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Course, academic year 2018/2019
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Representation Theory of Finite-Dimensional Algebras - NMAG442
Title in English: Teorie reprezentací konečně-dimenzionálních algeber
Guaranteed by: Department of Algebra (32-KA)
Faculty: Faculty of Mathematics and Physics
Actual: from 2016
Semester: summer
E-Credits: 6
Hours per week, examination: summer s.:3/1 C+Ex [hours/week]
Capacity: unlimited
Min. number of students: unlimited
State of the course: taught
Language: English
Teaching methods: full-time
Guarantor: doc. RNDr. Jan Šťovíček, Ph.D.
Class: M Mgr. MSTR
M Mgr. MSTR > Povinně volitelné
Classification: Mathematics > Algebra
Incompatibility : NALG022
Interchangeability : NALG022
Annotation -
Last update: T_KA (14.05.2013)
The lecture is meant as an introduction to representation theory of finite dimensional algebras. The focus is put on path algebras, Auslander-Reiten theory, representation types and basics of tilting theory.
Course completion requirements -
Last update: doc. RNDr. Jan Šťovíček, Ph.D. (11.10.2017)

The credit will be granted on the basis of handed in homework. The homework will consist of three sets of problems published on the web page of the lecturer. At least 50 % of points from solutions of the problems handed in within given deadlines are required. If the conditions are not met, it is still possible to have the credit granted, where the exact form of updated conditions (a new deadline for solving the problems and/or extending the homework sets) is decided by the lecturer.

Literature -
Last update: T_KA (14.05.2013)
  • I. Assem, D. Simson and A. Skowroński, Elements of the Representation Theory of Associative Algebras I, Cambridge University Press, 1997.
  • M. Auslander, I. Reiten and S. O. Smalo, Representation Theory of Artin Algebras, Cambridge University Press, 2006.

Requirements to the exam -
Last update: doc. RNDr. Jan Šťovíček, Ph.D. (11.10.2017)

The course is completed with an oral exam. The requirements for the exam correspond to the syllabus and will be applied to the extent to which the topic was presented in lectures. It will be also demanded that the student is able to work with particular examples and do computations to the extent exercised at problem sessions or in given homework.

Syllabus -
Last update: T_KA (14.05.2013)

1. Path algebras, representations of quivers as modules over path algebras.

2. Projective and injective modules, indecomposable modules, Krull-Schmidt theorem.

3. Irreducible morphisms and almost split sequences, Auslander-Reiten quiver.

4. Finite representation type, the first Brauer-Thrall conjecture.

5. Representations of hereditary algebras, Gabriel's theorem.

6. Tilting and cotilting modules.

Entry requirements -
Last update: doc. RNDr. Jan Šťovíček, Ph.D. (11.10.2017)

Basics of theory of modules (to the extent of lecture NMAG333) and basic homological algebra (the Ext functor).

 
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