Lifting algebras via Hochschild cohomology

这个项目是关于群和代数的表示理论，纯数学的一个分支。它的目的是研究电梯的有限维代数领域F到订单完成离散赋值环O形式幂级数的环或环威特向量在F的动机这是有限群的表示理论,自然群代数/ O在哪里搭车的代数在F,问题在多大程度上后者决定前者是基本的重要性。在这个项目中，学生将尝试利用Hochschild上同调来发展代数升子的结构理论。最初的方法是这样的：-我们可以写出所有的Brauer树或图代数到一个离散验证环的提拉吗？-我们能否根据Hochschild上同调参数化这些提升，如果可以，我们能否从这样的参数化中读出这些提升的代数理论性质？-研究更一般的Brauer图代数和加权曲面代数的升力。这包括四元数缺陷块，其中存在关于未确定的标量的长期开放问题（与多诺万猜想有关）。-识别具有唯一（或接近唯一）升程的有限维代数的其他类别，不一定与群代数相关。-发展了升降机的适当结构理论，扩展了Gerstenhaber关于Hochschild上同调的工作。目前还不清楚这是否会成功，这无疑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，并阐明是什么将局部环上的群代数块与具有相似性质的其他代数类区分开来。

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.