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Vladimir Rubtsov

Publications and source records attributed to Vladimir Rubtsov.

29 records · Page 2Linked to original sources
arXiv2013

Generalizations of Poisson structures related to rational Gaudin model

The Poisson structure arising in the Hamiltonian approach to the rational Gaudin model looks very similar to the so-called modified Reflection Equation Algebra. Motivated by this analogy, we realize a braiding of the mentioned Poisson structure, i.e. we introduce a "braided Poisson" algebra associated with an involutive solution to the quantum Yang-Baxter equation. Also, we exhibit another generalization of the Gaudin type Poisson structure by replacing the first derivative in the current parameter, entering the so-called local form of this structure, by a higher order derivative. Finally, we introduce a structure, which combines both generalizations. Some commutative families in the corresponding braided Poisson algebra are found.

math.QA↗
arXiv2013

Parameter-dependent associative Yang-Baxter equations and Poisson brackets

We discuss associative analogues of classical Yang-Baxter equation meromorphically dependent on parameters. We discover that such equations enter in a description of a general class of parameter-dependent Poisson structures and double Lie and Poisson structures in sense of M. Van den Bergh. We propose a classification of all solutions for one-dimensional associative Yang-Baxter equations.

math-ph↗
arXiv2013

Reconstruction of the Tunguska Event of 1908: Neither an Asteroid, Nor a Comet Core

The Tunguska explosion occurred in the morning of June 30, 1908, in Central Siberia, some 800 km NNW from Lake Baikal. It devastated the forested area of 2150 sq. km, flattening and scorching some 30 million trees. Before this, a luminous body flew overhead in the cloudless sky. The air waves from the explosion were recorded as far as in London. The object that flew that morning over Siberia is usually designated the "Tunguska meteorite" or - more cautiously - the "Tunguska space body" (TSB). Certainly, this body was dangerous: the taiga was leveled over an area twice as large as New York City. The whole number of Tunguska hypotheses reaches a hundred, or so. But few of them have been built according to the standards of science and with due consideration of empirical data. There is also a serious methodological problem that is, as a rule, overlooked: the need to take into consideration all empirical data and to reconstruct the Tunguska event before building any models of it. Such a reconstruction is essential - since the consequences of this event are many and varied. The main Tunguska traces may be grouped and listed as follows: (a) material traces; (b) instrumental traces; (c) informational traces. To be sure that a proposed theory is correct, the scientist must check it against all the three types of Tunguska evidence. Having reconstructed the Tunguska event with due attention to all the evidence, we have to conclude that it could not have been an asteroid or a comet core. There seems to exist in space another type of dangerous space objects, whose nature still remains unknown.

physics.gen-ph↗
arXiv2012

Confluence on the Painlevé Monodromy Manifolds, their Poisson Structure and Quantisation

In this paper we obtain a system of flat coordinates on the monodromy manifold of each of the Painlevé equations. This allows us to quantise such manifolds. We produce a quantum confluence procedure between cubics in such a way that quantisation and confluence commute. We also investigate the underlying cluster algebra structure and the relation to the versal deformations of singularities of type $D_4,A_3,A_2$, and $A_1$.

math-ph↗
arXiv2012

On localization in holomorphic equivariant cohomology

We prove a localization formula for a "holomorphic equivariant cohomology" attached to the Atiyah algebroid of an equivariant holomorphic vector bundle. This generalizes Feng-Ma, Carrell-Liebermann, Baum-Bott and K. Liu's localization formulas.

math.CV↗
arXiv2011

Bi-Hamiltonian ODEs with matrix variables

We consider a special class of linear and quadratic Poisson brackets related to ODE systems with matrix variables. We investigate general properties of such brackets, present an example of a compatible pair of quadratic and linear brackets and found the corresponding hierarchy of integrable models, which generalizes the two-component Manakov's matrix system in the case of arbitrary number of matrices.

nlin.SI↗
arXiv2010

Cohomology of skew-holomorphic Lie algebroids

We introduce the notion of skew-holomorphic Lie algebroid on a complex manifold, and explore some cohomologies theories that one can associate to it. Examples are given in terms of holomorphic Poisson structures of various sorts.

math.CV↗
arXiv2009

Manin Matrices, Quantum Elliptic Commutative Families and Characteristic Polynomial of Elliptic Gaudin Model

In this paper we construct the quantum spectral curve for the quantum dynamical elliptic gl(n) Gaudin model. We realize it considering a commutative family corresponding to the Felder's elliptic quantum group and taking the appropriate limit. The approach of Manin matrices here suits well to the problem of constructing the generation function of commuting elements which plays an important role in SoV and Langlands concept.

math-ph↗
arXiv2008

Compatible structures on Lie algebroids and Monge-Ampère operators

We study pairs of structures, such as the Poisson-Nijenhuis structures, on the tangent bundle of a manifold or, more generally, on a Lie algebroid or a Courant algebroid. These composite structures are defined by two of the following, a closed 2-form, a Poisson bivector or a Nijenhuis tensor, with suitable compatibility assumptions. We establish the relationships between such composite structures. We then show that the non-degenerate Monge-Ampère structures on 2-dimensional manifolds satisfying an integrability condition provide numerous examples of such structures, while in the case of 3-dimensional manifolds, such Monge-Ampère operators give rise to generalized complex structures or generalized product structures on the cotangent bundle of the manifold.

math.DG↗