Relativistic classical integrable tops and quantum R-matrices

Abstract

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.

Planck constant as spectral parameter in integrable systems and KZB equations

“…First, we construct the rational analogue of the (elliptic) KZB equations. For this purpose we find τ deformation of the quantum R-matrix suggested in [39,40]. Second, we show that integrable systems of Calogero-Moser type admit higher rank Lax representations which generalize the Krichever's one [30] in the same way as (1.10) generalize (1.9).…”

“…Its trigonometric analogue was found in [4,16] (we are going to consider it in separate publications). At last the rational case is known from [16,39,40,[58][59][60]. In the simplest cases one gets the ordinary XXZ and XXX Yang's R-matrices.…”

“…Besides the standard trigonometric and rational versions of the elliptic R-matrix there are more sophisticated degenerations. In this paper we consider one of them [39,40] and show that it leads to some modifications of the standard Gaudin and Schlesinger systems and the KZ (KZB) equations.…”

“…The natural (noncommutative) analogue of ϑ-function is the modification of bundle Ξ(z, τ ). In the elliptic case it was found in [26] in the context of the IRF-Vertex transformation, and then described in [35] (see also [37][38][39][40]) as an example of the Symplectic Hecke Correspondence for integrable systems. Its rational analogue was suggested in [3] and was know to be free of τ dependence.…”

“…Our general idea is that the both Planck constants can play different roles, i.e. each of the constants can be either the spectral parameter in a "classical-quantum" glÑ system (of (1.19) type) or the Planck constant in a quantum gl N system or the relativistic deformation parameter in a classical relativistic gl N model (see [39,40]). 4 We hope that this can shed light on numerous dualities in integrable systems mentioned in [19, 43-45, 48, 62-64].…”