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.

建议的博士项目是关于在推导代数几何的背景下研究位移辛和泊松结构及其量子化的例子。这是一个及时的研究项目，连接到代数几何，量子代数和数学物理领域的现代发展。在这个项目中使用的技术是从纯数学获得的。在本计画的第一阶段，学生将学习交换阶微分代数的几何，以及n型的定义和在这些结构上的衍生。这些工具可以用来定义n移辛结构和泊松结构。要探讨的一个主要例子是如何从这个框架中恢复李代数和拟李双代数结构上的非退化配对。在这些主要目标实现之后，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和更丰富的例子，例如微分分级李代数。