arXiv subjects

G. I. Sharygin

Publications and source records attributed to G. I. Sharygin.

7 recordsLinked to original sources

arXiv2026

Double Poisson brackets on low dimensional algebras

In this paper, we describe double Poisson brackets in the sense of M. Van den Bergh on certain finite-dimensional algebras. In particular we prove that all possible double Poisson brackets on matrix algebras are "inner", i.e. given by some commutators in bimodules. As a corollary of this result, we see that all possible double Poisson brackets in any finite-dimensional semisimple algebras over algebraically closed fields are also given by inner derivations. We further give a description of all double Poisson brackets on the algebra of 2x2 upper triangular matrices. We further discuss Poisson structures induced from the double Poisson brackets in its representation spaces of rank two and three. In the appendix, we describe modified double Poisson brackets (in the sense of S. Arthamonov) on this algebra.

math-ph↗

arXiv2024

Full symmetric Toda system and vector fields on the group $SO_n(\R)$

In this paper we discuss the relation between the functions that give first integrals of full symmetric Toda system (an important Hamilton system on the space of traceless real symmetric matrices) and the vector fields on the group of orthogonal matrices: it is known that this system is equivalent to an ordinary differential equation on the orthogonal group, and we extend this observation further to its first integrals. As a by-product we describe a representation of the Lie algebra of $B^+(\R)$-invariant functions on the dual space of Lie algebra $\mathfrak{sl}_n(\R)$ (under the canonical Poisson structure) by vector fields on $SO_n(\R)$.

nlin.SI↗

arXiv2023

The argument shift method in universal enveloping algebra $U\mathfrak{gl}_d$

We prove the conjecture that allows one extend the argument shifting procedure from symmetric algebra $S\mathfrak{gl}_d$ of the Lie algebra $\mathfrak{gl}_d$ to the universal enveloping algebra $U\mathfrak{gl}_d$. Namely, it turns out that the iterated quasi-derivations of the central elements in $U\mathfrak{gl}_d$ commute with each other. Here quasi-derivation is a linear operator on $U\mathfrak{gl}_d$, constructed by Gurevich, Pyatov and Saponov. This allows one better understand the structure of \textit{argument shift algebras} (or \textit{Mishchenko-Fomenko algebras}) in the universal enveloping algebra of $\mathfrak{gl}_d$.

math.RT↗

arXiv2017

Bruhat order in the Toda system on $\mathfrak{so}(2,4)$: an example of non-split real form

In our previous papers we described the structure of trajectories of the symmetric Toda system on normal real forms of various Lie algebras and showed that it was totally determined by the Hasse diagram of the Bruhat order on the corresponding Weil group. This note deals with the simplest non-split real Lie algebra, $\mathfrak{so}(2,4)$. It turns out, that the phase diagram in this case is also closely related with the Bruhat order of the relative Weyl group, but a bit more information is necessary to describe the dimensions of the trajectory spaces.

nlin.SI↗

arXiv2015

Cohomology of the tetrahedral complex and quasi-invariants of 2-knots

This paper explores a particular statistical model on 6-valent graphs with special properties which turns out to be invariant with respect to certain Roseman moves if the graph is the singular point graph of a diagram of a 2-knot. The approach uses the technic of the tetrahedral complex cohomology. We emphasize that this model considered on regular 3d-lattices appears to be integrable. We also set out some ideas about the possible connection of this construction with the area of topological quantum field theories in dimension 4.

math-ph↗

arXiv2013

Bruhat Order in Full Symmetric Toda System

In this paper we discuss some geometrical and topological properties of the full symmetric Toda system. We show by a direct inspection that the phase transition diagram for the full symmetric Toda system in dimensions $n=3,4$ coincides with the Hasse diagram of the Bruhat order of symmetric groups $S_3$ and $S_4$. The method we use is based on the existence of a vast collection of invariant subvarieties of the Toda flow in orthogonal groups. We show how one can extend it to the case of general $n$. The resulting theorem identifies the set of singular points of $\mathrm{dim}=n$ Toda flow with the elements of the permutation group $S_n$, so that points will be connected by a trajectory, if and only if the corresponding elements are Bruhat comparable. We also show that the dimension of the submanifolds, spanned by the trajectories connecting two singular points, is equal to the length of the corresponding segment in the Hasse diagramm. This is equivalent to the fact, that the full symmetric Toda system is in fact a Morse-Smale system.

nlin.SI↗

arXiv2010

Holonomy, twisting cochains and characteristic classes

The primary interest of this paper is to discuss the role of twisting cochains in the theory of characteristic classes. We begin with the homological description of monodromy map, associated with a connection on a trivial bundle over a 1-connected manifold. We regard it as a homomorphism from the algebra of differential forms on the structure group to the algebra of differential forms on the based loopspace of the base, represented by the (reduced) bar-complex of differential forms on it. Next we discuss the notion of "twisting cochains", or more generally "twisting maps", their equivalence relation and give various examples. We show that every twisting map gives rise to a map from the coalgebra to the bar-resolution of the algebra. Further we show that in the case of genuine twisting cochains one can obtain a map from the differential forms on the gauge bundle, associated with the given principal one, to the reduced Hochschild complex of the algebra, of differential forms of the base. Then we discuss a concrete example of a twisting cochain that is defined on the polynomial de Rham forms on an algebraic group and takes values in Cech complex of the base. We show how it can be used to obtain explicit formulas for the Chern classes. We also discuss few modifications of this construction. In the last section we discuss the construction, similar to the one, used by Getzler, Jones and Petrack in their 1991 paper. We show that the map we call "Getzler-Jones-Petrack's map" is homotopy-equivalent to the map that one obtains from a twisting cochain. This enables us to find a generalization of the Bismut's class, which we regard as an image of a suitable element in the differential forms on the group under the Getzler-Jones-Petrack's map.

math.KT↗