Let $X$ be a manifold with a bi-Poisson structure $\{η^t\}$ generated by a pair of $G$-invariant symplectic structures $ω_1$ and $ω_2$, where the Lie group $G$ acts properly on $X$. Let $H$ be some isotropy subgroup for this action representing the principle orbit type and $X^r_\mathfrak{h}$ be the submanifold of $X$ consisting of the points in $X$ with the stabilizer algebra equal to the Lie algebra $\mathfrak{h}$ of $H$ and with the stabilizer group conjugated to $H$ in $G$. We prove that the pair of symplectic structures $ω_1|_{X^r_\mathfrak{h}}$ and $ω_2|_{X^r_\mathfrak{h}}$ generates an $N(H^0)/H^0$-invariant bi-Poisson structure on $X^r_\mathfrak{h}$, where $N(H^0)$ is the normalizer in $G$ of the identity component $H^0$ of $H$. The action of $\widetilde G=N(H^0)/H^0$ on $X^r_\mathfrak{h}$ is locally free and proper and, moreover, the spaces $A^G$ of $G$-invariant functions on $X$ and $A^{\widetilde G}$ of $\widetilde G$-invariant functions on $X^r_\mathfrak{h}$ can be canonically identified and therefore the bi-Poisson structure $\{(η^t)'\}$ induced on $A^G\simeq A^{\widetilde G}$ can be treated as the reduction with respect to a {\em locally free} action of a Lie group which essentially simplifies the study of $\{(η^t)'\}$.
Poisson brackets with prescribed family of functions in involution
众所周知，关于泊松括号的对合函数在完全可积系统理论中具有特殊的作用。在Poisson流形上寻找与给定函数$h$对合的函数无关的函数是该理论的一个基本问题，并且对于由$h$定义的运动方程的显式积分非常有用.在本文中，我们提出了我们的研究结果的逆，可以这么说，问题。通过发展一种类似于P. Damianou和F. Petalidou，具有指定Casimirs的泊松括号，Canad. J. Math.，2012，第64卷，991-1018，为了建立具有规定的Casimir不变量的泊松括号，我们构造了一种算法，该算法产生具有给定的对合函数族的泊松括号。我们的方法使我们能够建设性地处理双汉密尔顿结构，因此也使我们能够处理在这样的框架中出现的完全可积系统。
We construct a symplectic realization and a bi-hamiltonian formulation of a 3-dimensional system whose solution are the Jacobi elliptic functions. We generalize this system and the related Poisson brackets to higher dimensions. These more general systems are parametrized by lines in projective space. For these rank 2 Poisson brackets the Jacobi identity is satisfied only when the Pl\" ucker relations hold. Two of these Poisson brackets are compatible if and only if the corresponding lines in projective space intersect. We present several examples of such systems.
Given a symplectic form and a pseudo-riemannian metric on a manifold, a non degenerate even Poisson bracket on the algebra of differential forms is defined and its properties are studied. A comparison with the Koszul-Schouten bracket is established.
We give a method to construct Poisson brackets $\{\cdot,\cdot\}$ on Banach manifolds~$M$, for which the value of $\{f,g\}$ at some point $m\in M$ may depend on higher order derivatives of the smooth functions $f,g\colon M\to{\mathbb R}$, and not only on the first-order derivatives, as it is the case on all finite-dimensional manifolds. We discuss specific examples in this connection, as well as the impact on the earlier research on Poisson geometry of Banach manifolds. Those brackets are counterexamples to the claim that the Leibniz property for any Poisson bracket on a Banach manifold would i…
We construct the family of algebroid brackets $[\cdot,\cdot]_{c,v}$ on the tangent bundle $T^*M$ to a Poisson manifold $(M,π)$ starting from an algebroid bracket of differential forms. We use these brackets to generate Poisson structures on the tangent bundle $TM$. Next, in the case when $M$ is equipped with a bi-Hamiltonian structure $(M,π_1, π_2)$ we show how to construct another family of Poisson structures. Moreover we present how to find Casimir functions for those structures and we discuss some particular examples.
We introduce a natural class of multicomponent local Poisson structures $\mathcal P_k + \mathcal P_1$, where $\mathcal P_1$ is a local Poisson bracket of order one and $\mathcal P_k$ is a homogeneous Poisson bracket of odd order $k$ under assumption that is has Darboux coordinates (Darboux-Poisson bracket) and non-degenerate. For such brackets we obtain the general formulas in arbitrary coordinates, find normal forms (related to Frobenius triples) and provide the description of the Casimirs, using purely algebraic procedure. In two-component case we completely classify such brackets up to the …
We propose an extension of n-ary Nambu-Poisson bracket to superspace R^{n|m} and construct by means of superdeterminant a family of Nambu-Poisson algebras of even degree functions, where the parameter of this family is an invertible transformation of Grassmann coordinates in superspace R^{n|m}. We prove in the case of the superspaces R^{n|1} and R^{n|2} that our n-ary bracket, defined with the help of superdeterminant, satisfies the conditions for n-ary Nambu-Poisson bracket, i.e. it is totally skew-symmetric and it satisfies the Leibniz rule and the Filippov-Jacobi identity (fundamental ident…
