Pure Maths Colloquium: Rosanna Laking

Representation theory aims to study an algebraic object, often a ring, by understanding how it acts on any given abelian group. In this talk we will be interested in studying a finite-dimensional algebra \(A\) from the viewpoint of representation theory. An abelian group together with an action of \(A\) defines an \(A\)-module and so we consider the collection of all (finite-dimensional) \(A\)-modules and the maps between them. This is the category \(\mathrm{mod}(A)\).

A classical problem, considered by Morita in 1958, is how to determine when we have an equivalence between the category \(\mathrm{mod}(A)\) and the category \(\mathrm{mod}(B)\) for two algebras \(A\) and \(B\). It turns out that such an equivalence can be detected by the presence of certain modules \(P\) called progenerators. We will take this classical theorem as the starting point of our talk and go on to review how this idea has developed in representation theory up until the present day. We will end by presenting some joint work with Karin Baur, in which we classify the cosilting modules over cluster-tilted algebras of type \(\tilde{\mathbb{A}}\).