 A Short Course on AuslanderReiten Theory (6 Sessions)
Helmut Lenzing Paderborn University Germany February 16 & 17, 2011 School of Mathematics, IPM
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 AuslanderReiten theory. The explanation for their importance is that AuslanderReiten 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 AuslanderReiten theory. We will start by defining the concept of almostsplit sequences (=AuslanderReiten sequences) forming a special class of short exact sequences. We discuss the uniqueness and the existence of such sequences, and the relationship between their endterms, leading to the concept of the AuslanderReiten translation (dual of the transpose). A combinatorial shadow of the module category (always of finite dimensional modules) is given by the concept of the AuslanderReiten 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 AuslanderReiten quivers, and discuss their shapes.
Another highlight of the lectures is to reformulate the existence of almostsplit sequences as AuslanderReiten duality, which provides a condensed form of AuslanderReiten theory and serves as a bridge to other contexts (some quite unexpected) where AuslanderReiten theory exists.
Information:
Date:  Wednesday, February 16, 2011, 10:0011:00 and 11:3012:30
Wednesday, February 16, 2011, 15:0016:00
Thursday, February 17, 2011, 10:0011:00 and 11:3012:30
 Place:  Niavaran Bldg., Niavaran Square, Tehran, Iran 

Lectures at the University of Isfahan
1. Frobenius and almost Frobenius Categories
Date and Time: Monday, February 7, 2011 at 16:0017:00
2. Two Lectures on Hereditary Categories
Date and Time: Wednesday, February 9, 2011 at 10:0011:00 and 11:3012:30
3. Stable Categories of Vector Bundles
Date and Time: Monday, February 14, 2011 at 16:0017:00

 
