Abstract

One particular reason for the success of the representation theory of finite dimensional algebras is that for the categories of finite dimensional modules over finite dimensional algebras one has Auslander-Reiten theory. The explanation for their importance is that Auslander-Reiten theory reveals combinatorial insight having its roots in deep homological features of such module categories (and categories with similar properties).
We intention of my lectures is to give an elementary introduction to Auslander-Reiten theory. We will start by defining the concept of almost-split sequences (=Auslander-Reiten sequences) forming a special class of short exact sequences. We discuss the uniqueness and the existence of such sequences, and the relationship between their end-terms, leading to the concept of the Auslander-Reiten translation (dual of the transpose). A combinatorial shadow of the module category (always of finite dimensional modules) is given by the concept of the Auslander-Reiten quiver. In good cases this will allow to construct all (indecomposable) modules from the indecomposable projective (or injective) ones. We will provide a number of examples of such Auslander-Reiten quivers, and discuss their shapes.
Another highlight of the lectures is to reformulate the existence of almost-split sequences as Auslander-Reiten duality, which provides a condensed form of Auslander-Reiten theory and serves as a bridge to other contexts (some quite unexpected) where Auslander-Reiten theory exists.



Information:

Lectures at the University of Isfahan


1. Frobenius and almost Frobenius Categories
Date and Time: Monday, February 7, 2011 at 16:00-17:00

2. Two Lectures on Hereditary Categories
Date and Time: Wednesday, February 9, 2011 at 10:00-11:00 and 11:30-12:30

3. Stable Categories of Vector Bundles
Date and Time: Monday, February 14, 2011 at 16:00-17:00