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\(R\)-matrix quantization of the elliptic Ruijsenaars-Schneider model. (English. Russian original) Zbl 0978.81509
Theor. Math. Phys. 111, No. 2, 536-562 (1997); translation from Teor. Mat. Fiz. 111, No. 2, 182-217 (1997).
Summary: It is shown that the classical \(L\)-operator algebra of the elliptic Ruijsenaars-Schneider model can be realized as a subalgebra of the algebra of functions on the cotangent bundle over the centrally extended current group in two dimensions. It is governed by two dynamic \(r\) and \(\overline r\)-matrices satisfying a closed system of equations. The corresponding quantum \(R\)- and \(\overline R\)-matrices are found as solutions to quantum analogues of these equations. We present the quantum \(L\)-operator algebra and show that the system of equations for \(R\) and \(\overline R\) arises as the compatibility condition for this algebra. It turns out that the \(R\)-matrix is twist-equivalent to the Felder elliptic \(R^F\)-matrix, with \(\overline R\) playing the role of the twist. The simplest representation of the quantum \(L\)-operator algebra corresponding to the elliptic Ruijsenaars-Schneider model is obtained. The connection of the quantum \(L\)-operator algebra to the fundamental relation \(R^BLL=LLR^B\) with the Belavin elliptic \(R\)-matrix is established. As a by-product of our construction, we find a new \(N\)-parameter elliptic solution to the classical Yang-Baxter equation.

MSC:
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
82B23 Exactly solvable models; Bethe ansatz
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