Representation Theory of Finite-Dimensional Algebras - NMAG442
Title: Teorie reprezentací konečně-dimenzionálních algeber
Guaranteed by: Department of Algebra (32-KA)
Faculty: Faculty of Mathematics and Physics
Actual: from 2023
Semester: summer
E-Credits: 6
Hours per week, examination: summer s.:3/1, C+Ex [HT]
Capacity: unlimited
Min. number of students: unlimited
4EU+: no
Virtual mobility / capacity: no
State of the course: taught
Language: Czech
Teaching methods: full-time
Additional information: https://www.karlin.mff.cuni.cz/~stovicek/index.php/cs/2324ls-nmag442
Guarantor: doc. RNDr. Jan Šťovíček, Ph.D.
Teacher(s): Mgr. Kateřina Fuková
doc. RNDr. Jan Šťovíček, Ph.D.
Class: M Mgr. MSTR
M Mgr. MSTR > Povinně volitelné
Classification: Mathematics > Algebra
Incompatibility : NALG022
Interchangeability : NALG022
Is interchangeable with: NALG022
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Annotation -
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. The course may not be taught every academic year, it will be taught at least once every two years.
Last update: Žemlička Jan, doc. Mgr. et Mgr., Ph.D. (14.05.2019)
Course completion requirements -

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 65 % 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.

Last update: Šťovíček Jan, doc. RNDr., Ph.D. (11.02.2023)
Literature -
  • 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.
  • H. Krause, Representations of quivers via reflection functors, https://arxiv.org/abs/0804.1428
Last update: Šťovíček Jan, doc. RNDr., Ph.D. (04.03.2019)
Requirements to the exam -

The course is completed with an oral exam. The requirements for the exam correspond to the syllabus and and correspond to the first 3 chapters of the monograph by Assem, Simson and Skowroński and Sections 3 to 5 in the paper by Krause. These requirements will be applied to the extent to which the topic was presented in lectures (including possible on-line ones). 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.

Last update: Šťovíček Jan, doc. RNDr., Ph.D. (14.02.2022)
Syllabus -

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

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

3. Representations of hereditary algebras, finite representation type, Gabriel's theorem.

Last update: Šťovíček Jan, doc. RNDr., Ph.D. (04.03.2019)
Entry requirements -

Basics of theory of modules (to the extent of lecture NMAG339) and basic homological algebra (the Ext functor).

Last update: Žemlička Jan, doc. Mgr. et Mgr., Ph.D. (08.11.2021)