Rational top and its classical r -matrix

2014

We construct special rational gl N Knizhnik-Zamolodchikov-Bernard (KZB) equations withÑ punctures by deformation of the corresponding quantum gl N rational R-matrix. They have two parameters. The limit of the first one brings the model to the ordinary rational KZ equation. Another one is τ . At the level of classical mechanics the deformation parameter τ allows to extend the previously obtained modified Gaudin models to the modified Schlesinger systems. Next, we notice that the identities underlying generic (elliptic) KZB equations follow from some additional relations for the properly normalized R-matrices. The relations are noncommutative analogues of identities for (scalar) elliptic functions. The simplest one is the unitarity condition. The quadratic (in R matrices) relations are generated by noncommutative Fay identities. In particular, one can derive the quantum Yang-Baxter equations from the Fay identities. The cubic relations provide identities for the KZB equations as well as quadratic relations for the classical r-matrices which can be treated as halves of the classical Yang-Baxter equation. At last we discuss the R-matrix valued linear problems which provide glÑ CM models and Painlevé equations via the above mentioned identities. The role of the spectral parameter plays the Planck constant of the quantum R-matrix. When the quantum gl N R-matrix is scalar (N = 1) the linear problem reproduces the Krichever's ansatz for the Lax matrices with spectral parameter for the glÑ CM models. The linear problems for the quantum CM models generalize the KZ equations in the same way as the Lax pairs with spectral parameter generalize those without it.

“…At the same time the rest of our construction works for ordinary XXX (and XXZ) R-matrices as well. 3 The purpose of the paper. is twofold.…”

Section: Jhep10(2014)109mentioning

confidence: 99%

Section: Jhep10(2014)109mentioning

confidence: 99%

Planck constant as spectral parameter in integrable systems and KZB equations

Levin

Olshanetsky

2014

“…The first one is to give description of the relativistic classical tops arising from the quantum R-matrices following (1.5). The second -is to derive new rational quantum R-matrix from the corresponding relativistic top via (1.9), which we obtain in a different way -by applying gauge transformation of Hecke type [6,[47][48][49][50][51][52][53]75] to the rational Ruijsenaars-Schneider (RS) model [69,70].…”

Section: Jhep07(2014)012mentioning

confidence: 99%

“…The latter was introduced in [6], where the non-relativistic rational top was constructed similarly starting from the rational Calogero-Moser (CM) model [19][20][21][22]. The transformation g(z) is known for the quantum elliptic and trigonometric RS models [7,36] (where the quantum IRF-Vertex JHEP07(2014)012…”

Section: Jhep07(2014)012mentioning

confidence: 99%

See 1 more Smart Citation

Relativistic classical integrable tops and quantum R-matrices

Levin

Olshanetsky

2014

We describe classical top-like integrable systems arising from the quantum exchange relations and corresponding Sklyanin algebras. The Lax operator is expressed in terms of the quantum non-dynamical R-matrix even at the classical level, where the Planck constant plays the role of the relativistic deformation parameter in the sense of Ruijsenaars and Schneider (RS). The integrable systems (relativistic tops) are described as multidimensional Euler tops, and the inertia tensors are written in terms of the quantum and classical R-matrices. A particular case of gl N system is gauge equivalent to the N -particle RS model while a generic top is related to the spin generalization of the RS model. The simple relation between quantum R-matrices and classical Lax operators is exploited in two ways. In the elliptic case we use the Belavin's quantum R-matrix to describe the relativistic classical tops. Also by the passage to the noncommutative torus we study the large N limit corresponding to the relativistic version of the nonlocal 2d elliptic hydrodynamics. Conversely, in the rational case we obtain a new gl N quantum rational non-dynamical R-matrix via the relativistic top, which we get in a different way -using the factorized form of the RS Lax operator and the classical Symplectic Hecke (gauge) transformation. In particular case of gl 2 the quantum rational R-matrix is 11-vertex. It was previously found by Cherednik. At last, we describe the integrable spin chains and Gaudin models related to the obtained R-matrix.

Generalized model of interacting integrable tops

Grekov

Сечин

2019

We introduce a family of classical integrable systems describing dynamics of M interacting gl N integrable tops. It extends the previously known model of interacting elliptic tops. Our construction is based on the GL N R-matrix satisfying the associative Yang-Baxter equation. The obtained systems can be considered as extensions of the spin type Calogero-Moser models with (the classical analogues of) anisotropic spin exchange operators given in terms of the R-matrix data. In N = 1 case the spin Calogero-Moser model is reproduced. Explicit expressions for gl N M -valued Lax pair with spectral parameter and its classical dynamical r-matrix are obtained. Possible applications are briefly discussed.