COURSE UNIT TITLE

: ALGEBRAS AND QUIVER REPRESENTATIONS

DEGREE PROGRAMMES

Description of Individual Course Units

Course Unit Code Course Unit Title Type Of Course D U L ECTS
MAT 5067 ALGEBRAS AND QUIVER REPRESENTATIONS ELECTIVE 3 0 0 8

Offered By

Graduate School of Natural and Applied Sciences

Level of Course Unit

Second Cycle Programmes (Master's Degree)

Course Coordinator

ASSOCIATE PROFESSOR ENGIN MERMUT

Offered to

Mathematics
Mathematics

Course Objective

The aim of this course is to introduce the fundamental concepts in algebras and quiver representations.

Learning Outcomes of the Course Unit

1   Will be able to identify main kinds of algebras like semisimple algebras, central simple algebras, separable algebras, Quasi-Frobenious algebras and serial algebras.
2   Will be able to use representations of algebras and groups.
3   Will be able to understand how the study of finite-dimensional algebras over a field reduces to the study of bound quiver algebras.
4   Will be able to understand how Gabriel s Theorem gives the indecomposable representations by roots of associated quadratic forms for a connected quiver with finite representation type.
5   Will be able to use Auslander-Reiten quiver of a finite dimensional algebra A to approximate the category of A-modules.

Mode of Delivery

Face -to- Face

Prerequisites and Co-requisites

None

Recomended Optional Programme Components

None

Course Contents

Week Subject Description
1 Associative algebras. Group algebras, endomorphism algebras, matrix algebras, finite-dimensional algebras over a fields. Quaternion algebras.
2 Simple modules, Semisimple modules, Radical. The structure of semisimple algebras; Wedderburn-Artin theorem. Maschke s Theorem.
3 The radical of an algebra. Artinian algebras. Nilpotent algebras.
4 Indecomposable modules. Local algebras. Fitting s lemma. The Krull-Schmidt Theorem. Representation of algebras. Indecomposable and irreducible representations.
5 Projective modules over Artinian algebras. Basic algebras. Finite Representation Type.
6 Central simple algebras. Separable Algebras.
7 The Brauer group.
8 Midterm
9 Representations of Quivers. Projective and injective representations. Auslander-Reiten translation.
10 Auslander-Reiten Quivers. Examples. The Knitting algorithm.
11 Gabriel's Theorem for finite represention type. Quadratic forms, roots and Gabriel s Theorem.
12 Auslander-Reiten quivers of type Dn. Bound quivers, quivers with relations.Bound quiver algebras.
13 New algebras from old: Tilted algebras, trivial extensions, self-injective algebras, etc. Quasi-Frobenious algebras and serial algebras.
14 Auslander-Reiten Theory. Almost split sequences, Auslander-Reiten translation, Auslander-Reiten formulas. Auslander-Reiten Theory. Almost split sequences, Auslander-Reiten translation, Auslander-Reiten formulas.

Recomended or Required Reading

Textbook(s):
[1] Pierce, R. S. Associative Algebras. Springer, 1982.
[2] Schiffler, R. Quiver Representations. Springer, 2014.

Supplementary Book(s):
[3] Ibrahim Assem, I., Simson, D. And Skowronski, A. Elements of the Representation Theory of Associative Algebras, Volume 1 Techniques of Representation Theory. Cambridge, 2006.
[4] Drozd, Y. A. and Kirichenko, V. V. Finite Dimensional Algebras. Springer, 1994.
[5] Farb, B. and Dennis, R. K. Noncommutative algebra. Springer, 1993.
[6] Hazewinkel, M., Gubareni, N. and Kirichenko, V. V. Algebras, Rings and Modules. Volumes 1 and 2. Kluwer, 2005 and 2007.

References:

Materials:
Instructor s notes and presentations

Planned Learning Activities and Teaching Methods

Lecture notes
Presentation
Problem solving

Assessment Methods

SORTING NUMBER SHORT CODE LONG CODE FORMULA
1 MTE MIDTERM EXAM
2 ASG ASSIGNMENT
3 FIN FINAL EXAM
4 FCG FINAL COURSE GRADE MTE * 0.30 + ASG * 0.40 + FIN * 0.30
5 RST RESIT
6 FCGR FINAL COURSE GRADE (RESIT) MTE * 0.30 + MAKRASG * 0.40 + MAKRRST * 0.30


*** Resit Exam is Not Administered in Institutions Where Resit is not Applicable.

Further Notes About Assessment Methods

None

Assessment Criteria

To be announced.

Language of Instruction

English

Course Policies and Rules

Attending at least 70 percent of lectures is mandatory.

Contact Details for the Lecturer(s)

e-mail: engin.mermut@deu.edu.tr
Office: (232) 301 85 82

Office Hours

To be announced.

Work Placement(s)

None

Workload Calculation

Activities Number Time (hours) Total Work Load (hours)
Lectures 13 3 39
Preparations before/after weekly lectures 13 5 65
Preparation for midterm exam 1 15 15
Preparation for final exam 1 25 25
Preparing assignments 10 5 50
Final 1 3 3
Midterm 1 3 3
TOTAL WORKLOAD (hours) 200

Contribution of Learning Outcomes to Programme Outcomes

PO/LOPO.1PO.2PO.3PO.4PO.5PO.6PO.7PO.8PO.9PO.10PO.11
LO.14332224333
LO.24332224333
LO.34332224333
LO.44332224333
LO.54332224333

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