Lifting algebras via Hochschild cohomology

通过 Hochschild 上同调提升代数

• 批准号: 2666022
• 金额: --
• 依托单位: University of Manchester
• 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-2666022
• 关键词: Lifting; algebras; via; Hochschild; cohomology

This project is concerned with the representation theory of groups and algebras, a branch of Pure Mathematics. Its aim is the study of lifts of finite-dimensional algebras over a field F to orders over complete discrete valuation rings O like the ring of formal power series or the ring of Witt vectors over F. The motivation for this is the representation theory of finite groups, where the group algebra over O is naturally a lift of the group algebra over F, and the question to what extent the latter determines the former is of fundamental importance. In this project the student will try to develop a structure theory of lifts of algebras using Hochschild cohomology. The initial approach will look as follows:- Can we write down all lifts of Brauer tree or graph algebras to a discrete valiation ring?- Can we parametrise such lifts in terms of Hochschild cohomology, and, if so, can we read off algebra-theoretic properties of the lifts from such a parametrisation?- Study lifts of more general Brauer graph algebras and weighted surface algebras. This includes blocks of quaternion defect where there is a long-standing open problem regarding an undetermined socle scalar (related to Donovan's conjecture). - Identify other classes of finite-dimensional algebras which possess unique (or near unique) lifts, not necessarily related to group algebras. - Develop a proper structure theory of lifts, extending Gerstenhaber's work on Hochschild cohomology. It is unclear whether this will succeed, and this is certainly the most ambitious aim of the project.The hope is that the results of this thesis will feed back into ongoing work on Donovan's conjecture and shed some light on what sets blocks of group algebras over local rings apart from other classes of algebras with similar properties.

这个项目是关于群和代数的表示论，纯数学的一个分支。其目的是研究域F上的有限维代数到完全离散赋值环O（例如形式幂级数环或F上的Witt向量环）上的阶的提升。这样做的动机是有限群的表示论，其中O上的群代数自然是F上的群代数的提升，而后者在多大程度上决定前者的问题具有根本的重要性。 在这个项目中，学生将尝试使用Hochschild上同调来发展代数提升的结构理论。最初的方法如下：-我们可以写下Brauer树或图代数的所有提升到一个离散的valation环吗？我们能用Hochschild上同调来参数化这样的提升吗？如果能，我们能从这样的参数化中读出提升的代数理论性质吗？研究更一般的Brauer图代数和加权曲面代数的提升。这包括四元数缺陷块，其中有一个长期悬而未决的问题，关于一个未定的socle标量（有关多诺万猜想）。- 识别其他类具有唯一（或接近唯一）提升的有限维代数，不一定与群代数相关。- 发展一个适当的结构理论的升降机，延长Gerstenhaber的工作Hochschild上同调。目前还不清楚这是否会成功，这肯定是最雄心勃勃的目标的项目。希望是，这篇论文的结果将反馈到正在进行的工作多诺万的猜想，并阐明了什么组块的群代数在当地环除了其他类别的代数具有类似的性质。