scholarly journals Classical r-Matrices and Compatible Poisson Structures for Lax Equations on Poisson Algebras

1999 ◽  
Vol 203 (3) ◽  
pp. 573-592 ◽  
Author(s):  
Luen-Chau Li
Keyword(s):  
Poisson Structures ◽  
Poisson Algebras ◽  
Compatible Poisson Structures ◽  
Lax Equations

Compatible poisson structures for lax equations: An r-matrix approach

1988 ◽  
Vol 130 (8-9) ◽  
pp. 456-460 ◽  
Author(s):  
A.G. Reyman ◽  
M.A. Semenov-Tian-Shansky
Keyword(s):  
Matrix Approach ◽  
Poisson Structures ◽  
R Matrix ◽  
Compatible Poisson Structures ◽  
Lax Equations

scholarly journals Applications of Nijenhuis geometry: non-degenerate singular points of Poisson–Nijenhuis structures

2021 ◽  
Author(s):  
Alexey V. Bolsinov ◽  
Andrey Yu. Konyaev ◽  
Vladimir S. Matveev
Keyword(s):  
Singular Points ◽  
Recursion Operator ◽  
Poisson Structures ◽  
Degeneracy Condition ◽  
Compatible Poisson Structures

AbstractWe study and completely describe pairs of compatible Poisson structures near singular points of the recursion operator satisfying natural non-degeneracy condition.

scholarly journals COMPATIBLE POISSON STRUCTURES OF TODA TYPE DISCRETE HIERARCHY

2005 ◽  
Vol 20 (07) ◽  
pp. 1367-1388 ◽  
Author(s):  
HENRIK ARATYN ◽  
KLAUS BERING
Keyword(s):  
Poisson Structure ◽  
Integrable Hierarchy ◽  
Difference Operators ◽  
Baxter Equation ◽  
Poisson Structures ◽  
Algebra Isomorphism ◽  
R Matrix ◽  
Compatible Poisson Structures ◽  
Special Value

An algebra isomorphism between algebras of matrices and difference operators is used to investigate the discrete integrable hierarchy. We find local and nonlocal families of R-matrix solutions to the modified Yang–Baxter equation. The three R-theoretic Poisson structures and the Suris quadratic bracket are derived. The resulting family of bi-Poisson structures include a seminal discrete bi-Poisson structure of Kupershmidt at a special value.

scholarly journals A NEW FAMILY OF POISSON ALGEBRAS AND THEIR DEFORMATIONS

2017 ◽  
Vol 233 ◽  
pp. 32-86 ◽  
Author(s):  
CESAR LECOUTRE ◽  
SUSAN J. SIERRA
Keyword(s):  
Positive Integer ◽  
Characteristic Zero ◽  
Semiclassical Limit ◽  
Poisson Structures ◽  
Prime Spectrum ◽  
Poisson Algebras ◽  
New Family ◽  
Regular Algebras ◽  
Point Modules

Let $\Bbbk$ be a field of characteristic zero. For any positive integer $n$ and any scalar $a\in \Bbbk$, we construct a family of Artin–Schelter regular algebras $R(n,a)$, which are quantizations of Poisson structures on $\Bbbk [x_{0},\ldots ,x_{n}]$. This generalizes an example given by Pym when $n=3$. For a particular choice of the parameter $a$ we obtain new examples of Calabi–Yau algebras when $n\geqslant 4$. We also study the ring theoretic properties of the algebras $R(n,a)$. We show that the point modules of $R(n,a)$ are parameterized by a bouquet of rational normal curves in $\mathbb{P}^{n}$, and that the prime spectrum of $R(n,a)$ is homeomorphic to the Poisson spectrum of its semiclassical limit. Moreover, we explicitly describe $\operatorname{Spec}R(n,a)$ as a union of commutative strata.

scholarly journals SYSTEMS OF HESS–APPEL'ROT TYPE AND ZHUKOVSKII PROPERTY

2009 ◽  
Vol 06 (08) ◽  
pp. 1253-1304 ◽  
Author(s):  
VLADIMIR DRAGOVIĆ ◽  
BORISLAV GAJIĆ ◽  
BOŽIDAR JOVANOVIĆ
Keyword(s):  
Symplectic Manifold ◽  
General Characteristic ◽  
Poisson Manifold ◽  
Partial Reduction ◽  
Poisson Structures ◽  
Remarkable Property ◽  
Compatible Poisson Structures ◽  
Casimir Functions ◽  
Crucial Assumption ◽  
Body Case

We start with a review of a class of systems with invariant relations, so called systems of Hess–Appel'rot type that generalizes the classical Hess–Appel'rot rigid body case. The systems of Hess–Appel'rot type have remarkable property: there exists a pair of compatible Poisson structures, such that a system is certain Hamiltonian perturbation of an integrable bi-Hamiltonian system. The invariant relations are Casimir functions of the second structure. The systems of Hess–Appel'rot type carry an interesting combination of both integrable and non-integrable properties. Further, following integrable line, we study partial reductions and systems having what we call the Zhukovskii property: These are Hamiltonian systems on a symplectic manifold M with actions of two groups G and K; the systems are assumed to be K-invariant and to have invariant relation Φ = 0 given by the momentum mapping of the G-action, admitting two types of reductions, a reduction to the Poisson manifold P = M/K and a partial reduction to the symplectic manifold N0 = Φ-1(0)/G; final and crucial assumption is that the partially reduced system to N0 is completely integrable. We prove that the Zhukovskii property is a quite general characteristic of systems of Hess–Appel'rot type. The partial reduction neglects the most interesting and challenging part of the dynamics of the systems of Hess–Appel'rot type — the non-integrable part, some analysis of which may be seen as a reconstruction problem. We show that an integrable system, the magnetic pendulum on the oriented Grassmannian Gr+(n, 2) has a natural interpretation within Zhukovskii property and that it is equivalent to a partial reduction of certain system of Hess–Appel'rot type. We perform a classical and algebro-geometric integration of the system in dimension four, as an example of a known isoholomorphic system — the Lagrange bitop. The paper presents a lot of examples of systems of Hess–Appel'rot type, giving an additional argument in favor of further study of this class of systems.

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