Series of lectures on Quiver Representations

May 28, 31, June 4, 7, 2013 h. 14:30 - Giovanni Cerulli Irelli


In the framework of the "Representation theory of groups" course, Dr. Giovanni Cerulli Irelli (Bonn) will deliver a series of lectures on "Quiver Representations" aimed at (Master) students and beyond according to the following schedule:

May 28th, 14:30-17:30 - Room 1BC45
May 31st, 14:30-16:30 - Room 2AB40
June 4th, 14:30-17:30 - Room 1BC45
June 7th, 14:30-16:30 - Room 2AB40

Representation theory of quivers plays a central role in different areas of mathematics: quantum groups, derived categories, finite dimensional algebras, combinatorics... In this ten-hours-long course, I will provide an introduction to the subject. The mini--course is organized around three main topics: basics on quiver representations, integral quadratic forms and root systems, basics on Auslander--Reiten theory. The main goal of the course is the proof of the fundamental theorem of Gabriel (1972): a quiver admits finitely many indecomposable representations if and only if it is an orientation of a simply-laced Dynkin diagram. But we will learn many more things, not only related to the proof of this theorem. Indeed, the course is not intended only for people with interests in representation theory and no previous familiarity with the subject is required.

I will mainly use the following three references:
Ringel, "Tame Algebras and Integral Quadratic Forms", Lecture Notes in Mathematics 1099, 1984. (This is a book; we will use some results from the 1st chapter).
Bernstein, Gel'fand and Ponomarev; "Coxeter Functors and Gabriel's theorem"; Uspehi Mat. Nauk 28 (1973), no. 2(170), 19-33. (This is a famous paper, but the journal is hard to find. I will put a copy of it in my webpage).
Assem, Simson, Skowronski; "Elements of the Representation Theory of Associative Algebras. Part 1: Techniques of Representation Theory" (This is an excellent introductory book; look at the first four chapters).