Surfaces and the Sklyanin bracket

Polynomilal Poisson algebras with regular structure of symplectic leaves

We study polynomial Poisson algebras with some regularity conditions. Linear (Lie-Berezin-Kirillov) structures on dual spaces of semi-simple Lie algebras, quadratic Sklyanin elliptic algebras of \cite{FO1},\cite{FO2} as well as polynomial algebras recently described by Bondal-Dubrovin-Ugaglia (\cite{Bondal},\cite{Ug}) belong to this class. We establish some simple determinantal relations between the brackets and Casimirs in this algebras. These relations imply in particular that for Sklyanin elliptic algebras the sum of Casimir degrees coincides with the dimension of the algebra. We are discus…

del Pezzo曲面上的例外对与兼容的Feigin-Odesskii括号空间

Exceptional pairs on del Pezzo surfaces and spaces of compatible Feigin-Odesskii brackets

本文研究了del Pezzo曲面上的例外对与兼容Feigin-Odesskii括号空间的关系，证明了某些条件下的存在性和构造新例子。

Poisson surfaces and algebraically completely integrable systems

One can associate to many of the well known algebraically integrable systems of Jacobians (generalized Hitchin systems, Sklyanin) a ruled surface which encodes much of its geometry. If one looks at the classification of such surfaces, there is one case of a ruled surface that does not seem to be covered. This is the case of projective bundle associated to the first jet bundle of a topologically nontrivial line bundle. We give the integrable system corresponding to this surface; it turns out to be a deformation of the Hitchin system.

Compatible Feigin-Odesskii Poisson brackets

We prove that several Feigin-Odesskii Poisson brackets associated with normal elliptic curves in ${\mathbb P}^n$ are compatible if and only if they are contained in a scroll or in a Veronese surface in ${\mathbb P}^5$ (with an exception of one case when $n=3$). In the case $n=3$ we determine the quartic corresponding to the Schouten bracket of two (non-compatible) Poisson brackets associated with normal elliptic curves $E_1$ and $E_2$.

与 G(2,5) 中椭圆曲线相关的相容泊松括号

Compatible Poisson Brackets Associated with Elliptic Curves in $G(2,5)$

本文证明了 P^4 上与 G(2,5) 线性截面椭圆曲线相关的一对 Feigin-Odesskii 泊松括号相容的充要条件是这对椭圆曲线包含于 G(2,5) 的线性截面 del Pezzo 曲面中。

The family of analytic Poisson brackets for the Camassa--Holm hierarchy

We consider the integrable Camassa--Holm hierarchy on the line with positive initial data rapidly decaying at infinity. It is known that flows of the hierarchy can be formulated in a Hamiltonian form using two compatible Poisson brackets. In this note we propose a new approach to Hamiltonian theory of the CH equation. In terms of associated Riemann surface and the Weyl function we write an analytic formula which produces a family of compatible Poisson brackets. The formula includes an entire function $f(z)$ as a parameter. The simplest choice $f(z)=1$ or $f(z)=z$ corresponds to the rational or…