A family of Poisson non-compact symmetric spaces

Compact symmetric spaces, triangular factorization, and Poisson geometry

Let X be a simply connected compact Riemannian symmetric space, let U be the universal covering group of the identity component of the isometry group of X, and let \g denote the complexification of the Lie algebra of U, \g=\u^\C. Each \u-compatible triangular decomposition \g=\n_- + \h + \n_+ determines a Poisson Lie group structure π_U on U. The Evens-Lu construction produces a (U,π_U)-homogeneous Poisson structure on X. By choosing the basepoint in X appropriately, X is presented as U/K where K is the fixed point set of an involution which stabilizes the triangular decomposition of \g. With …

一类 Poisson 问题的族上的一致估计：“磨圆角点”

Uniform estimates for a family of Poisson problems: `rounding off the corners'

本文针对一族有界区域上的椭圆问题证明了一致可解性估计，通过共形修改度量将区域并集化为带边界的相对有界几何流形，从而实现对多边形极限区域角点的“磨圆”处理。

Integrable systems associated with the Bruhat Poisson structures

The purpose of this note is to give a simple description of a (complete) family of functions in involution on certain hermitian symmetric spaces. This family, obtained via bi-hamiltonian approach using the Bruhat Poisson structure, is especially simple for the projective spaces, where the formulas in terms of the moment map coordinates are presented. We show how these functions are related to the Gelfand-Tsetlin coordinates. We also show how the Lenard scheme can be applied.

Regular Poisson structures on massive non-rotating BTZ black holes

We revisit the non-rotating massive BTZ black hole within a pseudo-Riemannian symmetric space context. Using classical symmetric space techniques we find that every such space intrinsically carries a regular Poisson structure whose symplectic leaves are para-hermitian symmetric surfaces. We also obtain a global expression of the metric yielding a dynamical description of the black hole from its initial to its final singularity.

Deformation of algebroid bracket of differential forms and Poisson manifold

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.

Poisson Reduction of Controlled Hamiltonian System by Controllability Distribution

In this paper, we first study the Poisson reductions of controlled Hamiltonian (CH) system and symmetric CH system by controllability distributions. These reductions are the extension of Poisson reductions by distribution for Poisson manifolds to that for phase spaces of CH systems with external force and control. We give Poisson reducible conditions of CH system by controllability distribution, and prove that the Poisson reducible property for CH systems leaves invariant under the CH-equivalence. Moreover, we study the Poisson reduction of symmetric CH system by G-invariant controllability di…

A nonlinear Poisson transform for Einstein metrics on product spaces

We consider the Einstein deformations of the reducible rank two symmetric spaces of noncompact type. If $M$ is the product of any two real, complex, quaternionic or octonionic hyperbolic spaces, we prove that the family of nearby Einstein metrics is parametrized by certain new geometric structures on the Furstenberg boundary of $M$.

Quasi-Poisson actions and massive non-rotating BTZ black holes

Using ideas from an article of P. Bieliavsky, M. Rooman and Ph. Spindel on BTZ black holes, I construct a family of interesting examples of quasi-Poisson actions as defined by A. Alekseev and Y. Kosmann-Schwarzbach. As an application, I obtain a genuine Poisson structure on $SL(2,R)$ which induces a Poisson structure on a BTZ black hole.

On a family of integral operators on the ball

In this work, we transform the equation in the upper half space first studied by Caffarelli and Silvestre to an equation in the Euclidean unit ball $\mathbb{B}^n$. We identify the Poisson kernel for the equation in the unit ball. Using the Poisson kernel, we define the extension operator. We prove an extension inequality in the limit case and prove the uniqueness of the extremal functions in the limit case using the method of moving spheres. In addition we offer an interpretation of the limit case inequality as a conformally invariant generalization of Carleman's inequality.

Jacobi-左对称代数胚上的 Jacobi-Koszul-Vinberg 结构

Jacobi-Koszul-Vinberg structures on Jacobi-left-symmetric algebroids

本文在 Jacobi-左对称代数胚上引入 Jacobi-Koszul-Vinberg 结构，作为 Jacobi 结构的对称类比，并研究其性质，进而定义 Jacobi-Koszul-Vinberg 流形。

Riemann 流形纤维化的 Poisson 求和公式

The Poisson summation formula for fibrations of Riemannian manifolds

本文研究 Riemann 纤维丛上广义 Laplace 算子族的波迹奇性，建立相应公式并说明其推广了族的 Poisson 求和公式。

Nambu-Poisson Bracket on Superspace

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…

Generalized n-Poisson brackets on a symplectic manifold

On a symplectic manifold a family of generalized Poisson brackets associated with powers of the symplectic form is studied. The extreme cases are related to the Hamiltonian and Liouville dynamics. It is shown that the Dirac brackets can be obtained in a similar way.