## UNDERGRADUATE SEMINAR IN REPRESENTATION THEORY

## Practical information

The seminar is scheduled for the Summer 2018 semester at University of Bonn.
Meetings will take place Tuesdays, 12:00-14:00 in room SR N0.003 in the Endenicher Allee math building.
This is a parallel session of the Hauptseminar led by Professor Franzen, of which a German description is
here, and we will follow the same schedule as Professor Franzen's section of the seminar.
Student talks may be given in English or German; I will communicate in English.
It is expected that every student will give at least one talk. If there are more weeks than people registered, students have the option to give a second talk (but this is not required!).
Contact me by email if you are a student interested in registering for the seminar.

The course "Einfuehrung in die Algebra" is sufficient preparation for participation in the seminar.

## Description

In this seminar we will be concerned with the representation theory of quivers. We will see that quiver representations coincide with left modules over the path algebra. Particular types of modules which we will study include indecomposable, projective, and injective modules. Further topics include homological properties and root systems.
## Textbook

"Quiver Representations" by Ralf Schiffler.
## Schedule of the seminar

All chapter numbers refer to Schiffler's book.

(1) Definitions and examples. 1.1 and 1.2. Quivers, representations, morphisms, direct sums, indecomposability, Krull-Schmidt theorem.

(2) Kernels and cokernels. 1.3. Kernels, cokernels, exactness, (split) short exact sequences.

(3) Hom-functors. 1.4. Categories and functors, Hom-functors.

(4) Projectives and injectives. 2.1 and 2.2. Simple, projective, and injective representations, from 2.2 only the definition of projective and injective resolutions and construction of the projective resolution

(5) 2.4. Ext-groups.

(6) Rings and algebras. 4.1 and 4.2. Ring, ideal, radical, algebra, path algebra.

(7) Modules. 4.3. Module, Nakayama's lemma, the five lemma

(8) Idempotents. 4.4 and 4.5. Itempotent, indecomposability criterion.

(9) Representations are modules. 5.2. Equivalence between modules over a path algebra and quiver representations in the hereditary case (so when I=0).

(10) Quadratic forms. 8.2. Tits form, classification of quivers with positive definite Tits form (Dynkin diagrams).

(11) Roots. 8.3. Roots, root systems from Dynkin diagrams.

(12) The representation variety. 8.1. The variety of representations of dimension d, group action via change of basis, orbit dimension.

(13) Gabriel's theorem. 8.4. Gabriel's theorem with proof.