Selected Topics in Representation Theory
The selected topics started as part of the internal seminar of the BIREP group during 2005–2008. The aim was to create resource and make them available for the everyone interested. This was reinstated in the winter semester 2020/21.
Below you can find a list of topics and short documents about those topics, which include a list of references. For now, they are ordered by the date the corresponding talks were given.
 Bill CrawleyBoevey: Sylvester rank functions for rings and universal localization
 Julia Sauter: Exact structures on exact categories
 Henning Krause: Chase's Lemma and its Context
 Marc Stephan: Quillen's Stratification Theorem
 Andrew Hubery: Projective modules over hereditary tensor algebras
 Rolf Farnsteiner: Algebras of finite global dimension:
 Philipp Fahr: Algebraic aspects of stability
 Angela Holtmann: Finitistic dimension conjecture: a reduction formula
 Rolf Farnsteiner: Induced modules: The Mackey decomposition theorem
 Philipp Fahr: The HarderNarasimhan filtration
 Philipp Fahr: Torsion Theory and Stability
 Rolf Farnsteiner: Induced modules:
 Claus M. Ringel: Artin algebras of dominant dimension ≥ 2
 Angela Holtmann: Families of representations for wild quivers
 Claus M. Ringel: The torsionless modules of an artin algebra
 Rolf Farnsteiner: Cartier's Theorem
 Rolf Farnsteiner: Quasihereditary algebras:
 Philipp Fahr: Elementary Modules
 Angela Holtmann: Properties and use of the characteristic tilting module
 Reinhard Waldmüller: The Theorems of Maschke and ArtinWedderburn
 Rolf Farnsteiner: Hopf modules and integrals:
 Claus M. Ringel: Degenerations using RiedtmannZwara sequences
 Claus M. Ringel: The ladder construction of Prüfer modules
 Reinhard Waldmüller: The orthogonality relations
 Claus M. Ringel: Prüfer modules of finite type (If there are no generic modules, then all Prüfer modules are of finite type,)
 Reinhard Waldmüller: Burnside's $p^a q^b$theorem
 Reinhard Waldmüller: Symmetries of flat manifolds
 Angela Holtmann: General representations of quivers and canonical decompositions:
 Reinhard Waldmüller: ladic cohomology
 Claus M. Ringel: Pillars, see also BrauerThrall I (A module which is not of finite type contains indecomposable submodules of arbitrarily large finite length)
 Reinhard Waldmüller: DeligneLusztig characters
 Rolf Farnsteiner: Nakayama algebras: Kupisch series and Morita type

Claus M. Ringel: GabrielRoiter inclusions and irreducible monomorphisms
If there is a GabrielRoiter inclusion $X \to Y$, there is an irreducible monomorphism $X \to M$ and an epimorphism $M \to Y$ such that the composition $X \to M \to Y$ is a monomorphism.  Rolf Farnsteiner: Simple modules and pregular classes
 Claus M. Ringel: Takeoff subcategories: Any infinite cogenerationclosed subcategory contains a minimal infinite cogeneration closed subcategory.
 Rolf Farnsteiner: Complexity and Krull dimension
 Claus M. Ringel: The real root modules for some quivers
 Rolf Farnsteiner: Separated quivers and representation type
 Philipp Fahr: Infinitedimensional Modules over Wild Algebras:

Rolf Farnsteiner: Self injective algebras:
 The structure of the projective indecomposable modules
For group algebras of finite groups, the socles and the tops of the principal indecomposable modules are isomorphic. We shall look at Nakayama's generalization concerning self injective algebras.  Self injective algebras vs. Frobenius algebras.
 Examples and Morita equivalence
We are going to give examples for symmetric, weaklysymmetric and Frobenius algebras, and discuss stability of these classes of algebras under Morita equivalence.  Frobenius algebras and coalgebras
 The structure of the projective indecomposable modules

Angela Holtmann: What are tilting modules?  A very brief introduction to tilting theory.
The main topic will be the BrennerButler theorem: If $B = {\rm End}({}_AT)$ is the endomorphism ring of an $A$tilting module ${}_AT$, then "certain subcategories" of the module categories of $A$ and $B$ are equivalent. 
Claus M. Ringel: If R is a local artinian ring, then there are no nontrivial tilting complexes.
A tilting complex is trivial if it is homotopy equivalent to a complex $C^\bullet$ with all but at most one $C^i$ zero, and the only nonzero $C^i$ being projective.
A stronger assertion: If $T$ is a bounded complex of free modules of finite rank and any map $f \colon T \to T[n]$ with $n \neq 0$ is homotopic to zero, then $T$ is homotopy equivalent to a trivial complex.  Lutz Hille: Mutations of exceptional sequences
 Angela Holtmann: Resolution of simple modules over the polynomial ring k[X_{1},...,X_{n}] (and the exterior algebra of a vector space)
 Lutz Hille: The braid group action on the set of exceptional sequences

Rolf Farnsteiner: The Burnside theorem
If G is a finite group, and M is a faithful representation, then any irreducible representation occurs as a subfactor of some tensor product of copies of M. 
Angela Holtmann: Modules with standard filtration:
 Part I
Let A be an Artin algebra, mod A the category of finitely generated Amodules, and Θ={Θ(1),...,Θ(n)} a sequence of Amodules with Ext^{1}_{A}(Θ(j),Θ(i))=0 for all j≥i. Denote by F(Θ) the full subcategory of mod A of modules filtered by modules from Θ. Then F(Θ) is functorially finite in mod A.  Part II
The subcategories F(Δ) and F(∇) of mod A have (relative) almost split sequences.
 Part I
 Lutz Hille: Quadratic algebras and Koszul algebras

Lutz Hille: Koszul algebras and distributive lattices
Koszul algebras are characterized by the fact that the relations yield distributive lattices.  Rolf Farnsteiner: Burnside's Theorem for Hopf Algebras

Angela Holtmann: Examples:
 Covariantly finite, but not contravariantly finite full subcategories of a module category
 Right F(Δ)approximations
 Rolf Farnsteiner: The Theorem of WedderburnMalcev:

Lutz Hille: The volume of a tilting module
This notion allows to show easily that an almost complete partial tilting module has at most two complements, and at most one in case the module is not sincere. 
Angela Holtmann: Torsion theories and tilting modules
Let (T, F) be a torsion theory on mod A such that D(A_{A})∈T and either T or F contain only finitely many isomorphism classes of indecomposable Amodules. Let T be the sum of all Extprojectives in T. Then T is a tilting module such that (T(T), F(T))=(T, F). 
Rolf Farnsteiner: Stable Representation Quivers:
 The Riedtmann Structure Theorem.
Using covering techniques we shall show that stable representation quivers are determined by certain trees and certain subgroups of the universal coverings defined by these trees.  Growth Numbers and Zhang's Theorem.
If Q is a nonperiodic, tree infinite, regular component of the ARquiver of an Artin algebra whose growth number is small, then the tree class of Q is an infinite Dynkin diagram.  Subadditive Functions and Webb's Theorem.
We shall show that the tree classes of the components of the stable AuslanderReiten quiver associated to the group algebra of a finite group are finite or infinite Dynkin diagrams, or Euclidean diagrams.
 The Riedtmann Structure Theorem.
 Lutz Hille: The volume of a tilting module for $A_n$