zotov, “1+1 Gaudin model”, SIGMA, 7 (2011), 067, 26 pp. ; (2010 (v2 – 2011)), 26 pp., arXiv: 1012.1072 | DOI | MR | Zbl

[106] A. V. Zotov, “Classical integrable systems and their field-theoretical generalizations”, Phys. Particles Nuclei, 37:3 (2006), 400–443 | DOI

[107] G. Aminov, S. Arthamonov, A. Levin, M. Olshanetsky, A. Zotov, Painlevé field theory, 2013, 67 pp., arXiv: 1306.3265

[108] V. I. Arnold, B. A. Khesin, Topological methods in hydrodynamics, Appl. Math. Sci., 125, Springer-Verlag, New York, 1998, xvi+374 pp. | MR | Zbl

[109] V. I. Arnold, “Gamiltonovost uravnenii Eilera dinamiki tverdogo tela i idealnoi zhidkosti”, UMN, 24:3(147) (1969), 225–226 | MR | Zbl

[110] B. Khesin, A. Levin, M. Olshanetsky, “Bihamiltonian structures and quadratic algebras in hydrodynamics and on non-commutative torus”, Comm. Math. Phys., 250:3 (2004), 581–612 | DOI | MR | Zbl

[111] N. E. Zhukovskii, “O dvizhenii tverdogo tela, imeyuschego polosti, napolnennye odnorodnoi kapelnoi zhidkostyu. I, II, III”, Zhurnal Russkogo fiziko-khimicheskogo obschestva, 17 (1885), 81–113, 145–199, 231–280

[112] V. Volterra, “Sur la théorie des variations des latitudes”, Acta Math., 22:1 (1899), 201–357 | DOI | MR | Zbl

[113] E. K. Sklyanin, “Separation of variables. New trends”, Quantum field theory, integrable models and beyond (Kyoto, 1994), Progr. Theoret. Phys. Suppl., 1995, no. 118, 35–60 | DOI | MR | Zbl

[114] A. V. Zotov, Yu. B. Chernyakov, “Integrable many-body systems via the Inosemtsev limit”, Theoret. and Math. Phys., 129:2 (2001), 1526–1542 | DOI | DOI | MR | Zbl

[115] A. V. Smirnov, “Integrable $sl(N,\mathbb C)$ tops as Calogero–Moser systems”, Theoret. and Math. Phys., 158:3 (2009), 300–312 ; Correspondence between Calogero–Moser systems and integrable SL(N,$\mathbb C$) Euler–Arnold tops, 2008, 14 pp., arXiv: 0809.2187 | DOI | DOI | MR | Zbl

[116] G. Aminov, S. Arthamonov, New $2\times 2$-matrix linear problems for the Painlevé equations, 2011 (v2 – 2012), 18 pp., arXiv: 1112.4688

[117] G. Aminov, S. Artamonov, “Degenerating the elliptic Schlesinger system”, Theoret. and Math. Phys., 174:1 (2013), 1–20 | DOI | DOI

[118] A. Levin, A. Zotov, “On rational and elliptic forms of Painlevé VI equation”, Moscow Seminar on mathematical physics. II, Amer. Math. Soc. Transl. Ser. 2, 221, Amer. Math. Soc., Providence, RI, 2007, 173–183 | MR | Zbl

[119] A. L. Onishchik, È. B. Vinberg, Lie groups and algebraic groups, Springer Ser. Soviet Math., Springer-Verlag, Berlin, 1990, xx+328 pp. | DOI | MR | MR | Zbl | Zbl

[120] D. B. Fairlie, P. Fletcher, C. K. Zachos, “Infinite dimensional algebras and a trigonometric basis for the classical Lie algebras”, J. Math. Phys., 31:5 (1990), 1088–1094 | DOI | MR | Zbl

[121] D. Mumford, Tata lectures on theta, v. I, Progr. Math., 28, Birkhäuser Boston, Inc., Boston, MA, 1983, xiii+235 pp. ; v. II, Progr. Math., 43, 1984, xiv+272 pp. | DOI | MR | Zbl | MR | Zbl

[122] A. Weil, Elliptic functions according to Eisenstein and Kronecker, Ergeb. Math. Grenzgeb., 88, Springer-Verlag, Berlin–New York, 1976, ii+93 pp. | MR | MR | Zbl

[123] N. L. Gordeev, V. L. Popov, “Automorphism groups of finite dimensional simple algebras”, Ann. of Math. (2), 158:3 (2003), 1041–1065 | DOI | MR | Zbl