Speaker
Description
Koszul algebras were first defined by Priddy in 1970. These algebras have been extensively studied and arise naturally in various fields of mathematics, such as algebraic geometry, noncommutative geometry, topology and number theory. In this lecture series, we give an introduction to Koszul algebras and the role they play in representation theory. Many central classes of algebras in representation theory turn out to be Koszul. Examples include hereditary algebras, gentle algebras, quadratic monomial algebras, polynomial algebras and exterior algebras, as well as certain preprojective algebras and trivial extensions.
A main reason for the importance of Koszul algebras is their duality theory, as studied in the influential paper "Koszul duality patterns in representation theory" by Beilinson, Ginzburg and Soergel. To any Koszul algebra, there is an associated Koszul dual algebra. A key topic in the mini-course is the Koszul duality equivalence and how it reflects the strong connection between a Koszul algebra and its dual. As a motivating example, we look at how Koszul duality manifests in the geometric model for gentle algebras due to Opper, Plamondon and Schroll.
A core perspective in the lecture series is the notion of higher Koszul algebras, or n-T-Koszul algebras, which yields a natural connection to Iyama's higher Auslander–Reiten theory. This framework builds on a generalization of T-Koszul algebras, as introduced by Madsen and Green, Reiten and Solberg. We discuss a higher version of classical Koszul duality and sketch some applications for n-hereditary algebras. This part of the lecture series builds on joint work with Mads H. Sandøy, who is also responsible for the exercise classes in the course.