In this paper we expound systematically an invariant approach to quantum integrable models in wholly discrete 2+1-dimensional space-time. We formulate an auxiliary linear problem on two-dimensional lattice which generalizes the notion of the quantum chains. We give a complete set of integrals of motion derivation method. We formulate and solve a zero curvature representation for two-dimensional lattice, it allows one to construct integrable evolution mappings. As the main example, we investigate finite-dimensional representations of the algebra of observables existing when Weyl algebra's parameter lies in a rational point on the unit circle, the so-called root of unity. For this case we derive a universal functional equation for eigenvalues of the integrals of motion. Besides, for the finite dimensional representation of the algebra of observables we construct a gruppoid of isospectral transformations. The fact that at the root of unity the algebra of observables is finite-dimensional allows one to interpret an integrable system as a model of statistical mechanics. We formulate a method for construction of eigenvectors for investigated models, this method is based on isospectral deformations (the method of quantum separation of variables for (2+1)-dimensional models).

PDF (603 Kb)