We propose the systematic construction of classical and quantum two-dimensional space-time lattices primarily based on algebraic considerations, i.e. on the existence of associated r-matrices and underlying spatial and temporal classical and quantum algebras. This is a novel construction that leads to the derivation of fully discrete integrable systems governed by sets of consistent integrable non-linear space-time difference equations. To illustrate the proposed methodology, we derive two versions of the fully discrete non-linear Schrdinger type system. The first one is based on the existence of a rational r-matrix, whereas the second one is the fully discrete Ablowitz–Ladik model and is associated to a trigonometric r-matrix. The Darboux-dressing method is also applied for the first discretization scheme, mostly as a consistency check, and solitonic as well as general solutions, in terms of solutions of the fully discrete heat equation, are also derived. The quantization of the fully discrete systems is then quite natural in this context and the two-dimensional quantum lattice is thus also examined.

我们主要基于代数考虑，即基于相关r矩阵的存在以及潜在的时空古典和量子代数，提出了经典和量子二维时空格的系统构造。这是一种新颖的结构，可导致推导完全离散的可积分系统，该系统由一组一致的可积分非线性时空差分方程控制。为了说明所提出的方法，我们导出了完全离散的非线性Schrdinger类型系统的两个版本。第一个是基于有理r矩阵的存在，而第二个是完全离散的Ablowitz-Ladik模型，并且与三角r相关。-矩阵。Darboux修正方法也用于第一个离散化方案，主要是作为一致性检查，并且还推导了完全离散热方程解的孤子以及一般解。因此，在这种情况下，完全离散系统的量化非常自然，因此还检查了二维量子晶格。