Poisson几何及其在数学物理中的应用

Poisson geometry is an important geometric branch. It has deep and wide influence on the theory and application of both classical mechanics and modern mathematical physics. In recent years, it is found to have close relation with the active mathematical fields cluster algebras and Stokes matrices. Lie groups and Lie algebras, to describe symmetry and conservative property, play important roles in many branches of mathematics and physics. With the development of advanced physics such as string theory and the need of math itself, Lie theory has been extended greatly. Higher Lie theory and nonlinear Lie theory become very popular research fields. The applicant has achieved a series of results on the research of topics related to Lie 2-algebras, Lie groupoids and Lie algebroids, which lays a solid foundation for the future work. This project focuses on the following three topics: (a) studies on the Poisson Lie groupoids and dual Poisson Lie groupoids coming from the solutions of classical dynamical Yang-Baxter equation and the applications in the linearization problem, cluster algebras and Stokes matrices; (b) generalized holomorphic vector bundles, generalized holomorphic connections, and the construction of the obstruction class for the existence of these connections; (c) studies on the deformations of Lie groupoids and groupoid convolution algebras.

Poisson几何是一个重要的几何分支，在经典力学与现代数学物理的理论和应用中有着深刻而广泛的影响。近些年，Poisson几何被发现与cluster代数和Stokes矩阵理论等热门领域有着紧密的联系。李群和李代数，用来描述对称和守恒性质，在数学和物理的许多分支中极为重要。由于弦理论等前沿物理学以及数学本身发展的需要，李理论被广泛地拓广，高阶李理论和非线性李理论成为非常活跃的研究领域。申请人已经在李2-代数、李群胚和李代数胚等若干课题研究中，取得了一些成果，有扎实的工作基础。本项目主要研究三方面的内容：(a)经典动力Yang-Baxter方程的解给出的Poisson李群胚和它的对偶Poisson李群胚的结构以及在线性化、cluster代数和Stokes矩阵理论中的应用研究；(b)广义全纯向量丛和广义全纯联络，构造这个联络存在的障碍类；(c)李群胚和它的群胚卷积代数的形变。

由于弦理论等前沿物理学以及数学本身发展的需要，李理论被广泛地拓广，高阶李理论和非线性李理论成为非常活跃的研究领域。在本项目中，我们 1）在李群和李群胚上分别引入Rota-Baxter算子，实现Rota-Baxter李代数的积分化和几何化；2）研究李群胚上仿射几何结构，并发现其中的高阶代数结构；3）研究高阶Courant代数胚的线性化问题，引入高阶omni-李代数胚的概念；研究李双代数胚交叉模结构，证明其与余二次Manin三元组一一对应。；4）证明double向量丛的标架丛是double主丛，且任何double向量丛都可以实现成它的标架丛的配丛。..这些结果丰富了高阶李理论和非线性李理论的研究内容。特别地，李群上Rota-Baxter算子的成功引入突破了Rota-Baxter算子对线性结构的依赖，为相关方向提供了新的思想，获得了良好的反响和多次引用。