Representation Theory 1 (Winter semester 2020/21)

Master course:
This is a 9 credit points course, available for master students in mathematics and also for bachelor students. The department has not yet processed the module description, which therefore is not yet visible in Campus.
However, registering to the course in Ilias / Campus is possible.
Responsible for this course: Steffen Koenig / Anne Henke

Information on the second part of this course is available here.

Prerequisites: Linear Algebra 1 and 2, Algebra. Basic representation theory (parts of Grundlagen der Darstellungstheorie or of Algebra 2) - material for the topics needed will be provided, see below and in Ilias. Not being afraid of abstract structures. Enjoying mathematics.

Format: Currently, the format of the course is quite unclear, unfortunately. The university has published new corona guidelines advising against classroom teaching, and the current situation in and around Stuttgart is not promising.
It is very unlikely that lectures can happen in the traditional format. Depending on the general situation and on the number of participants, some problem classes or discussion meetings may be possible. Otherwise, webex meetings or recordings will be used and lecture notes and further material will be made available on this website and in Ilias.
In order to stimulate interaction, lecture notes will contain questions to be discussed in personal or online meetings. Similarly, quizzes are supposed to make clearer what needs to be discussed.
Lecture notes will be posted below. In addition, for each section, there will be a preview text introducing into the topics for this section, also including some motivating examples that may help to get into these topics.
Moreover, below also some recap material will be posted, in a similar style as the lecture notes. This is material about prerequisites, and reading it may help recalling some basics of representation theory. If participants are unfamilar with such things, additional problem sessions or discussions can be set up.
Further reading material and some ebooks provided by the library will be made available in Ilias.

Recaps:
This course assume some familiarity with basic concepts of algebra and representation theory. For the participants' convenience, some short texts are posted here to help recalling such concepts. When there is interest, more details or examples can be discussed during the course.
Rings, algebras, modules and representations.
Simple modules, composition series and the theorem of Jordan and Hoelder.
Unique decompositions: the theorem of Krull, Remak and Schmidt.

Contents:
Chapter 1. Short exact sequences and the first extension group.
Preview. (A preview is meant as a teaser, to raise interest, and for preparation, by giving some relevant examples. It is not meant to replace the chapter - fully understanding the preview allows for skipping the chapter.)
Lecture notes with questions. Key concepts: Exact sequences, short exact sequences, extensions, equivalence of extensions, first extension group. Pullback, pushout, Baer sum.
Appendix A on diagram lemmas. Here you find some diagram lemmas. Proving these gives you practise in diagram chasing. The Snake Lemma will become important on several occasions later on.
This chapter corresponds to about three to four lectures. So you should have worked through this material by November 13.
Exercises on universal properties. We will see frequently that universal properties are extremely useful tools. These exercises (which you can try before having worked through this chapter) are supposed to help you getting into this way of thinking, starting with an example you know from linear algebra.
More exercises on pullback and pushout in other situations, in order to better understand the abstract definitions.

Chapter 2. Projective and injective modules
Preview.
Lecture notes with questions. Key concepts: Projective modules, injective modules.
This material, including some recap material maybe, should take you about one week and you should have worked through it by November 20.
Appendix on free modules.
Exercises on projective and injective modules.

Chapter 3: Homomorphisms, extensions and resolutions
Preview.
Lecture notes with questions, first part. Key concepts: Covariant Hom and contravariant Hom, left and right exact. The maps relating Ext1 and a quotient of Hom.
Exercises on computing Ext1 by using Theorem 3.3.
Lecture notes with questions, second part. Proof of theorem 3.3. Relating pullbacks and pushouts with exact sequences.
Exercises on determining some pullbacks and pushouts.
Exercises on monomorphisms, epimorphisms, products and coproducts.
Lecture notes with questions, third part. Key concepts: Hom-Ext sequences. Projective and injective resolutions and dimensions.
Working through the lecture notes (three parts) of this chapter may take you until December 11.
More exercises on Ext1.

Chapter 4: Examples (path algebras and bound quiver algebras)
Preview.
Lecture notes with questions, first part. Key topics: Path algebras of quivers, simple representations associated with vertices and their projective resolutions and projective dimensions. Connection between extensions of simples and the quiver.
Exercises on quiver representations.
Lecture notes with questions, second part. On bound quiver algebras.
Working through the lecture notes of this chapter (two parts) may take you until December 22.
Exercises on Hom-Ext sequences.

Chapter 5: Categories and functors
Preview.
Lecture notes with questions. Key concepts: Categories, functors and natural transformations. Characterisation of equivalences as fully faithful and essentially surjective functors.
Exercises on natural transformations.
Working through the lecture notes of this chapter (two parts) may take you until January 11.

Chapter 6: Complexes and derived functors
Preview.
Lecture notes with questions, first part. Key concepts: Chain and cochain complexes, homology and cohomology. quasi-isomorphisms. Main result: Long exact (co)homology sequence.
Exercises on complexes.
Lecture notes with questions, second part. Key concepts: Projective cover, injective hull, null-homotopic, homotopy equivalence. Main results: Horseshoe lemma, Comparison theorem.
Lecture notes with questions, third part. Key concepts: Extn. Universal (co)homological δ-functors. Dimension shift.
Appendix to the third part, containing the proof of Theorem 6.17.
Working through the lecture notes of this chapter may take you until January 29.
The next set of exercises (for the problem class on January 28) is contained in the text of chapter 7; the theory of cohomological δ-functors is not required, only the basic properties of Extn.

Chapter 7: Homological dimensions
Preview.
Lecture notes with questions and exercises. This short text is an illustration of the concepts of chapter 6 and contains exercises for the second and third part of chapter 6. It should be read before January 28.

Chapter 8: Tensor product and Tor
Preview.
Lecture notes with questions. Key concepts: Adjoint pairs of functors. Yoneda's lemma. Tensor product and its derived functors.
Exercises on adjoint functors.
Exercises on Yoneda's lemma.
Exercises on tensor products.

Chapter 9: Morita equivalences
Preview.
Lecture notes with questions. Key results: Properties of equivalences. Morita equivalences. Basic algebras, elementary algebras and bound quiver algebras. The theorem of Eilenberg and Watts.
Exercises on Morita equivalences.

Literature:
The following books can be obtained electronically from the university library:
Ibrahim Assem and Flavio Coelho, Basic Representation Theory of Algebras
Ibrahim Assem, Andrzej Skowronski and Daniel Simson, 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
Charles Weibel, An Introduction to Homological Algebra

Printed copies are available in the library of:
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



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