Shifted Symplectic & Poisson Structures and their Quantisations in the context of Derived Algebraic Geometry

移辛

• 批准号: 2747173
• 金额: --
• 依托单位: University of Nottingham
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2022
• 资助国家: 英国
• 起止时间: 2022 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2747173
• 关键词: Shifted; Symplectic; Poisson; Structures; their

The proposed PhD project is about studying examples of shifted symplectic and Poisson structures and their quantisations in the context of derived algebraic geometry. This is a timely research project connecting to modern developments in the areas of algebraic geometry, quantum algebra and mathematical physics. The techniques used in this project are obtained from pure mathematics. In the first stage of this project, the student will learn about the geometry of commutative differential graded algebras and the definitions of n-forms and derivations on these structures. These tools can then be used to define n-shifted symplectic and Poisson structures. A main example to be explored is how non-degenerate pairings on a Lie algebra and quasi-Lie bialgebra structures can be recovered from this framework. After these primary goals have been achieved, the next step will be to apply the above techniques to investigate more novel and richer examples involving higher algebraic structures, such as differential graded Lie algebras.

拟议的博士项目是关于在派生代数几何背景下研究移位辛结构和泊松结构及其量子化的例子。这是一个与代数几何、量子代数和数学物理领域的现代发展相关的及时的研究项目。该项目中使用的技术来自纯数学。在本项目的第一阶段，学生将学习交换微分分级代数的几何以及 n 型的定义和这些结构的推导。然后，这些工具可用于定义 n 平移辛结构和泊松结构。要探索的一个主要例子是如何从该框架中恢复李代数和拟李双代数结构上的非简并配对。实现这些主要目标后，下一步将是应用上述技术来研究涉及更高代数结构的更新颖、更丰富的示例，例如微分分级李代数。