Higher Structures in Representation Theory 1 (Winter semester 2023/24)

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 October 16th.)
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 October 20th, 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). 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 0, recalling some basic material.

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 will focus on such hidden structures behind rings and modules and on applications of these. Later on, in particular in part 2 of the course, the focus will shift to higher structures behind module categories.
Some of these hidden structures are known as higher structures and will be studied in particular detail.

Chapter 0. Recap of basic material.
Monday, October 16. Rings and algebras.
Wednesday, October 18. Modules, representations and functors.
Friday, October 20. Natural transformations.

Chapter 1. Frobenius algebras.
Friday, October 20. Regular, free and projective modules.
Monday, October 23. Duality. Injective modules.
Wednesday, October 25. More examples. Frobenius algebras. Bilinear forms.
Monday, October 30. Characterising Frobenius algebras by linear forms and by bilinear forms. Symmetric algebras.
Wednesday, November 1: All Saints.
Monday, November 6. Trivial extensions.

Chapter 2. Coalgebras.
Monday, November 6. Tensor products.
Wednesday, November 8. Tensor-hom adjunction. Definitions of coalgebras and cocommutative coalgebras.
Monday, November 13. Examples. Fundamental theorem of coalgebras.
Wednesday, November 15. Simple coalgebras. Characterisation of Frobenius algebras in terms of coalgebra structures.
Monday, November 20. Continuing the proof of the characterisation. Another characterisation in terms of Frobenius pairs. Quantum Yang Baxter equation and braid equation.
Wednesday, November 22. Applications: Statistical mechanics, braid groups, knot theory, topological quantum field theory.

Chapter 3. Equivalences of categories.
Monday, November 27. Characterising equivalences of categories. Yoneda's Lemma.
Wednesday, November 29. Yoneda's Lemma, continued. Yoneda embedding. Projectivisation.
Monday, December 4. Global dimension of functor categories and almost split sequences. Uniqueness of adjoints.
Wednesday, December 6. Morita equivalence and Morita invariance. Generator and progenerator. Trace ideal.
Monday, December 11. Morita context and Morita context ring.
Wednesday, December 13. Morita context continued. Double centraliser property.
Monday, December 18. Morita's theorems.

Chapter 4. Corings.
Wednesday, December 20. Ring extensions. Corings. Base ring extensions of corings.
Monday, January 8. Corings associated with projective modules. Coring morphisms. Comodules, comodule morphisms.
Wednesday, January 10. Comodules of trivial corings.

Chapter 5. Differential biquivers.
Wednesday, January 10. Partitioned matrix problems. Biquivers. Graded algebras. Differential biquivers.
Monday, January 15. Representations of differential biquivers.
Wednesday, January 17. Shocking examples of differential biquivers and representations.
Monday, January 22. Derivations. Bimodule problems. A matrix problem as a bimodule problem.
Wednesday, January 24. Another matrix problem as a bimodule problem. Representations of algebras, projective presentations and bimodule problems.
Monday, January 29. From bimodule problems to differential biquivers. From differential biquivers to Roiter bocses.
Wednesday, January 31. Representations of bocses and of corings.

Chapter 6. Bocses and representation types.
Wednesday, January 31. Tame and wild representation type. Tame-wild dichotomy.
Monday, February 5. Examples of tame and wild algebras. Induced bocses and an induced fully faithful functor.
Wednesday, February 7. Sketch of proof of tame-wild dichotomy.

Problem sheets: On Wednesday, or occasionally on Friday, a new problem sheet will be handed out, for Friday a week later, and. Problem sheets also get posted in Ilias.

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

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 subsequent course Higher structures 2 can be found here.


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