ИСТИНА

The talk is based on the recent paper in co-authorship with V. Gorbounov [1]. We propose a new approach to studying electrical networks interpreting the Ohm law as the operator which solves certain Local Yang-Baxter equation. Using this operator and the medial graph of the electrical network we define a vertex integrable statistical model and its boundary partition function. As an evidence that this gives an equivalent description of electrical networks, we show that in the important case of an electrical network on the standard graph introduced in [2], the response matrix of an electrical network, its most important feature, and the boundary partition function of our statistical model, can be recovered from each other. We show that this approach provides a natural deformation for the Lusztig decomposition of the Borel unipotent subgroup [3]. This also pay attention on the essential action of the affine Weil group on some particular electrical varieties which produces some discrete integrable models deforming the Toda system analogs of [4].