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Georgy Sharygin

Publications and source records attributed to Georgy Sharygin.

7 recordsLinked to original sources
arXiv2024

Non-Abelian discrete Toda chains and related lattices

We have derived a non-abelian analog for the two-dimensional discrete Toda lattice which possesses solutions in terms of quasideterminants and admits Lax pairs of different forms. Its connection with non-abelian analogs for several well-known (1+1) and one-dimensional lattices is discussed. In particular, we consider a non-commutative analog of the scheme: discrete Toda equations $\rightarrow$ Somos-$N$ sequences $\rightarrow$ discrete Painlevé equations.

nlin.SI↗
arXiv2023

On the quantum argument shift method

In a recent work by two of us the argument shift method was extended from the symmetric algebra ${\rm S}({\mathfrak g})$ of the general linear Lie algebra ${\mathfrak g}$ to the universal enveloping algebra ${\rm U}({\mathfrak g})$. We show in this paper that some features of this 'quantum argument shift method' can be applied to the remaining classical matrix Lie algebras ${\mathfrak g}$. We prove that a single application of the quasi-derivation to central elements of ${\rm U}({\mathfrak g})$ yields elements of the corresponding quantum Mishchenko-Fomenko subalgebra. We show that generators of this subalgebra can be obtained by iterated application of the quasi-derivation to generators of the center of ${\rm U}({\mathfrak g})$.

math.RT↗
arXiv2019

Noncommutative cross-ratio and Schwarz derivative

We present here a theory of noncommutative cross-ratio, Schwarz derivative and their connections and relations to the operator cross-ratio. We apply the theory to "noncommutative elementary geometry" and relate it to noncommutative integrable systems. We also provide a noncommutative version of the celebrated "pentagramma mirificum".

math.RA↗
arXiv2016

On local combinatorial formulas for Chern classes of triangulated circle bundle

Principal circle bundle over a PL polyhedron can be triangulated and thus obtains combinatorics. The triangulation is assembled from triangulated circle bundles over simplices. To every triangulated circle bundle over a simplex we associate a necklace (in combinatorial sense). We express rational local formulas for all powers of first Chern class in the terms of mathematical expectations of parities of the associated necklaces. This rational parity is a combinatorial isomorphism invariant of triangulated circle bundle over simplex, measuring mixing by triangulation of the circular graphs over vertices of the simplex. The goal of this note is to sketch the logic of deduction these formulas from Kontsevitch's cyclic invariant connection form on metric polygons.

math.AT↗
arXiv2014

Cohomologies of $n$-simplex relations

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.

math-ph↗
arXiv2013

Deformation quantization of integrable systems

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.

math.QA↗