Higher Structures in Representation Theory 2 (Summer semester 2024)

Master course:
This is a 9 credit points course.
Registering to the course in C@mpus is possible.
Responsible for this course: Steffen Koenig

Lectures and problem classes:
Lectures:
Monday 9:45 to 11:15 in seminar room 7.527. (Starting April 8th.)
Wednesday 9:45 to 11:15 in seminar room 7.527.
Problem classes:
Friday 9:45 to 11:15 in seminar room 7.527.
(In the first teaching week, on Friday April 12th, there will be a lecture instead of a problem class.)

Prerequisites: Linear Algebra 1 and 2, Algebra. Basic knowledge in representation theory (Grundlagen der Darstellungstheorie, Algebra 2), and knowledge about some parts of Higher Structures 1. Not being afraid of abstract structures. Enjoying mathematics.

Introductory material on representation theory also can be found on the web sites of previous courses such as:
Representation Theory 1 (winter semester 2020/21)
Representation theory 2 (summer semester 2021).

The course will start with chapter 7, continuing some material covered in Higher Structures 1.

Contents: Behind familiar structures such as rings and modules there are further structures that provide additional information and allow for a deeper understanding. The first part of this two semester course has been focussing on such hidden structures behind rings and modules and on applications of these. In part 2 of the course, the focus will shift to higher structures behind module categories and other categories.

Chapter 7. Differential tensor algebras.
Monday, April 8. Tensor algebras and freely generated algebras.
Wednesday, April 10. Non-uniqueness. Differentials.
Friday, April 12. Differential tensor algebras. Normal bocses.
Monday, April 15. Roiter correspondence: From normal bocses to ditalgebras.
Wednesday, April 17. From ditalgebras to normal bocses.
Monday, April 22. Isomorphisms of categories of representations of normal bocses and of ditalgebras.
Wednesday, April 24. Differential biquivers and free normal bocses.
Monday, April 29. From bocses to differential biquivers.

Chapter 8. Complexes.
Monday, April 29. (Co)chain complexes. Morphisms, quasi-isomorphisms.
Monday, May 6. Category of (co) chain complexes. Snake lemma.
Wednesday, May 8. Long exact (co)homology sequence. Homotopy.
Monday, May 13. Homotopy category. Comparison theorem.
Wednesday, May 15. Triangulated categories. Basic properties.
Monday, May 27. Long exact sequences. Stable categories.
Wednesday, May 29. Stable categories are triangulated.
Monday, June 3. Exact categories. Frobenius categories. Additive categories with translation. Mapping cone.
Wednesday, June 5. Mapping cone triangle. Homotopy categories of additive categories with translation.
Monday, June 10. Triangulated structure on homotopy categories of additive categories with translation.
Wednesday, June 12. Triangulated structure, continued. Associated graded category. Triangulated structure on homotopy categories of (co)chain complexes. Localisation. Derived categories.

Chapter 9. Differential graded categories.
Monday, June 17. Differential graded k-modules. Tensor products, Koszul sign rule. Differential graded k-categories and dg k-algebras. Examples.
Wednesday, June 19. Cocycles and cohomologies. Enriched morphisms. Dg functors.
Monday, June 24. Acyclic, zero, contractible. Shifts. Mapping cones.
Wednesday, June 26. Representable functors. Pretriangulated categories and dg enhancements.
Monday, July 1. From a pretriangulated category over a Frobenius category to a homotopy category. Algebraic triangulated categories.
Wednesday, July 3. The dg category of dg modules.
Monday, July 8. Deriving dg categories. Keller's Morita theorems.
Wednesday, July 10. Rickard's Morita theorems.
Monday, July 15. Dg bimodules. Derived functors.

Chapter 10. A∞-algebras.
Monday, July 15. A theorem of Bongartz. Yoneda extension algebras. An example.
Wednesday, July 17. A∞-algebras, morphisms and quasi-isomorphisms. Kadeishvili's theorem. Coderivations. The result by Keller and Lefèvre-Hasegawa.


Literature:
The following books can be obtained electronically from the university library:
Ibrahim Assem and Flavio Coelho, Basic Representation Theory of Algebras
Ibrahim Assem, Daniel Simson and Andrzej Skowronski, Elements of the Representation Theory of Associative Algebras, Volume 1
Maurice Auslander, Idun Reiten and Sverre Smalø, Representation Theory of Artin Algebras
Michael Barot, Introduction to the Representation Theory of Algebras
Karin Erdmann and Thorsten Holm, Algebras and Representation Theory
Daniel Simson and Andrzej Skowronski, Elements of the Representation Theory of Associative Algebras, Volumes 2 and 3
Andrzej Skowronski and Kunio Yamagata, Frobenius Algebras I.
Andrzej Skowronski and Kunio Yamagata, Frobenius Algebras II.
Charles Weibel, An Introduction to Homological Algebra

Printed copies are available in the library of:
Raymundo Bautista, Leonardo Salmeron and Rita Zuazua, Differential Tensor Algebras and Their Module Categories.
Tomasz Brzezinski and Robert Wisbauer, Corings and Comodules.
Charles Curtis and Irving Reiner, Representation theory of finite groups and associative algebras.
Charles Curtis and Irving Reiner, Methods of representation theory, volumes I and II.
Masaki Kashiwara and Pierre Schapira, Sheaves on manifolds.
Masaki Kashiwara and Pierre Schapira, Categories and sheaves.

Many good books are available online (legally, in not necessarily final versions). A good collection of freely available material on category theory can be found on Logic matters (section on category theory).
Good books you can find there are for instance
Tom Leinster, Basic Category Theory.
Emily Riehl, Category Theory in Context.

Further literature will be listed here later on.

The web page of the preceding course Higher structures 1 can be found here.


Data protection information Impressum (legal notice)