# Math 7362- Topics in Algebra: Quiver representations (Spring 2013)

## Course Information (updated Jan. 7, 2013)

 Time and place: Mon 3:10-4:50, Wed 4:10 pm - 5:50 pm, 544 Nightingale Textbook: Various resources are given below. Instructor: Ryan Kinser Office and phone: 550 Nightingale Hall, (617) 373-5028. (From the Nightingale entrance go to the 5th floor, then from the stairway/elevator head straight until you get to Jerzy Weyman's office, then my office is the one to your right.) Email: r.kinser (at) neu (dot) edu Office hours: TBA

### Course Description

We will cover basic theory of representations of quivers, both algebraic and geometric aspects. The algebraic side approaches representations as modules over the path algebra, utilizing techniques of homological algebra to understand the structure of representations. A primary goal would be to understand how Auslander-Reiten theory gives a combinatorial picture of the module category, especially in Dynkin types.

On the other hand, the geometric side views representations as orbits in a matrix variety that is equipped with a group action. Here the goal is to relate the geometry of orbit closures (these include, for example, varieties such as the nilpotent cone in gl(n) and Buchsbaum-Eisenbud varieties of complexes) to representation theoretic properties. We can also apply GIT to try and construct moduli spaces of certain representations.

The exact pace and special topics covered will be tailored to the interests and background of the students. I will post my personal class notes as we go, but these are not a textbook and may contain typos or vague parts that I modify in real time during class. The structure basically follows old class notes of Harm Derksen and Jerzy Weyman.

• Fundamentals (the category of representations of a quiver, Path algebra, Representation spaces) Notes 1
• Homological tools (projectives and injectives, hereditary, Ext, Euler form) Notes 2
• Gabriel's Theorem (mainly geometric proof, following Brion's notes) My notes
• Examples with Dynkin quivers, Kronecker quivers Ringel's preprint
• GIT and moduli spaces associated to quivers, probably follow Reineke's survey
• Quivers with relations (algebraic and geometric basics)
• ...

The grading will be based on a few homework sets and student presentations.

## Homework:

Survey on background. Please indicate any special interests or remarks at the bottom of the page. You can leave it in my mailbox if you didn't give me one already.
First assignment
Second
Third

## Talk schedule:

• Wed. March 27: Dylan Rupel (preview of his Fall 2013 grad course)
• Mon. April 1: Floran and José
• Wed. April 3: Anupam and Chen
• Mon. April 8: Huijun and Gufang
• Wed. April 10: András and Rahul