CENTRAL EXTENSION APPROACH TO INTEGRABLE FIELD AND LATTICE-FIELD SYSTEMS IN (2+1)-DIMENSION.pdf

Vol. 44 (1999) REPORTS ON MATHEMATICAL PHYSIC’S No. l/2 CENTRAL EXTENSION APPROACH TO INTEGRABLE FIELD AND LATTICE-FIELD SYSTEMS IN (2+1)-DIMENSIONS * MACIEJ BL,ASZAK, ANDRZEJ SZUM Physics Department, A. Mickiewicz University, Umultowska 85. 61-614 Poznan, Poland (e-mail: blaszakm@main.amu.edu.pl; szumQmain.amu.edu.pl) ANATOLIJ PRYKARPATSKY Department of Nonlinear Analysis at the IAPMM of the NAS, Lviv 290601, Ukraina and Department of Applied Mathematics. AGH, 30-059 Krak6w. Poland (Received September 4, 1999) The so-called central extension approach is applied t,o ‘R.-matrix formalism to extend it from (l+l) to (2+1)-dimensions. Two-dimensional integrable field and lattice-field systems are constructed. 1. Introduction In this work we present a systematic method for constructing (2+1)-dimensional integrable Hamiltonian field and lattice-field dynamic systems. In this paper by integrable systems we understand these possessing infinite hierarchy of commuting Hamiltonian symmetries. The method is based on an extension of the R-matrix approach from (l+l) to (2+1)-dimensions. For (l+l)-dimension R-matrix approach is a simple and powerful algebraic tool which allows us to construct integrable Hamiltonian dynamical systems from scratch [I-G]. Although this approach is formulated in a rather abstract algebraic way, but it offers the advantage of getting a simple and effective method for generating infinite hierarchies of commuting symmetries for constructed basic systems. The crucial point of this approach is the observation that the Lax equation Lt = [L, A] (1) can be treated as an abstract dynamical system from which “physical” dynamical systems are obtained by introducing suitable charts. Hence, the phase space for most of these equations can be regarded as given by the set of Lax operators taking values from some Lie algebra. The construction of Hamiltonian equations and the related hierarchy of symmetries, becomes quite transparent when