scholarly journals Complete commutative subalgebras in polynomial poisson algebras: A proof of the Mischenko-Fomenko conjecture

2016 ◽  
Vol 43 (2) ◽  
pp. 145-168 ◽  
Author(s):  
Alexey Bolsinov
Keyword(s):  
Poisson Bracket ◽  
Lie Algebras ◽  
Hamiltonian Systems ◽  
Dual Space ◽  
State University ◽  
Integrable Hamiltonian Systems ◽  
Finite Dimensional ◽  
Complete Set ◽  
Commutative Subalgebras ◽  
Moscow State University

The Mishchenko-Fomenko conjecture says that for each real or complex finite-dimensional Lie algebra g there exists a complete set of commuting polynomials on its dual space g*. In terms of the theory of integrable Hamiltonian systems this means that the dual space g* endowed with the standard Lie-Poisson bracket admits polynomial integrable Hamiltonian systems. This conjecture was proved by S. T. Sadetov in 2003. Following his idea, we give an explicit geometric construction for commuting polynomials on g* and consider some examples. (This text is a revised version of my paper published in Russian: A. V. Bolsinov, Complete commutative families of polynomials in Poisson?Lie algebras: A proof of the Mischenko?Fomenko conjecture in book: Tensor and Vector Analysis, Vol. 26, Moscow State University, 2005, 87?109.)

DEFORMATIONS OF LOOP ALGEBRAS AND CLASSICAL INTEGRABLE SYSTEMS: FINITE-DIMENSIONAL HAMILTONIAN SYSTEMS

2004 ◽  
Vol 16 (07) ◽  
pp. 823-849 ◽  
Author(s):  
T. SKRYPNYK
Keyword(s):  
Lie Algebras ◽  
Hamiltonian Systems ◽  
Spectral Parameter ◽  
Integrable Systems ◽  
Integrable Hamiltonian Systems ◽  
Infinite Dimensional ◽  
Loop Algebras ◽  
Finite Dimensional ◽  
Quasigraded Lie Algebras ◽  
Classical Integrable

We construct a family of infinite-dimensional quasigraded Lie algebras, that could be viewed as deformation of the graded loop algebras and admit Kostant–Adler scheme. Using them we obtain new integrable hamiltonian systems admitting Lax-type representations with the spectral parameter.

scholarly journals Singularities of integrable Liouville systems, reduction of integrals to lower degree and topological billiards: Recent results

2019 ◽  
Vol 46 (1) ◽  
pp. 47-63 ◽  
Author(s):  
A.T. Fomenko ◽  
V.V. Vedyushkina
Keyword(s):  
Differential Geometry ◽  
Hamiltonian Systems ◽  
Degrees Of Freedom ◽  
Lower Degree ◽  
State University ◽  
Integrable Hamiltonian Systems ◽  
Two Degrees Of Freedom ◽  
Moscow State University

In the paper we present the new results in the theory of integrable Hamiltonian systems with two degrees of freedom and topological billiards. The results are obtained by the authors, their students, and participants of scientific seminars of the Department of Differential Geometry and Applications, Faculty of Mathematics and Mechanics at Lomonosov Moscow State University.

scholarly journals PERIODIC AUTOMORPHISMS, COMPATIBLE POISSON BRACKETS, AND GAUDIN SUBALGEBRAS

2021 ◽  
Author(s):  
DMITRI I. PANYUSHEV ◽  
OKSANA S. YAKIMOVA
Keyword(s):  
Poisson Bracket ◽  
Lie Algebra ◽  
Invariant Theory ◽  
Cyclic Permutation ◽  
Poisson Brackets ◽  
Enveloping Algebra ◽  
Order Automorphism ◽  
Finite Dimensional ◽  
Compatible Poisson Brackets ◽  
Commutative Subalgebras

AbstractLet 𝔮 be a finite-dimensional Lie algebra. The symmetric algebra (𝔮) is equipped with the standard Lie–Poisson bracket. In this paper, we elaborate on a surprising observation that one naturally associates the second compatible Poisson bracket on (𝔮) to any finite order automorphism ϑ of 𝔮. We study related Poisson-commutative subalgebras (𝔮; ϑ) of 𝒮(𝔮) and associated Lie algebra contractions of 𝔮. To obtain substantial results, we have to assume that 𝔮 = 𝔤 is semisimple. Then we can use Vinberg’s theory of ϑ-groups and the machinery of Invariant Theory.If 𝔤 = 𝔥⊕⋯⊕𝔥 (sum of k copies), where 𝔥 is simple, and ϑ is the cyclic permutation, then we prove that the corresponding Poisson-commutative subalgebra (𝔮; ϑ) is polynomial and maximal. Furthermore, we quantise this (𝔤; ϑ) using a Gaudin subalgebra in the enveloping algebra 𝒰(𝔤).

AN INTEGRABLE DECOMPOSITION OF THE SYMMETRIC MATRIX KdV EQUATION

2008 ◽  
Vol 22 (13) ◽  
pp. 1307-1315
Author(s):  
RUGUANG ZHOU ◽  
ZHENYUN QIN
Keyword(s):  
Hamiltonian Systems ◽  
Matrix Method ◽  
Kdv Equation ◽  
Symmetric Matrix ◽  
Lax Pair ◽  
Integrable Hamiltonian Systems ◽  
R Matrix ◽  
Finite Dimensional ◽  
Higher Rank ◽  
Gaudin Models

A technique for nonlinearization of the Lax pair for the scalar soliton equations in (1+1) dimensions is applied to the symmetric matrix KdV equation. As a result, a pair of finite-dimensional integrable Hamiltonian systems, which are of higher rank generalization of the classic Gaudin models, are obtained. The integrability of the systems are shown by the explicit Lax representations and r-matrix method.

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