  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. This gives an
equivalent description of electrical networks We show that in the important
case of an electrical network on the standard graph introduced by Curtis,
Ingerman and Morrow, 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.
  Defining the electrical varieties in the usual way we compare them to the
theory of the Lusztig varieties developed by Berenstein, Fomin and Zelevinsky.
In our picture the former turns out to be a deformation of the latter. We
describe how our approach produces new interesting mathematical structures on
the electrical varieties. Our results should be compared to the earlier work
started in Lam and Pylyavskyy on the connection between the Lusztig varieties
and the electrical varieties.
