Seminar on Representation Theory (WiSe 2015/2016)

( = Seminar zur Darstellungstheorie )
Wednesday 10-12 h in V5-227 (2 SWS)
Organizer: Prof. Dr. Henning Krause, Dr. Julia Sauter , Ögmundur Eiriksson
Content: Selected topics from representation theory of finite dimensional algebras
This is a seminar with varying topics, suggestions are always welcome. In particular, phd students are encouraged to formulate their research needs and interests and we will try to incorporate this in the seminar. Usually we speak English and in exceptional cases German.
At the moment, we plan the following topics (with the person in charge of planning the talks):

Submodule-closed categories and the Weyl group (Apolonia Gottwald)
21. Oktober 2015
Homological study of Nakajima quiver varieties (Julia Sauter)
28. Oktober 2015
Review work of Keller-Scherotzke and Scherotzke, see here and here.
Cohen-Macaulay Representations November 2015
A series of talks with the aim to introduce connections of Cohen-Macaulay modules with matrix factorizations and classification of simple singularities. A suggestion for schedule for the topic can be found here.
Hopf algebras (Philipp Lampe) December 2015
A series of talks introducing Hopf algebras with some examples and properties. The following talks are planned.
- Introduction to Hopf algebras
  Define co/bi/Hopf algebras [GR 1.1-1.5, HGK 3.1-3.3] and introduce the notion of cocommutativity [GR 1.6, HGK 3.3] as well as primitive and group-like elements.
  Give the examples of group algebras [GR 1.5, HGK 3.3.12, C 3.1], universal enveloping algebras [GR 1.38, HGK 3.2.15, C 3.6], tensor/shuffle algebras [GR 1.40, HGK 3.2.6], the Sweedler Hopf algebra [HGK 3.4.3] and the cohomology of an algebraic group [GR 1.20].
- Applications of Cartier-Gabriel's structure theorem for convolution algebras
  Prove the structure theorem of Cartier-Gabriel for a cocommutative Hopf algebra over an algebraically closed field of characteristic zero [C 3.8.2]. Introduce convolution algebras and explain Ringel's [R] and Schofield's [S] theorems relating convolution algebras with universal enveloping algebras.
- The Hopf algebra of symmetric functions
  The aim of the talk is to introduce the Hopf algebra of symmetric polynomials and to explain why it is positive and self-dual with respect the basis of Schur functions.

Schedule of talks

The schedule will be updated when more talks have been organized.

Okt 21: Submodule-closed categories and the Weyl group (Apolonia Gottwald)
Okt 28: Homological study of Nakajima quiver varieties (Julia Sauter)
Nov 04: Modules over skew group algebras (Andre Beineke)
Nov 11: Kleinian singularities and finite CM type (Ögmundur Eiriksson)
Nov 18: Matrix factorizations and the double branched cover (Manuel Flores Galicia)
Nov 25: Cancelled
Dec 02: Introduction to Hopf algebras (Florian Gellert)
Dec 09: Applications of Cartier-Gabriel's structure theorem for convolution algebras (Philipp Lampe)
Dec 16: The Hopf algebra of symmetric functions (Rebecca Reischuk)
Jan 06: Hypersurfaces with finite CM type (Baolin Xiong)
Jan 13: Dg Hopf Algbras (Fajar Yuliawan)
Jan 20: Hereditary triangulated categories (Andrew Hubery)
Jan 27: Tilting in exact categories (Henning Krause)
Feb 03: Hopf algebra structure on strict polynomial functors (Cosima Aquilino)
Feb 10: Discussion about next semester's seminar (Takes place in V3-201)

Literature

[C] Pierre Cartier. A primer of Hopf algebras.
[GR] Darij Grinberg and Victor Reiner. Hopf algebras in combinatorics.
[HGK] Michiel Hazewinkel, Nadiya Gubareni, V.V. Kirichenko. Algebras, Rings and Modules, volume 168 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2010.
[Knõ86] Horst Knõrrer. Cohen-Macaulay modules on hypersurface singularities, In Representations of algebras (Durham 1985), volume 116 of London Math. Soc. Lecture note Ser., pages 147-164. Cambridge Univ. Press, Cambridge, 1986.
[LW12] Graham J. Leuschke and Roger Wiegand. Cohen-Macaulay Representations, volume 181 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2012.
[R] Claus Michael Ringel: Hall algebras and quantum groups, Inventiones mathematicae 101, 583-592 (1990).
[S] Aidan Schofield: Quivers and Kac-⁠Moody Lie algebras, Manuscript.