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 in [Curtis E B et al 1998 Linear Algebr. Appl. 283 115–50], 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 in [Berenstein A et al 1996 Adv. Math. 122 49–149]. In our picture the former turns out to be a deformation of the later. Our results should be compared to the earlier work started in [Lam T and Pylyavskyy P 2015 Algebr. Number Theory 9 1401–18] on the connection between the Lusztig varieties and the electrical varieties. There the authors introduced a one-parameter family of Lie groups which are deformations of the unipotent group. For the value of the parameter equal to 1 the group in the family acts on the set of response matrices and is related to the symplectic group. Using the data of electrical networks we construct a representation of the group in this family which corresponds to the value of the parameter −1 in the symplectic group and show that our boundary partition functions belong to it. Remarkably this representation has been studied before in the work on six vertex statistical models and the representations of the Temperley–Lieb algebra.
|Number of pages
|Journal of Physics. A, Mathematical and theoretical
|Early online date
|21 Oct 2020
|Published - 21 Oct 2020
Bibliographical noteOpen Access via the IOP open access agreement
The authors are grateful to Arkady Berenstein, Azat Gainutdinov, Michael Gekhtman, Gleb Koshevoy and Vladimir Roubtsov for generously sharing knowledge and ideas and to Thomas Lam and Pavlo Pylavskyy who read the first draft of the paper and made a number of critical but helpful remarks and comments.
The work of VG has been funded within the framework of the HSE University Basic Research Programme and the Russian Academic Excellence Project ‘5-100’. The work of DT was partially supported by RFBR Grant 20-01-00157,this work was carried out within the framework of a development programme for the Regional Scientific and Educational Mathematical Centre of the Yaroslavl State University with financial support from the Ministry of Science and Higher Education of the Russian Federation (Agreement No. 075-02-2020-1514/1 additional to the agreement on provision of subsidies from the federal budget No. 075-02-2020-1514). DT is grateful for the Th´el`eme atmosphere of the IHES where part of this work was done.
- electrical networks
- quantum intergable models
- response matrix
- boundary partition functions