Last updated: October 2, 2006
Northeastern
University
Mathematics Department
Instructional Workshop
October 2 - 6, 2006
Dieter Happel (TU Chemnitz)
Representations of
Quivers, Past and Present
Abstract
The module theory of finite dimensional hereditary
algebras over an algebraically closed field is equivalent to the
representation theory of finite quivers without oriented cycles. A
quiver is a directed graph. A representation of a quiver is a
collection of vector spaces associated with the vertices and linear
maps associated with the arrows. The aim of these lectures is to
outline some of the information available on the category of finite
dimensional representations of a finite quiver without oriented cycles.
The first main result is the Theorem of Gabriel which states that this
category has finitely many indecomposable objects up to isomorphism if
and only if the underlying graph of the quiver is a disjoint union of
the Dynkin graphs occurring in the theory of finite dimensional
semisimple complex Lie algebras. There is a bijection between the
isomorphism classes of indecomposable representations and the positive
roots of the root system associated with the Dynkin graph. This result
was later generalized by Kac to all quivers using the root system
associated with Kac-Moody Lie algebras. Using
Auslander-Reiten theory the components of the Auslander-Reiten quiver
of category of finite
dimensional representations can be described. During the lectures we
will also outline tilting theory which allows to transfer some of the
information on quiver representations to more general finite
dimensional algebras. If time permits we will also indicate a different
aspect of tilting theory, namely that tilting theory can be used to
study the category of finite dimensional representation of a given
quiver by combinatorial methods.
Schedule
| Monday |
Tuesday |
Wednesday |
Thursday |
Friday |
| 12 - 1, 544 NI |
11-12, 544 NI |
12-1, 511 Lake |
12-1, 544 NI |
3:30-4:10, 320 Behrakis
|
Archive:
Fall
2005, Matrix Problems and Conformal Algebras
Spring
2005, Thom Polynomials for Group Actions
For further info,
contact Alex
Martsinkovsky >alexmart >at< neu >dot< edu<