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.