arXiv subjects
Publications and source records attributed to Dmitry Talalaev.
This note is a modest addition to the work arXiv:2109.13952. Here we construct an embedding of the space of electrical networks to the totally non-negative Lagrangian Grassmannian in a generic situation with the help of the technique of vertex integrable models of statistical mechanics.
We refine the result of T. Lam \cite{L} on embedding the space $E_n$ of electrical networks on a planar graph with $n$ boundary points into the totally non-negative Grassmannian $\mathrm{Gr}_{\geq 0}(n-1,2n)$ by proving first that the image lands in $\mathrm{Gr}(n-1,V)\subset \mathrm{Gr}(n-1,2n)$ where $V\subset \mathbb{R}^{2n}$ is a certain subspace of dimension $2n-2$. The role of this reduction in the dimension of the ambient space is crucial for us. We show next that the image lands in fact inside the Lagrangian Grassmannian $\mathrm{LG}(n-1,V)\subset \mathrm{Gr}(n-1,V)$. As it is well known $\mathrm{LG}(n-1)$ can be identified with $\mathrm{Gr}(n-1,2n-2)\cap \mathbb{P} L$ where $L\subset \bigwedge^{n-1}\mathbb R^{2n-2}$ is a subspace of dimension equal to the Catalan number $C_n$, moreover it is the space of the fundamental representation of the symplectic group $Sp(2n-2)$ which corresponds to the last vertex of the Dynkin diagram. We show further that the linear relations cutting the image of $E_n$ out of $\mathrm{Gr}(n-1,2n)$ found in \cite{L} define that space $L$. This connects the combinatorial description of $E_n$ discovered in \cite{L} and representation theory of the symplectic group.
We explore several types of functional relations on the family of multivariate Tutte polynomials: the Biggs formula and the star-triangle ($Y-\Delta$) transformation at the critical point $n=2$. We deduce the theorem of Matiyasevich and its inverse from the Biggs formula, and we apply this relation to construct the recursion on the parameter $n$. We provide two different proofs of the Zamolodchikov tetrahedron equation satisfied by the star-triangle transformation in the case of $n=2$ multivariate Tutte polynomial, we extend the latter to the case of valency 2 points and show that the Biggs formula and the star-triangle transformation commute.
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.
The notion of compatible braidings was introduced by Isaev, Ogievetsky and Pyatov. On the base of this notion they defined certain quantum matrix algebras generalizing the RTT algebras and Reflection Equation ones. They also defined analogs of some symmetric polynomials in these algebras and showed that these polynomials generate commutative subalgebras, called Bethe. By using a similar approach we introduce certain new algebras called generalized Yangians and define analogs of some symmetric polynomials in these algebras. We claim that they commute with each other and thus generate a commutative Bethe subalgebra in each generalized Yangian. Besides, we define some analogs (also arising from couples of compatible braidings) of the Knizhnik-Zamolodchikov equation--classical and quantum.
Applying the method of the paper [CT], we perform a quantum version of the Drinfeld-Sokolov reduction in Reflection Equation algebras and braided Yangians, associated with involutive and Hecke symmetries of general forms. This reduction is based on the Cayley-Hamilton identity valid for the generating matrices of these algebras.
A theory of (co)homologies related to set-theoretic $n$-simplex relations is constructed in analogy with the known quandle and Yang--Baxter (co)homologies, with emphasis made on the tetrahedron case. In particular, this permits us to generalize Hietarinta's idea of "permutation-type" solutions to the quantum (or "tensor") $n$-simplex equations. Explicit examples of solutions to the tetrahedron equation involving nontrivial cocycles are presented.
In this paper we address the following question: is it always possible to choose a deformation quantization of a Poisson algebra A so that certain Poisson-commutative subalgebra C in it remains commutative? We define a series of cohomological obstructions to this, that take values in the Hochschild cohomology of C with coefficients in A. In some particular case of the pair (A,C) we reduce these classes to the classes of the Poisson relative cohomology of the Hochschild cohomology. We show, that in the case, when the algebra C is polynomial, these obstructions coincide with the previously known ones, those which were defined by Garay and van Straten.