Poisson structures whose Poisson diffeomorphism group is not locally path-connected

Realization of tangent perturbations in discrete and continuous time conservative systems

We prove that any perturbation of the symplectic part of the derivative of a Poisson diffeomorphism can be realized as the derivative of a $C^1$-close Poisson diffeomorphism. We also show that a similar property holds for the Poincaré map of a Hamiltonian on a Poisson manifold. These results are the conservative counterparts of the Franks lemma, a perturbation tool used in several contexts most notably in the theory of smooth dynamical systems.

Universal cocycles and the graph complex action on homogeneous Poisson brackets by diffeomorphisms

The graph complex acts on the spaces of Poisson bi-vectors $P$ by infinitesimal symmetries. We prove that whenever a Poisson structure is homogeneous, i.e. $P = L_{\vec{V}}(P)$ w.r.t. the Lie derivative along some vector field $\vec{V}$, but not quadratic (the coefficients of $P$ are not degree-two homogeneous polynomials), and whenever its velocity bi-vector $\dot{P}=Q(P)$, also homogeneous w.r.t. $\vec{V}$ by $L_{\vec{V}}(Q)=n\cdot Q$ whenever $Q(P)= Or(γ)(P^{\otimes^n})$ is obtained using the orientation morphism $Or$ from a graph cocycle $γ$ on $n$ vertices and $2n-2$ edges in each term, t…

The momentum map in Poisson geometry

Every action on a Poisson manifold by Poisson diffeomorphisms lifts to a Hamiltonian action on its symplectic groupoid which has a canonically defined momentum map. We study various properties of this momentum map as well as its use in reduction.

哈密顿微分同胚群的光滑完美性

Smooth perfectness of Hamiltonian diffeomorphism groups

本文细化了 Banyaga 关于哈密顿微分同胚群完美性的经典结果，证明了在 $C^\infty$ 拓扑中局部表示所需交换子数量有界且光滑依赖，并将该光滑完美性结果推广到开辛流形和带边辛流形。

Poisson geometry and first integrals of geostrophic equations

We describe first integrals of geostrophic equations, which are similar to the enstrophy invariants of the Euler equation for an ideal incompressible fluid. We explain the geometry behind this similarity, give several equivalent definitions of the Poisson structure on the space of smooth densities on a symplectic manifold, and show how it can be obtained via the Hamiltonian reduction from a symplectic structure on the diffeomorphism group.

Ginzburg-Weinstein via Gelfand-Zeitlin

Let U(n) be the unitary group, and $u(n)^*$ the dual of its Lie algebra, equipped with the Kirillov Poisson structure. In their 1983 paper, Guillemin-Sternberg introduced a densely defined Hamiltonian action of a torus of dimension $(n-1)n/2$ on $u(n)^*$, with moment map given by the Gelfand-Zeitlin coordinates. A few years later, Flaschka-Ratiu described a similar, `multiplicative' Gelfand-Zeitlin system for the Poisson Lie group $U(n)^*$. By the Ginzburg-Weinstein theorem, $U(n)^*$ is isomorphic to $u(n)^*$ as a Poisson manifold. Flaschka-Ratiu conjectured that one can choose the Ginzburg-We…

Integrability of central extensions of the Poisson Lie algebra via prequantization

We present a geometric construction of central S^1-extensions of the quantomorphism group of a prequantizable, compact, symplectic manifold, and explicitly describe the corresponding lattice of integrable cocycles on the Poisson Lie algebra. We use this to find nontrivial central S^1-extensions of the universal cover of the group of Hamiltonian diffeomorphisms. In the process, we obtain central S^1-extensions of Lie groups that act by exact strict contact transformations.

Deformations of log-symplectic structures

We describe the space of Poisson bivectors near a log-symplectic structure up to small diffeomorphisms.

Linearization of Poisson groupoids

Motivated by a search for Lie group structures on groups of Poisson diffeomorphisms [24], we investigate linearizability of Poisson structures of Poisson groupoids around the unit section. After extending the Lagrangian neighbourhood theorem to the setting of cosymplectic Lie algebroids, we establish that dual integrations of triangular bialgebroids are always linearizable. Additionally, we show that the (non-dual) integration of a triangular Lie bialgebroid is linearizable whenever the $r$-matrix is of so-called cosymplectic type. The proof relies on the integration of a triangular Lie bialge…

Generalized Hunter-Saxton equation and the geometry of the group of circle diffeomorphisms

We study an equation lying `mid-way' between the periodic Hunter-Saxton and Camassa-Holm equations, and which describes evolution of rotators in liquid crystals with external magnetic field and self-interaction. We prove that it is an Euler equation on the diffeomorphism group of the circle corresponding to a natural right-invariant Sobolev metric. We show that the equation is bihamiltonian and admits both cusped, as well as smooth, traveling-wave solutions which are natural candidates for solitons. We also prove that it is locally well-posed and establish results on the lifespan of its soluti…

Diffeomorphisms of 7-Manifolds with Closed G_2-Structure

We introduce G_2-vector fields, Rochesterian 1-forms and Rochesterian vector fields on manifolds with a closed G_2-structure as analogues of symplectic vector fields, Hamiltonian functions and Hamiltonian vector fields respectively, and we show that the spaces of G_2-vector fields and of Rochesterian vector fields are Lie subalgebras of the Lie algebra of vector fields with the standard Lie bracket. We also define, in analogy with the Poisson bracket on smooth real-valued functions from symplectic geometry, a bracket operation on the space of Rochesterian 1-forms associated to the space of Roc…

非挤压及其他局部共形辛几何中的全局刚性结果

Non-squeezing and other global rigidity results in locally conformal symplectic geometry

本文通过二次无穷远生成函数定义了扭切丛中紧支持lcs哈密顿同胚的谱选择器，获得非挤压定理等应用，并引入了本质Lee弦和本质翻译点的概念。

Poisson cohomology in dimension two

It is known that the computation of the Poisson cohomology is closely related to the classification of singularities of Poisson structures. In this paper, we will first look for the normal forms of germs at (0,0) of Poisson structures on the real (or complex) plane and recall a result given by Arnold. Then, we will compute locally the Poisson cohomology of a particular type of Poisson structure.

Picard groups of Poisson manifolds

For a Poisson manifold $M$ we develop systematic methods to compute its Picard group $Pic(M)$, i.e., its group of self Morita equivalences. We establish a precise relationship between $Pic(M)$ and the group of gauge transformations up to Poisson diffeomorphisms showing, in particular, that their connected components of the identity coincide; this allows us to introduce the Picard Lie algebra of $M$ and to study its basic properties. Our methods lead, in particular, to the proof of a conjecture from [BW04] stating that for any compact simple Lie algebra $\mathfrak{g}$ the group $Pic(\mathfrak{g…

Hamiltonian pseudo-representations

The question studied here is the behavior of the Poisson bracket under C^0-perturbations. In this purpose, we introduce the notion of pseudo-representation and prove that for a normed Lie algebra, it converges to a representation. An unexpected consequence of this result is that for many non-closed symplectic manifolds (including cotangent bundles), the group of Hamiltonian diffeomorphisms (with no assumptions on supports) has no C^{-1} bi-invariant metric. Our methods also provide a new proof of Gromov-Eliashberg Theorem, it is to say that the group of symplectic diffeomorphisms is C^0-closed…

Examples of non $d_ω$-exact locally conformal symplectic forms

We exhibit two three-parameter families of locally conformal symplectic forms on the solvmanifold $M_{n,k}$ considered in [1], and show, using the Hodge-de Rham theory for the Lichnerowicz cohomology that that they are not $d_ω$ exact, i.e. their Lichnerowicz classes are non-trivial (Theorem 1). This has several important geometric consequences (corollary 2, 3). This also implies that the group of automorphisms of the corresponding locally conformal symplectic structures behaves much like the group of symplectic diffeomorphisms of compact symplectic manifolds. We initiate the classification of…

Poisson 结构空间上的一个 Poisson 括号

A Poisson bracket on the space of Poisson structures

本文在流形的 Poisson 结构空间上构造依赖体积形式的 Poisson 括号，提出流方程不动点不变量，并将其用于检测正则 Poisson 3-流形的幺模性及联系辛上同调。

A decoupling property of some Poisson structures on ${\rm Mat}_{n\times d}(\mathbb{C}) \times {\rm Mat}_{d\times n}(\mathbb{C})$ supporting ${\rm GL}(n,\mathbb{C}) \times {\rm GL}(d,\mathbb{C})$ Poisson-Lie symmetry

We study a holomorphic Poisson structure defined on the linear space $S(n,d):= {\rm Mat}_{n\times d}(\mathbb{C}) \times {\rm Mat}_{d\times n}(\mathbb{C})$ that is covariant under the natural left actions of the standard ${\rm GL}(n,\mathbb{C})$ and ${\rm GL}(d,\mathbb{C})$ Poisson-Lie groups. The Poisson brackets of the matrix elements contain quadratic and constant terms, and the Poisson tensor is non-degenerate on a dense subset. Taking the $d=1$ special case gives a Poisson structure on $S(n,1)$, and we construct a local Poisson map from the Cartesian product of $d$ independent copies of $S…