Lifting algebras via Hochschild cohomology

这个项目是关于群和代数的表示理论，它是纯数学的一个分支。它的目的是研究域F上有限维代数到完全离散赋值环O上的阶的提升，如F上的形式幂级数环或Witt向量环。这方面的动机是有限群的表示理论，其中O上的群代数自然是F上的群代数的提升，而F上的群代数在多大程度上决定了前者是基本重要的问题。在这个项目中，学生将尝试利用Hochschild上同调来发展代数的Lifts的结构理论。最初的方法如下：-我们能把Brauer树或图代数的所有提升写到一个离散验证环上吗？-我们能用Hochschild上同调来参数化提升吗？如果能，我们能从这样的参数化中读出提升的代数论性质吗？-研究更一般的Brauer图代数和加权曲面代数的提升。这包括四元数块缺陷，其中存在关于未确定的基标量的长期悬而未决的问题(与多诺万猜想有关)。-识别具有唯一(或接近唯一)提升的其他有限维代数类，这些提升不一定与群代数相关。-发展了Lifts的适当结构理论，扩展了Gertenhaber在Hochschild上同调方面的工作。目前还不清楚这是否会成功，这肯定是该项目最雄心勃勃的目标。希望这篇论文的结果将反馈到正在进行的多诺万猜想的工作中，并揭示局部环上的群代数块与其他具有类似性质的代数类的区别。

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.