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Journal of Physics A Mathematical and General

Classical r-matrix, new integrable system and finite boundary condition

A Gouse Choudhury, Asesh Roy Chowdhury

Published Jan 21, 1995• 1 citation
Publisher

Abstract

We have analysed the classical r-matrix structure of a new integrable model in two-dimensional coupled Liouville-Thirring model. Due to the non-ultralocal character of the system, a new form of (r, -s) structure is obtained. It is also proved that the integrability of the model is not destroyed if non-trivial finite boundary conditions are imposed. An equation determining the form of the matrices K + and K - is deduced which is a simple generalization of that of Sklyanin for the ultralocal case.

DOI
10.1088/0305-4470/28/2/020

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We have analysed the classical r-matrix structure of a new integrable model in two-dimensional coupled Liouville-Thirring model. Due to the non-ultralocal character of the system, a new form of (r, -s) structure is obtained. It is also proved that the integrability of the model is not destroyed if non-trivial finite boundary conditions are imposed. An equation determining the form of the matrices K + and K - is deduced which is a simple generalization of that of Sklyanin for the ultralocal case.
We study two-dimensional classically integrable field theory with independent boundary conditions at each end, and obtain three possible generating functions for integrals of motion when this model is an ultralocal one. Classically integrable boundary conditions can be found when solving boundary equations. In the quantum case, we also find that the unitarity condition of the quantum R -matrix is sufficient to construct commutative quantities with boundaries, and its reflection equations are obtained.
Abstract We reconcile the Hamiltonian formalism and the zero curvature representation in the approach to integrable boundary conditions for a classical integrable system in 1 + 1 space-time dimensions. We start from an ultralocal Poisson algebra involving a Lax matrix and two (dynamical) boundary matrices. Sklyanin’s formula for the double-row transfer matrix is used to derive Hamilton’s equations of motion for both the Lax matrix and the boundary matrices in the form of zero curvature equations. A key ingredient of the method is a boundary version of the Semenov-Tian-Shansky formula for the generating function of the time-part of a Lax pair. The procedure is illustrated on the finite Toda chain for which we derive Lax pairs of size for previously known Hamiltonians of type BC N and D N corresponding to constant and dynamical boundary matrices respectively.
We reconcile the Hamiltonian formalism and the zero curvature representation in the approach to integrable boundary conditions for a classical integrable system in 1 + 1 space-time dimensions. We start from an ultralocal Poisson algebra involving a Lax matrix and two (dynamical) boundary matrices. Sklyanin's formula for the double-row transfer matrix is used to derive Hamilton's equations of motion for both the Lax matrix and the boundary matrices in the form of zero curvature equations. A key ingredient of the method is a boundary version of the Semenov-Tian-Shansky formula for the generating function of the time-part of a Lax pair. The procedure is illustrated on the finite Toda chain for which we derive Lax pairs of size for previously known Hamiltonians of type BC N and D N corresponding to constant and dynamical boundary matrices respectively.
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 gl2 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.
A simple relation between inhomogeneous transfer matrices and boundary quantum Knizhnik-Zamolodchikov (KZ) equations is exhibited for quantum integrable systems with reflecting boundary conditions, analogous to an observation by Gaudin for periodic systems. Thus, the boundary quantum KZ equations receive a new motivation. We also derive the commutativity of Sklyanin’s boundary transfer matrices by merely imposing appropriate reflection equations, in particular without using the conditions of crossing symmetry and unitarity of the R-matrix.