The Universal Generating Function of Analytical Poisson Structures

Generating functions for local symplectic groupoids and non-perturbative semiclassical quantization

This paper contains three results about generating functions for Lie-theoretic integration of Poisson brackets and their relation to quantization. In the first, we show how to construct a generating function associated to the germ of any local symplectic groupoid and we provide an explicit (smooth, non-formal) universal formula $S_π$ for integrating any Poisson structure $π$ on a coordinate space. The second result involves the relation to semiclassical quantization. We show that the formal Taylor expansion of $S_{tπ}$ around $t=0$ yields an extract of Kontsevich's star product formula based o…

Universal star products

One defines the notion of universal deformation quantization: given any manifold $M$, any Poisson structure $¶$ on $M$ and any torsionfree linear connection $\nabla$ on $M$, a universal deformation quantization associates to this data a star product on $(M,¶)$ given by a series of bidifferential operators whose corresponding tensors are given by universal polynomial expressions in the Poisson tensor $¶$, the curvature tensor $R$ and their covariant iterated derivatives. Such universal deformation quantization exist. We study their unicity at order 3 in the deformation parameter, computing the …

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…

Formal symplectic groupoid

The multiplicative structure of the trivial symplectic groupoid over $\mathbb R^d$ associated to the zero Poisson structure can be expressed in terms of a generating function. We address the problem of deforming such a generating function in the direction of a non-trivial Poisson structure so that the multiplication remains associative. We prove that such a deformation is unique under some reasonable conditions and we give the explicit formula for it. This formula turns out to be the semi-classical approximation of Kontsevich's deformation formula. For the case of a linear Poisson structure, t…

辛群胚乘法与泊松积分器

Symplectic groupoid multiplication and Poisson integrators

本文研究局部辛群胚乘法在泊松流形哈密顿流近似方法构造中的作用，证明其可用于构造拉格朗日二分法的新型乘法型近似序列，并给出理论结果、实现方式及示例。

Rigidity of quadratic Poisson tori

We prove a rigidity theorem for the Poisson automorphisms of the function fields of tori with quadratic Poisson structures over fields of characteristic 0. It gives an effective method for classifying the full Poisson automorphism groups of ${\mathbb{N}}$-graded connected cluster algebras equipped with Gekhtman-Shapiro-Vainshtein Poisson structures. Based on this, we classify the groups of algebraic Poisson automorphisms of the open Schubert cells of the full flag varieties of semisimple algebraic groups over fields of characteristic 0, equipped with the standard Poisson structures. Their coor…