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Singularities of bi-Hamiltonian systems
Cornell University
,
2013.
Bolsinov A., Izosimov A.
We study the relationship between singularities of bi-Hamiltonian systems and algebraic properties of compatible Poisson brackets. As the main tool, we introduce the notion of linearization of a Poisson pencil. From the algebraic viewpoint, a linearized Poisson pencil can be understood as a Lie algebra with a fixed 2-cocycle. In terms of such linearizations, we give a criterion for non-degeneracy of singular points of bi-Hamiltonian systems and describe their types.
Language:
English
Keywords: exactly solvable and integrable systems
A. Levin, Olshanetsky M., Zotov A., Journal of High Energy Physics 2014 Vol. 2014 No. 7:12 P. 1-39
We describe classical top-like integrable systems arising from the quantum exchange relations and corresponding Sklyanin algebras. The Lax operator is expressed in terms of the quantum non-dynamical R-matrix even at the classical level, where the Planck constant plays the role of the relativistic deformation parameter in the sense of Ruijsenaars and Schneider (RS). The integrable ...
Added: January 23, 2015
Izosimov A., Journal of Geometry and Physics 2012 Vol. 62 No. 12 P. 2414-2423
The presence of two compatible hamiltonian structures is known to be one of the main, and the most natural, mechanisms of integrability. For every pair of hamiltonian structures, there are associated conservation laws (first integrals). Another approach is to consider the second hamiltonian structure on its own as a tensor conservation law. The latter is ...
Added: November 18, 2013
Derbyshev A. E., Poghosyan S. S., Povolotsky A. M. et al., Journal of Statistical Mechanics: Theory and Experiment 2012 Vol. P05014 P. 1-13
We consider the totally asymmetric exclusion process in discrete time with generalized updating rules. We introduce a control parameter into the interaction between particles. Two particular values of the parameter correspond to known parallel and sequential updates. In the whole range of its values the interaction varies from repulsive to attractive. In the latter case ...
Added: February 12, 2013
М. : МАИК "Наука/Интерпериодика", 2018
A specific model of statistical mechanics on graphs with vertices of valence 6 and 1. It is shown that the model under consideration is invariant with respect to certain Rozman movements if the graph is interpreted as the graph of singular points of the 2-node diagram. The approach uses the technique of cohomology of a ...
Added: August 14, 2020
Krasnov T., Zotov A., Annales Henri Poincare. A Journal of Theoretical and Mathematical Physics 2019 Vol. 20 No. 8 P. 2671-2697
We consider a special class of quantum non-dynamical R -matrices in the fundamental representation of GL N with spectral parameter given by trigonometric solutions of the associative Yang-Baxter equation. In the simplest case N=2 these are the well-known 6-vertex R -matrix and its 7-vertex deformation. The R -matrices are used for construction of the classical relativistic ...
Added: July 12, 2019
Grekov A., A. Zabrodin, A. Zotov, Nuclear Physics B 2019 Vol. 939 P. 174-190
We describe the correspondence of the Matsuo-Cherednik type between the quantum nn -body Ruijsenaars-Schneider model and the quantum Knizhnik-Zamolodchikov equations related to supergroup GL(N|M)GL(N|M) . The spectrum of the Ruijsenaars-Schneider Hamiltonians is shown to be independent of the {\mathbb Z}_2 -grading for a fixed value of N+M , so that N+M+1 different qKZ systems of ...
Added: May 24, 2019
Izosimov A., / Cornell University. Series math "arxiv.org". 2013.
It is shown that a generic bihamiltonian structure on an odd-dimensional manifold is flat if and only if it is locally unimodular. ...
Added: November 19, 2013
Gavrylenko P., Iorgov N., Lisovyy O., Journal of Physics A: Mathematical and Theoretical 2012 Vol. 45 No. 2 P. 025402
We study U(1) twist fields in a two-dimensional lattice theory of massive Dirac fermions. Factorized formulas for finite-lattice form factors of these fields are derived using elliptic parametrization of the spectral curve of the model, elliptic determinant identities and theta functional interpolation. We also investigate the thermodynamic and infinite-volume scaling limit, where the corresponding expressions ...
Added: October 20, 2014
Levin A., Olshanetsky M., Zotov A., / Cornell University. Series math "arxiv.org". 2014.
e describe classical top-like integrable systems arising from the quantum
exchange relations and corresponding Sklyanin algebras. The Lax operator is
expressed in terms of the quantum non-dynamical $R$-matrix even at the
classical level, where the Planck constant plays the role of the relativistic
deformation parameter in the sense of Ruijsenaars and Schneider (RS). The
integrable systems (relativistic tops) are described ...
Added: January 23, 2015
Aminov S., Arthamonov S., A. Levin et al., / Cornell University. Series math "arxiv.org". 2013.
We propose multidimensional versions of the Painleve VI equation and its degenerations. These field theories are related to the isomonodromy problems on flat holomorphic infinite rank bundles over elliptic curves and take the form of non-autonomous Hamiltonian equations. The modular parameter of curves plays the role of "time". Reduction of the field equations to the ...
Added: December 27, 2013
Izosimov A., Differential Geometry and its Application 2013 Vol. 31 P. 557-567
A Poisson pencil is called flat if all brackets of the pencil can be simultaneously locally brought to a constant form. Given a Poisson pencil on a 3-manifold, we study under which conditions it is flat. Since the works of Gelfand and Zakharevich, it is known that a pencil is flat if and only if ...
Added: November 18, 2013
Levin A., Olshanetsky M., Zotov A., / Cornell University. Series math "arxiv.org". 2014.
In our recent paper we suggested a natural construction of the classical relativistic integrable tops in terms of the quantum R-matrices. Here we study the simplest case -- the 11-vertex R-matrix and related gl_2 rational models. The corresponding top is equivalent to the 2-body Ruijsenaars-Schneider (RS) or the 2-body Calogero-Moser (CM) model depending on its ...
Added: January 23, 2015
Izosimov A., / Cornell University. Series math "arxiv.org". 2013.
It is a classical result of Euler that the rotation of a torque-free three-dimensional rigid body about the short or the long axis is stable, and the rotation about the middle axis is unstable. This result is generalized to the case of a multidimensional body. ...
Added: November 19, 2013
Boiti M., Pempinelli F., Pogrebkov A., Journal of Physics A: Mathematical and Theoretical 2017 Vol. 50 No. 30 P. 1-23
Direct definition of the Cauchy-Jost (known also as Cauchy-Baker-Akhiezer) function in the case of pure solitonic solution is given and properties of this function are discussed in detail using the Kadomtsev-Petviashvili II equation as example. This enables formulation of the Darboux transformations in terms of the Cauchy-Jost function and classification of these transformations. Action of ...
Added: September 4, 2017
Zotov A., Atalikov K., JETP Letters 2022 Vol. 115 No. 12 P. 809-810
We propose a construction of 1 + 1 integrable Heisenberg–Landau–Lifshitz type equations in the glN case. The dynamical variables are matrix elements of N × N matrix S with the property S2 = const · S. The Lax pair with spectral parameter is constructed by means of a quantum R-matrix satisfying the associative Yang–Baxter equation. Equations of motion for glN Landau–Lifshitz model are derived from the Zakharov–Shabat ...
Added: June 20, 2022
Khoroshkin S. M., Shapiro A., Journal of Geometry and Physics 2010 Vol. 60 No. 11 P. 1833-1851
In this article, we give an explicit formula for the universal weight function of the quantum twisted affine algebra Uq(A(2)2 ). The calculations use the technique of projecting products of Drinfeld currents onto the intersection of Borel subalgebras of different types. ...
Added: September 26, 2012
A. Levin, Olshanetsky M., Zotov A., / Cornell University. Series math "arxiv.org". 2013.
We consider the isomonodromy problems for flat $G$-bundles over punctured
elliptic curves $\Sigma_\tau$ with regular singularities of connections at
marked points. The bundles are classified by their characteristic classes.
These classes are elements of the second cohomology group
$H^2(\Sigma_\tau,{\mathcal Z}(G))$, where ${\mathcal Z}(G)$ is the center of
$G$. For any complex simple Lie group $G$ and arbitrary class we define ...
Added: December 27, 2013
A. Levin, Olshanetsky M., Zotov A., Nuclear Physics B 2014 Vol. 887 P. 400-422
In our recent paper we suggested a natural construction of the classical relativistic integrable tops in terms of the quantum R -matrices. Here we study the simplest case – the 11-vertex R -matrix and related gl2 rational models. The corresponding top is equivalent to the 2-body Ruijsenaars–Schneider (RS) or the 2-body Calogero–Moser (CM) model depending ...
Added: January 22, 2015
Levin A., Olshanetsky M., Zotov A., / Cornell University. Series math "arxiv.org". 2014.
We construct special rational ${\rm gl}_N$ Knizhnik-Zamolodchikov-Bernard
(KZB) equations with $\tilde N$ punctures by deformation of the corresponding
quantum ${\rm gl}_N$ rational $R$-matrix. They have two parameters. The limit
of the first one brings the model to the ordinary rational KZ equation. Another
one is $\tau$. At the level of classical mechanics the deformation parameter
$\tau$ allows to extend the ...
Added: January 23, 2015
Izosimov A., / Cornell University. Series math "arxiv.org". 2013.
In 1970s, a method was developed for integration of nonlinear equations by means of algebraic geometry. Starting from a Lax representation with a spectral parameter, the algebro-geometric method allows to solve the system explicitly in terms of Theta functions of Riemann surfaces. However, the explicit formulas obtained in this way fail to answer such natural ...
Added: November 19, 2013