Electrical varieties as vertex integrable statistical modelsстатья
Статья опубликована в высокорейтинговом журнале
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Дата последнего поиска статьи во внешних источниках: 23 декабря 2020 г.
Аннотация: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 [BerensteinAet 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.
Keywords: electrical networks, quantum intergable models, response matrix,
boundary partition functions.